To determine the height of a cone with a given radius and volume, we will use the formula for the volume of a cone:
[tex]\[ V = \frac{1}{3} \pi r^2 h \][/tex]
Where:
- [tex]\( V \)[/tex] is the volume of the cone,
- [tex]\( r \)[/tex] is the radius of the cone's base,
- [tex]\( h \)[/tex] is the height of the cone.
Given:
- The radius [tex]\( r = 2.5 \)[/tex] units,
- The volume [tex]\( V = 19 \)[/tex] cubic units.
We need to find the value of [tex]\( h \)[/tex]. First, let's rearrange the formula to solve for the height [tex]\( h \)[/tex]:
[tex]\[ h = \frac{3V}{\pi r^2} \][/tex]
Now, we substitute the given values into the equation:
[tex]\[ h = \frac{3 \times 19}{\pi \times 2.5^2} \][/tex]
Let's calculate the denominator first:
[tex]\[ 2.5^2 = 6.25 \][/tex]
So the expression becomes:
[tex]\[ h = \frac{3 \times 19}{\pi \times 6.25} \][/tex]
Next, simplify the numerator and the denominator:
[tex]\[ \text{Numerator} = 3 \times 19 = 57 \][/tex]
[tex]\[ \text{Denominator} = \pi \times 6.25 \][/tex]
Thus, the expression that represents the height of the cone is:
[tex]\[ h = \frac{57}{\pi \times 6.25} \][/tex]
By substituting the values, we find that the height [tex]\( h \)[/tex] is approximately [tex]\( 2.902986161996171 \)[/tex] units.
