A constant volume of pizza dough is formed into a cylinder with a relatively small height and large radius. The dough is spun and tossed into the air in such a way that the height of the dough decreases as the radius increases, but it retains its cylindrical shape. At time t=k, the height of the dough is 13 inch, the radius of the dough is 12 inches, and the radius of the dough is increasing at a rate of 2 inches per minute.

(a) At time t=k, at what rate is the area of the circular surface of the dough increasing with respect to time? Show the computations that lead to your answer. Indicate units of measure.

(b) At time t=k, at what rate is the height of the dough decreasing with respect to time? Show the computations that lead to your answer. Indicate units of measure. (The volume V of a cylinder with radius r and height h is given by V=πr2h.)

(c) Write an expression for the rate of change of the height of the dough with respect to the radius of the dough in terms of height h and radius r.

Answer:

a) [tex]\frac{dA}{dt} = 48 \pi\frac{in^{2}}{min}[/tex]

b) [tex] \frac{dh}{dt} = - \frac{13}{3} \frac{in}{min}[/tex]

c) [tex]\frac{dh}{dt} = - 2\frac{h}{r} \frac {dr}{dt}[/tex]

Step-by-step explanation:

In order to solve this problem, we must first picture a cylinder of height h and radius r (see attached picture).

a) So, in order to find the rate at which the area of the circular surface of the dough is increasing with respect to time, we need to start by using the are formula for a circle:

[tex]A=\pi r^{2}[/tex]

So, to find the rate of change of the area, we can now take the derivative of this formula with respect to the radius r:

[tex]dA = \pi(2) r dr[/tex]

and divide both sides into dt so we get:

[tex]\frac{dA}{dr} = 2\pi r \frac{dr}{dt}[/tex]

and now we can substitute:

[tex]\frac{dA}{dr} = 2\pi(12in)(2\frac{in}{min})[/tex]

[tex]\frac{dA}{dt} = 48\pi\frac{in^{2}}{min}[/tex]

b) In order to solve part b, we can start with the formula for the volume:

[tex]V=\pi r^{2} h[/tex]

and solve the equation for h, so we get:

[tex]h=\frac{V}{\pi r^{2}}[/tex]

So now we can rewrite the equation so we get:

[tex]h=\frac{V}{\pi}r^{-2}[/tex]

and now we can take its derivative so we get:

[tex]dh=\frac{V}{\pi} (-2) r^{-3} dr[/tex]

we can rewrite the derivative so we get:

[tex]\frac{dh}{dt}=-2\frac{V}{\pi r^{3}}\frac{dr}{dt}[/tex]

we can take the original volume formula and substitute it into our current derivative, so we get:

[tex]\frac{dh}{dt}= -2\frac{\pi r^{2} h}{\pi r^{3}} \frac{dr}{dt}[/tex]

and simplify:

[tex]\frac{dh}{dt} =-2\frac{h}{r} \frac{dr}{dt}[/tex]

so now we can go ahead and substitute the values  provided by the problem:

[tex]\frac{dh}{dt} =-2\frac{13in}{12in} (2\frac{in}{min})[/tex]

Which simplifies to:

[tex] \frac{dh}{dt} = - \frac{13}{3} \frac{in}{min}[/tex]

c)

Part c was explained as part of part b where we got the expression for the rate of change of the height of the dough with respect to the radius of the dough in terms of the height h and the radius r:

[tex]\frac{dh}{dt} =-2\frac{h}{r} \frac{dr}{dt}[/tex]

The rate of change of the height of the pizza with respect to (w.r.t.) time

can be found given that the volume of the pizza is constant.

Reasons:

The height of the dough when t = k is 13 inches

Radius of the dough = 12 inches

Rate at which the radius of the dough is increasing, [tex]\dfrac{dr}{dt}[/tex] = 2 in.²/min

(a) Required: The rate of increase of the surface area with time

Solution:

The circular surface area, A = π·r²

By chain rule of differentiation, we have;

[tex]\dfrac{dA}{dt} = \mathbf{\dfrac{dA}{dr} \times \dfrac{dr}{dt}}[/tex]

[tex]\dfrac{dA}{dt} = \dfrac{d ( \pi \cdot r^2)}{dr} \times \dfrac{dr}{dt} = 2 \cdot \pi \times 2 = 4 \cdot \pi[/tex]

The rate of increase of the surface area with time, [tex]\mathbf{\dfrac{dA}{dt}}[/tex] = 4·π in.²/min.

(b) Required: The rate of decrease of the height with respect to time

The volume of the pizza is constant, given by; V = π·r² ·h

Therefore;

[tex]h = \mathbf{ \dfrac{V}{\pi \cdot r^2}}[/tex]

[tex]\dfrac{dh}{dt} = \dfrac{d \left( \dfrac{V}{\pi \cdot r^2} \right)}{dr} \times \dfrac{dr}{dt} = \dfrac{-2 \cdot V}{\pi \cdot r^3} = \dfrac{-2 \cdot \pi \cdot r^2 \cdot h}{\pi \cdot r^3} \times \dfrac{dr}{dt} = \mathbf{-2 \cdot \dfrac{h}{r} \times \dfrac{dr}{dt}}[/tex]

[tex]\dfrac{dh}{dt} = -2 \cdot \dfrac{h}{r} \times \dfrac{dr}{dt} = -2 \times \dfrac{13}{12} \times 2 = \dfrac{13}{3} = 4. \overline 3[/tex]

The rate at which the height of the dough is decreasing, [tex]\mathbf{\dfrac{dh}{dt}}[/tex]= [tex]\underline{4.\overline 3 \ in./min}[/tex]

(c) Required:]The expression for the rate of change the height of the dough with respect to the radius of the cone.

