Question

1. find the height of the cylinder given the volume is 254 in³. round to the nearest inch. we know
2. find the length of the sides of the cube given the volume is 1815.8 mm³. round to one decimal place. we know

Explanation:

Step1: Recall volume formula for cylinder

The volume formula of a cylinder is $V=\pi r^{2}h$, where $V$ is volume, $r$ is radius and $h$ is height. Given diameter $d = 6$ in, so radius $r=\frac{d}{2}=3$ in and $V = 254$ in³. We need to solve for $h$.
\[h=\frac{V}{\pi r^{2}}\]

Step2: Substitute values and calculate

Substitute $V = 254$, $r = 3$ into the formula:
\[h=\frac{254}{\pi\times3^{2}}=\frac{254}{9\pi}\approx\frac{254}{9\times3.14}\approx\frac{254}{28.26}\approx9\] in

Step3: Recall volume formula for cube

The volume formula of a cube is $V = s^{3}$, where $V$ is volume and $s$ is the side - length. Given $V=1815.8$ mm³, we need to solve for $s$.
\[s=\sqrt[3]{V}\]

Step4: Calculate side - length of cube

Substitute $V = 1815.8$ into the formula:
\[s=\sqrt[3]{1815.8}\approx12.2\] mm

Answer:

The height of the cylinder is approximately 9 in.
The length of the side of the cube is approximately 12.2 mm.