You are blowing air into a spherical balloon at a rate of 33 cubic inches per second. Given that the radius of the balloon is 22 inches when t=4t=4 seconds answer the following questions:

a. How fast is the radius of the balloon growing at t=4 seconds.
b. What is the rate of change of the surface area at t =4 seconds?

a) the radius of the balloon increases at a rate of 5.42 in/s

b) the surface area of the balloon increases at a rate of 3 in²/s

Step-by-step explanation:

a) since the volume of a sphere V is

V= 4/3*π*R³

where R= radius , then the rate of change of the volume is

V' = dV/dR= 4*π*R²

using the chain rule

dV/dt = dV/dR*dR/dt

thus

k = 4*π*R² * dR/dt

dR/dt = k/(4*π*R²)

replacing values

dR/dt = k/(4*π*R²) = (33 in³/s) /(4*π*(22 in)²] = 5.42 in/s

then the radius of the balloon increases at a rate of 5.42 in/s

b) since the surface area is

S=4*π*R²

then

S' = dS/dR= 8*π*R

and

dS/dt = dS/dR*dR/dt = 8*π*R * k/(4*π*R²) = 2*k/R

replacing values

dS/dt = 2*k/R = 2*(33 in³/s)/( 22 in) = 3 in²/s

then the surface area of the balloon increases at a rate of 3 in²/s

answer: there are x+184 pages in olivia's book.

step-by-step explanation:

let the total number of pages in the book which olive has already read be x.

then if olive has 184 pages left to read, then the total number of pages are in olive's book= pages she has read+pages she has left to read

therefore, the total number of pages in olivia's book=

hence, there are x+184 pages in olivia's book.