The statements that are always true regarding the given diagram of the triangle are m∠2 + m∠3 = m∠6, m∠3 + m∠4 + m∠5 = 180°, m∠5 + m∠6 = 180°. Therefore, options (4) and (5) are not always true.
What is an exterior angle?An exterior angle of a polygon is an angle that forms a linear pair with an interior angle of the polygon. In other words, it is an angle formed by extending one of the sides of the polygon. For any given vertex of the polygon, the exterior angle is the angle between a line containing the side of the polygon next to the vertex and a line that is an extension of the adjacent side. The measure of an exterior angle is equal to the sum of the measures of its corresponding remote interior angles (interior angles that are not adjacent to the exterior angle). The sum of the exterior angles of any polygon, including a triangle, is always equal to 360 degrees. Exterior angles are used in a variety of geometric proofs and constructions.
We know that the sum of all the interior angles of any triangle is 180 degrees. Therefore, we can use this fact to determine which statements are always true regarding the given diagram.
Now, we can use the following relationships between the interior and exterior angles of a triangle:
Exterior angle = Sum of interior angles adjacent to it
Interior angle = 180 - Exterior angle
Using these relationships, we can determine which statements are always true:
m∠5 + m∠3 = m∠4: This is true because the exterior angle at angle 3 is equal to the sum of angles 3 and 4, and the exterior angle at angle 5 is equal to the sum of angles 5 and 6. Therefore, m∠5 + m∠3 + m∠6 = m∠4 + m∠3, which simplifies to m∠5 + m∠3 = m∠4.
Given m∠3 + m∠4 + m∠5 = 180°: This is true because the sum of all the interior angles of a triangle is 180 degrees.
m∠5 + m∠6 = 180°: This is not always true because sum of all the angles should be 180. It is true in this specific case because angle 1 is a straight angle, which means that m∠5 + m∠6 = 180°. However, in general, this statement is not always true.
m∠2 + m∠3 = m∠6: This is true because the exterior angle at angle 2 is equal to the sum of angles 2 and 6. Therefore, m∠2 + m∠3 = m∠6.
Given m∠2 + m∠3 + m∠5 = 180°: This is may not always true. It is true in this specific case because angle 1 is a straight angle, which means that m∠2 + m∠3 + m∠5 = 180°. However, in general, this statement is not always true.
Therefore, the three statements that are always true from the diagram are:
m∠5 + m∠3 = m∠4
m∠3 + m∠4 + m∠5 = 180°
m∠2 + m∠3 = m∠6
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What is the mean of the values in the stem-and-leaf plot?
Enter your answer in the box.
Answer:
mean = 24
Step-by-step explanation:
the mean is calculated as
mean = [tex]\frac{sum}{count}[/tex]
the sum of the data set is
sum = 12 + 13 + 15 + 28 + 28 + 30 + 42 = 168
there is a count of 7 in the data set , then
mean = [tex]\frac{168}{7}[/tex] = 24
What are answers to these questions?
1. f is concave up on the intervals = ?
2. f is concave down on the intervals = ?
3. The inflection points occur at x = ?
f(x) is concave up on the interval (-∞, √(6/7)) U (√(6/7), ∞),f(x) is concave down on the interval (-√(6/7), √(6/7)) ,The inflection points occur at x = -√(6/7) and x = √(6/7).
What is inflection Point?An inflection point is a point on a curve where the concavity changes, from concave up to concave down or vice versa, indicating a change in the curvature of the curve.
According to the given information:
To determine the intervals where f(x) is concave up or down, we need to find the second derivative of f(x) and determine its sign.
First, we find the first derivative of f(x):
f'(x) = (14x)/(7x²+6)²
Then, we find the second derivative of f(x):
f''(x) = [28(7x²+6)²- 28x(7x²+6)(4x)] / (7x²+6)^4
Simplifying the expression, we get:
f''(x) = 28(42x² - 72) / (7x^2+6)³
To determine where f(x) is concave up or down, we need to find the intervals where f''(x) is positive or negative, respectively.
Setting f''(x) = 0, we get:
42x² - 72 = 0
Solving for x, we get:
x = ±√(6/7)
These are the possible inflection points of f(x). To determine if they are inflection points, we need to check the sign of f''(x) on both sides of each point.
We can use a sign chart to determine the sign of f''(x) on each interval.
Intervals where f''(x) > 0 are where f(x) is concave up, and intervals where f''(x) < 0 are where f(x) is concave down.
