18. shrinking cube the volume of a cube decreases at a rate of 0.5 ft³/min. what is the rate of change of…

Explanation:

Step1: Write the volume formula for a cube

The volume (V) of a cube with side length (x) is (V = x^{3}).

Step2: Differentiate the volume formula with respect to time (t)

Using the chain - rule, (\frac{dV}{dt}=3x^{2}\frac{dx}{dt}).

Step3: Substitute the given values into the differentiated equation

We know that (\frac{dV}{dt}=- 0.5) (negative because the volume is decreasing) and (x = 12). Substitute into (\frac{dV}{dt}=3x^{2}\frac{dx}{dt}), we get (-0.5=3\times(12)^{2}\times\frac{dx}{dt}).

Step4: Solve for (\frac{dx}{dt})

First, calculate (3\times(12)^{2}=3\times144 = 432). Then (\frac{dx}{dt}=\frac{-0.5}{432}=-\frac{1}{864}\approx - 0.00116)

Answer:

The rate of change of the side length is (-\frac{1}{864}\text{ ft/min}\approx - 0.00116\text{ ft/min})