It took Rapunzel approximately 14.58 minutes to clean her room.
What is cleaning ?
Cleaning refers to the process of removing dirt, dust, and other unwanted substances from a surface or an object.
Let's start by converting the time to minutes.
15 1/2 minutes = 15.5 minutes
Let's now subtract the time she finished cleaning from the time she started cleaning:
15.5 - 0.92 = 14.58 minutes
Therefore, it took Rapunzel approximately 14.58 minutes to clean her room.
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what is the answer of 10.9% of $8.85
Answer:
To find 10.9% of $8.85 we can convert the percentage into a decimal:
10.9% = 0.109
0.109x8.85 = $0.964
=$0.96 (Rounded to 2 decimal places)
Find the area of the shaded sector of the circle
Answer:
32.67 square meters
Step-by-step explanation:
finding the area of the shaded region.
area of sector = (θ/360°) x πr²
where "θ" is the central angle of the sector in degrees, "r" is the radius of the sector, and π is a mathematical constant approximately equal to 3.14.
Substituting the given values into the formula, we get:
area of sector = (60°/360°) x π(14m)²
area of sector = (1/6) x 3.14 x 196m²
area of sector = 32.67m² (rounded to two decimal places)
Therefore, the area of the section is 32.67 square meters
State any excluded values of a domain and identify the type of break in the graph at the value of x
f(x)=x+7/x-2
Answer:
The excluded value in the domain is x = 2 because the denominator of the rational function becomes zero at this value of x, which results in division by zero.
At x = 2, there is a vertical asymptote, which means that the graph of the function approaches positive or negative infinity as x approaches 2 from either side.
A circular flower bed is 20 m in diameter and has a circular sidewalk around it that is 2m wide. Find the area of the sidewalk in square meters. Use 3.14 for
According to the solving the area of the sidewalk is 138.16 square meters.
What is an area?Size of a location on a surface is expressed in terms of area. As opposed to surface area, which refers to the area of an open surface or the border of a three-dimensional object, plane region or plane size refers to the area of a shape or planar lamina.
According to the given information:The width of the circular sidewalk is 2 m, which means that the radius of the entire area, including the flower bed and the sidewalk, is 10 m + 2 m = 12 m.
To find the area of the sidewalk, we need to subtract the area of the flower bed from the area of the larger circle.
Area of larger circle = πr²
= 3.14 x 12²
= 452.16 square meters (rounded to two decimal places)
Area of flower bed = πr²
= 3.14 x 10²
= 314 square meters
Area of the sidewalk = Area of larger circle - Area of flower bed
= 452.16 - 314
= 138.16 square meters (rounded to two decimal places)
Therefore, the area of the sidewalk is 138.16 square meters.
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is 180. The sum of the measures of the second and third angles is five times the measure of the first angle. The third angle is 26 more than the second. Let x, y, and z represent the measures of the first, second, and third angles, respectively. Find the measures of the three angles.
Answer:
x is first angle
y is second angle
and z is third angle
Step-by-step explanation:
This question is solved by a system of equations. We have that:x is the first angle.y is the second angle.z is the third angle.Doing this, we get that:The first angle measures 30º.The second angle measures 67º.The third angle measures 83º.The sum of the measures of the angles of a triangle is 180. This means that The sum of the measures of the second and third angles is five times the measure of the first angle.This means that:From this, the first angle can be found:The measure of the first angle is of 30º.The third angle is 16 more than the second.This means that:Since We get that the second angle is:The second angle measures 67º.For the third angle:The third angle measures 83º.
please help immediately
please go to my profile and answer the other I need them asap.
10³•10⁵•10³
Step-by-step explanation:
10³means 10×3,10⁵means 10×5and 10³means 10×3 30+50+30=110
Jen is studying how years of drought conditions have caused the water level of Richland Reservoir to drop. At the start of the study, the water in the reservoir was 65 meters deep. Jen observed that the depth of the water dropped by about 0.8 meters the first month of the study. She wants to know what the depth of the water will be if it continues dropping at the same rate. You can use a function to approximate the depth of the water in the reservoir x months after the start of the study. Write an equation for the function.
