Is my solution correct?

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My physics teacher gave us a maths problem to solve. I solved the problem, but the number was a long decimal, which seems wrong to me. I was wondering if my method could be checked. Thanks

Question:
There is a cone, with a frustum indicated.

The volume of the frustum is $488\pi$; the diameter of the base of the entire cone is $\frac{15}{4}$; the vertical height of the top part of the cone (above the frustum) is $96$. The top part of the frustum (base of top part of cone) has a radius of $r$.

We are asked to find $r$.

My solution:
The radius of the base of the entire cone is $\frac{15}{8}$.

Now, call the volume of the frustum $V_f$, for the cone $V_c$ and $V_t$ for the top part.

Then $V_c = V_f + V_t$

We know the volume of the frustum, so

$V_c = 488\pi + V_t \Longrightarrow 488\pi = V_c - V_t$

We look for an expression for $h$ (the vertical height of the entire cone) in terms of $r$. The ratio of $r$ to $\frac{15}{8}$ is equal to that of $h$ to $96$.

So $\frac{r}{\frac{15}{8}} = \frac{h}{96} \Longrightarrow h = \frac{180}{r}$

Then
$V_c = \frac{h}{3}\pi\cdot (\frac{15}{8})^2 = \frac{225h}{192}\pi = \frac{10125}{48r}\pi$

Now, $V_t = \frac{96}{3}\pi r^2 = 32\pi r^2$

So $488\pi = \frac{10125}{48r}\pi - 32\pi r^2 \Longrightarrow 488 = \frac{10125}{48r} - 32 r^2$

Multiplying through by $48r$ we have that
$23424r = 10125 - 1536r^3$

Hence
$1536r^3 + 23424r - 10125 = 0$

We may divide through by three to give smaller numbers
$512r^3 + 7808r - 3375 = 0$

Using a cubic solver, we find that the only real solution is that $r \approx 0.42714$.

Your solution is correct.

I cheated and used the formula for the volume of a frustrum.

I then plugged everything into Wolfram Alpha and got the same as you.

I don't think there's a nice way to do this question (that doesn't involve a cubic). What level of education are you at? Did your Physics teacher make this question up and why did they give it to you?

Looks right to me!!

I'm currently AS level and my teacher likes giving us maths problems to solve. I don't think he made it up - I think someone else did. Thanks for the response

Weird question to come from a Physics teacher