Question

A pyramid has height h and a square base with side x. The volume of a pyramid is V = 1 3 x2h. If the height remains fixed and the side of the base is decreasing by 0.004 meter/yr, what rate is the volume decreasing when the height is 120 meters and the width is 150 meters?The volume is decreasing at a rate of ___________meters^3/year.

Answer 1

Answer:

The volume is decreasing at a rate of 48 cubic meters per year.

Step-by-step explanation:

We are given the following information in the question:

A pyramid has height h and a square base with side x.

$\text{Volume of square pyramid} = \displaystyle\frac{1}{3}x^2 h$

The height of the square pyramid remains constant.

$\displaystyle\frac{dh}{dt} = 0$

The side of the base is decreasing by 0.004 meter/yr

$\displaystyle\frac{dx}{dt} = -0.004\text{ meter per year}$

We have to find the rate at which the volume is decreasing when the height is 120 meters and the width is 150 meters.

$V = \displaystyle\frac{1}{3}x^2 h\\\\\frac{dV}{dt} = \frac{1}{3}\bigg(2x\frac{dx}{dt}h + x^2 \frac{dh}{dt}\bigg)\\\\\text{Putting h = 120 meters and x = 150 meters}\\\\\frac{dV}{dt} = \frac{1}{3}\bigg(2(150)(-0.004)(120) + (150)^2(0)\bigg) = -48\text{ cubic meter per year}$

The volume is decreasing at a rate of 48 cubic meters per year.