To derive f(x) = -67x, we can use the Power Rule.
[tex]x^n\Rightarrow nx^{n-1}[/tex]In the given term, our n = 1 since x¹ = x. So, following the power rule, we will multiply the exponent 1 to the constant term -67, then subtract 1 from the exponent 1, hence x¹ will become x⁰.
[tex]-67x^1\Rightarrow1(-67)(x^{1-1})[/tex]Then, simplify.
[tex]-67x^0\Rightarrow-67(1)=-67[/tex]Therefore, the first derivative of f(x) = -67x is -67.
[tex]f^{\prime}(x)=-67[/tex]Use the strategy to simplify 4/576Write the prime factorization of the radicand.442834O42/2832O 4./283²O4. 2882
To simplify the fraction we will need to facto
Two Way Tables, URGENT
Step-by-step explanation:
a) modal number is 3
b) mean is x = ∑fx/n
= ((5•1)+ (2•10)+(3•15)+(7•4)+(3•5))/(5+10+15+7+3)
= 113/40
= (Decimal: 2.825)
What are the explicit and recursive formulas for the sequence 540, 180, 60, 20, ...?
Here we have a geometric sequence, the recursive formula is:
Aₙ = (1/3)*Aₙ₋₁
And the explicit formula is:
Aₙ = (1/3)*ⁿ⁻¹*540
How to get the recursive formula?
Here we have the following sequence:
540, 180, 60, 20, ...
This seems to be a geometric sequence, to check this, we need to take the quotients between consecutive terms and see if we get the same thing.
180/540 = 1/3
60/180 = 1/3
20/60 = 1/3
So yes, this is a geometric sequence where the common ratio is 1/3, so each term is (1/3) times the previous one, so the recursive formula is:
Aₙ = (1/3)*Aₙ₋₁
And the explicit formula is:
Aₙ = (1/3)*ⁿ⁻¹*A₁
Where A₁ is the first term, in this case 540, so the formula becomes:
Aₙ = (1/3)*ⁿ⁻¹*540
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6. A profit function for a new business follows the functionP(x) = 1/3x^2 - 6x, where x represents the number of months.After how many months will the company begin to make aprofit?A. 2B. 9C. 12D. 18
ANSWER
It will take 18 months before the company starts making a profit.
STEP-BY-STEP EXPLANATION
Given information
[tex]P(x)\text{ = }\frac{1}{3}x^2\text{ - 6x}[/tex]Where x is the number of months.
Step 1: Make P(x) = 0
[tex]\begin{gathered} \text{ p(x) = }\frac{1}{3}x^2\text{ - 6}x \\ 0\text{ = }\frac{1}{3}x^2\text{ - 6}x \end{gathered}[/tex]Step 2: Find x from the above equation
[tex]\begin{gathered} 0\text{ = }\frac{1}{3}x^2\text{ - 6x} \\ \text{Add 6x to the both sides} \\ 0\text{ + 6x = }\frac{1}{3}x^2\text{ - 6x + 6x} \\ 6x\text{ = }\frac{1}{3}x^2 \\ \text{cross multiply} \\ 6x\text{ }\times3=x^2 \\ 18x=x^2 \\ \text{Divide both sides by x} \\ \frac{18\cancel{x}}{\cancel{x}}\text{ = }\frac{\cancel{x^2}}{\cancel{x}} \\ x\text{ = 18 months} \end{gathered}[/tex]Therefore, it will take 18 months before the company starts making a profit.
