Here is a question that I have, but I have no idea where to do go from here. Here is the question:
The vase company designs a new vase that is shaped like a cylinder on the bottom with a cone on top. The catalog states that the width is $12$ cm and the total height is $42$ cm. What would the height of the cylinder part have to be in order for the total volume to be $1224 \pi$ $\mathrm{cm}^3$.
The three equations that I got to help me solve this problem are:
Volume of a cylinder: $V=\pi r^2h$
Volume of a cone: $V=\frac{\pi r^2h}{3}$
Volume of a sphere: $V=\frac{4\pi r^3}{3}$
$h$ =height
$r$=radius
How can I start this problem off and solve it?
Best Answer
The diameter of the cylinder is $12$, so the radius is $6$. Let $a$ be the height of the cylinder, and $b$ the height of the cone. We are told that $$a+b=42.$$
The volume of the cylinder is, by one of the formulas you quoted, equal to $36\pi a$, and the volume of the cone, by another of the formulas, is $12\pi b$. Since the combined volume is $1224\pi$, we obtain $$36a+12b=1224.$$ We now have two linear equations in the two unknowns $a$ and $b$. Solve.