Help me solve them pls

Question

If a snowball melts so its surface area S decreases at a rate of 4 𝑐𝑚2 𝑠𝑒𝑐 ⁄ , find the rate at which the radius decreases when the diameter is 14 cm. Hint: 𝑆 = 4𝜋𝑟2 (surface area of a sphere)

a. What picture would you draw and label it?

b. How would you write down the numerical information ( in terms of the symbol you have chosen)

c. What are you asked to find? (the rate is expressed as a derivative)

d. What equation relates to the variables? Show algebraically how you would write this equation in terms of one variable

e. When you differentiate with respect to t, express the rate you want in terms of the rates and variables whose values you know.

f. What is the unknown rate to you are solving using the known quantities?

Answer 1

a) you're dealing with a sphere, so you should draw a sphere and label either the diameter, d, or the radius, r, or both;

b) I'm not sure what the "symbol" is that is being referred to, but you have the rate of change of the surface area, which should be represented as dS/dt, so dS/dt = -4 cm²/sec ( a negative rate because the sphere is decreasing in surface area ), you have the diameter measurement at the "freeze-frame" moment, which is 14, so d = 14, and you can deduce that the radius is 7, so r = 7.

c) you are asked to find the rate of change of the radius, so this can be represented as dr/dt (Note: this, too, will end up being negative)

d) The equation we use will be the surface area formula, so the equation, as stated in the problem, is S = 4πr².

e) Once we take the derivative of each side of the equation with respect to t, the equation will look like this:

dS/dt = 8πr dr/dt. If we fill in the given information for our "freeze-frame" moment, the equation becomes

-4 = 8π(7) dr/dt. Isolating the dr/dt in this equation, we have dr/dt = -4/(56π).

f) Doing the arithmetic here, we have dr/dt = -1/(16π) which is approximately -0.02 cm/sec ( Or, 0.02 cm/sec that the radius is decreasing ).