The equation for the function is D(x) = 65 - 0.8x. Where 65 is the initial depth of the water and 0.8x is the amount by which the depth drops after x months.
What is a linear function?
A linear function is a mathematical function that has a constant rate of change or slope between the independent variable (x) and the dependent variable (y). It is a function that can be graphically represented as a straight line.
We can use a linear function to approximate the depth of the water in the reservoir x months after the start of the study, since the depth is dropping at a constant rate of 0.8 meters per month. Let D(x) be the depth of the water in meters x months after the start of the study. Then we have:
D(x) = 65 - 0.8x
where 65 is the initial depth of the water and 0.8x is the amount by which the depth drops after x months.
Therefore, the equation for the function is D(x) = 65 - 0.8x.
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The two top concert tours in 2016 were concert A and concert B. Based on average ticket prices, it cost a total of $1707 to purchase six tickets for concert A and six tickets for concert B. Three tickets for concert B cost a total of $687. How much did an average ticket cost for each tour?
The average ticket cost for each concert is given as follows:
Concert A: $188.83.Concert B: $95.67.How to obtain the ticket costs?The ticket costs are obtained by a system of equations, for which the variables are given as follows:
Variable a: cost for Concert A.Variable b: cost for Concert B.It cost a total of $1707 to purchase six tickets for concert A and six tickets for concert B, hence:
6a + 6b = 1707
a + b = 284.5.
Three tickets for concert B cost a total of $687, hence the cost for concert B is of:
3b = 687
b = 287/3
b = $95.67.
Replacing into the first equation, the cost for concert A is given as follows;
a = 284.5 - 95.67
a = $188.83.
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A line has the equation y - 6 = 5x + 9 . work out the gradient and the y intercept of the line.
The gradient of the line is 5, and the y-intercept of the line is 15.
EquationsThe given equation is in the form of y = mx + c, where m is the gradient (slope) of the line and c is the y-intercept of a straight line represented in 2D plane.
Rearranging the given equation, we get:
y - 6 = 5x + 9
Adding 6 to both sides, we get:
y = 5x + 15
Now we can see that the equation is in the required form of y = mx + c. The gradient (slope) of the line is 5, which is the coefficient of x in the equation. The y-intercept is 15, which is the constant term in the equation.
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COMM 291C Applications of Statistics in Business
Winter 2023
Assignment 07
1. A fish tank is filled with the fish of three different species, and each fish comes in three sizes. The
counts of all the fish are summarized in the following contingency table:
Small Medium Large Total
Goldfish 48 45 19 112
Guppy 70 42 19 131
Gourami 40 9 10 59
Total 158 96 48 302
a. If I randomly catch any one fish from this tank, what is the probability that the caught
fish is medium-sized? Show your calculations. (1)
b. If I randomly catch one fish from this tank, what is the probability that the caught fish is a
medium-sized guppy? Show your calculations. (1)
c. If I randomly catch one medium-sized fish from this tank, what is the probability that the
caught fish is a guppy? Show your calculations. (1)
d. If I randomly catch one guppy from this tank, what is the probability that the caught fish
is medium-sized? Show your calculations. (1)
e. If I randomly catch one fish from this tank, what is the probability that the caught fish is a
not a goldfish? Show your calculations. (1)
f. Are two events, randomly catching a guppy and randomly catching a medium-sized fish,
independent? Explain how you know. (2)
a) If I randomly catch any one fish from this tank, the probability of catching a medium-sized fish from the tank is 0.3182.
b) the probability of catching a medium-sized guppy from the tank is 0.1391.
c) the probability of catching a guppy if we randomly select a medium-sized fish from the tank is 0.4375.
d) the probability of catching a medium-sized guppy if we randomly select a guppy from the tank is 0.3206.
e) the probability of catching a fish that is not a goldfish if we randomly select a fish from the tank is 0.6291.
f) 0.1384 is not equal to 0.1391, which indicates that the two events are not independent
Probability is a measure of the likelihood or chance of an event occurring, expressed as a number between 0 and 1, with 0 indicating impossibility and 1 indicating certainty.
