Question

I know that I would use dv/dt but I’m confused on what to do #39 and 41

Answer 1

Starting with the volume of a cube with side length
`x`,

`V(x) = x^3`

differentiate both sides with respect to time
`t`. By the chain rule,

`(dV)/(dt) = 3x^2 (dx)/(dt)`

We're given that
`(dx)/(dt) = 2(\rm in)/(\rm min)`.

39. At the moment
`x=8\,\rm in`, solve for
`(dV)/(dt)`.

`(dV)/(dt) = 3 (8\,\mathrm{in})^2 \left(2(\rm in)/(\rm min)\right) = 384 (\rm in^3)/(\rm min)`

41. Find
`x` when
`V = 55\,\rm in^3`.

`55\,\mathrm{in}^3 = x^3 \implies x = 55^(1/3) \,\rm in`

Solve for
`(dV)/(dt)`.

`(dV)/(dt) = 3 \left(55^(1/3) \,\mathrm{in}\right)^2 \left(2(\rm in)/(\rm min)\right) = 6\cdot55^(2/3)(\rm in^3)/(\rm min)`