Problem 41 - College Physics

Question

You have two cylindrical tanks. The tank with the greater volume is 1.20 times the height of the smaller tank. It takes 218 gallons of water to fill the larger tank and 150 gallons to fill the other. What is the ratio of the radius of the larger tank to the radius of the smaller one?

Step-by-Step Solution

Answer

The radius of the larger tank is 1.1 times the radius of the smaller tank.

Step 1: Understand the Problem

We are given two cylindrical tanks. The larger tank is 1.20 times the height of the smaller one, and it takes 218 gallons to fill, whereas the smaller tank takes 150 gallons to fill. We need to find the ratio of their radii.

Step 2: Convert Gallons to Cubic Feet

Since 1 gallon is approximately 0.133681 cubic feet, we convert the volumes of the tanks into cubic feet. For the larger tank: \( 218 \times 0.133681 \approx 29.1192 \) cubic feet.For the smaller tank: \( 150 \times 0.133681 \approx 20.05215 \) cubic feet.

Step 3: Write the Formula for Volume of a Cylinder

The formula for the volume of a cylinder is given by: \[ V = \pi r^2 h \] where \( V \) is the volume, \( r \) is the radius, and \( h \) is the height.

Step 4: Express the Heights in Terms of Each Other

Let the height of the smaller tank be \( h_s \). Then, the height of the larger tank is \( 1.20 \times h_s \).

Step 5: Formulate Volume Equations

For the smaller tank, the volume equation is: \[ \pi r_s^2 h_s = 20.05215 \]For the larger tank, the volume equation is:\[ \pi r_l^2 (1.20h_s) = 29.1192 \]

Step 6: Eliminate Height from Equations

Dividing the equation for the larger tank by the equation for the smaller tank:\[ \frac{\pi r_l^2 (1.20h_s)}{\pi r_s^2 h_s} = \frac{29.1192}{20.05215} \]Simplifying the left side gives us:\[ 1.20 \cdot \left(\frac{r_l}{r_s}\right)^2 = \frac{29.1192}{20.05215} \]

Step 7: Solve for the Ratio of Radii

First, divide both sides of the equation by 1.20:\[ \left(\frac{r_l}{r_s}\right)^2 = \frac{29.1192}{20.05215 \times 1.20} \]Calculate the right-hand side:\[ \frac{29.1192}{24.06258} \approx 1.210 \]Take the square root of both sides to find \( \frac{r_l}{r_s} \):\[ \frac{r_l}{r_s} \approx \sqrt{1.210} \approx 1.10 \]

Step 8: Conclude

The ratio of the radius of the larger tank to the radius of the smaller tank is approximately 1.10.

Key Concepts

Cylindrical Tanks

Cylindrical tanks are used in a variety of applications, such as storing water or other liquids. They are essentially 3D shapes with circular bases and straight, parallel sides. Understanding the structure of a cylindrical tank is crucial when solving problems related to volume, as it allows straightforward application of mathematical formulas.

• The size of a cylindrical tank is defined by its radius (distance from the center of the base to its edge) and its height (distance between the two circular bases).
• Tanks with larger radii or taller heights can hold more volume.
• In exercises involving tanks, typical tasks include calculating capacity, converting measurements, or comparing different dimensions like in the problem above.

Volume of Cylinder Formula

The volume of a cylinder is a key concept in understanding how much a cylindrical tank can hold. We calculate this using the formula: \[V = \pi r^2 h\] where:

• \(V\) is the volume of the cylinder.
• \(\pi\) is a mathematical constant approximately equal to 3.14159.
• \(r\) is the radius of the base of the cylinder.
• \(h\) is the height of the cylinder.

In solving problems, make sure each dimension unit is consistent. This formula is crucial for determining volume in both theoretical problems and real-world applications. By arranging the formula, you can also solve for either the radius or the height if the volume and one dimension are known.
This flexibility helps in comparing tanks or designing them to specific volume requirements.

Converting Gallons to Cubic Feet

In many physics and engineering problems, converting gallons to cubic feet is essential, especially in regions using the imperial system. As given in the problem, 1 gallon is approximately equivalent to 0.133681 cubic feet. This conversion is vital because it aligns with the standard units used in the volume formula (cubic feet), enabling seamless calculations.
Understanding how to convert these units correctly is important for solving practical problems that involve measures of capacity, such as filling tanks or pools. Here’s how you can convert gallons to cubic feet step-by-step:

• Multiply the number of gallons by 0.133681 to convert it to cubic feet.
• For instance, a larger tank with 218 gallons converts to \(218 \times 0.133681 \approx 29.1192\) cubic feet.
• A smaller tank taking up 150 gallons becomes \(150 \times 0.133681 \approx 20.05215\) cubic feet.

These calculated values directly feed into the volume equations, assisting in finding other unknown variables like radius ratios.

Radius Ratio Calculation

Calculating the ratio of two radii involves understanding the proportions between different dimensions of cylinders. In this instance, it’s about comparing two tanks where the height of one is a known multiple of the other, and their volumes are given.
To find the radius ratio:

• Formulate the volume equations for each tank using the volume formula \(V = \pi r^2 h\), incorporating their specific dimensions.
• Express the known relation between their heights, here the larger tank’s height is 1.20 times the smaller one's height.
• Simplify and compare both volume equations, allowing you to eliminate redundant variables like height.

For example, by dividing, you’ll root out the heights and focus on the radius, ultimately determining the ratio through algebraic manipulation, like taking square roots. In this scenario, the calculation yielded a radii ratio of approximately 1.10. This step-by-step manipulation is essential for deriving the dimensions of cylindrical tanks based on given parameters.