To determine the height of a cone with a given base radius and volume, we use the volume formula for a cone:
\(\[ V = \frac{1}{3} \pi r^2 h \]\)
where:
- \(\( V \)\) is the volume of the cone,
- \(\( r \)\) is the radius of the base, and
- \(\( h \)\) is the height of the cone.
Given:
- The radius \(\( r = 4 \)\) meters,
- The volume \(\( V = 40 \)\) cubic meters.
We need to solve for the height \(\( h \)\). First, we rearrange the formula to solve for \(\( h \)\):
\(\[ h = \frac{3V}{\pi r^2} \]\)
Substitute the given values into the equation:
\(\[ h = \frac{3 \times 40}{\pi \times 4^2} \]\)
Calculate the denominator:
\(\[ \pi \times 4^2 = \pi \times 16 \]\)
Then the equation for \(\( h \)\) becomes:
\(\[ h = \frac{120}{16\pi} \]\)
Next, calculate the value of \(\( h \)\):
\(\[ h = \frac{120}{16 \times 3.141592653589793} \]\)
Simplify this further:
\(\[ h = \frac{120}{50.26548245743669} \]\)
\(\[ h \approx 2.39 \, \text{meters} \]\)
To the nearest tenth, the height \(\( h \)\) is:
\(\[ h \approx 2.4 \, \text{meters} \]\)
So, the height of the cone is approximately 2.4 meters.