Here is the sign chart for f''(x):
x | -∞ | -√(6/7) | √(6/7) | ∞
f''(x)| - | + | - | +
From the sign chart, we can see that:
a) f(x) is concave up on the interval (-∞, √(6/7)) U (√(6/7), ∞).
b) f(x) is concave down on the interval (-sqrt(6/7), √(6/7)).
c) The inflection points occur at x = -√(6/7) and x = √(6/7).
Therefore, the open intervals where f(x) is concave up are (-∞, -√(6/7)) and (√(6/7), ∞), and the open interval where f(x) is concave down is (-√(6/7), √(6/7)). The inflection points occur at x = -√(6/7) and x = √(6/7).
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The f(x) is concave up on the interval (-∞, √(6/7)) U (√(6/7), ∞),f(x) is concave down on the interval (-√(6/7), √(6/7)) ,The inflection points occur at x = -√(6/7) and x = √(6/7).
What is inflection Point?
An inflection point is a point on a curve where the concavity changes, from concave up to concave down or vice versa, indicating a change in the curvature of the curve.
According to the given information:
To determine the intervals where f(x) is concave up or down, we need to find the second derivative of f(x) and determine its sign.
First, we find the first derivative of f(x):
f'(x) = (14x)/(7x²+6)²
Then, we find the second derivative of f(x):
f''(x) = [28(7x²+6)²- 28x(7x²+6)(4x)] / (7x²+6)^4
Simplifying the expression, we get:
f''(x) = 28(4x² - 72) / (7x^2+6)³
To determine where f(x) is concave up or down, we need to find the intervals where f''(x) is positive or negative, respectively.
Setting f''(x) = 0, we get:
42x² - 72 = 0
Solving for x, we get:
x = ±√(6/7)
These are the possible inflection points of f(x). To determine if they are inflection points, we need to check the sign of f''(x) on both sides of each point.
We can use a sign chart to determine the sign of f''(x) on each interval.
Intervals where f''(x) > 0 are where f(x) is concave up, and intervals where f''(x) < 0 are where f(x) is concave down.
Here is the sign chart for f''(x):
x | -∞ | -√(6/7) | √(6/7) | ∞
f''(x)| - | + | - | +
From the sign chart, we can see that:
a) f(x) is concave up on the interval (-∞, √(6/7)) U (√(6/7), ∞).
b) f(x) is concave down on the interval (-sqrt(6/7), √(6/7)).
c) The inflection points occur at x = -√(6/7) and x = √(6/7).
Therefore, the open intervals where f(x) is concave up are (-∞, -√(6/7)) and (√(6/7), ∞), and the open interval where f(x) is concave down is (-√(6/7), √(6/7)). The inflection points occur at x = -√(6/7) and x = √(6/7).
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Given that sin 0 = 20/29 and that angle terminates in quadrant III, then what is the value of tan 0?
The value of Tanθ is 20/21.
What is Pythagorean Theorem?
A right triangle's three sides are related in Euclidean geometry by the Pythagorean theorem, also known as Pythagoras' theorem. According to this statement, the areas of the squares on the other two sides add up to the size of the square whose side is the hypotenuse.
Here, we have
Given: sinθ = -20/29 and that angle terminates in quadrant III.
We have to find the value of tanθ.
Using the definition of sine to find the known sides of the unit circle right triangle. The quadrant determines the sign on each of the values.
Sinθ = Perpendicular/hypotenuse
Find the adjacent side of the unit circle triangle. Since the hypotenuse and opposite sides are known, use the Pythagorean theorem to find the remaining side.
Adjacent = - √Hypotenuse² - perpenducualr²
Replace the known values in the equation.
Adjacent = -√29² - (-20)²
Adjacent = -21
Find the value of tangent.
Tanθ = Perpendicular/base
Tanθ = -20/(-21)
Hence, the value of Tanθ is 20/21.
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The line plot displays the number of roses purchased per day at a grocery store.
A horizontal line starting at 0 with tick marks every one unit up to 10. The line is labeled Number of Rose Bouquets, and the graph is titled Roses Purchased Per Day. There is one dot above 10. There are two dots above 1 and 4. There are three dots above 2 and 5. There are 4 dots above 3.
Which of the following is the best measure of variability for the data, and what is its value?
The IQR is the best measure of variability, and it equals 3.
The IQR is the best measure of variability, and it equals 9.
The range is the best measure of variability, and it equals 3.
The range is the best measure of variability, and it equals 9.
The range is the best measure of variability for this data, and its value is 4.
Which of the following is the best measure of variability for the data, and what is its value?The line plot displays the number of roses purchased per day at a grocery store, with the data values ranging from 0 to 4 (since there are no dots above 4).