Which of the following is equal to the rational expression below when x+112x² – 121x +11A. +11B.X+ 11c. -11XD. X-11
SOLUTION
From the question we have
[tex]\frac{x^2-121}{x+11}[/tex]from difference of two squares, we have
[tex]\begin{gathered} \frac{(x-11)(x+11)}{x+11} \\ x+11\text{ above cancels the one below, we have } \\ x-11 \end{gathered}[/tex]Hence the answer is option D
Please help I need to graph this and i can only have two points
Given the function:
[tex]f\mleft(x\mright)=\mleft(x+2\mright)\mleft(x-4\mright)[/tex]You can rewrite it as follows:
[tex]y=\mleft(x+2\mright)\mleft(x-4\mright)[/tex]You need to remember that the y-value is zero when the function intersects the x-axis. Then, you need to make it equal to zero, in order to find the x-intercepts:
[tex]\begin{gathered} 0=\mleft(x+2\mright)\mleft(x-4\mright) \\ (x+2)(x-4)=0 \end{gathered}[/tex]Solving for "x", you get these two values:
[tex]\begin{gathered} x+2=0\Rightarrow x_1=-2 \\ \\ x-4=0\Rightarrow x_2=4 \end{gathered}[/tex]In order to find the vertex, you can follow these steps:
1. Find the x-coordinate of the vertex with this formula:
[tex]x=-\frac{b}{2a}[/tex]To find the value of "a" and "b", you need to multiply the binomials of the equation using the FOIL Method. This states that:
[tex]\mleft(a+b\mright)\mleft(c+d\mright)=ac+ad+bc+bd[/tex]Then, in this case, you get:
[tex]\begin{gathered} y=(x)(x)-(x)(4)+(2)(x)-(2)(4) \\ y=x^2-4x+2x-8 \end{gathered}[/tex]Add the like terms:
[tex]y=x^2-2x-8[/tex]Notice that, in this case:
[tex]\begin{gathered} a=1 \\ b=-2 \end{gathered}[/tex]Then, you can substitute values into the formula and find the x-coordinate of the vertex of the parabola:
[tex]x=-\frac{(-2)}{2\cdot1}=-\frac{(-2)}{2}=1[/tex]2. Substitute that value of "x" into the function and then evaluate, in order to find the y-coordinate of the vertex:
[tex]\begin{gathered} y=x^2-2x-8 \\ y=(1)^2-2(1)-8 \\ y=1-2-8 \\ y=-9 \end{gathered}[/tex]Therefore, the vertex of the parabola is:
[tex](1,-9)[/tex]Knowing the x-intercepts and the vertex of the parabola, you can graph it.
Hence, the answer is:
Given ARPMAYC. complete each of the following statementsAYC9) AMPKAYAa) A d) 4YEb) CYe) EKc) PAZACYh) Al Aca
The triangles KMP and AYC, are congruent angles, because we are told that:
[tex]\text{KMP}\cong AYC[/tex]Thus, to find our answers we compare both triangles.
a) KM is approximately equal to AC.
That is because, as we can see in the following image, they are corresponding sides:
[tex]KM\cong AC[/tex]b) CY is approximately equal to MP.
That is because they are corresponding sides, they are the short side of each triangle. And since the triangles are equal, CY and MP are also equal:
[tex]CY\cong MP[/tex]c) for the same reasons as the previous two, PK anf AY, are corresponding sides:
[tex]PK\cong AY[/tex]d) and e)
The corresponding angles of Y and K are represented in the following image:
rrespon
What is the position of see on the number line belowWrite your answer as a fraction or mixed number
Answer:
1/3
Explanation:
We can see that from 0 to 1 the number line is divided into 6 parts and the point is right after the second part. Therefore, the fraction that represents point C is 2/6
This fraction is also equal to 1/3 because we can divide the line from 0 to 1 into 3 parts and take the first. The point will be at the exact same position of C.
Therefore, the answer is:
1/3
A population grows according to an exponential growth model. The initial population is 224 and the population after one year is 263. Complete the formula where P is the population and n is the number of years.: P=224*(___)n
Round your answer to three decimal places.
The equation of the population function is is P = 224(1.17)ⁿ
How to complete the equation?From the question, the given parameters are:
Initial population = 224Population after one year = 263The above parameters imply that the rate of change of the population every year is
Rate = Population after one year/Initial population
Substitute the known values in the above equation
So, we have
Rate = 263/224
Evaluate the quotient
Rate = 1.17
The exponential function can be represented as
P = Initial population * (Rate)ⁿ
So, we have
P = 224(1.17)ⁿ
Hence, the complete equation is P = 224(1.17)ⁿ
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The required rate of the increase in the population is 17% and the equation is fulfilled as P = 224 (1.17)ⁿ.
As per the question, the population of the 2 years are given as 224 and the preceding year's population is 263 and equation is illustrated the exponential growth is given as P = 224(__)ⁿ. The blank space in the equation is to be filled.
