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
[tex]\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}[/tex]
We have to find the rate of increase of length.
Volume of cuboid =
[tex]V = l\times w\times h[/tex]
Differentiating we get,
[tex]\displaystyle\frac{dV}{dt} = \frac{dl}{dt}wh + \frac{dw}{dt}lh +\frac{dh}{dt}lw[/tex]
Putting values, we get,
[tex]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[/tex]
Thus, the length of rectangular box is increasing at a rate 0.225 meters per hour.
