Question

A cylindrical glass has a radius of 4 centimetres and a height of 6 centimetres. A large cylindrical jar full of water is a similar shape to the glass. The glass can be filled with water from the jar exactly 216 times.
Work out the radius and height of the jar.

Answer 1

Answer on Question#37695-Math-Geometry

Question

A cylindrical glass has a radius of 4 centimetres and a height of 6 centimetres. A large cylindrical jar full of water is a similar shape to the glass. The glass can be filled with water from the jar exactly 216 times. Work out the radius and height of the jar.

Solution

Volume of the glass

$$V_g = \pi \cdot R_g^2 \cdot H_g = 3.1416 \cdot 4^2 \cdot 6 = 302 \, \text{cm}^3$$

Volume of the jar

$$V_j = 216 \cdot V_g = 216 \cdot 302 = 65232 \, \text{cm}^3$$

Since the jar is a similar shape to the glass, we know that

$$\frac{R_j}{H_j} = \frac{R_g}{H_g} = \frac{4}{6} = \frac{2}{3}$$

or

$$H_j = 1.5 \cdot R_j$$

Volume of the jar

$$V_j = \pi \cdot R_j^2 \cdot H_j = 65232 \, \text{cm}^3$$

Let us design $R_j = x$ and $H_j = y$.

Thus, we have a set of two equations with two unknown values:

$$\left\{ \begin{array}{l} y = 1.5 \cdot x \\ 3.1416 \cdot x^2 \cdot y = 65232 \end{array} \right.$$ $$3.1416 \cdot x^2 \cdot (1.5 \cdot x) = 65232$$ $$x^3 = \frac{65232}{3.1416 \cdot 1.5} = 13843$$ $$x = \sqrt[3]{13843} = 24$$ $$y = 1.5 \cdot x = 1.5 \cdot 24 = 36$$

So, the radius of the jar $R_j = 24 \, \text{cm}$ and the height of the jar $H_j = 36 \, \text{cm}$.

**Answer**: the radius of the jar is 24 cm and the height of the jar is 36 cm