The function which is in format f(x) =aˣ where a is constant and x is variable, the domain of this exponential function lies (-∞, ∞).
Here,
The given function of the population is incomplete as rate of growth is missing, So,
The rate is given as,
Rate = 263 - 224 / 224
Rate = 0.17 or 17%
Growth = 1 + 0.17 = 1.17
now, put this growth rate in the blank space.
So,
P = 224 (1.17)ⁿ
Thus, the required rate of the increase in the population is 17% and the equation is fulfilled as P = 224 (1.17)ⁿ.
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A soup can has a radius of 4.3 cm and a height of 11.6 cm. What is the volume of the soup can to the nearest tenth of a cubic centimeter?A. 1816.8B. 49.9C. 168.4D. 673.8
hello
to solve this problem, we need to identify the shape of the soup can first since soup is a liquid and carries the shape of whatever container its in.
volume of a cylinder is given as
[tex]\begin{gathered} V=\pi r^2h \\ \pi=3.142 \\ r=\text{radius} \\ h=\text{height} \end{gathered}[/tex][tex]\begin{gathered} v=\text{ ?} \\ r=4.3\operatorname{cm} \\ h=11.6\operatorname{cm} \\ \pi=3.142 \\ v=\pi r^2h \\ v=3.142\times4.3^2\times11.6 \\ v=673.9\operatorname{cm}^3 \end{gathered}[/tex]from the calculations above, the volume of the soup is equal to 673.9cm^3 which corresponds with option D
I need help I need help I need help I need help I need help i need help I need help
Answer:
5) The midrange is 19.5ºF
6) The midrange is 67.5º
Explanation:
The problem tell us how to calculate the midrange.
In (5) the minimum and maximum values are given (-6ºF and 45ºF, respectively). Using the formula:
[tex]Midrange=\frac{-6+45}{2}=\frac{39}{2}=19.5ºF[/tex]In (6), we need to find the minimum and maximum values from a list of them. We can see that the minimum is 58º and the maximum 77º
Then:
[tex]Midrange=\frac{58+77}{2}=\frac{135}{2}=67.5º[/tex]lineal or no?1) 2x+y=52) y= x + 6 --- 2thanks
1) 2x+y=5 ...... It is a linear equation
2) y= x + 6 ....... It is a linear equation
Because they are of first degree and they contain x and y (equations of a line)
Determine the value of b for which x = 1 is a solution of the equation shown.
2x + 14 = 10x + b
B=
The linear equation has the solution x = 1 only if the value of b is 6
For which value of b is x = 1 a solution?
Here we have the linear equation:
2x + 14 = 10x + b
If we replace x by 1 in that equation, we will get:
2*1 + 14 = 10*1 + b
2 + 14 = 10 + b
16 = 10 + b
To find the value of b such that x = 1 is a solution, we need to isolate b, to do so we need to subtract 10 in both sides.
16 - 10 = 10 + b - 10
6 = b
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15.) In the accompanying diagram, ABC is a straight line and BE bisects 4DBC. If m4ABD = 2x and m4DBE = 2x + 15, find m&ABD.
Using bisection, the measure of angle ABD is of m<ABD = 50º.
What is the bisection of an angle?The bisection of an angle is when the angle is divided into two angles of equal measure.
In the context of this problem, we have that the angle BE bisects the angle DBC, hence the measures of these angles are given as follows:
mDBE = mEBC = 2x + 15.
As shown in the diagram, the entire line forms a ray, meaning that the sum of the measures of the angles is of 180º, hence we can solve for x as follows:
2x + 2(2x + 15) = 180º
2x + 4x + 30 = 180º
6x = 150º
x = 150º/6
x = 25º.
Then the measure of angle ABD is found as follows:
m<ABD = 2x = 2(25) = 50º.
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determine whether AB and AC are parallel,perpendicular,or neither.A(9,-3) , B(9,4), C(-2,10), D(-2,6)
We first determine the value of AB & CD:
AB (0, 7)
CD (0, -4)
We will calculate first if they are perpendicular:
[tex](0,7)\cdot(0,-4)=0\cdot0+(7)(-4)=-28\ne0[/tex]From this, we know AB and CD are not perpendicular.