The best measure of variability for this data is the range, which is the difference between the maximum and minimum values in the data set. In this case, the minimum value is 0 and the maximum value is 4, so the range is:
Range = Maximum value - Minimum value = 4 - 0 = 4
Therefore, the range is the best measure of variability for this data, and its value is 4.
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Given the expression 3x+2 evaluate the expression for the given values of x when x=(-2)
Answer:
...............................
The diameter of a circle is 38 feet.what is the circles circumfrence. Use 3.14 for pi
Answer:
The circumference of the circle is 119.32 ft.
Step-by-step explanation:
The circumference of a circle can be solved through the formula:
C = πd
where d is the diameter
Given: d = 38 ft
π = 3.14
Solve:
C = πd
C = 3.14 (38 ft)
C = 119.32 ft
0\left\{-10\le x\le10\right\}
Describe the transformations (vertical translation, horizontal translation, and dilation/reflection) from the parent function that happened to these formulas
The formula g(x) = 2 * 0{-10≤x+3≤10} represents a function that has been horizontally shifted left by 3 units, reflected about the y-axis, and vertically scaled by a factor of 2.
What is Function?A function is a mathematical rule that assigns each input from a set (domain) a unique output from another set (range), typically written as y = f(x).
The notation "0{-10≤x≤10}" typically represents the domain of a function or an inequality. It means that the function is defined only for the values of x that are between -10 and 10 (including -10 and 10).
Assuming that the function in question is a constant function equal to zero, the parent function is f(x) = 0.
To describe the transformations that happened to this function, we need more information about the specific formula. For example, if the formula is:
g(x) = 0{-10≤x≤10}
Then there are no transformations from the parent function. The function is simply a constant function that is equal to zero over the interval [-10, 10].
However, if the formula is something like:
g(x) = 2 * 0{-10≤x+3≤10}
Then we can describe the transformations as follows:
Horizontal translation: The function has been shifted horizontally to the left by 3 units. This means that the point (3, 0) on the parent function is now located at the origin (0, 0) on the transformed function.
Dilation/reflection: The function has been reflected about the y-axis and vertically scaled by a factor of 2. This means that the point (-1, 0) on the parent function is now located at (-4, 0) on the transformed function, and the point (1, 0) on the parent function is now located at (2, 0) on the transformed function.
Vertical translation: There is no vertical translation in this case, since the constant function is already centered at y = 0.
To summarize, the formula g(x) = 2 * 0{-10≤x+3≤10} represents a function that has been horizontally shifted left by 3 units, reflected about the y-axis, and vertically scaled by a factor of 2.
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The formula g(x) = 2 * 0{-10≤x+3≤10} represents a function that has been horizontally shifted left by 3 units, reflected about the y-axis, and vertically scaled by a factor of 2.
What is Function?A function is a mathematical rule that assigns each input from a set (domain) a unique output from another set (range), typically written as y = f(x).
The notation "0{-10≤x≤10}" typically represents the domain of a function or an inequality. It means that the function is defined only for the values of x that are between -10 and 10 (including -10 and 10).
Assuming that the function in question is a constant function equal to zero, the parent function is f(x) = 0.
To describe the transformations that happened to this function, we need more information about the specific formula. For example, if the formula is:
g(x) = 0{-10≤x≤10}
Then there are no transformations from the parent function. The function is simply a constant function that is equal to zero over the interval [-10, 10].
However, if the formula is something like:
g(x) = 2 * 0{-10≤x+3≤10}
Then we can describe the transformations as follows:
Horizontal translation: The function has been shifted horizontally to the left by 3 units. This means that the point (3, 0) on the parent function is now located at the origin (0, 0) on the transformed function.
Dilation/reflection: The function has been reflected about the y-axis and vertically scaled by a factor of 2. This means that the point (-1, 0) on the parent function is now located at (-4, 0) on the transformed function, and the point (1, 0) on the parent function is now located at (2, 0) on the transformed function.
Vertical translation: There is no vertical translation in this case, since the constant function is already centered at y = 0.
To summarize, the formula g(x) = 2 * 0{-10≤x+3≤10} represents a function that has been horizontally shifted left by 3 units, reflected about the y-axis, and vertically scaled by a factor of 2.
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Please help me solve and show my work
The degree measure of the angles are;
1. 5π/3 = 300°
2 3π/4 = 135°
3. 5π/6 = 150°
4. -3π/2 = 90°
What is degree and radian?A degree is a unit of measurement which is used to measure circles, spheres, and angles while a radian is also a unit of measurement which is used to measure angles.
