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Answer :
We start with the formula for the volume of a cone:
[tex]$$
V = \frac{1}{3}\pi r^2 h.
$$[/tex]
Given that the volume is [tex]$19$[/tex] cubic units and the radius is [tex]$2.5$[/tex], we substitute these values into the formula:
[tex]$$
19 = \frac{1}{3}\pi (2.5)^2 h.
$$[/tex]
First, compute the square of the radius:
[tex]$$
(2.5)^2 = 6.25.
$$[/tex]
Now the equation becomes:
[tex]$$
19 = \frac{1}{3}\pi (6.25) h.
$$[/tex]
To solve for [tex]$h$[/tex], multiply both sides of the equation by [tex]$3$[/tex]:
[tex]$$
3 \times 19 = \pi (6.25) h.
$$[/tex]
This simplifies to:
[tex]$$
57 = 6.25\pi\, h.
$$[/tex]
Finally, isolate [tex]$h$[/tex] by dividing both sides by [tex]$6.25\pi$[/tex]:
[tex]$$
h = \frac{57}{6.25\pi}.
$$[/tex]
Thus, the expression representing the height of the cone is:
[tex]$$
\boxed{\frac{57}{6.25\pi}}.
$$[/tex]
[tex]$$
V = \frac{1}{3}\pi r^2 h.
$$[/tex]
Given that the volume is [tex]$19$[/tex] cubic units and the radius is [tex]$2.5$[/tex], we substitute these values into the formula:
[tex]$$
19 = \frac{1}{3}\pi (2.5)^2 h.
$$[/tex]
First, compute the square of the radius:
[tex]$$
(2.5)^2 = 6.25.
$$[/tex]
Now the equation becomes:
[tex]$$
19 = \frac{1}{3}\pi (6.25) h.
$$[/tex]
To solve for [tex]$h$[/tex], multiply both sides of the equation by [tex]$3$[/tex]:
[tex]$$
3 \times 19 = \pi (6.25) h.
$$[/tex]
This simplifies to:
[tex]$$
57 = 6.25\pi\, h.
$$[/tex]
Finally, isolate [tex]$h$[/tex] by dividing both sides by [tex]$6.25\pi$[/tex]:
[tex]$$
h = \frac{57}{6.25\pi}.
$$[/tex]
Thus, the expression representing the height of the cone is:
[tex]$$
\boxed{\frac{57}{6.25\pi}}.
$$[/tex]
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