To solve this problem, we need to start with the given formula for the volume of a cylinder, which is:
[tex]\[ V = \pi r^2 H \][/tex]
We are asked to express the height [tex]\( H \)[/tex] in terms of the volume [tex]\( V \)[/tex], the radius [tex]\( r \)[/tex], and [tex]\( \pi \)[/tex]. We can do this by isolating [tex]\( H \)[/tex] on one side of the equation.
To isolate [tex]\( H \)[/tex], divide both sides of the equation by [tex]\( \pi r^2 \)[/tex]:
[tex]\[ H = \frac{V}{\pi r^2} \][/tex]
Now, we need to use this formula to find the height [tex]\( H \)[/tex] given that the volume [tex]\( V \)[/tex] is [tex]\( 45\pi \)[/tex] and the radius [tex]\( r \)[/tex] is 3.
Substitute [tex]\( V = 45\pi \)[/tex] and [tex]\( r = 3 \)[/tex] into the formula:
[tex]\[ H = \frac{45\pi}{\pi \cdot 3^2} \][/tex]
Simplify the expression:
[tex]\[ H = \frac{45\pi}{\pi \cdot 9} \][/tex]
The [tex]\( \pi \)[/tex] terms cancel each other out:
[tex]\[ H = \frac{45}{9} \][/tex]
Perform the division:
[tex]\[ H = 5 \][/tex]
So, the height [tex]\( H \)[/tex] is [tex]\( 5 \)[/tex].
Therefore, the correct expression and value are:
[tex]\[ H = \frac{V}{\pi r^2} ; H = 5 \][/tex]
