Question

A rectangular prism has a width of 16 inches and a length of 10 inches. The volume of the rectangular prism is decreasing at a rate of 555 cubic inches per minute, with the width and length being held constant. What is the rate of change, in inches per minute, of the height when the height is 3 inches?

Answer 1

Answer:

-3.47 inches per minute

Step-by-step explanation:

Let the height be 'h' for any time 't'.

Given:

Length of the prism (L) = 10 in

Width of the prism (w) = 16 in

The rate of change of volume of prism is, $\frac{dV}{dt}=-555\ in^3/min$ (The negative sign implies the volume decreases with time.)

Now, rate of change of height of prism = $\frac{dh}{dt}=?$

Length and width remains constant.

Now, we know that, volume of a rectangular prism is given as:

Volume = Length × width × Height

$V=Lwh\\\\V=10\times 16\times h\\\\V=160h$

Now, differentiating both sides with respect to time 't', we get:

$\frac{dV}{dt}=\frac{d}{dt}(160h)$

Plug in the given values and solve for $\frac{dh}{dt}$. This gives,

$-555=160\times \frac{dh}{dt}\\\\\frac{dh}{dt}=\frac{-555}{160}\\\\\frac{dh}{dt}=-3.47\ in/min$

The negative sign implies the height decreases with time.

Therefore, the the rate of change, in inches per minute, of the height when the height is 3 inches is -3.47 inches per minute.