The explanation to the above probability answer are given below:
a)
From the contingency table, we can see that the total number of medium-sized fish is 96. The total number of fish in the tank is 302. Therefore, the probability of catching a medium-sized fish is:
Probability of catching a medium-sized fish = Total number of medium-sized fish / Total number of fish
Probability of catching a medium-sized fish = 96 / 302
Probability of catching a medium-sized fish = 0.3182
Therefore, the probability of catching a medium-sized fish from the tank is 0.3182
b)
From the contingency table, we can see that the total number of medium-sized guppies is 42. The total number of fish in the tank is 302. Therefore, the probability of catching a medium-sized guppy is:
Probability of catching a medium-sized guppy = Total number of medium-sized guppies / Total number of fish
Probability of catching a medium-sized guppy = 42 / 302
Probability of catching a medium-sized guppy = 0.1391
Therefore, the probability of catching a medium-sized guppy from the tank is 0.1391.
c)
From the contingency table, we can see that the number of medium-sized guppies is 42, and the number of medium-sized fish is 96. Therefore, the probability of catching a guppy if we randomly select a medium-sized fish is:
Probability of catching a guppy given a medium-sized fish = Number of medium-sized guppies / Total number of medium-sized fish
Probability of catching a guppy given a medium-sized fish = 42 / 96
Probability of catching a guppy given a medium-sized fish = 0.4375
Therefore, the probability of catching a guppy if we randomly select a medium-sized fish from the tank is 0.4375.
d)
To calculate the probability of catching a medium-sized guppy if we randomly select a guppy from the tank, we need to find the number of medium-sized guppies in the tank and divide it by the total number of guppies in the tank.
From the contingency table, we can see that the number of medium-sized guppies is 42, and the total number of guppies is 131. Therefore, the probability of catching a medium-sized guppy if we randomly select a guppy from the tank is:
Probability of catching a medium-sized guppy given a guppy = Number of medium-sized guppies / Total number of guppies
Probability of catching a medium-sized guppy given a guppy = 42 / 131
Probability of catching a medium-sized guppy given a guppy = 0.3206
Therefore, the probability of catching a medium-sized guppy if we randomly select a guppy from the tank is 0.3206.
e)
From the contingency table, we can see that the total number of non-goldfish is 190 (45 medium goldfish + 70 small guppy + 42 medium guppy + 9 medium gourami + 10 large gourami + 14 small gourami). The total number of fish in the tank is 302. Therefore, the probability of catching a fish that is not a goldfish is:
Probability of catching a fish that is not a goldfish = Total number of non-goldfish / Total number of fish
Probability of catching a fish that is not a goldfish = 190 / 302
Probability of catching a fish that is not a goldfish = 0.6291
Therefore, the probability of catching a fish that is not a goldfish if we randomly select a fish from the tank is 0.6291.
f)
From the contingency table, we can see that the probability of randomly catching a guppy is 131/302, which is approximately 0.4344. Similarly, the probability of randomly catching a medium-sized fish is 96/302, which is approximately 0.3182.
Now, let's consider the joint probability of randomly catching a medium-sized guppy, which we calculated earlier to be 42/302, or approximately 0.1391. To determine if the events are independent, we need to compare the product of the probabilities of the individual events (catching a guppy and catching a medium-sized fish) to the joint probability of the events occurring together.
