Question

A spherical exercise ball has a maximum diameter of 30 inches when filled with air. The ball was completely empty at the start, and an electric air pump is filling it with air at the rate of 1600 cubic Inches per minute. The formula for the volume of a sphere is 4pi (r^3/3)

Part A Enter an equation for the amount of air still needed to fill the ball to its maximum volume, y, with respect to the number of minutes the pump has been pumping air into the ball, X.

Part B Enter the total amount of air, in cubic inches, still needed to fill the ball after the pump has been running for 4 minutes.

Part C Enter the estimated number of minutes it takes to pump up the ball to its maximum volume.

Answer 1

A) A(t) = 4500*π - 1600*t

B) A(4) = 7730 in³

C) t = 8,8 sec

Explanation:

The volume of the sphere is:

d max = 30 r max = 15 in

V(s) = (4/3)*π*r³ V(s) = (4/3)*π* (15)³

V(s) = 4500*π

A) Amount of air needed to fill the ball A(t)

A(t) = Total max. volume of the sphere - rate of flux of air * time

A(t) = 4500*π - 1600*t in³

B) After 4 minutes

A(4) = 4500*π - 6400

A(4) = 14130 - 6400

A(4) = 7730 in³

C) A(t) = 4500*π - 1600*t

when A(t) = 0 the ball got its maximum volume then:

4500*π - 1600*t = 0

t = 14130/1600

t = 8,8 sec