A circle has 360 degrees which are its full area while its radian is only half of it which is 180 degrees or one pi radian.
therefore π = 180°
1. 5π/ 3 = 5×180/3 = 300°
2. 3π/4 = 3× 180/4 = 540/4 = 135°
3. 5π/6 = 5×180/6 = 150°
4. - 3π/2 = -3 × 180/2 = -270° = 90°
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Write an equation for the polynomial graphed below
The polynomial in factor form is y(x) = - (1 / 6) · (x + 3) · (x + 1) · (x - 2) · (x - 3).
How to derive the equation of the polynomial
In this problem we find a representation of polynomial set on Cartesian plane, whose expression is described by the following formula in factor form:
y(x) = a · (x - r₁) · (x - r₂) · (x - r₃) · (x - r₄)
Where:
x - Independent variable.r₁, r₂, r₃, r₄ - Roots of the polynomial.a - Lead coefficient.y(x) - Dependent variable.Then, by direct inspection we get the following information:
y(0) = - 3, r₁ = - 3, r₂ = - 1, r₃ = 2, r₄ = 3
First, determine the lead coefficient:
- 3 = a · (0 + 3) · (0 + 1) · (0 - 2) · (0 - 3)
- 3 = a · 3 · 1 · (- 2) · (- 3)
- 3 = 18 · a
a = - 1 / 6
Second, write the complete expression:
y(x) = - (1 / 6) · (x + 3) · (x + 1) · (x - 2) · (x - 3)
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Complete the truth table for (A ⋁ B) ⋀ ~(A ⋀ B).
The truth table for (A ⋁ B) ⋀ ~(A ⋀ B) is:
A B (A ⋁ B) ⋀ ~(A ⋀ B)
0 0 0
0 1 0
1 0 0
1 1 0
The truth table is what?A truth table is a table that displays all possible combinations of truth values (true or false) for one or more propositions or logical expressions, as well as the truth value of the resulting compound proposition or expression that is created by combining them using logical operators like AND, OR, NOT, IMPLIES, etc.
The columns of a truth table reflect the propositions or expressions themselves as well as the compound expressions created by applying logical operators to them. The rows of a truth table correspond to the various possible combinations of truth values for the propositions or expressions.
To complete the truth table for (A ⋁ B) ⋀ ~(A ⋀ B), we need to consider all possible combinations of truth values for A and B.
A B A ⋁ B A ⋀ B ~(A ⋀ B) (A ⋁ B) ⋀ ~(A ⋀ B)
0 0 0 0 1 0
0 1 1 0 1 0
1 0 1 0 1 0
1 1 1 1 0 0
So, the only case where the expression is true is when both A and B are true, and for all other cases it is false.
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Find the length of the triangle.
The length of the unknown side of the triangle is __________
Answer:
The answer is 2√10
Step-by-step explanation:
Hyp²=opp²+adj²
let hyp be x
x²=6²+2²
x²=36+4
x²=40
square root both sides
√x²=√40
x=2√10
7. A group of students wants to demonstrate that sunlight provides the energy for plants to grow. What is
the control group for the experiment?
A. Some plants will receive less water
B. Some plants will receive fertilizer
C. Some plants will receive no sunlight
D. Some plants will receive no water
I
The control group for the experiment would be option C: some plants will receive no sunlight.
What is control group for experiment ?The control group in an experiment is the group that does not receive the treatment or intervention being tested, so that the effects of the treatment can be compared to a baseline or reference point. In this experiment, the treatment being tested is the provision of sunlight as an energy source for plant growth.
Therefore, the control group should not receive sunlight, so that the effects of sunlight can be compared to the baseline of plant growth without sunlight.
Therefore, the control group for the experiment would be option C: some plants will receive no sunlight.
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PLEASE HELP
Find the Area
2cm
___cm^2
Answer:
3.14 cm^2
Step-by-step explanation:
1. Find radius:
If diameter is 2, divide it by 2 to get radius = 1
2. Find formula:
A=πr^2
3. Plug in:
A = π(1)^2
4. Solve (multiply):
A = π(1)^2:
3.14159265359
Or
3.14 cm^2
Answer:
3.14 cm^2
Step-by-step explanation:
A=[tex]\pi[/tex]r^2
r=2
2/2=1
A=[tex]\pi[/tex](1)^2
=[tex]\pi[/tex]1
≈3.14x1
≈3.14cm^2
6. Two out of every five Canadians read at least 10 books a year. What percent of Canadians read at least 10 books a year?
40% of Canadians read at least 10 books a year.