If the events are independent, then the product of their individual probabilities should be equal to the joint probability:
P(catching a guppy) x P(catching a medium-sized fish)
= P(catching a medium-sized guppy)
0.4344 x 0.3182 = 0.1391
0.1384 is not equal to 0.1391, which indicates that the two events are not independent.
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PLS HELP MEEEEEEEEEEEEEEEEEE I CAN'T ANSWER
A. Tell whether distance, speed, or time is asked in each given situation.
_____1. How long will it take Zihann to arrive at school if she walks a constants 2 kilometers per hour?
_____2. How far do you need to travel at an average speed for 5 hours?
_____3. How fast should you travel to arrive at your destination that is 2.5 hours away?
_____4. If Carla travels a distance of 150 kilometers and arrive after 3 hours, how fast did she drive?
_____5. If Alwyn drove at an average speed to travel a certain distance, how long did it take him to travel?
what is required for a current to flow?
A. Electrons, protons, and an energy source
B. An insulator, battery, and recipient
C. A recipient, a connection, and something made out of rubber
D. An Energy source, a recipient, and a connection
Therefore , the solution of the given problem of unitary method comes out to be enable the current to travel, a connection, such as a wire or conductor, is required.
What is a unitary method?Utilizing previously well-known variables, this uniform convenience, or all crucial elements from a prior flexible study that followed a particular methodology event can all be used to achieve the goal. It will be possible to contact the entity again if the anticipated assertion outcome actually happens; if it doesn't, both important systems will surely miss the statement.
Here,
D. In order for a current to move, there must be an energy source, a recipient, and a connection.
Current is the movement of charged ions through a conductor, most frequently electrons.
The energy required to transport the charged particles is provided by an energy source, such as a battery or generator.
The current has a route through a recipient, which can be a machine or a circuit.
In order to finish the circuit and enable the current to travel, a connection, such as a wire or conductor, is required.
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Use the quadratic formula to solve the equation 2 - 5x-9=0
The answer is x = 5±√61/2, I really hope this helps (:
the length of a rectangle is 9 centimeters less than its width. what are the rectangles dimension if its area is 90.
Length of rectangle in cm:
Width of rectangle in cm:
Let the width of the rectangle be "X". Then the length of the rectangle will be (X-9).
Formula to find the area of a rectangle is given by-
[tex] \small \underline{ \boxed{ \sf{ \pmb{Area_{(rectangle)} = Length\times Width }}}}\\[/tex]
On substituting the values-
[tex] \:\:\:\:\:\:\longrightarrow \sf {Area_{(rectangle)} = X \times \bigg(X-9\bigg)}\\[/tex]
[tex] \:\:\:\:\:\:\longrightarrow \sf {90 = X^2 -9X}\\[/tex]
[tex] \:\:\:\:\:\:\longrightarrow \sf {X^2-9X =90}\\[/tex]
[tex] \:\:\:\:\:\:\longrightarrow \sf {X^2-9X -90=0}\\[/tex]
[tex] \:\:\:\:\:\:\longrightarrow \sf {X^2-15X+6X -90=0}\\[/tex]
[tex] \:\:\:\:\:\:\longrightarrow \sf {X\bigg(X-15\bigg) +6\times \bigg(X-15\bigg)=0}\\[/tex]
[tex] \:\:\:\:\:\:\longrightarrow \sf {\bigg(X-15\bigg) \times \bigg(X+6\bigg) =0}\\[/tex]
[tex] \:\:\:\:\:\:\longrightarrow \boxed{ \tt{ \pmb{ \red{X = 15\: Or,\: -6}}}}\\[/tex]
Since,width cannot be (Negative) , so the width will be 15cm.And the length will be (X-9)= (15-9)=6cm.
Solve the problems. Show your work.
7
1. Mr. Nguyen had 7/8
pint of water in his water bottle. Then, he drank 2/3
pint. How much water is left in the bottle?
Answer:
7/8 - 2/3 = 5/8 pint of water left in the bottle.
