Question

A rectangular box initially has length 3 meters, width 4 meters and height 2 meters. Its volume is increasing at a rate of 9 cubic meters per hour. If the width and height are both increasing at a rate of 40 centimeters per hour, how quickly does the length increase by_____________

Answer 1

The length of rectangular box is increasing at a rate 0.225 meters per hour.

Explanation:

We are given the following in the question:

Initial dimensions of rectangular box:

Length,l = 3 m

Width,w = 4 m

Height,h = 2 m

`(dV)/(dt) = 9\text{ cubic meters per hour}\\\\(dw)/(dt) = (dh)/(dt) = 40\text{ centimeters per hour} =0.4\text{ meters per hour}`

We have to find the rate of increase of length.

Volume of cuboid =

`V = l* w* h`

Differentiating we get,

`\displaystyle(dV)/(dt) = (dl)/(dt)wh + (dw)/(dt)lh +(dh)/(dt)lw`

Putting values, we get,

`9 = (dl)/(dt)(4)(2) + (0.4)(3)(2) + (0.4)(3)(4)\\\\(dl)/(dt)(4)(2) = 9 -7.2\\\\(dl)/(dt)=0.225`

Thus, the length of rectangular box is increasing at a rate 0.225 meters per hour.