Now, in order to know if they are parallel, we will do as follows:
[tex](0,7)x=(0,-4)[/tex]From this, we will have:
[tex](0,7x)=(0,-4)\Rightarrow7x=-4\Rightarrow x=-\frac{4}{7}[/tex]From this we have that they are multiple of each other, therefore they are parallel.
Write the equation of a line the goes through point
(3,-4) and is perpendicular to the line x = 1.
keeping in mind that x = 1 is just a vertical line, Check the picture below.
How many solutions does the equation 5(m + 3) = 6-7m have? Explain how you found your answer.
Expand the left hand side using distributive property:
[tex]\begin{gathered} 5\cdot m+5\cdot3=6-7m \\ 5m+15=6-7m \\ \text{Add 7m to both sides:} \\ 5m+15+7m=6-7m+7m \\ 12m+15=6 \\ \text{subtract 15 from both sides:} \\ 12m+15-15=6-15 \\ 12m=-9 \\ \text{divide both sides by 12:} \\ \frac{12}{12}m=-\frac{9}{12} \\ m=-\frac{3}{4} \end{gathered}[/tex]In 2009, there were 6.1 million females enrolled in degree granting institutions of higher education. over the next several years this number increased at a rate of 400,000 per year. estimate the number of females enrolled in 2024. y = ______ millionthe equation of the line that models this information is;y = 0.4t + 6.1Determine what year 12.9 million females will be enrolled.
Notice that
400,000 = 0.4 million
That's why the equation that models that information has the factor 0.4, since it expresses the result in millions of females.
Now, we need to notice that t, in the expression 0.4t + 6.1, is the number of years passed since 2009. So, in the year 2024, we have:
t = 2024 - 2009 = 15
Therefore, the number of females enrolled in 2024 can be estimated to be:
y = (0.4 * 15 + 6.1) million
y = (6 + 6.1) million
y = 12.1 million
Now, to determine the year when 12.9 million females will be enrolled, we first need to find t corresponding to y = 12.9, and then add it to the year 2009.
y = 0.4t + 6.1
12.9 = 0.4t + 6.1
12.9 - 6.1 = 0.4t
6.8 = 0.4t
t = 6.8/0.4
t = 68/4
t = 17
Therefore, the year when it happens will be:
2009 + 17 = 2026
In a factory, the profit, P, varies directly with the inventory, I. If the factory has a profit of $60,000 when their inventory is 1,500 units, find the profit for an inventory of 50 units.
The factory has a profit of $60 000when their inventory is 1, 500 units
Let x be the profit for an inventory of 50 units
$60 000 = 1,500 units
X = 50 units
cross-multiply
1500X = $60 000 x 50
1500X =3,000,000
Divide both-side of the equation by 1500
1500X/1500 = 3,000,000/1500
x= $2000
The factory has a profit of $2000 for an inventory of 50 units
The Brock family uses up a
1
2
-gallon jug of milk every 3 days. At what rate do they drink milk?
Simplify your answer and write it as a proper fraction, mixed number, or whole number.
The rate at which the Brock family drinks milk is 4 gallon per day.
According to the question,
We have the following information:
The Brock family uses 12 gallon jug of milk every 3 days.
Now, in order to find the rate at which they drink milk, we will have to divide the amount of milk by the total number of days.
So, we have the following expression:
Rate at which they drink milk = 12/3 galloon per day
Rate at which they drink milk = 4 galloon per day
Now, this the simplified answer because this is a whole number and we can not solve it further.
Hence, the rate at which they drink milk is 4 galloon per day.
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Figure L is the result of a transformation on Figure K. Which transformation would accomplish this? FigureL Figurek 5 4 2 2 -4
Answer is the first option, a reflection over the y-axis
Consider the line y=7x-1Find the equation of the line that is perpendicular to this line and passes through the point −2, 3.Find the equation of the line that is parallel to this line and passes through the point −2, 3.Note that the ALEKS graphing calculator may be helpful in checking your answer.Equation of per pendicular line:Equation of parallel line:
Algebra / Graphs and Functions / Equations of Parallel and Perpendicular Lines
We have the line:
[tex]y=7x-1.[/tex]We must find the equation:
0. of the perpendicular line,
,1. and the parallel line,
to the given line that passes through the point (-2, 3).