Interpret the data in the circle graph. If 560 books were sold at the book fair, find the number of the books that were mystery books.
If 560 books were sold at the book fair,
(Type a whole number.)
of the books were mystery books.
Circle graph
Fantasy 8%
Science
Fiction
12%
Comic 15%
Other 5%
Mystery 20%
-Fictic
Answer:
112
Step-by-step explanation:
According to the circle graph, the mystery books make up 20% of all books sold. So, we can calculate the number of mystery books sold as follows:
Number of mystery books = 20% of 560
= (20/100) x 560
= 112
Therefore, the number of mystery books sold at the book fair was 112.
In one town 44% of voters are democrats if two voters are randomly selected for a survey find the probability that they are both Democrats assume events are independent round to the nearest thousand if necessary
the probability that they are both Democrats. round to the nearest thousandth if necessary is 0.194
The probability that BOTH is democrats means the probability of "one being democrat" AND "another also being democrat".
The AND means we need to MULTIPLY the individual probability of a person being a democrat.
The probability that a voter is democrat is 44% (0.44) -- stated in the problem
Now, the Probability of BOTH being Democrats is simply MULTIPLYING 0.44 with 0.44
Rounded to the nearest thousandth, 0.194
The last answer choice is correct.
the complete question is-
In one town 44% of all voters are Democrats if two voters are randomly selected for a survey find the probability that they are both Democrats. round to the nearest thousandth if necessary.
0.189
0.880
0.440
0.194
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Lola hosts a podcast about computers, and Franco hosts a podcast about parenting. Both podcasts have a set duration for each episode. Last year, Lola released 17 episodes and Franco released 43 episodes, for a total of 5,645 minutes of content. This year, Lola released 36 episodes and Franco released 43 episodes, which lasted a total of 6,956 minutes. How long is each episode?
Answer:
Let's denote the duration of each episode of Lola's podcast by "L" and the duration of each episode of Franco's podcast by "F".
From the information given, we can write two equations based on the total duration of content for the two years:
17L + 43F = 5,645 (equation 1)
36L + 43F = 6,956 (equation 2)
To solve for L and F, we can eliminate F by subtracting equation 1 from equation 2:
(36L + 43F) - (17L + 43F) = 6,956 - 5,645
Simplifying, we get:
19L = 1,311
Therefore, the duration of each episode of Lola's podcast (L) is:
L = 1,311 / 19
L ≈ 69.0 minutes
To find the duration of each episode of Franco's podcast (F), we can substitute the value of L into either equation 1 or 2:
17L + 43F = 5,645
17(69.0) + 43F = 5,645
1,173 + 43F = 5,645
43F = 4,472
F = 4,472 / 43
F ≈ 104.0 minutes
Therefore, each episode of Lola's podcast is approximately 69.0 minutes long, and each episode of Franco's podcast is approximately 104.0 minutes long.
48805 rounded to the nearest thousand
Answer: 49,000
48805 is greater than 48500, so it rounds to 49,000
what is the average allowance for each student?
$7,$12,$8.50,$10,$7,$8,$10.50,$11
The average allowance for each student is $9.25.
What is the average allowance for each student?To find the average allowance for each student, we need to add up all the allowances and divide by the total number of students.
Adding up the allowances:
$7 + $12 + $8.50 + $10 + $7 + $8 + $10.50 + $11 = $74
There are 8 students, so we divide the total allowance by 8 to get:
$74 / 8 = $9.25
Therefore, the average allowance for each student is $9.25.
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A line has a slope of 5/6
and passes through the point (7,6). Write its equation in slope-intercept form.
Answer:
y = 5/6x + 1/6
Step-by-step explanation:
m = 5/6, x = 7, y = 6
y = mx + b
6 = 5/6(7) + b
b = 6 - 35/6
b = 36/6 - 35/6 = 1/6
y = 5/6x + 1/6
Answer:
Below
Step-by-step explanation:
Here is another way:
Start with point ( 7,6) slope ( m= 5/6) form :
(y-6) = 5/6 ( x-7) expand
y-6 = 5/6 x - 35/6 add 6 to both sides
y = 5/6 x + 1/6 Done.
