Understanding Cylinder Volume Ratios

This problem asks us to find the ratio of the volumes of two cylinders given the ratios of their radii and heights. To solve this, we need to recall the formula for the volume of a cylinder and apply the given ratios.

Cylinder Volume Formula

The volume ($V$) of a cylinder is calculated using the formula:

$$V = \pi r^2 h$$

where $r$ is the radius of the base and $h$ is the height of the cylinder. $\pi$ is a mathematical constant, approximately equal to 3.14159.

Setting Up the Ratios

Let the radii of the two cylinders be $r_1$ and $r_2$, and their heights be $h_1$ and $h_2$.

• The ratio of radii is given as $r_1 : r_2 = 3 : 4$. We can express this as $\frac{r_1}{r_2} = \frac{3}{4}$. This means we can assume $r_1 = 3k$ and $r_2 = 4k$ for some positive constant $k$.
• The ratio of heights is given as $h_1 : h_2 = 8 : 5$. We can express this as $\frac{h_1}{h_2} = \frac{8}{5}$. This means we can assume $h_1 = 8m$ and $h_2 = 5m$ for some positive constant $m$. Note that the constants $k$ and $m$ can be different because the relationships between radii and heights are independent.

Calculating Individual Volumes

Now, let's calculate the volume of each cylinder using the formula $V = \pi r^2 h$ and the expressions for $r_1, h_1, r_2, h_2$ based on the ratios:

Volume of the first cylinder ($V_1$):

$$V_1 = \pi r_1^2 h_1$$

Substitute $r_1 = 3k$ and $h_1 = 8m$:

$$V_1 = \pi (3k)^2 (8m)$$

$$V_1 = \pi (9k^2) (8m)$$

$$V_1 = 72 \pi k^2 m$$

Volume of the second cylinder ($V_2$):

$$V_2 = \pi r_2^2 h_2$$

Substitute $r_2 = 4k$ and $h_2 = 5m$:

$$V_2 = \pi (4k)^2 (5m)$$

$$V_2 = \pi (16k^2) (5m)$$

$$V_2 = 80 \pi k^2 m$$

Finding the Ratio of Volumes

The ratio of the volumes of the two cylinders is $V_1 : V_2$, which is $\frac{V_1}{V_2}$.

$$\frac{V_1}{V_2} = \frac{72 \pi k^2 m}{80 \pi k^2 m}$$

We can cancel out the common terms $\pi$, $k^2$, and $m$ from the numerator and denominator:

$$\frac{V_1}{V_2} = \frac{72}{80}$$

Simplifying the Ratio

Now, we simplify the fraction $\frac{72}{80}$. Both numbers are divisible by 8:

• $72 \div 8 = 9$
• $80 \div 8 = 10$

So, the simplified ratio is $\frac{9}{10}$.

Therefore, the ratio of their volumes is $9 : 10$.

Conclusion

Based on our calculations, the ratio of the volumes of the two cylinders is $9 : 10$. We found this by using the given ratios for radii and heights, calculating the volume for each cylinder using the standard formula, and then simplifying the resulting ratio of the volumes.

Revision Table: Cylinder Formulas

Additional Information: Impact of Dimension Ratios on Volume

Understanding how changes in dimensions affect volume is crucial. For a cylinder, the volume $V$ is proportional to the square of the radius ($r^2$) and directly proportional to the height ($h$).

• If the radius is doubled while the height remains constant, the volume becomes $V_{new} = \pi (2r)^2 h = \pi (4r^2) h = 4 (\pi r^2 h) = 4V$. The volume increases by a factor of $2^2 = 4$.
• If the height is doubled while the radius remains constant, the volume becomes $V_{new} = \pi r^2 (2h) = 2 (\pi r^2 h) = 2V$. The volume doubles.
• In our problem, we had ratios for both radius and height. The radius ratio was $3:4$. The square of this ratio is $3^2:4^2 = 9:16$. The height ratio was $8:5$. The volume ratio is a combination of the square of the radius ratio and the height ratio: $(r_1/r_2)^2 \times (h_1/h_2) = (3/4)^2 \times (8/5) = (9/16) \times (8/5) = (9 \times 8) / (16 \times 5) = 72 / 80 = 9/10$. This confirms our detailed calculation and shows a quicker way to think about such ratio problems once the formula is understood.