a.write an equation to model the relationship
b.explain how you know if an equation or a graph represents a proportional relationship
Answers
Answer:
a) 12 · y = x
b) If the points show equivalent fractions and the line goes through (0,0)
Step-by-step explanation:
a1) divide any numbers from the x and y line (for example, 2 ÷ 24 would give us 12)
a2) This equation makes sense because any number from the x line (for example, 2) multiplied by 12 would give us the number from the y line (in this example, 24).
Related Questions
eacher: ""the first triangle is an isosceles triangle. the second triangle is an isosceles triangle. the third triangle is not an isosceles triangle. based on your observations alone what should be the definition of an isosceles triangle?""what does this question illustrate about the geometry teacher’s questioning techniques?
Answers
We have demonstrated the definition of an isosceles triangle.
What is an isosceles triangle?
An isosceles triangle is a triangle with at least two equal-length sides in geometry. It is sometimes specified as having exactly two equal-length sides, and sometimes as having at least two equal-length sides, with the latter version including the equilateral triangle as a special case.
The isosceles triangle has both two equal sides and two equal angles. The name derives from the Greek iso (same) and Skelos (leg). A triangle with all sides equal is called an equilateral triangle, and a triangle with no sides equal is called a scalene triangle.
The given diagram shows an isosceles triangle.
Hence, we have demonstrated the definition of an isosceles triangle.
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A right triangle has a leg of length 13 yards and a hypotenuse of
length 42 yards. Find the length of the other leg. Provide an answer
accurate to the nearest tenth.
Answers
Answer:
39.9 yards
Step-by-step explanation:
Pythagorean theorem is a^2+b^2=c^2
Where a and b are the legs and c is the hypotheneus
We set this so it will be equal to a leg
a^2+b^2=c^2
-b^2. -b^2
a^2=c^2-b^2
42^2-13^2=a^2
1764-169=a^2
1595=a^2
√ √
39.9=a
Hopes this helps please mark brainliest
What is the number n of the highest harmonic that could be heard by a person who is capable of hearing frequencies up to f = 16 khz?.
Answers
The Frequency will be "266.14 Hz" and the number of loops are "60".
Explain the term harmonic motion?A simple periodic motion (such a pendulum swinging or a violin string making music) which has a specific frequency and amplitude or is made up of two or more of these motions.For the given question-
Tension T = 765 NLength L = 0.600 mMass m =4.50 gThe formula for the linear velocity is given as;
Linear density; μ = mass/length
Put the values,
μ = 4.5 x 10⁻³/0.6
μ = 7.5 x 10⁻³ kg/m
The formula for the fundamental frequency is;
F = (1/2L)√(T/ μ)
F = (1/2x0.6)√(765/ 7.5 x 10⁻³)
F = 266.14 hz
The number of loops is;
n = f'/f
n = 16000/266.14
n = 60
Thus, a person who can hear frequencies up to f = 16 kHz would be able to hear the highest harmonic with number n equal to 60.
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The complete question is-
A piano tuner stretches a steel piano wire with a tension of 765 N . The steel wire has a length of 0.600 m and a mass of 4.50g .
What is the frequency (f_1) of the string's fundamental mode of vibration?
What is the number n of the highest harmonic that could be heard by a person who is capable of hearing frequencies up to f = 16 kHzc?
please help.. just how many times am i gonna have to post the same question over and over again to get an actual answer
Answers
A number a is more than 6. An inequality that represents this word sentence is..
Answers
Answer:
x>
Step-by-step explanation:
x>
A number a:
a
is more than 6
< 6
Solution:a < 6I hope my answer helped you! If you need more information or help, comment down below and I will be sure to respond if I am online. Have a wonderful rest of your day!
assume that you have a labeled data set . explain how can you use only k-mean clusterign to build a classification model fot this dataset
Answers
K-means clustering is a type of unsupervised learning which uses unlabeled data, or data without clearly defined categories or groupings.
Explain the term K-means clustering?Data is divided into k clusters using K-means clustering, with the goal of making the clusters closer together and the individual data points more dispersed.
The distance between two points determines how similar they are.The distance can be measured in a variety of ways. One of the most often used distance metrics is the distance measure (minkowski distance for p=2). The calculation of the Euclidean separation distance in a two-dimensional space is shown in the image below.The square of a difference between the points' x and y dimensions is used to calculate it.K-means clustering seeks to maximize distances across clusters while minimizing distances within each cluster.K-means clustering is a sort of unsupervised learning which uses unlabeled data, or data without clearly defined categories or groupings.
