Answer:
5571.99
Step-by-step explanation:
We need to use the Pythagorean theorem to solve the problem.
The theorem indicates that,
[tex]r^2+h^2=24^2 \\r^2+h^2=576\\r^2=576-h^2[/tex]
Once this is defined, we proceed to define the volume of a cone,
[tex]v=\frac{1}{3}\pi r^2 h[/tex]
Substituting,
[tex]v=\frac{1}{3} \pi (576-h^2)h\\v=\frac{1}{3} \pi (576h-h^3)[/tex]
We need to find the maximum height, so we proceed to calculate h, by means of its derivative and equalizing 0,
[tex]\frac{dv}{dh} = \frac{1}{3} \pi (576-3h^2)[/tex]
[tex]\frac{dv}{dh} = 0[/tex] then [tex]\rightarrow \frac{1}{3}\pi(576-3h^2)=0[/tex]
[tex]h_1=-8\sqrt{3}\\h_2=8\sqrt{3}[/tex]
We select the positiv value.
We have then,
[tex]r^2 = 576-(8\sqrt3)^2 = 384\\r=\sqrt{384}[/tex]
We can now calculate the maximum volume,
[tex]V_{max}= \frac{1}{3}\pi r^2 h = \frac{1}{3}\pi (\sqrt{384})^2 (8\sqrt{3}) = 5571.99[/tex]