Define percentageA percentage is a way of expressing a portion or a part of a whole as a fraction of 100. It is represented by the symbol "%". Percentages are often used to compare different quantities or to describe how much of a total is made up by a specific amount or group. For example, if you score 80 out of 100 on a test, your score can be expressed as 80%, meaning you got 80 out of 100 possible points.
If two out of every five Canadians read at least 10 books a year, then we can write this as a fraction:
2/5
We can multiply this fraction by 100 to turn it to a percentage:
(2/5) x 100 = 40%
Therefore, 40% of Canadians read at least 10 books a year.
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how am i supposed to prove that theyre collinear
Answer:
They are collinear if they are on the same line
Consider the verbal phrase.
Two and nineteen hundredths times a number g, plus fifty-nine hundredths
Part A
Enter an expression to represent the verbal phrase.
g +
Part B
Evaluate the expression when g = 3.
3.96
7.16
8.34
8.93
verbal phrase can be represented by 7.16.
What is the verbal phrase?Part A:
The verbal phrase can be represented by the following expression:
[tex]2.19g + 0.59[/tex]
Here, "Two and nineteen hundredths times a number g" can be written as 2.19g, and "plus fifty-nine hundredths" can be written as [tex]0.59[/tex] .
Part B:
To evaluate the expression when [tex]g = 3[/tex], we substitute 3 for g in the expression and simplify:
[tex]2.19g + 0.59[/tex]
[tex]= 2.19(3) + 0.59 [\ Substitute g = 3][/tex]
[tex]= 6.57 + 0.59[/tex]
[tex]= 7.16[/tex]
Therefore, the correct answer is [tex]7.16.[/tex]
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The given question is incomplete. the complete question is given below:
Consider the verbal phrase.
Two and nineteen hundredths times a number g, plus fifty-nine hundredths
Part A
Enter an expression to represent the verbal phrase.
g +
Part B
Evaluate the expression when g = 3.
3.96
7.16
8.34
8.93
I need helppp!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Answer:
Step-by-step explanation:
The distance formula is
[tex]d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]
x1 is a,
x2 is 0,
y1 is 0 and
y2 is b. Fitting those into the formula where they belong:
[tex]d=\sqrt{(0-a)^2+(b-0)^2}[/tex] and
[tex]d=\sqrt{(-a)^2+(b)^2}[/tex]
Since a negative squared is a positive, then
[tex]d=\sqrt{a^2+b^2}[/tex]
which is the second choice down.
Oliver and Mike each place some apples and oranges into the same bowl.
• The ratio of apples to oranges Oliver places in the bowl is 2:3
• The ratio of apples to oranges Mike places in the bowl is 1:2
• They each place 6 oranges in the bowl.
Write the total number of apples and oranges they place in the bowl.
Answer: 5 * 5 = 20
20 divided by 1 = 20
If there are 5 apples with an apple:orange ratio of 1:4, there are 20 oranges.
Suppose that the function h is defined as follows.
if
-2
-1
h(x)= 0
1
2
Graph the function h.
-3.5
if-2.5
if-1.5
if -0.5
if 0.5 ≤x≤1.5
+
X
Ś
The required graph of the function given; h (x) has been attached.
Define a graph?In mathematics, graph theory is the study of graphs, which are mathematical structures used to represent pairwise interactions between objects. In this definition, a network is made up of nodes or points called vertices that are connected by edges, also called links or lines. In contrast to directed graphs, which have edges that connect two vertices asymmetrically, undirected graphs have edges that connect two vertices symmetrically. Graphs are one of the primary areas of study in discrete mathematics.
Here as per the question the graph of the function, h (x) has been attached.
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You can afford a $1000 per month mortgage payment. You've found a 30 year loan at 5.3% interest.
a) How big of a loan can you afford? (Round to the nearest cent, as needed.)
$
b) How much total money will you pay the loan company? (Round to the nearest cent, as needed.)
$
c) How much of that money is interest? (Round to the nearest cent, as needed.)
Answer:
a) To find out how big of a loan you can afford, we can use the formula for the monthly payment of a mortgage:
M = P [ i(1 + i)^n ] / [ (1 + i)^n - 1 ]
where M is the monthly payment, P is the principal (the amount borrowed), i is the monthly interest rate (which is the annual interest rate divided by 12), and n is the number of monthly payments (which is the number of years times 12).
In this case, we know that M = $1,000, i = 0.053/12, and n = 30 x 12 = 360. We want to solve for P, the principal we can afford.