Thus this algorithm's goal is to identify groups among the data, where K stands for the set of organizations.
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The complete question is-
Assume that you have a labeled dataset. Explain how you can use only K-means clustering to build a classification model for this dataset. What can go wrong?
find an explicit description of nul a by listing vectors that span the null space. a question content area bottom part 1 a spanning set for nul a is
Answers
By using the concept of null space of a matrix, it can be calculated that
Nullspace of A = [tex]\{ \begin{bmatrix} 3 \\ 1 \\0 \\ 0\\0 \end{bmatrix} , \begin{bmatrix} 3 \\ 0 \\-4 \\ 1\\0 \end{bmatrix}\}[/tex]
What is Nullspace of a matrix?
Nullspace of a matrix A is the solution of the equation AX= 0 where 0 is the zero matrix
[tex]\begin{bmatrix} 1 & -3 & 0 & -3 & 0 \\ 0 & 0 & 1 & 4 & 0 \\0 & 0 & 0 & 0 & -1 \end{bmatrix} \begin{bmatrix} x \\ y \\z \\w \\t \end{bmatrix}[/tex]
We need to find Ax = 0
[tex]\begin{bmatrix} 1 & -3 & 0 & -3 & 0 \\ 0 & 0 & 1 & 4 & 0 \\0 & 0 & 0 & 0 & -1 \end{bmatrix} \begin{bmatrix} x \\ y \\z \\w \\t \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\0 \end{bmatrix}[/tex]
x - 3y - 3w = 0
x = 3y + 3w
z + 4w = 0
z = -4w
-t = 0
t = 0
[tex]\begin{bmatrix} x \\ y \\z \\w \\t \end{bmatrix} = \begin{bmatrix} 3y + 3w\\ y \\-4w \\ w\\0 \end{bmatrix}\\\\y \begin{bmatrix} 3 \\ 1 \\0 \\ 0\\0 \end{bmatrix} + w \begin{bmatrix} 3 \\ 0 \\-4 \\ 1\\0 \end{bmatrix}[/tex]
Nullspace of A = [tex]\{ \begin{bmatrix} 3 \\ 1 \\0 \\ 0\\0 \end{bmatrix} , \begin{bmatrix} 3 \\ 0 \\-4 \\ 1\\0 \end{bmatrix}\}[/tex]
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Complete Question
The complete question has been attached
What are the zeros of the function f(x) = x2 + 5x + 5 written in simplest radical form? quadratic formula: x = startfraction negative b plus or minus startroot b squared minus 4 a c endroot over 2 a endfraction.
Answers
Answer:
Below
Step-by-step explanation:
a = 1 b = 5 c = 5
Using these values in the Quadratic Formula given results in zeroes of
x = -5/2 ± sqrt (5) / 2
if the video is shot at 190 frames per second, how many wing beats does the dragonfly perform per second?
Answers
After solving, 41.67 Hz wing beats the dragonfly performs per second.
In the given question, if the video is shot at 190 frames per second, then we have to find how many wing beats the dragonfly performs per second.
From the given question,
Frames per one wing beat = 6 frames
Video shot at = 250 frames per second
We firstly find the time period
So, Time period (T) = 6/250
T= 0.024 s
Snce we have to find how many wing beats the dragonfly performs per second.
Per second number of wing beats = 1/T
Per second number of wing beats = 1/0.024
Per second number of wing beats = 41.67 Hz
Hence, 41.67 Hz wing beats the dragonfly performs per second.
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The right question is;
A frame by frame analysis of a slowmotion video shows that a hovering dragonfly takes 6 frames to complete one wing beat.
If the video is shot at 190 frames per second, how many wing beats does the dragonfly perform per second?
Expand and simplify if possible (√2+5)(1+√2)
Answers
Answer:
7 + 6[tex]\sqrt{2}[/tex]
Step-by-step explanation:
note that [tex]\sqrt{a}[/tex] × [tex]\sqrt{a}[/tex] = a
([tex]\sqrt{2}[/tex] + 5)(1 + [tex]\sqrt{2}[/tex])
each term in the second factor is multiplied by each term in the first factor, that is
[tex]\sqrt{2}[/tex](1 + [tex]\sqrt{2}[/tex]) + 5(1 +[tex]\sqrt{2}[/tex]) ← distribute parenthesis
= [tex]\sqrt{2}[/tex] + 2 + 5 + 5[tex]\sqrt{2}[/tex] ← collect like terms
= 7 + 6[tex]\sqrt{2}[/tex]
Which ordered pair would make this set a function when it replaces (x, y)?
