Question

The volume of a spherical hot air balloon v(r) = 4 3 πr3 changes as its radius changes. the radius is a function of time given by r(t) = 3t. find the average rate of change of the volume with respect to t as t changes from t = 1 to t = 4.

Answer 1

we are given

volume of sphere as

$V=\frac{4}{3} \pi r^3$

$r(t)=3t$

(a)

we can plug r into V

we get

$V=\frac{4}{3} \pi (3t)^3$

$V(t)=36\pi t^3$

(b)

we can use average rate of change formula

$A=\frac{f(b)-f(a)}{b-a}$

here , we are given t=1 to t=4

so, we get formula as

$A=\frac{V(4)-V(1)}{4-1}$

now, we can plug values

$A=\frac{36 \pi (4)^3-36 \pi (1)^3}{4-1}$

$A=756\pi$.............Answer