Substituting these values into the formula, we get:
$1,000 = P [ 0.004416(1 + 0.004416)^360 ] / [ (1 + 0.004416)^360 - 1 ]
Simplifying and solving for P, we get:
P = $183,928.72
Therefore, you can afford a loan of approximately $183,928.72.
b) The total money paid to the loan company will be the monthly payment multiplied by the number of payments over the life of the loan. In this case, we have:
Total money paid = $1,000 x 360 = $360,000
Therefore, the total amount of money paid to the loan company will be $360,000.
c) To find out how much of that money is interest, we can subtract the principal from the total amount paid. In this case, we have:
Interest paid = Total money paid - Principal = $360,000 - $183,928.72 = $176,071.28
Therefore, the amount of money paid in interest will be $176,071.28.
Given F(x) = 4x - 8 and g(x) = -3x + 1, what is (f - g)(x)?
A) 7x-9
B) 7x - 7
C) x-9
D) x - 7
Therefore, the answer is (A) 7x-9 when it is given that function F(x) = 4x - 8 and g(x) = -3x + 1.
What is function?In mathematics, a function is a relationship between two sets of values, where each input (or domain element) is associated with a unique output (or range element). In other words, a function is a rule or a process that takes an input (or inputs) and produces a corresponding output. Functions can be expressed using various mathematical notations, such as algebraic formulas, graphs, tables, or even verbal descriptions. They are widely used in many fields of mathematics, science, engineering, economics, and computer science, to model and solve problems that involve relationships between variables or quantities.
Here,
To find (f - g)(x), we need to subtract g(x) from f(x), so we get:
(f - g)(x) = f(x) - g(x)
Substituting the given functions, we get:
(f - g)(x) = (4x - 8) - (-3x + 1)
Simplifying the expression by distributing the negative sign, we get:
(f - g)(x) = 4x - 8 + 3x - 1
Combining like terms, we get:
(f - g)(x) = 7x - 9
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find the radius of a cylinder if the volume is 2,035.75 in^3 and the height is 3 times the radius. use the formula V= pi r^2h
The radius of the cylinder is approximately 6.75 inches.
How to calculate volume of cylinder?We can use the formula for the volume of a cylinder to solve this problem:
[tex]V = \pi r^2h[/tex]
We know that the volume V is 2,035.75 [tex]in^3[/tex], and the height h is 3 times the radius r. So we can write:
[tex]V = \pi r^2(3r)[/tex]
Simplifying this expression, we get:
V = 3π[tex]r^3[/tex]
To solve for r, we can divide both sides of the equation by 3π[tex]r^2[/tex]:
V/(3π[tex]r^2[/tex]) = r
Substituting the given value for V, we have:
2,035.75/(3π[tex]r^2[/tex]) = r
Multiplying both sides by 3πr^2, we get:
2,035.75 = 3π[tex]r^3[/tex]
Dividing both sides by 3π, we have:
[tex]r^3[/tex] = 2,035.75 / (3π)
Taking the cube root of both sides, we get:
r = [tex](2,035.75 / (3\pi ))^{1/3[/tex]
Using a calculator, we find that:
r ≈ 6.75 inches
Therefore, The cylinder has a radius of about 6.75 inches.
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La Suma delos cuadrados de dos números naturales consecutivos es 181 halla dichos numeros
The two consecutive natural numbers whose sum of squares is 181 are 9 and 10
Let's assume that the two consecutive natural numbers are x and x+1. Then, we can write an equation based on the given information:
x² + (x+1)² = 181
Expanding the equation:
x² + x² + 2x + 1 = 181
Combining like terms:
2x² + 2x - 180 = 0
Dividing both sides by 2:
x² + x - 90 = 0
Now, we can use the quadratic formula to solve for x:
x = (-b ± √(b² - 4ac)) / 2a
where a = 1, b = 1, and c = -90
x = (-1 ± √(1 + 360)) / 2
x = (-1 ± √(361)) / 2
x = (-1 ± 19) / 2
We discard the negative value, as it does not correspond to a natural number:
x = 9
Therefore, the two consecutive natural numbers are 9 and 10, and their sum of squares is 81 + 100 = 181.
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Write an equation in point-slope form. Part I: Create an equation of a line in point-slope form. Be sure to identify all parts of the equation before writing the equation. (3 points) Part II: Using the equation of the line you wrote in Part I, write an equation of a line that is perpendicular to this line. Show your work. (3 points)
The line's equation in point-slope form is shown here. Point (2, 5) is the given point on the line, and slope 2 is the given slope of the line. The slope of this line is -1/2, which is the negative reciprocal of the slope.
How do you formulate an equation in point-slope form?A line's point slope form equation is [tex]y - y_1 = m(x - x_1)[/tex]. Consequently, y - 0 = m(x = 0), or y = mx, is the equation of a line passing through the origin with a slope of m.
