I need help with two related rate questions!?

Question

1. Assume that the radius r of a sphere is expanding at a rate of 14in./min. It's surface area is 4π r^2. Determine the rate at which the surface area is changing with respect to time at t=2min, assuming that r=3 at t=0

2.At a given moment, a plane passes directly above a radar station at an altitude of 6 miles. (after drawing a diagram) Suppose that the line through the radar station and the plane makes an angle θ with the horizontal. How fast is θ changing 10 min after the plane passes over the radar station?

Answer 1

Only one problem per question please.

1. Assume that the radius r of a sphere is expanding at a rate of 14in./min. It's surface area is 4πr². Determine the rate at which the surface area is changing with respect to time at t=2min, assuming that r=3 at t=0.

Let
A = surface area
r = radius sphere
t = time in minutes

Given
dr/dt = 14 in/min

Find
dA/dt when t = 2 min

We have:

r = 3 + 14t = 3 + 14*2 = 3 + 28 = 31 in
at t = 2 min.

A = 4πr²
dA/dr = 8πr

dA/dt = (dA/dr)(dr/dt) = (8πr)(14) = 112πr
dA/dt = 112π(31) = 3472π in²/min

at t = 2 minutes

Answer 2

1. Assume that the radius r of a sphere is expanding at a rate of 14in./min. It's surface area is 4π r^2. Determine the rate at which the surface area is changing with respect to time at t=2min, assuming that r=3 at t=0
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Give the surface area a function name, G(x)

So G(r)=4π r^2 and r is changing as you presented at
14in /min this rate.

The the rate at which the surface area is changing with respect to time would be the derivative of the G(x)
Here is G(r) since the variable is radius, in stead of x.

Which is G'(r)= ?
G'(r)= { ( 2*4π )*r^1 } dr/dt

we can simplify the equation of G'(r) to
G'(r)=
8*π*r dr/dt

Well, you also know the rate for dr/ dt which is 14 in. /min., so what you want to do is replace dr/dt as 14

Now you have G'(r)=

8*π*r*14 = 112*π*r

You knew t = 0 , r = 3, now you want to know t = 2, r =?
r = 14*2 = 28

For surface area, the rate would be 112*π*28 =

3136*π

Answer: 3136*π