1) Perpendicular line
The equation of the perpendicular line has the form:
[tex]y=m_p\cdot(x-x_0)+y_0.[/tex]Where mₚ is the slope, and (x₀, y₀) = (-2, 3).
From the equation of the given line, we see that its slope is m = 7. The slope of the perpendicular line mₚ is given by the equation:
[tex]\begin{gathered} m\cdot m_p=-1, \\ 7\cdot m_p=-1, \\ m_p=-\frac{1}{7}. \end{gathered}[/tex]Replacing mₚ = -1/7 and (x₀, y₀) = (-2, 3) in the equation of the perpendicular line, we get:
[tex]y=-\frac{1}{7}\cdot(x-(-2))+3=-\frac{1}{7}\cdot(x+2)+3=-\frac{1}{7}\cdot x-\frac{2}{7}+3=-\frac{1}{7}\cdot x+\frac{19}{7}.[/tex]2) Parallel line
The equation of the perpendicular line has the form:
[tex]y=m_p\cdot(x-x_0)+y_0.[/tex]Where mₚ is the slope, and (x₀, y₀) = (-2, 3).
From the equation of the given line, we see that its slope is m = 7. The parallel line has the same slope as the given line, so we have:
[tex]\begin{gathered} m_p=m, \\ m_p=7. \end{gathered}[/tex]Replacing mₚ = 7 and (x₀, y₀) = (-2, 3) in the equation of the parallel line, we get:
[tex]y=7\cdot(x-(-2))+3=7\cdot(x+2)+3=7x+14+3=7x+17.[/tex]3) Graph
Plotting the equations obtained, we get the following graph:
Answer1) Equation of the perpendicular line:
[tex]y=-\frac{x}{7}+\frac{19}{7}[/tex]2) Equation of the parallel line:
[tex]y=7x+17[/tex]Do the side measures 35 mm, 53 mm and 70 mm create a triangle?Yes, there are infinitely many triangles that can be created.No, it is impossible to create a triangle with the given measures.Yes, there is a unique triangle that can be created.Yes, there are two triangles that can be created.
Hello!
Let's call these sides a, b, and c:
• a = 35mm
,• b = 53mm
,• c = 70mm
To be a triangle, it must satisfy the existence condition of triangles, that is:
Let's check each of them:
[tex]\begin{gathered} |b-c|Answer:
Yes! There are infinitely many triangles that can be created.
Please helpwhat does A∩B=∅ mean. Thus, please help with:Suppose Pr(A)=0.3, Pr(B)=0.4 and A∩B=∅. Find:a- Pr(A∩B)b- Pr(A∪B)
Given: A and B are two sets such that-
[tex]\begin{gathered} A\cap B=\phi \\ Pr(A)=0.3 \\ Pr(B)=0.4 \end{gathered}[/tex]Required: To determine-
[tex]\begin{gathered} Pr(A\cap B) \\ Pr(A\cup B) \end{gathered}[/tex]Explanation: Since A and B have no common elements, the events are independent events or disjoints or mutually exclusive.
For independent events, we have-
[tex]Pr(A\cap B)=Pr(A).Pr(B)[/tex]Substituting the values into the formula-
[tex]\begin{gathered} Pr(A\cap B)=0.3\times0.4 \\ =0.12 \end{gathered}[/tex]Recall that-
[tex]Pr(A\cup B)=Pr(A)+Pr(B)-Pr(A\cap B)[/tex]Substituting the values into the formula and further solving as-
[tex]\begin{gathered} Pr(A\cup B)=0.3+0.4-0.12 \\ =0.7-0.12 \\ =0.58 \end{gathered}[/tex]Final Answer: a)
[tex]Pr(A\cap B)=0.12[/tex]b)
[tex]Pr(A\cup B)=0.58[/tex]I need help with this question Write and expression that models the situation:Sarah has spent x dollars out of the 30 dollars she started with.