If the graph of a polynomial function P(x) has -intercepts at x = - 4, x = 0, x * 1 point
= 5, which of the following must be true for P(x)?
• (x + 5) is a factor of the polynomial.
• (x-4) is a factor of the polynomial.
•' The degree of the polynomial is 3.
• The degree of the polynomial is greater than or equal to 3.
(x + 5) is nοt necessarily a factοr οf the pοlynοmial, (x-4) is a factοr οf the pοlynοmial are cοrrect statement.
What is a functiοn ?Functiοn can be define in which it relates an input tο οutput.
If the graph οf a pοlynοmial functiοn P(x) has x-intercepts at x = -4, x = 0, and x = 5, then we knοw that the factοrs οf P(x) are (x + 4), x, and (x - 5). This is because a pοlynοmial has x-intercepts where the value οf P(x) is equal tο zerο, and this οccurs when each factοr is equal tο zerο.
Therefοre, we can cοnclude that (x + 4) and (x - 5) are factοrs οf the pοlynοmial P(x), but x is nοt necessarily a factοr. This is because x is a linear factοr with a zerο intercept, but it cοuld be cancelled οut by anοther factοr in the pοlynοmial.
Thus, the cοrrect statement is:
(x + 5) is nοt necessarily a factοr οf the pοlynοmial.
(x-4) is a factοr οf the pοlynοmial.
The degree οf the pοlynοmial is 3 οr greater since the pοlynοmial has three x-intercepts. Hοwever, we cannοt determine the exact degree οf the pοlynοmial withοut additiοnal infοrmatiοn.
Therefοre, (x + 5) is nοt necessarily a factοr οf the pοlynοmial, (x-4) is a factοr οf the pοlynοmial are cοrrect statement.
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In PQR, PQ= 5.4, QR= 3.6, and PR=6.2. To the nearest Tenth, what is M∠R
Therefore , the solution of the given problem of angles comes out to be M∠R measured at 45.4 degrees, to the closest tenth.
An angle meaning is what?The intersection of the lines that form a skew's ends determines the size of its biggest and smallest walls. There's a possibility that two paths will intersect at a junction. Angle is another outcome of two things interacting. They mirror dihedral forms the most. A two-dimensional curve can be created by placing two line beams in various configurations between their ends.
Here,
To determine the size of angle R in triangular PQR, we can apply the Law of Cosines:
=> cos(R) = (PQ₂ + PR₂ - QR₂) / (2 * PQ * PR)
=> cos(R) = (5.4₂ + 6.2₂ - 3.6₂) / (2 * 5.4 * 6.2)
=> cos(R) = 0.6960917
When we calculate the inverse cosine of both sides, we obtain:
=> R = cos⁻¹(0.6960917)
=> R equals 45.4 degrees
Angle R in triangle PQR is therefore measured at 45.4 degrees, to the closest tenth.
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Simplify the expression below.
5^2 + 9^2
To simplify the expression 5^2 + 9^2, we can evaluate the squares of 5 and 9 and then add the results:
5^2 = 5 x 5 = 25
9^2 = 9 x 9 = 81
So, 5^2 + 9^2 = 25 + 81 = 106
Therefore, the simplified form of the expression 5^2 + 9^2 is 106.
PLEASE HELP! Find missing side lengths of B and C. Explain
Answer:
b=7 & c=7√2
Step-by-step explanation:
b=7 as it is an isosceles triangle
now using Pythagoras theorem,
(c)^2= (7)^2+(7)^2
⇒(c)^2= 49+49
⇒(c)^2= 98
⇒c= √98
⇒c=7√2
Answer:
b = 7c = 7√2Step-by-step explanation:
You want the missing side lengths in an isosceles right triangle with one side given as 7.