Solution:

[tex]\dfrac{dh}{dr} = \dfrac{d \left( \dfrac{V}{\pi \cdot r^2} \right)}{dr} = \dfrac{-2 \cdot V}{\pi \cdot r^3} = \dfrac{-2 \cdot \pi \cdot r^2 \cdot h}{\pi \cdot r^3} = -2 \cdot \dfrac{h}{r}[/tex]

[tex]\dfrac{dh}{dr} = \mathbf{ -2 \cdot \dfrac{h}{r}}[/tex]

The rate of change the height of the dough w.r.t. the radius is [tex]\underline{\dfrac{dh}{dr} = -2 \cdot \dfrac{h}{r}}[/tex]

Learn more here:

https://brainly.com/question/20489729

Split 36 into the ratio 1:2

a real estate agent expected 18 people to show up for an open house but 25 attended

Kylie drove 406 miles in 7 hours, how far would she travel in 19 hours

Which relations are functions?
Select Function or Not a function for each graph.

Bfksksbckdnxjmcbssoxonxnf

You pay 6% tax on a $30 portable speaker. What is the amount of tax?

what is 0.000 002 in standard form​

To be a member of the choir, you must go to an audition. Henry's cousin is a member of the choir. What conjecture can you make about Henry's cousin?

A.) Henry’s cousin practices with the choir
B.) Henry is in the choir
C.) Henry’s cousin went to an audition

Find the slope of a line parallel to 3y - 6x = 9
A. 2
B. 6
C.9
D.-6

You have $100 in your savings account and plan to deposit $20 each month. Write a linear equation to model the situation.

y= 100x + 20
y= 80x
y= 20x + 100
y= 120x

A teacher’s monthly gross pay is $5,321. The federal government withholds 15% of the teacher’s monthly gross pay for income taxes. How much does the teacher pay in income tax every month?

The middle-school camera club sold 240 tulip bulbs and 360 daffodil bulbs. Students divided the bulbs into 100 bags to sell at the school fair. Write an expression to show how many bulbs went into each of the 100 bags if students put the same total number of bulbs in each bag.

I literally had one more then someone gave the wrong answer so will give crown

in a survey of 3,200 people who owned a certain type of car, 1,760 said they would buy that type of car again. What percent of the people surveyed were satisfied with the car?​

Please find X! plspls​

Write the following number in word form: 735.009Immersive Reader
(2 Points)

pls help

due at 11:59

i need the equation in slope intercept in fraction form.
(-3,18),(3,2)

what is the lowest term of 35/63

​

the half life of radium is 1690 years. if 100 grams are present now, when will there be 20% of the original amount​

14. The diagram in Fig. 1.34 shows a uniform metre rule
weighing 100 gf, pivoted at its centre 0. Two weights
150 gf and 250 gf hang from the points A and B
respectively of the metre rule such that OA = 40 cm
and OB = 20 cm. Calculate : (i) the total anticlockwise
moment about O, (ii) the total clockwise moment about
O, (iii) the difference of anticlockwise and clockwise
moments, and (iv) the distance from O where a 100 gf
weight should be placed to balance the metre rule.
7-20 cm
А
승
B
40 cm
)
150 gf
Fig. 1.34
250 gf

(-9,-11) , (6,3) helppppppp

hihi! can you help me please? Giving brainleist to the best/correct awnser :))

Solve for x
9(3−2x)=2(10−8x)

Adriana can read 6 pages in 10 minutes. At this rate, how many pages can
she read in 25 minutes *
Your answer

HELP PLEASE ASAPPP I WILL GIVE BRAINLIEST

Pls help me ASAP I have other homework to do and I don’t have time for this. It also detects if it’s right or wrong:(

Beth was working at the grocery store, in the quick check-out line. A customer bought cereal for $3.29 and milk for $3.45. The customer paid with $10.00. How much change does Beth need to give the customer?
$3.36
$3.26
$6.74
None of these is correct.

Please helpp!!

A model car is for sale online. The owner discounts the price by 5% each day until it sells. On the third day, the car sells. If the original price of the model car was $40, how much did the car sell for to the nearest cent? What percent discount does this represent from the original price? Round to the nearest whole percent.

1) 15 = 2W - 5
2) 24 = 2x - 9

A zoologist is recording the loss of wolves in her state. She notes the number of wolves, w, in the state on January 1. One year later, there were 84 wolves in the state, which is 23 fewer wolves than were in the state a year earlier. What is the number of wolves in the state on January 1?

Which equation can be used to find the number of wolves in the state on January 1?


w + 23 = 84

w−23=84

w−84=−23

23−w=84