{(0, 1), (2, 6), (1, 7), (5, 7), (x, y)}
ANSWER CHOICES
(2, 4)
(5, 9)
(1, 6)
(4, 8)
Answers
Hence, the ordered pair which replace [tex](x,y)[/tex] is [tex](4,8)[/tex].
What is the function?
Functions are often defined by a formula that describes a combination of arithmetic operations and previously defined functions; such a formula allows computing the value of the function from the value of any element of the domain.
Here given function is
[tex]{(0, 1), (2, 6), (1, 7), (5, 7), (x, y)}[/tex]
So, the value which replace [tex](x,y)[/tex] is [tex](4,8)[/tex], because for the function every element have a single image.
Hence, the ordered pair which replace [tex](x,y)[/tex] is [tex](4,8)[/tex].
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3. Se tiene un auto pesado y un auto ligero, Ex-
plica qué ocurrirá en los siguientes casos:
a. Si los dos carros aceleran igualmente en el
arranque, ¿cuál gana?
b. Si ambos desaceleran igual, ¿cuál se detie-
ne en menos tiempo?
Answers
Answer:
a. El auto pesado ganará, ya que debido a su mayor masa tendrá una mayor fuerza de arrastre.
b. El auto ligero se detendrá en menos tiempo, debido a que su masa es menor, lo cual significa que tendrá una menor resistencia al movimiento.
Step-by-step explanation:
let g be a random graph generated using the gn,p model. a clique of k vertices in a graph is a subset of k vertices such that all (k 2 )edges lie in the graph. for what value of p as a function of n, is the expected number of cliques in g equal to 1?
Answers
At p = Θ(1/√n) the expected number of cliques in G will be going to be equal to 1.
What is meant by Probability?The likelihood that a random event will occur is defined by probability. There are several real-life circumstances in which we must forecast the result of an occurrence. We might be certain or uncertain about the outcome of an event. In these circumstances, we state that there is a likelihood that this occurrence will occur or not. Generally speaking, probability has many useful uses in games, business (to make forecasts based on probability), and this emerging branch of artificial intelligence.
By simply dividing the favorable number of possibilities by the entire number of potential outcomes, the probability of an occurrence may be computed using the probability formula. Probability values range from 0 to 1, with a value of 1 being the most likely.
How to solve?
Fix any S-vertex subgroup that contains less than 5. Given that S includes (5, 2)=10 edges, the probability of S being a clique in G is p^10. Given that S has (n, 5) options, the predicted number of 5 cliques in G is (n, 5)p^10 due to linearity of expectation. The result of solving for (n, 5)p10 = 1 is^ p10 = (1/n^5).
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Select all the equations that have g = 6
the solution.
Og+2=10
0g-1=10
09-2=4
58+ g = 60
44 - g = 38
Answers
if two erminette chickens were crossed, what is the probability that: a. they would have a black chick?
Answers
Probability of two black Erminette chickens crossed is equal to 1/4.
As given in the question,
Erminette chickens are basically a chicken with a gene of black and white feathers.
There are two groups of chicken black feather chicken and white feather chicken.
All possible outcomes when Erminette chicken crossed each other are:
= { BB , BW, WB, WW}
Total number of outcomes are = 4
Favourable outcomes are ={ BB }
Number of favourable outcomes = 1
Probability of two black Erminette chickens crossed each other
= ( Favourable outcomes) / ( Total number of outcomes )
= 1 / 4
Therefore, probability of two black Erminette chickens crossed each other with the given condition is equal to 1/ 4.
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25 POINTS! PLEASE HELP QUICKLY.
Simplify the expression.
six tenths squared minus 18 plus the quantity negative 3 divided by the absolute value of 5.6 minus 7.1 end quantity times 4.19
A: −25.18
B: −26.02
C: −8.42
D: −9.26
Answers
Answer:
D
Step-by-step explanation:
Exam test completed. Your welcome
from a point on a river, two boats are driven in opposite directions, one at 10 miles per hour and the other at 14 miles per hour. in how many hours will they be 120 miles apart?
Answers
The two boats will be 120 miles apart in five hours.
We need to find the time at which the boats will be 120 miles apart from each other.
It has been given that the total distance is 120 miles.
Now, let's take up the following assumptions in order to arrive at the time taken by two boats to cover the requisite distance:-
Let x= distance traveled by boat 1 at 10 miles per hour.