Isosceles right triangleThe two congruent acute angles tell you this right triangle is isosceles. That means sides 7 and b are the same length:
b = 7
The hypotenuse of an isosceles right triangle is √2 times the side length:
c = 7√2
__
Additional comment
You can figure the hypotenuse using the Pythagorean theorem if you haven't memorized the side relations of this "special" right triangle.
c² = 7² + b²
c² = 7² +7² = 2·7²
c = √(2·7²) = 7√2
The side length ratios for an isosceles right triangle (angles 45°-45°-90°) are 1 : 1 : √2.
The other "special" right triangle is the 30°-60°-90° triangle, which has side length ratios 1 : √3 : 2.
Find the product……..
The product of the expressions are;
Step 1: (x + 2)(x + 3) and 3(x + 3)/4(x + 5)
Step 3: 4(x + 3) × 3(x + 3)/4(x + 5)
Step 4: 3/x + 5
How to determine the productIt is important to note that algebraic expressions are described as expressions that are composed of variables, terms, coefficients, constants and factors.
From the information given, we have the fraction;
4x + 8/x² + 5x + 6 × 3x + 9/4x + 20
To determine the product, let us reduce the expressions to their lowest forms, we have;
4x + 8 = 4(x + 2)
x² + 5x + 6 = (x + 2)(x + 3)
3x + 9 = 3(x +3)
4x + 20 = 4(x + 5)
Substitute the expressions
4(x + 2)/(x + 2)(x + 3) × 3(x +3)/4(x + 5)
divide the common terms
4(x + 3) × 3(x + 3)/4(x + 5)
Divide further, we have.
3/x + 5
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Your tank has a volume of 10 L at the surface (1 atm pressure). You reach a depth of 66 ft. What is the
pressure? What is the volume?
the pressure at a depth of 66 ft is 197,580 Pa, and the volume of the tank at this depth is 0.000505 L.
EquationsTo find the pressure at a depth of 66 ft in a liquid, we can use the formula:
pressure = density x gravity x depth
Assuming the liquid in the tank is water, the density is 1000 kg/m³, and gravity is 9.81 m/s².
First, we need to convert 66 ft to meters:
66 ft x 0.3048 m/ft = 20.1168 m
Then, we can find the pressure at this depth:
pressure = 1000 kg/m³ x 9.81 m/s² x 20.1168 m = 197,580 Pa
To find the volume of the tank at this depth, we can use Boyle's Law, which states that the pressure and volume of a gas are inversely proportional at a constant temperature:
P₁V₁ = P₂V₂
where P₁ and V₁ are the initial pressure and volume (1 atm and 10 L, respectively), and P₂ and V₂ are the final pressure and volume.
We can rearrange this equation to solve for V₂:
V₂ = (P₁ x V₁) / P₂
Substituting the values, we get:
V₂ = (1 atm x 10 L) / (197,580 Pa / 1 atm) = 0.000505 L
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1. Identify and clearly label the slope and y-intercept for each equation in slope intercept form. Choose the correct answer from the choices below.
Y=-5
A. Slope is-5 and the y-intercept is (0,0)
B.Slope is zero and the y-intercept is (0,-5)
C. Slope is zero and the y-intercept is (0,0)
D. Slope is -5 and the y-intercept is (0,-5)
Slope is zero and the y-intercept is (0,-5)
What is slope ?
In mathematics, slope is a measure of the steepness of a line. It is defined as the ratio of the vertical change (rise) between two points on the line to the horizontal change (run) between the same two points.
In other words, the slope of a line is the change in the y-coordinate divided by the change in the x-coordinate between any two points on the line. It can also be thought of as the rate at which the line rises or falls as it moves horizontally.
The formula for calculating slope is:
slope = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are two points on the line.