Okay, here we have this:
Considering that it says "spent", it represents an outflow of money, therefore we take it as negative, so we obtain:
Actual Situation: Initial money - money spent
Actual Situation: 30 - x
Solve the inequality and graph the solution set.3 ≤ 4x + 1 < 9
Okay, here we have this:
Considering the provided inequality, we are going to solve it and graph the solution set, so we obtain the following:
3 ≤ 4x + 1 < 9
3 -1≤ 4x + 1 -1< 9-1
2 ≤ 4x < 8
2/4 ≤ 4x/4 < 8/4
1/2 ≤ x < 2
In interval notation the solution set will be: [1/2, 2)
And if we plot this solution interval we get:
Where the solution set will be the purple part.
Evaluate.
34+(12+14)2⋅2
Enter your answer as a mixed number in simplest form by filling in the boxes.
Answer:
=1.875
Step-by-step explanation:
Add: 1/
2
+ 1/
4
= 1 · 2/
2 · 2
+ 1/
4
= 2/
4
+ 1/
4
= 2 + 1/
4
= 3/
4
It is suitable to adjust both fractions to a common (equal, identical) denominator for adding, subtracting, and comparing fractions. The common denominator you can calculate as the least common multiple of both denominators - LCM(2, 4) = 4. In practice, it is enough to find the common denominator (not necessarily the lowest) by multiplying the denominators: 2 × 4 = 8. In the following intermediate step, it cannot further simplify the fraction result by canceling.
In other words - one half plus one quarter is three quarters.
Exponentiation: No. 1 ^ 2 = 3/
4
^ 2 = 32/
42
= 9/
16
In other words - three quarters raised to the power of squared is nine sixteenths.
Multiple: No. 2 * 2 = 9/
16
* 2 = 9 · 2/
16 · 1
= 18/
16
= 9 · 2/
8 · 2
= 9/
8
Multiply both numerators and denominators. Result fraction keep to lowest possible denominator GCD(18, 16) = 2. In the following intermediate step, cancel by a common factor of 2 gives 9/
8
.
In other words - nine sixteenths multiplied by two is nine eighths.
Add: 3/
4
+ the result of step No. 3 = 3/
4
+ 9/
8
= 3 · 2/
4 · 2
+ 9/
8
= 6/
8
+ 9/
8
= 6 + 9/
8
= 15/
8
It is suitable to adjust both fractions to a common (equal, identical) denominator for adding, subtracting, and comparing fractions. The common denominator you can calculate as the least common multiple of both denominators
The vertex of a quadratic function is (2, -1) and its y-intercept is 7. Find the function,
Given:-
[tex]\text{vertex}=(2,-1),y-intercept=7[/tex]To find:-
The function.
So the formula is,
[tex]y=a\mleft(x-h\mright)^{2}+k[/tex]So substituting we get,
[tex]y=7(x-2)^2-1[/tex]So the value. we get,
[tex]\begin{gathered} y=7(x-2)^2-1 \\ y=7(x^2-4x+4)-1 \end{gathered}[/tex]Since the value of x is,
[tex]\begin{gathered} y=7x^2-28x+28-1 \\ y=7x^2-28x+27 \end{gathered}[/tex]So the value,
[tex]y=7x^2-28x+27[/tex]-ractions:
On a website, there is an ad for jeans every 5 minutes, an ad for sneakers
every 10 minutes, and an ad for scarves every 45 minutes.
If they all appeared together at 9:00 P.M., when is
the next time they will all appear together?
ICM to solve the problem
Answer:
Step-by-step explanation:
Determine the reasonableness of a solution to a logarithmic equation
SOLUTION
Given the question in the image, the following are the solution steps to answer the question.
STEP 1: Write the given equation
[tex]\log_3x=7[/tex]STEP 2: State the law of logarithm
[tex]\begin{gathered} If\text{ }\log_ab=c \\ \Rightarrow b=a^c \\ By\text{ substitution,} \\ \therefore\log_aa^c=c \end{gathered}[/tex]STEP 3: Substitute the given values in the question to get the correct answer
[tex]\begin{gathered} \log_3x=7 \\ x=3^7 \\ By\text{ substitution,} \\ \log_3(3^7)=7 \end{gathered}[/tex]Hence, Answer is:
[tex]\log_3(3^7)=7[/tex]OPTION A