Let y= distance traveled by boat 2 at 14 miles per hour.
Thus, x+y=120(given)------(i) [That is, distance travelled by boat 1+ Distance travelled by boat 2= Total Distance]
Let t denote the time factor,
then, Boat 1 will travel x miles in T hours, and the Boat 2 will travel y miles in T hours.
We know, Distance= Rate * Time
Thus, we can say, x= 10*T---------------(ii)
y=14*T-----------------(iii)
Adding equations ii and iii, we get
x+y=24T----------------(iv)
Now, subtracting equation (iv) from (i), we get
24T-120=0
24T= 120
thus, we get t=5
Hence, the time taken by two boats, to be 120 miles apart is five hours.
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jacked biked 28 miles in 3.5 hours
(question one )whats the unit rate per hour?
(question two) how far can he bike in 9 hours
Answers
(1). The unit rate of per hour is 8 hours.
(2). He bikes 72 miles in 9 hours.
What do you mean by unit rate?
An item's unit rate is its price for one of it. This is expressed as a ratio with a one as the denominator. For instance, if you covered 70 yards in 10 seconds, you covered 7 yards on average every second. Seven yards in one second and 70 yards in ten seconds are both ratios, but only one of them is a unit rate.
According to data in the given question,
We have :
Distance travelled in 3.5 hours = 28 miles
Now, firstly we need to find unit rate per hour,
Distance travelled in 1 hours = 28/3.5 miles
= 8 miles
So, the unit rate of per hour is 8 hours.
Then, we will find the how far can he bike in 9 hours,
Distance travelled in 9 hours = 9*8 miles
= 72 miles
Therefore, he bikes 72 miles in 9 hours.
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Evaluate the given integral by making an appropriate change of variables, where R is the region in the first quadrant bounded by the ellipse 100x2 + 49y2 = 1
Answers
On solving tthe question, we got by double integration [tex]\int\limits^0_1 \int\limits^1_2{100x^2+49y^2} \, dx dy = x(\frac{100x^2 + 147y^2}{3} )[/tex]
What is double integration?A function in two variables over a region in R2, or the real number plane, is referred to in mathematics as a double integral. It is possible to write the double integral of a function of two variables, such as f(x, y), over a rectangular region as follows: ∬ R f ( x , y ) d A = ∬ R f ( x , y ) d x d y.
g[tex]\int\limits^0_1 \int\limits^1_2{100x^2+49y^2} \, dx dy \\ At first,\\ \int\limits^{} \,100x^2 dx = 100\int\limits^ {} \,x^2 dx \\ \int\limits^ {} \,x^2 dx = \frac{x^3}{3} \\[/tex]
[tex]the result - \frac{100x^3}{3}\\ now,\\\int\limits \, 49y^2dx = 49xy^2\\\\result - \frac{100x^3}{3} + 49xy^2\\\\On simplyfing-\\[/tex]
[tex]x(\frac{100x^2 + 147y^2}{3} )[/tex]
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Jewels has $6. 75 to ride the ferry around connecticut. It will cost her $0. 45 every time she rides. Identify the dependent variable and the independent variable in this scenario.
Answers
The number of rides is the dependent variable, while the total cost is the independent variable.
What is dependent variable?In mathematical modeling, statistical modeling, and experimental sciences, there are dependent and independent variables. Dependent variables get their name because, during an experiment, their values are investigated under the presumption or requirement that they are dependent on the values of other variables according to some law or rule.
What is independent variable?The variable that the researcher modifies or manipulates in an experiment is known as the independent variable. The measured reaction is the dependent variable. The dependent variable is dependent on the change in the independent variable, to put it another way.
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Bittersweet bakery normally sells out wers
of loaves of bread quickly. The graph
shows the loaves of bread that the
bakery has left. Label each segment
of the graph with letters A through
E, beginning from the left. Record the
time interval for each segment. Then,
describe how fast the number of
loaves changes.
Answers
Answer:
Step-by-step explanation:
Location & Hours
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K B.
Fort Collins, CO
0223
6/18/2015
wonderful cinnamon rolls and sticky buns, so glad we stopped on our way thru town.
It's a small little place in residential area, not far from a main road, definitely worth stopping by for a delicious treat!
Kaitlin J.
