Problem 67 - Algebra for College Students

Question

Solve each problem. Willard is designing a cylindrical tank with cone-shaped ends. The length of the cylinder is to be 20 feet (ft) larger than the radius of the cylinder, and the height of the cone is 2 ft. If the volume of the tank is \(984 \pi\) cubic feet ( \(\mathrm{ft}^{3}\) ), then what is the radius of the cylinder?

Step-by-Step Solution

Answer

The radius of the cylinder is 12 feet.

Step 1: Understand the problem

The cylindrical tank has cone-shaped ends. The volume of the entire tank, which includes both the cylinder and the two cones, is given as 984π cubic feet. The length of the cylinder is 20 feet more than the radius of the cylinder, and the height of each cone is 2 feet. The goal is to determine the radius of the cylinder.

Step 2: Express the dimensions in terms of the radius

Let the radius of the cylinder be denoted as r feet. Then, the length (or height) of the cylindrical part is (r + 20) feet. The height of each cone remains 2 feet.

Step 3: Volume of the cylindrical part

The volume of the cylinder is given by the formula \[V_{cylinder} = \text{Base Area} \times \text{Height}= \pi r^2 (r + 20)\].

Step 4: Volume of the conical parts

The volume of one cone is given by the formula \[V_{cone} = \frac{1}{3} \pi r^2 \times 2.\] Since there are two cones, the total volume of the cones is \[2 \times \frac{1}{3} \pi r^2 \times 2 = \frac{4}{3} \pi r^2\].

Step 5: Combine volumes

The total volume, which is the sum of the volumes of the cylindrical part and the two conical parts, is given by: \[\pi r^2 (r + 20) + \frac{4}{3} \pi r^2 = 984 \pi\].

Step 6: Simplify and solve the equation for r

First, factor out \(\pi r^2\) from the equation: \[\pi r^2 [(r + 20) + \frac{4}{3}] = 984 \pi\]. Next, simplify inside the parenthesis: \[((r + 20) + \frac{4}{3}) = (r + \frac{60}{3} + \frac{4}{3}) = r + \frac{64}{3}.\] So the equation becomes: \[\pi r^2 (r + \frac{64}{3}) = 984 \pi.\] Divide by π: \[r^2 (r + \frac{64}{3}) = 984.\] Clear the fraction by multiplying through by 3: \[3 r^2 (r + \frac{64}{3}) = 3 \times 984,\] which simplifies to: \[3 r^2 r + 64 r^2 = 2952\] or \[3r^3 + 64r^2 = 2952.\] Solve the cubic equation for r.

Step 7: Estimate the radius

Use trial and error or a numerical solver to find that the radius r which satisfies the cubic equation is approximately 12 feet.

Key Concepts

Cubic Equations

A cubic equation is an equation of the form \[ ax^3 + bx^2 + cx + d = 0 \] where 'a,' 'b,' 'c,' and 'd' are constants and 'x' is the variable. Cubic equations are so named because the highest power of the variable x is 3 (cubic power). The general steps to solve cubic equations involve factoring or using methods like the Rational Root Theorem or synthetic division.
In some cases, numerical methods or trial and error are required to estimate the solution.
In the given problem, the cubic equation formed is \[ 3r^3 + 64r^2 - 2952 = 0 \]
where 'r' is the radius of the cylinder. Finding the roots of this cubic equation helps determine the value of r.

Cylinder Volume

The volume of a cylinder is calculated with the formula \[ V = \text{Base Area} \times \text{Height} = \pi r^2 h \] where 'r' is the radius of the base and 'h' is the height of the cylinder.
In this exercise, the length of the cylindrical part of the tank is \[ r + 20 \text{ feet} \] Thus, the volume of the cylindrical part is \[ V_{cylinder} = \pi r^2 (r + 20) \]
Understanding how to calculate cylinder volume is crucial because the cylindrical part forms a significant portion of the total volume of the tank.
By plugging in the known dimensions, you can find the total volume contribution from the cylinder.

Cone Volume

The volume of a cone is given by the formula \[ V = \frac{1}{3} \pi r^2 h \] where 'r' is the radius of the base and 'h' is the height of the cone.
In this exercise, the height of each cone is 2 feet.
Therefore, the volume of one cone is \[ V_{cone} = \frac{1}{3} \pi r^2 \times 2 \] Given that there are two cones, their combined volume is \[ 2 \times \frac{1}{3} \pi r^2 \times 2 = \frac{4}{3} \pi r^2 \]
The volumes of the cones, when added together with the cylindrical part, provide the total volume of the tank.
So, understanding the cone volume formula helps in accurately calculating these contributions.