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ivolga24 [154]
3 years ago
5

The surface area of a cube is increasing at a rate of 15 square meters per hour. At a certain instant, the surface area is 24 sq

uare meters. What is the rate of change of the volume of the cube at that instant (in cubic meters per hour)?
Mathematics
1 answer:
algol133 years ago
4 0

Answer:

The volume of cube is increasing at a rate 7.5 cubic meter per hour.  

Step-by-step explanation:

We are given the following in the question:

Surface area of cube = 24 square meters.

Let l be the edge of cube.

Surface area of cube =

6l^2 = 24\\l^2 = 4\\l = 2

Thus, at that instant the edge of cube is 2 meters.

\dfrac{dS}{dt} = 15\text{ square meters per hour}\\\\\dfrac{d(6l^2)}{dt} = 15\\\\12l\dfrac{dl}{dt} = 15\\\\\dfrac{dl}{dt} = \dfrac{15}{12\times 2} = \dfrac{15}{24}

We have to find the rate of change in volume.

Volume of cube =

l^3

Rate of change of volume =

\dfrac{dV}{dt} = \dfrac{d(l^3)}{dt} = 3l^2\dfrac{dl}{dt}\\\\\dfrac{dV}{dt} = 3(2)^2\times \dfrac{15}{24} \\\\\dfrac{dV}{dt} = 7.5\text{ cubic meter per hour}

Thus, the volume of cube is increasing at a rate 7.5 cubic meter per hour.

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