Question

A block of ice in the form of a cube has one edge 10 feet long. It is melting uniformly so that its dimensions decrease at a rate of 1/10 fps (feet per second). At what rate is the volume of the block changing when the edge is 5 feet long?

Answer 1

Answer:

-7.5 ft^3/sec

OR

Volume is decreasing at the rate of 7.5 ft^3/sec

Step-by-step explanation:

Side of cube = 10 feet

Rate of decreasing of side of cube = $\frac{1}{10}\ feet/sec$

OR

Rate of change of side of cube = $-\frac{1}{10}\ feet/sec$

To find:

Rate of change in volume when the edge is 5 feet long = ?

Solution:

Volume of a cone is given by:

$V =Side^3$

If side is $a$ units, then Volume can be written as:

$V =a^3$

Differentiating w.r.to time:

$\dfrac{dV}{dt} = 3a^2\dfrac{da}{dt}\\\Rightarrow \dfrac{dV}{dt} = 3\times 5^2(-\frac{1}{10})\\\Rightarrow \dfrac{dV}{dt} = -7.5 ft^3/sec$

Negative sign indicates that the volume is decreasing at the rate of 7.5 ft^3/sec