The ratio of the volume of two cylinders is 13 ∶ 1 and the ratio of their heights is 13 ∶ 9. If the area of the base of the second cylinder is 154 cm2, then what will be the radius of the first cylinder?
21 cm
Solving for the Radius of the First Cylinder
This problem involves using the properties of cylinders, specifically their volume and base area, along with given ratios of volumes and heights to find the radius of one of the cylinders.
Understanding the Given Information
We are provided with the following details about two cylinders:
- The ratio of their volumes ($V_1$ and $V_2$) is 13 ∶ 1. This means $\frac{V_1}{V_2} = \frac{13}{1}$.
- The ratio of their heights ($h_1$ and $h_2$) is 13 ∶ 9. This means $\frac{h_1}{h_2} = \frac{13}{9}$.
- The area of the base of the second cylinder is 154 cm2.
Our goal is to determine the radius of the first cylinder ($r_1$).
Key Formulas for Cylinders
To solve this, we need the standard formulas for a cylinder:
- Volume of a cylinder ($V$) = $\pi r^2 h$, where $r$ is the radius of the base and $h$ is the height.
- Area of the base of a cylinder = $\pi r^2$.
Step-by-Step Calculation
Step 1: Find the radius of the second cylinder ($r_2$)
We are given the base area of the second cylinder is 154 cm2. Using the base area formula:
Area of base of second cylinder = $\pi r_2^2 = 154$ cm$^2$
Using the value of $\pi = \frac{22}{7}$:
\(\frac{22}{7} \times r_2^2 = 154\)
\(r_2^2 = 154 \times \frac{7}{22}\)
\(r_2^2 = (22 \times 7) \times \frac{7}{22}\)
\(r_2^2 = 7 \times 7 = 49\)
\(r_2 = \sqrt{49}\)
\(r_2 = 7\) cm
So, the radius of the second cylinder is 7 cm.
Step 2: Use the volume and height ratios to find the ratio of radii
The ratio of the volumes is given as $\frac{V_1}{V_2} = \frac{13}{1}$.
Substitute the volume formula ($V = \pi r^2 h$) for both cylinders:
\(\frac{\pi r_1^2 h_1}{\pi r_2^2 h_2} = \frac{13}{1}\)
The $\pi$ terms cancel out:
\(\frac{r_1^2 h_1}{r_2^2 h_2} = 13\)
This can be rewritten as:
\(\left(\frac{r_1}{r_2}\right)^2 \times \left(\frac{h_1}{h_2}\right) = 13\)
We are given the ratio of heights $\frac{h_1}{h_2} = \frac{13}{9}$. Substitute this into the equation:
\(\left(\frac{r_1}{r_2}\right)^2 \times \frac{13}{9} = 13\)
Now, isolate the term involving the radii ratio:
\(\left(\frac{r_1}{r_2}\right)^2 = 13 \times \frac{9}{13}\)
\(\left(\frac{r_1}{r_2}\right)^2 = 9\)
Taking the square root of both sides:
\(\frac{r_1}{r_2} = \sqrt{9}\)
\(\frac{r_1}{r_2} = 3\)
Step 3: Calculate the radius of the first cylinder ($r_1$)
From Step 2, we found that $\frac{r_1}{r_2} = 3$.
From Step 1, we know that $r_2 = 7$ cm.
Substitute the value of $r_2$ into the ratio equation:
\(\frac{r_1}{7} = 3\)
Solve for $r_1$:
\(r_1 = 3 \times 7\)
\(r_1 = 21\) cm
Therefore, the radius of the first cylinder is 21 cm.
Summary of Results
| Parameter | Cylinder 1 | Cylinder 2 | Ratio (C1 : C2) |
|---|---|---|---|
| Volume (V) | $V_1$ | $V_2$ | 13 : 1 |
| Height (h) | $h_1$ | $h_2$ | 13 : 9 |
| Radius (r) | $r_1$ | $r_2 = 7$ cm (calculated) | 3 : 1 (calculated) |
| Base Area | $\pi r_1^2$ | $\pi r_2^2 = 154$ cm$^2$ (given) |
Revision Table: Cylinder Calculations
| Concept | Formula | Application in Problem |
|---|---|---|
| Area of Base | $\pi r^2$ | Used to find $r_2$ from given base area of Cylinder 2. |
| Volume | $\pi r^2 h$ | Used in the volume ratio to relate $r_1, h_1, r_2, h_2$. |
| Ratio of Volumes | $\frac{V_1}{V_2} = \frac{\pi r_1^2 h_1}{\pi r_2^2 h_2} = \left(\frac{r_1}{r_2}\right)^2 \times \left(\frac{h_1}{h_2}\right)$ | Key relation used with height ratio to find radius ratio. |
| Ratio of Heights | $\frac{h_1}{h_2}$ | Given as 13:9, substituted into the volume ratio equation. |
Additional Information: Understanding Cylinders and Ratios
A cylinder is a three-dimensional solid shape with two parallel circular bases connected by a curved surface. Many real-world objects, like cans and pipes, are cylindrical.
Volume and Dimensions
The volume of a cylinder depends on its base radius ($r$) and its height ($h$). The formula $V = \pi r^2 h$ shows that the volume is directly proportional to the square of the radius and directly proportional to the height.
Working with Ratios
When dealing with ratios of geometric properties like volume, area, or dimensions (radius, height), it's helpful to set up proportions. For example, if the ratio of heights is $h_1:h_2 = a:b$, we can write $\frac{h_1}{h_2} = \frac{a}{b}$. When ratios of multiple dimensions are involved in a formula (like $V = \pi r^2 h$), the ratio of volumes will involve the product of the ratios of the dimensions raised to their respective powers in the formula. In this case, $\frac{V_1}{V_2} = \frac{\pi r_1^2 h_1}{\pi r_2^2 h_2} = \left(\frac{r_1}{r_2}\right)^2 \times \left(\frac{h_1}{h_2}\right)$. Understanding how these ratios combine is crucial for solving such problems.
In this problem, we used the given base area to find one radius, then used the relationship between volume ratio, height ratio, and radius ratio to solve for the unknown radius of the first cylinder.
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