Cedar Grove, Panama City, FL
016
8/27/2020
2 photos
Erin is so friendly and truly a master of her trade! We ordered a cake shaped like a suitcase (complete with fondant flight attendant) as a surprise for our mom's retirement party. It turned out even better than we could have ever expected with the best hand-painted intricate details. She really went above and beyond, and the carrot cake was absolutely delicious!! We are all still so impressed. We just had such a great experience from start to finish--Erin is so friendly and easy to work with. She made our vision come to life and at an incredibly reasonable price. I can't recommend her bakery enough!
Answer: 5.5
Step-by-step explanation:
An investor owns shares of Stock A and Stock B. The investor owns a total of 200 shares with a total value of $4000. How many shares of each stock does the investor own?
Stock: Price:
A $9.50
B $27.00
Answers
Taking into account the definition of a system of linear equations, Finally, the investor owns 80 and 120 shares of Stock A and B respectively.
System of linear equationsA system of linear equations is a set of two or more equations of the first degree, in which two or more unknowns are related.
A solution to a system is an assignment of values to the unknowns that makes each of the equations true. Solving a system means finding all the solutions of the system.
Amount of shares of each stockIn this case, a system of linear equations must be proposed taking into account that:
"A" is the amount of shares of Stock A."B" is the amount of shares of Stock B.You know:
An investor owns shares of Stock A and Stock B. The investor owns a total of 200 shares with a total value of $4000.The price of Stock A is $9.50The price of Stock B is $27.00The system of equations to be solved is:
A + B= 200
9.50A +27B= 4000
There are several methods to solve a system of equations, it is decided to solve it using the substitution method, which consists of clearing one of the two variables in one of the equations of the system and substituting its value in the other equation.
In this case, isolating the variable "A" from the first equation you get:
A= 200 - B
Substituting the expression in the second equation:
9.50× (200 - B) +27B= 4000
Solving:
9.50×200 - 9.50B +27B= 4000
1900 - 9.50B +27B= 4000
- 9.50B +27B= 4000 - 1900
17.5B= 2100
B= 2100÷ 17.5
B= 120
Substituting this value in the expression A=200 - B you get:
A= 200 - 120
A= 80
Finally, the investor owns 80 shares of Stock A and 120 shares of Stock B.
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Please Help!!!! Hurry!!!
Answers
Answer:
Step-by-step explanation:
x is increasing by 2
y is decreasing by 4
Rise over run:
-4/2 --> -2
Answer: I believe the answer is -2
a store owner currently sells bags of peanuts that cost $2.2 per pound and bags of almonds that cost $9.1 per pound. the owner wishes to make a 49 pound bag of a mixture of peanuts and almonds that costs $6.85 per pound. how many pounds of peanuts and almonds does the owner need?
Answers
To make a 49 pound bag of a mixture of peanuts and almonds that costs $6.85 per pound, the owner needs 15.98 pounds of peanuts and 33.02 pounds of almonds.
An algebraic expression is the combination of numbers and variables in expressing and solving a particular mathematical question. An equation is the equality of expressions.
Let x = pounds of peanuts in the mixture
y = pounds of almonds in the mixture
The owner wished to make a total of 49 pounds of mixture of peanuts ang almonds. Hence,
x + y = 49
x = 49 - y (equation 1)
Bags of peanuts cost $2.2 per pound and bags of almonds cost $9.1 per pound, and the mixture should cost $6.85 per pound. Hence,
2.2x + 9.1y = 6.85(49)
2.2x + 9.1y = 335.65
Substitute equation 1 and solve for y.
2.2x + 9.1y = 335.65
2.2(49 - y) + 9.1y = 335.65
107.8 - 2.2y + 9.1y = 335.65
6.9y = 227.85
y = 33.02
pounds of almonds in the mixture = 33.02
Using equation 1, solve for x.
x = 49 - y
x = 49 - 33.02
x = 15.98
pounds of peanuts in the mixture = 15.98
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What is the Slope-Intercept Formula if the line IS Proportional. (Slope-intercept form is y =
mx + b)
Type a response
Answers
Answer:
y = mx
Step-by-step explanation:
You want to know the slope-intercept formula for a line that represents a proportion.
ProportionA proportion is a linear function that includes the point (0, 0). The y-intercept is zero, so the slope-intercept equation is ...
y = mx
Find the asymptote(s) of the graph of the function. Select all that apply.
