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

What is the slope of the line 4x – 2y = 5? 4 O-2 2
​

transversals angles​

what's b ? plz help ​

Help me with this question! Will mark brainliest

What is the place the 8 is in in the number 38,341,097.54

Which equation has no solution? Select all that apply.
A 15 - 3x = 5(x + 4) - 8x
B) 2(x - 7) = 2x – 10
C) -2 - 2x = 6(2x + 2)
D) 24x – 22 = -4(1 - 6x)​

Write 6.25 in expanded notation

HELLLOOOOOO VERIFIED PEOPLE OR GENIUS PEOPLE PLEASE HELP ME OUT!!

Use the graph at the right. Find the vertices of the image of QRTW for a dilation with center (0,0) and a scale factor of 9.

Q' =
R' =
T' =
W' =

1. A right triangle is shown.
25 cm
7 cm
Based on this information, what is the length of the base of this triangle?
HELP PLS I BEG YOU

Jim and Carla are scuba diving. Jim started out 8 feet below the surface. He descended 20 feet, rose 7 feet, and descended 12 more feet. Then he rested. Carla started out at the surface. She descended 15 feet, rose 7 feet, and descended another 20 feet. Then she rested. Which person rested at a greater depth?

write two statements of inequality please!

During an airplanes ascent, the plane reaches a horizontal distance from the airport of 1 km and then 7 km. The plane ascends at a slope of 2/3. Find the vertical distance (y) in feet that the plane traveled during the indicated spots

John bought a play station game for $54.50. The tax on the game is 5%. What
is the total cost of the play station game, including the tax?

Which division problem can be described using this model?

2÷15

2÷5​

2÷110​

1÷110​
Two rectangular strips in a row, the top of each strip is labeled 1. Each strip is divided into 5 equal parts.

the movie theater has 250 seats 200 seats were sold for the current showing what percent of seats of seats are empty

Help with these questions I´ll CASHAPP YOU 55$$

PLS HELP 50 POINTS IF YOU DONT KNOW DONT ANSWER THE QUESTION

Need help on the first one! Will mark brainliest :)

What is the area of this triangle?

12 points
8. Javier worked out the following on his math homework: "What
equation best represents the relationship between x and y in the graph?"
Javier's answer was y= - 4x - 2. Is he correct? If not, explain what he did
wrong AND give the correct answer. (This question will be hand graded) *
1096
2716810
M

Micah who is not yet 15 years old, is two years older than DeMonte.


How old is DeMonte?
x = the age of Micah

y = the age of DeMonte

help plz bc idk what to do​

Suppose the linear regression line y = 2.1x + 130 predicts sales based on the
money spent on advertising. If x represents the dollars spent in advertising,
and y represents the company sales in dollars, about how much can the
company expect in sales if it spends $150 in advertising?

The sum of two numbers is 1633 and their difference is 35. Find the 2 numbers using simultaneous equation.
Please show working out

- x3 + 4x2 – 3x + 12 when x = -2​

what is the answer to 3(4x-5)

Help I can’t fail plz help

Which ratio is equivalent to 5:9?
9:45
25:81
27:15
30:54

what is the range of the data 79, 52, 85,94, 71, 82, 84, 66, 58, 98, 72, 100

​

Help? I would appreciate it