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

Answer:

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

Step-by-step 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

$\dfrac{dV}{dt} = 9\text{ cubic meters per hour}\\\\\dfrac{dw}{dt} = \dfrac{dh}{dt} = 40\text{ centimeters per hour} =0.4\text{ meters per hour}$

We have to find the rate of increase of length.

Volume of cuboid =

$V = l\times w\times h$

Differentiating we get,

$\displaystyle\frac{dV}{dt} = \frac{dl}{dt}wh + \frac{dw}{dt}lh +\frac{dh}{dt}lw$

Putting values, we get,

$9 = \dfrac{dl}{dt}(4)(2) + (0.4)(3)(2) + (0.4)(3)(4)\\\\\dfrac{dl}{dt}(4)(2) = 9 -7.2\\\\\dfrac{dl}{dt}=0.225$

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