We require a point on the line and the slope of the line in order to create a line equation in point-slope form. In point-slope form,
[tex]y - y1 = m(x - x1)[/tex]
As an illustration, suppose we want to formulate the equation of the line passing through the coordinates (2, 5) and having a slope of 2. The values can be entered into the point-slope form as follows:
y - 5 = 2(x - 2)Let's say the given line has the equation [tex]y - y1 = m(x - x1)[/tex], where (x1, y1) is a point on the line and m is the slope of the line.
we can use the given point (2, 5). Then we can plug in the values into the point-slope form:
[tex]y - 5 = (-1/2)(x - 2).[/tex]
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solve the equation
a) y''-2y'-3y= e^4x
b) y''+y'-2y=3x*e^x
c) y"-9y'+20y=(x^2)*(e^4x)
Answer:
a) To solve the differential equation y''-2y'-3y= e^4x, we first find the characteristic equation:
r^2 - 2r - 3 = 0
Factoring, we get:
(r - 3)(r + 1) = 0
So the roots are r = 3 and r = -1.
The general solution to the homogeneous equation y'' - 2y' - 3y = 0 is:
y_h = c1e^3x + c2e^(-x)
To find the particular solution, we use the method of undetermined coefficients. Since e^4x is a solution to the homogeneous equation, we try a particular solution of the form:
y_p = Ae^4x
Taking the first and second derivatives of y_p, we get:
y_p' = 4Ae^4x
y_p'' = 16Ae^4x
Substituting these into the original differential equation, we get:
16Ae^4x - 8Ae^4x - 3Ae^4x = e^4x
Simplifying, we get:
5Ae^4x = e^4x
So:
A = 1/5
Therefore, the particular solution is:
y_p = (1/5)*e^4x
The general solution to the non-homogeneous equation is:
y = y_h + y_p
y = c1e^3x + c2e^(-x) + (1/5)*e^4x
b) To solve the differential equation y'' + y' - 2y = 3xe^x, we first find the characteristic equation:
r^2 + r - 2 = 0
Factoring, we get:
(r + 2)(r - 1) = 0
So the roots are r = -2 and r = 1.
The general solution to the homogeneous equation y'' + y' - 2y = 0 is:
y_h = c1e^(-2x) + c2e^x
To find the particular solution, we use the method of undetermined coefficients. Since 3xe^x is a solution to the homogeneous equation, we try a particular solution of the form:
y_p = (Ax + B)e^x
Taking the first and second derivatives of y_p, we get:
y_p' = Ae^x + (Ax + B)e^x
y_p'' = 2Ae^x + (Ax + B)e^x
Substituting these into the original differential equation, we get:
2Ae^x + (Ax + B)e^x + Ae^x + (Ax + B)e^x - 2(Ax + B)e^x = 3xe^x
Simplifying, we get:
3Ae^x = 3xe^x
So:
A = 1
Therefore, the particular solution is:
y_p = (x + B)e^x
Taking the derivative of y_p, we get:
y_p' = (x + 2 + B)e^x
Substituting back into the original differential equation, we get:
(x + 2 + B)e^x + (x + B)e^x - 2(x + B)e^x = 3xe^x
Simplifying, we get:
-xe^x - Be^x = 0
So:
B = -x
Therefore, the particular solution is:
y_p = xe^x
The general solution to the non-homogeneous equation is:
y = y_h + y_p
y = c1e^(-2x) + c2e^x + xe^x
c) To solve the differential equation y" - 9y' + 20y = x^2*e^4x, we first find the characteristic equation:
r^2 - 9r + 20 = 0
Factoring, we get:
(r - 5)(r - 4) = 0
So the roots are r = 5 and r = 4.
The general solution to the homogeneous equation y" - 9y' + 20y = 0 is:
y_h = c1e^4x + c2e^5x
To find the particular solution, we use the method of undetermined coefficients. Since x^2*e^4x is a solution to the homogeneous equation, we try a particular solution of the form:
y_p = (Ax^2 + Bx + C)e^4x
Taking the first and second derivatives of y_p, we get:
y_p' = (2Ax + B)e^4x + 4Axe^4x
y_p'' = 2Ae^4x +
The count in a bateria culture was initially 300, and after 35 minutes the population had increased to 1600. Find the doubling period. Find the population after 70 minutes. When will the population reach 10000?