According to the question:
The equation Y = -5 is already in slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept.
Comparing the equation Y = -5 to y = mx + b, we can see that:
The slope, m, is 0, since there is no x-term in the equation.
The y-intercept, b, is -5, since that is the constant value in the equation.
Therefore, the correct answer is:
B. Slope is zero and the y-intercept is (0,-5)
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video
Let the region R be the area enclosed by the function f(x) = ln (x) + 1 and
g(x)=x-1. If the region R is the base of a solid such that each cross section
perpendicular to the a-axis is a semi-circle with diameters extending through the
region R, find the volume of the solid. You may use a calculator and round to the
nearest thousandth.
The volume of the solid is approximately 0.558 cubic units.
To find the volume of the solid, we need to integrate the area of the semi-circles along the a-axis.
We know that the diameter of each semi-circle is the distance between the functions f(x) and g(x), which is:
d(a) = f(a) - g(a) = ln(a) + 1 - (a-1) = ln(a) - a + 2
The radius of each semi-circle is half of the diameter, which is:
r(a) = (ln(a) - a + 2) / 2
The area of each semi-circle is π times the square of its radius, which is:
[tex]A(a) = πr(a)^2 = π/4 (ln(a) - a + 2)^2[/tex]
To find the volume of the solid, we integrate the area of each semi-circle along the a-axis, from a = e to a = 2:
V = ∫[e,2] A(a) da
V = ∫[e,2] π/4 [tex](ln(a) - a + 2)^2 da[/tex]
V ≈ 0.558 (rounded to the nearest thousandth)
Therefore, the volume of the solid is approximately 0.558 cubic units.
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13. The profit, in thousands of dollars, from the sale of x kilogram of coffee bean can be modelled by the function () = 5−400 +600 . a) State the asymptotes and the intercepts. Then, sketch a graph of this function using its key features. (5 pts) b) State the domain and range in this context. (2 points) c) Explain the significance of the horizontal asymptote. (1 point) d) Algebraically, find how much amount of tuna fish, in kg, should be sold to have a profit of exactly $4000? (4 points) SOLUTION
Answer: a) The profit function can be written as:
P(x) = 5x - 400x + 600
To find the asymptotes, we can look at the denominator of the second term, which is (x - 3). This means that there is a vertical asymptote at x = 3. To find the intercepts, we can set P(x) = 0:
5x - 400x + 600 = 0
Solving for x, we get:
x = 1.5 and x = 2.5
Therefore, there are x-intercepts at (1.5, 0) and (2.5, 0). To sketch the graph, we can also note that the coefficient of x^2 is negative, which means that the graph is a downward-facing parabola.
b) The domain of the function is the set of all possible values of x, which in this context represents the amount of coffee sold. Since we cannot sell a negative amount of coffee, the domain is x ≥ 0.
The range of the function is the set of all possible values of P(x), which represents the profit. Since the coefficient of x^2 is negative, the maximum profit occurs at the vertex of the parabola. The vertex has x-coordinate:
x = -b/(2a) = -(-400)/(2(-200)) = 1
Therefore, the maximum profit occurs when x = 1. The vertex has y-coordinate:
P(1) = 5(1) - 400(1) + 600 = 205
Since the coefficient of x^2 is negative, the range is (-∞, 205].
c) The horizontal asymptote of the function is y = -400, which represents the long-term average profit per kilogram of coffee sold. This means that as x gets very large, the profit per kilogram approaches -400. This could happen, for example, if the cost of producing the coffee increased significantly while the price remained the same.
d) To find the amount of coffee that must be sold to make a profit of $4000, we can set P(x) = 4000 and solve for x:
5x - 400x + 600 = 4000
Simplifying, we get:
-395x = -3400
Dividing both sides by -395, we get:
x ≈ 8.61
Therefore, approximately 8.61 kg of coffee must be sold to make a profit of $4000.
Step-by-step explanation: