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Express the formula [tex]V = \pi r^2 H[/tex] in terms of the height, [tex]H[/tex]. Use that formula to find the height when the volume is [tex]45 \pi[/tex] and the radius is 3.

A. [tex]H = \frac{V}{\pi r}[/tex] ; [tex]H = 15[/tex]
B. [tex]H = \frac{V}{2 r}[/tex] ; [tex]H = 7.5[/tex]
C. [tex]H = \frac{V}{\pi r^2}[/tex] ; [tex]H = 5[/tex]
D. [tex]H = \frac{V}{\pi} - r^2[/tex] ; [tex]H = 30[/tex]

Sagot :

GinnyAnsweridn
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]
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