please help!

[Muppers3262]

Please help me solve for the h of the cone. I know that the volume of a cone is (1/3)Pi*r^2*h and that the volume of a sphere is (4/3)Pi*r^3, but don't know how to solve for h after setting the volumes to equal each other. :?

A sphere and the base of a cone have the same radius, r. What is the height, h of the cone if the volumes of the cone and the sphere are the same?

Thanks for your help!

 

[stapel]

After you set the volume expressions equal to each other, you can divide out r² (since the radii are the same). You can also divide out the \(\displaystyle \pi\), and multiply through by 3 to cancel the denominators.

However, this gives you an equation in two variables. While you can solve for the height h in terms of the radius r, I don't see how one could obtain a numerical value for the height h.

Sorry.

Eliz.

 

[Daniel_Feldman]

Please help me solve for the h of the cone. I know that the volume of a cone is (1/3)Pi*r^2*h and that the volume of a sphere is (4/3)Pi*r^3, but don't know how to solve for h after setting the volumes to equal each other. :?

A sphere and the base of a cone have the same radius, r. What is the height, h of the cone if the volumes of the cone and the sphere are the same?

Thanks for your help!

So we have

(4/3)pi(r)^3=(1/3)pi(r)^2h


Let's do this step by step:

There is pi on each side, so it will cancel, and we are left with:

(4/3)r^3=(1/3)r^2(h)


Now, we have r^2 on on side, and r^3 on the other, so the r^2 goes away and the r^3 becomes r, and we have:

(4/3)r=(1/3)h

Multiplying both sides yields:

h=4r


and that's the answer.

 

[stapel]

h=4r and that's the answer.

Yes, that's the height "h" in terms of the radius "r". But the exercise seems to want an actual numerical value for the height: it doesn't ask for "an expression for the height in terms of the radius"; it asks for "the height".

Eliz.

 

[Mrspi]

It isn't going to be possible to find a numerical value for h as a solution for this problem (with the information given in the problem). As the previous responses have shown, when the radius of the base of the cone is equal to the radius of the sphere, the volumes of the cone and sphere will be the same whenever the height of the cone is 4 times the radius.

So, there would be infinitely many numerical values for h that would "work": pick any positive number for h. Then, use h for the height of a cone, and h/4 for the radius of the cone and a sphere. The two wiil have the same volume.