. Bert has a well-shuffled standard deck of 52 cards, from which he draws one card; Ernie has a 12-sided die, which he rolls at the same time Bert draws a card. Compute the probability that:
a. Bert gets a Jack and Ernie rolls a five.
b. Bert gets a heart and Ernie rolls a number less than six.
c. Bert gets a face card (Jack, Queen or King) and Ernie rolls an even number.
d. Bert gets a red card and Ernie rolls a fifteen.
e. Bert gets a card that is not a Jack and Ernie rolls a number that is not twelve.
Therefore , the solution of the given problem of probability comes out to be a)1/78 ,b)65/624 ,c)1/4 ,d)0 and e)12/13.
What is probability, exactly?The basic goal of any considerations technique is to assess the probability that a statement is accurate or that a specific incident will occur. Chance can be represented by any number range between 0 and 1, where 0 normally indicates a percentage but 1 typically indicates the level of certainty. An illustration of probability displays how probable it is that a specific event will take place.
Here,
a.
P(Bert gets a Jack and Ernie rolls a five) = P(Bert gets a Jack) * P(Ernie rolls a five)
= (4/52) * (1/12)
= 1/78
b.
P(Bert gets a heart and Ernie rolls a number less than six) = P(Bert gets a heart) * P(Ernie rolls a number less than six)
= (13/52) * (5/12)
= 65/624
c.
P(Bert gets a face card and Ernie rolls an even number) = P(Bert gets a face card) * P(Ernie rolls an even number)
= (12/52) * (6/12)
= 1/4
d.
P(Bert gets a red card and Ernie rolls a fifteen) = 0
e.
Ernie rolls a number that is not twelve, and Bert draws a card that is not a Jack:
A regular 52-card deck contains 48 cards that are not Jacks,
so the likelihood that Bert will draw one of those cards is 48/52, or 12/13.
On a 12-sided dice with 11 possible outcomes,
Ernie rolls a non-12th-number (1, 2, 3, etc.).
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The monthly cost (in dollars) of a long-distance phone plan is a linear function of the total calling time (in minutes). The monthly cost for 35 minutes of calls is $16.83 and the monthly cost for 52 minutes is $18.87. What is the monthly cost for 39 minutes of calls?
Answer: We can use the two given points to find the equation of the line and then plug in 39 for the calling time to find the corresponding monthly cost.
Let x be the calling time (in minutes) and y be the monthly cost (in dollars). Then we have the following two points:
(x1, y1) = (35, 16.83)
(x2, y2) = (52, 18.87)
The slope of the line passing through these two points is:
m = (y2 - y1) / (x2 - x1) = (18.87 - 16.83) / (52 - 35) = 0.27
Using point-slope form with the first point, we get:
y - y1 = m(x - x1)
y - 16.83 = 0.27(x - 35)
Simplifying, we get:
y = 0.27x + 7.74
Therefore, the monthly cost for 39 minutes of calls is:
y = 0.27(39) + 7.74 = $18.21
Step-by-step explanation:
The five number summary of a dataset was found to be:
45, 52, 56, 63, 66
An observation is considered an outlier if it is below:
An observation is considered an outlier if it is above:
An observation is considered an outlier if it is below 35.5 or above 79.5 in this dataset.
Identifying the outliers in the summaryTo determine the outliers in a dataset using the five-number summary, we need to calculate the interquartile range (IQR), which is the difference between the third quartile (Q3) and the first quartile (Q1).
IQR = Q3 - Q1
Where
Q1 = 52
Q3 = 63
So, we have
IQR = 63 - 52
IQR = 11
An observation is considered an outlier if it is:
Below Q1 - 1.5 × IQR
Above Q3 + 1.5 × IQR
Substituting the values, we get:
Below 52 - 1.5 × 11 = 35.5
Above 63 + 1.5 × 11 = 79.5
Therefore, an observation is considered an outlier if it is below 35.5 or above 79.5 in this dataset.
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See the photo below
This problem involves integration and algebraic manipulation, and belongs to the subject of calculus. The solutions are:
[tex]A) $\int_{0}^{2} (f(x) + g(x)) dx = -3$[/tex]
[tex]B) $\int_{0}^{3} (f(x) - g(x)) dx = -4$[/tex]
[tex]C) $\int_{2}^{3} (3f(x) + g(x)) dx = -32$[/tex]
This is a problem that asks us to find the values of some definite integrals using given values of other definite integrals. We are given three definite integrals, and we are asked to compute three other integrals involving the same functions, using the given values.
The problem involves some algebraic manipulation and the use of the linearity of the integral.
It also involves finding the constant "a" that makes a definite integral equal to zero. The integral involves two functions, "f(x)" and "g(x)," whose definite integrals over certain intervals are also given.
See the attached for the full solution.
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