Question

A cone (V=1/3 pi r^ 2 h) dagger has a height of 9 inches and a volume equal to (3pi * y ^ 2 + 30pi*y + 75pi) cubic inches What is the cone's radius () in terms of y? Show all steps.

Answer 1

Answer:

$\sqrt{y^2 + 10y + 25} = r$

Step-by-step explanation:

So we're given that the volume of a cone can be defined as:

$V=\frac{1}{3}\pi r^2h$

We're also given that:

$h=9 \text{ (since height = 9 inches)}$

So let's substitute in that nine in to the equation:

$V=\frac{1}{3}\pi r^2 * 9$

Let's simplify a bit by multiplying the 9 by 1/3 (same as dividing by 3)

$V=\frac{9}{3}\pi r^2\\\\V=3\pi r^2$

We're finally given that:

$V = 3\pi y^2 + 30\pi y + 75\pi$

so let's use this definition and substitute it in for our volume to get:

$3\pi y^2 + 30\pi y + 75\pi = 3\pi r^2$

From here we want to solve for r, since we want the radius in terms of y.

So let's first divide by $3\pi$ on both sides to get rid of the coefficient.

$\frac{3\pi y^2 + 30\pi y + 75\pi}{3\pi} = \frac{3\pi r^2}{3\pi}$

from here we can distribute the division across the terms (and simplify on the right) to get:

$\frac{3\pi y^2}{3\pi} + \frac{30\pi y}{3\pi} + \frac{75\pi}{3\pi} = r^2$

from here we can simplify the terms to get:

$y^2+10y+25=r^2$

Now the only thing that we need to remove is that square, which we can do by applying the square root to both sides of the function:

$\sqrt{y^2 + 10y + 25} = \sqrt{r^2}\\\\\sqrt{y^2 + 10y + 25} = r$

and now we're done isolating "r" and it's expressed in terms of y