Answers
The asymptotes of the graph of the function are given as follows:
x = -2.x = -1.y = 0.What are the asymptotes of the function?The function has two types of asymptotes, which are given as follows:
Vertical asymptotes.Horizontal asymptotes.The vertical asymptotes are the values of x for which the function is not defined, that is, empty spaces on the graph of the function, hence they are given as follows:
x = -2, as the function goes to infinity at both sides of x = -2.x = -1, as the function is not defined at x = -1 due to the open circle.At x = 1, there is a closed circle, hence the function is defined and it is not a vertical asymptote.
The horizontal asymptote is the limit of the function as it goes to infinity, as long as this limit is different of infinity.
The limits are given as follows:
x goes to negative infinity: y = 0 -> horizontal asymptote.x goes to positive infinity: negative infinity -> not an asymptote.More can be learned about the asymptotes of a function at https://brainly.com/question/1851758
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simplify the picture
Answers
When multiplying with exponents in this situation just add them together.
determine the type ii error if the null hypothesis, h0, is: the percentage of college students that will graduate with debt is no more than 65%.and, the alternative hypothesis, ha, is: the percentage of college students that will graduate with debt is more than 65%.
Answers
We know that in statistics, the Type II error happens when the null hypothesis is false but fails to get rejected.
Type II error in this scenario will be when the researcher claim 65% of college students will graduate with debt is false but fails to be rejected.
A type II error is a statistical term used within the context of hypothesis testing that describes the error that occurs when one fails to reject a null hypothesis that is actually false. A type II error produces a false negative, also known as an error of omission.
For example, a test for a disease may report a negative result when the patient is infected. This is a type II error because we accept the conclusion of the test as negative, even though it is incorrect.
A Type II error can be contrasted with a type I error is the rejection of a true null hypothesis, whereas a type II error describes the error that occurs when one fails to reject a null hypothesis that is actually false. The error rejects the alternative hypothesis, even though it does not occur due to chance.
Given : The null hypothesis, , is: researchers claim that 65% of college students will graduate with debt.
Then , Type II error in this scenario will be when the researcher claim 65% of college students will graduate with debt is false but fails to be rejected.
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Let ????=−5y????+5x????. Use the tangential vector form of Green's Theorem to compute the circulation integral ∫????????⋅???????? where C is the positively oriented circle x2+y2=25
Answers
The value of the required integral is 50π where C is the positively oriented circle x2+y2=25
We are given a vector F = -yi + xj. A vector is a quantity with magnitude and direction that is commonly represented by a directed line segment, with length representing magnitude and orientation in space representing direction. We need to use the tangential vector form of Green's Theorem to compute the circulation integral ∫CF⋅dr where C is the positively oriented circle x² + y² = 25
Let the value of the integral be represented by the variable "I".
I = ∫∫[(/x)(x) - (/y)(-y)]dA
I = ∫∫[(1 - (-1)]dA
I = ∫∫2dA
I = 2∫∫dA
I = 2×(Area of the circle)
I = 2*(πr²)
I = 2*(π*25)
I = 50π
Hence, the value of the required integral is 50π.
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the volume of an expanding sphere is increasing at a rate of 12 cubic feet per second. when the volume of the sphere is 36 pi cubic feet, how fast, in square feet per second, is the surface area increasing?
Answers
Surface are increasing at the rate of 8 square feet per second.
What is a sphere?
A sphere is a geometrical object that resembles a two-dimensional circle in three dimensions. In three-dimensional space, a sphere is a collection of points that are all located at the same r-distance from a single point. The radius of the sphere is equal to r, and the provided point is its centre.
Volume(V) = [tex]4\pi r^3[/tex]
[tex]\frac{dV}{dt} =\frac{4\pi }{3} \times3r^2\times\frac{dr}{dt} =4\pi r^2.\frac{dr}{dt} \\\\\boxed{\frac{dr}{dt} =\frac{1}{4\pi r^2} (\frac{dV}{dt} )}[/tex]
[tex]V=36\pi \\\\\frac{4}{3} \pi r^3=36\pi \\\\r^3=\frac{108}{4} =27\\\\\boxed{r=3}\\\\\boxed{\frac{dV}{dt} =12}[/tex]
Surface area(S) = [tex]4\pi r^2[/tex]
[tex]\frac{dS}{dr} =8\pi r\times\frac{dr}{dt} \\\\\frac{dS}{dt} =\frac{8\pi r}{4\pi r^2} \times\frac{dr}{dt}\\\\ \frac{dS}{dt}=\frac{2}{r} \times \frac{dV}{dt} =\frac{2}{3} \times12=8\\\\\boxed{\frac{dS}{dt} =8}[/tex]
Hence, the surface area increasing at the rate of 8 square feet per second.
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