Respuesta :
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, [tex]\frac{dV}{dt}=-555\ in^3/min[/tex] (The negative sign implies the volume decreases with time.)
Now, rate of change of height of prism = [tex]\frac{dh}{dt}=?[/tex]
Length and width remains constant.
Now, we know that, volume of a rectangular prism is given as:
Volume = Length × width × Height
[tex]V=Lwh\\\\V=10\times 16\times h\\\\V=160h[/tex]
Now, differentiating both sides with respect to time 't', we get:
[tex]\frac{dV}{dt}=\frac{d}{dt}(160h)[/tex]
Plug in the given values and solve for [tex]\frac{dh}{dt}[/tex]. This gives,
[tex]-555=160\times \frac{dh}{dt}\\\\\frac{dh}{dt}=\frac{-555}{160}\\\\\frac{dh}{dt}=-3.47\ in/min[/tex]
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.
