Question

1. a customer comes to pick up some washed rock from als landscaping business. in the back of his truck, he has one large barrel in the shape of a cylinder. he wants to put the washed rock into the barrel. the dimensions of the barrel are 1.5 m diameter by 1.5 m high.

a. what is the capacity of the barrel in cubic metres? round to one decimal place.
b. the customer plans to divide the rock into 3 equal conical piles at his project location. using your knowledge of cylinders and cones, suggest the diameter and height of each pile that he should use. explain how you arrived at your suggestion.

Explanation:

Step1: Recall cylinder volume formula

The volume formula for a cylinder is $V=\pi r^{2}h$, where $r$ is the radius and $h$ is the height. Given the diameter $d = 1.5$ m, the radius $r=\frac{d}{2}=\frac{1.5}{2}= 0.75$ m and height $h = 1.5$ m.

Step2: Calculate cylinder volume

$V=\pi\times(0.75)^{2}\times1.5=\pi\times0.5625\times1.5\approx2.6$ m³ (rounded to one - decimal place).

Step3: Recall cone - cylinder volume relationship

The volume of a cone is $V_{cone}=\frac{1}{3}\pi r_{1}^{2}h_{1}$, and the volume of the cylinder is $V_{cylinder}=\pi r^{2}h$. If we divide the volume of the cylinder into 3 equal - volume cones, the volume of each cone $V_{cone}=\frac{V_{cylinder}}{3}$.
Since $V_{cylinder}=\pi r^{2}h$ and $V_{cone}=\frac{1}{3}\pi r_{1}^{2}h_{1}$, and $V_{cone}=\frac{V_{cylinder}}{3}$, we can assume that if the radius and height of the cone are the same as the radius and height of the cylinder, the volume of the cone will be one - third of the volume of the cylinder. So a possible suggestion is that the diameter of each conical pile is $1.5$ m and the height is $1.5$ m. Because when $r_{1}=r = 0.75$ m and $h_{1}=h = 1.5$ m, $V_{cone}=\frac{1}{3}\pi r_{1}^{2}h_{1}=\frac{1}{3}\times\pi\times(0.75)^{2}\times1.5$, and $V_{cylinder}=\pi\times(0.75)^{2}\times1.5$, and $V_{cone}=\frac{V_{cylinder}}{3}$.

Answer:

a. $2.6$ m³
b. Diameter: $1.5$ m, Height: $1.5$ m. The volume of a cone is one - third of the volume of a cylinder with the same radius and height. So if we want 3 equal - volume cones from the volume of the given cylinder, we can make the cones have the same radius and height as the cylinder.