Problem 74 - A Graphical Approach to College Algebra

Question

A storage tank for butane gas is to be built in the shape of a right circular cylinder having altitude 12 feet, as shown, with a half sphere attached to each end. If \(x\) represents the radius of each half sphere, what radius should be used to cause the volume of the tank to be \(144 \pi\) cubic feet?

Step-by-Step Solution

Answer

The radius should be 3 feet.

Step 1: Understand the Components of the Tank

The storage tank consists of a cylindrical section and two hemispherical sections at each end. The cylinder has a radius \(x\) and a height of 12 feet. Each hemisphere also has a radius of \(x\).

Step 2: Calculate the Volume of the Cylinder

The volume of the cylinder \(V_{\text{cyl}}\) is given by the formula \(V_{\text{cyl}} = \pi x^2 h\), where \(h\) is the height. Substituting \(h = 12\), the volume becomes \(V_{\text{cyl}} = \pi x^2 \times 12 = 12\pi x^2\).

Step 3: Calculate the Volume of the Hemispheres

The volume of a full sphere is \(\frac{4}{3}\pi x^3\). A hemisphere, being half of a sphere, has a volume of \(\frac{2}{3}\pi x^3\). Since there are two hemispheres, their total volume is \(2 \times \frac{2}{3}\pi x^3 = \frac{4}{3}\pi x^3\).

Step 4: Write the Equation for the Total Volume

The total volume \(V_{\text{total}}\) is the sum of the cylinder and the two hemispheres: \[ V_{\text{total}} = 12\pi x^2 + \frac{4}{3}\pi x^3 \].

Step 5: Set the Total Volume Equal to 144π

Since the problem states that the total volume is \(144\pi\) cubic feet, set the equation: \[ 12\pi x^2 + \frac{4}{3}\pi x^3 = 144\pi \].

Step 6: Solve for the Radius x

Cancel \(\pi\) from the equation to get: \[ 12x^2 + \frac{4}{3}x^3 = 144 \]. Multiply through by 3 to eliminate the fraction: \[ 36x^2 + 4x^3 = 432 \]. Rearrange to form a polynomial: \[ 4x^3 + 36x^2 - 432 = 0 \]. Factor the polynomial: \[ 4(x^3 + 9x^2 - 108) = 0 \]. Solving \(x^3 + 9x^2 - 108 = 0\) using the Rational Root Theorem or factoring, find \(x = 3\).

Step 7: Verify the Solution

Plug \(x = 3\) back into the volume formula: the cylinder's volume is \(12\pi (3)^2 = 108\pi\), and the two hemispheres' combined volume is \(\frac{4}{3}\pi (3)^3 = 36\pi\). Thus, the total volume is \(108\pi + 36\pi = 144\pi\), which matches the given volume.

Key Concepts

Right Circular Cylinder

A right circular cylinder is a 3-dimensional shape with two parallel circular bases connected by a curved surface. The axis of this cylinder is perpendicular to its bases. In the context of our exercise, it forms the main body of the storage tank. The volume of a right circular cylinder is calculated using the formula:

• \( V_{\text{cyl}} = \pi r^2 h \)

where \( r \) is the radius of the circular base, and \( h \) is the height of the cylinder.
In our specific problem, the radius \( x \) and the height of 12 feet are given, so the volume can be directly computed as \( 12\pi x^2 \).
Understanding this formula is key for solving volume-related questions involving right circular cylinders.

Hemisphere Volume

A hemisphere is half of a sphere. When calculating the volume of a hemisphere, it's important to modify the volume formula for a full sphere accordingly. The full sphere's volume is given by:

• \( V_{\text{sphere}} = \frac{4}{3}\pi r^3 \)

To find the volume of a hemisphere, you simply take half of this volume, leading to:

• \( V_{\text{hemi}} = \frac{2}{3}\pi r^3 \)

In our tank problem, each end of the cylinder is capped with a hemisphere of radius \( x \).
The exercise involves two hemispheres, thus their combined volume is \( \frac{4}{3}\pi x^3 \).
Understanding how to use these formulas is crucial when working with composite shapes like the tank.

Algebraic Equations

Algebraic equations involve expressions with variables, numbers, and operations. Solving these equations means finding the values of the variables that make the equation true.
In our butane storage tank problem, the equation set up to find the radius relates to the known total volume. Combining the cylinder and hemispheres, we get:

• \( 12\pi x^2 + \frac{4}{3}\pi x^3 = 144\pi \)

By dividing the entire equation by \( \pi \) and eliminating the constant, you simplify the problem to:

• \( 12x^2 + \frac{4}{3}x^3 = 144 \)

This expression allows us to isolate \( x \) by further simplifying.
Solving such equations allows us to understand the relationships between the different parts of composite solids.

Polynomial Factorization

Polynomial factorization refers to expressing a polynomial as a product of its factors. This technique simplifies solving polynomial equations.
For the equation resulting from our tank volume problem, factorizing the polynomial is crucial:

• Start with: \( 4x^3 + 36x^2 - 432 = 0 \)
• Factor out the greatest common factor (GCF): \( 4(x^3 + 9x^2 - 108) = 0 \)
• Further solve \( x^3 + 9x^2 - 108 = 0 \)

Using methods such as the Rational Root Theorem can help identify possible roots.
Once a root like \( x = 3 \) is found, it is used to check the problem's consistency. This ensures that the value aligns with conditions given in the problem, such as the tank's total volume.