A balloon (in the shape of a sphere) is being inflated. The radius is increasing at a rate of 4 cm per second. Find a function, r(t) , for the radius in terms of t . Find a function, V(r) , for the volume of the balloon in terms of r . Find $\left( \mathrm { V } { } ^ { \circ } \mathrm { r } \right) ( \mathrm { t } )$
A) $( \mathrm { V } \cdot \mathrm { r } ) ( \mathrm { t } ) = \frac { 320 \pi \mathrm { t } ^ { 2 } } { 3 }$
B) $\left( V { } ^ { \circ } r \right) ( t ) = \frac { 112 \pi t ^ { 3 } } { 3 }$
C) $( \mathrm { V } \circ \mathrm { r } ) ( \mathrm { t } ) = \frac { 256 \pi t ^ { 3 } } { 3 }$
D) $( \mathrm { V } \circ \mathrm { r } ) ( \mathrm { t } ) = \frac { 1024 \pi \sqrt { \mathrm { t } } } { 3 }$

Question

A balloon (in the shape of a sphere) is being inflated. The radius is increasing at a rate of  4 cm  per second. Find a function,  r(t) , for the radius in terms of  t . Find a function,  V(r) , for the volume of the balloon in terms of r . Find $\left( \mathrm { V } { } ^ { \circ } \mathrm { r } \right) ( \mathrm { t } )$
A) $( \mathrm { V } \cdot \mathrm { r } ) ( \mathrm { t } ) = \frac { 320 \pi \mathrm { t } ^ { 2 } } { 3 }$
B) $\left( V { } ^ { \circ } r \right) ( t ) = \frac { 112 \pi t ^ { 3 } } { 3 }$
C) $( \mathrm { V } \circ \mathrm { r } ) ( \mathrm { t } ) = \frac { 256 \pi t ^ { 3 } } { 3 }$
D) $( \mathrm { V } \circ \mathrm { r } ) ( \mathrm { t } ) = \frac { 1024 \pi \sqrt { \mathrm { t } } } { 3 }$

Quizplus · Accepted Answer

Given that the radius is increasing at a rate of 4 cm per second, the function for the radius in terms of time, $t$, is $r(t) = 4t$. The volume of a sphere is given by $V(r) = \frac{4}{3}\pi r^3$. Substituting $r(t) = 4t$ into $V(r)$ gives $V(r(t)) = \frac{4}{3}\pi (4t)^3 = \frac{256\pi t^3}{3}$, which corresponds to choice C.

A Balloon (In the Shape of a Sphere) Is Being (V∘r)(t)\left( \mathrm { V } { } ^ { \circ } \mathrm { r } \right) ( \mathrm { t } )(V∘r)(t)

A Balloon (In the Shape of a Sphere) Is Being $\left( \mathrm { V } { } ^ { \circ } \mathrm { r } \right) ( \mathrm { t } )$