Problem 25 - Precalculus Student Solutions Manual 5th

Question

A manufacturing firm wants to package its product in a cylindrical container 3 ft high with surface area \(8 \pi \mathrm{ft}^{2} .\) What should the radius of the circular top and bottom of the container be? (Hint: The surface area consists of the circular top and bottom and a rectangle that represents the side cut open vertically and unrolled.)

Step-by-Step Solution

Answer

The radius of the circular top and bottom is 1 ft.

Step 1: Understand the Surface Area Formula

The surface area of a cylinder includes the area of the two circular bases and the area of the rectangular side. The formula for the surface area of a cylinder is: \[ \text{Surface Area} = 2\text{πr}^2 + 2\text{πrh} \]where \( r \) is the radius and \( h \) is the height.

Step 2: Substitute Known Values

The height \( h \) is given as 3 ft and the total surface area is \( 8π \ \text{ft}^2 \). Substitute these values into the surface area formula: \[ 8π = 2πr^2 + 2πr(3) \]

Step 3: Simplify the Equation

First, factor out \( π \) from the right side of the equation: \[ 8π = 2π(r^2 + 3r) \]Next, divide both sides by \(2π\): \[ 4 = r^2 + 3r \]

Step 4: Solve the Quadratic Equation

To solve the quadratic equation \( r^2 + 3r - 4 = 0 \), use the quadratic formula: \[ r = \frac{-b \, \pm \, \sqrt{b^2 - 4ac}}{2a} \]Here, \(a = 1\), \(b = 3\), and \(c = -4\). So:\[ r = \frac{-3 \, \pm \, \sqrt{3^2 - 4 \cdot 1 \cdot (-4)}}{2 \cdot 1} \]\[ r = \frac{-3 \, \pm \, \sqrt{9 + 16}}{2} \]\[ r = \frac{-3 \, \pm \, 5}{2} \]

Step 5: Find the Possible Radii

This equation gives two possible values for \( r \): \[ r = \frac{-3 + 5}{2} = 1 \]\[ r = \frac{-3 - 5}{2} = -4 \]Since the radius cannot be negative, the valid solution is \( r = 1 \).

Key Concepts

Surface Area of a Cylinder

First, let's understand the formula for the surface area of a cylinder. A cylinder has two parts: the top and bottom circles, and the side, which is a rectangle when unrolled. The formula combines the area of these parts.To find the surface area, use:

• The area of the two circular bases: \[ 2\text{πr}^2 \] where π (pi) is a constant approximately 3.14 and r is the radius of the circles.
• The area of the rectangular side: \[ 2\text{πrh} \] where h is the height of the cylinder.

Combining these gives: \[ \text{Surface Area} = 2\text{πr}^2 + 2\text{πrh} \] Plugging the known values into the formula helps calculate the surface area, and eventually, solve for the radius. If any values are given, like the total surface area and height, substitute them into the formula to solve for r.

Quadratic Equation

To find the radius of a cylindrical container, we may need to solve a quadratic equation. In this problem, we reached a step where we had:\[ 4 = r^2 + 3r \]This is a quadratic equation because it’s in the form \[ ar^2 + br + c = 0 \] where a, b, and c are constants. Here, a = 1, b = 3, and c = -4.You solve a quadratic equation using the quadratic formula:\[ r = \frac{-b \, \, \sqrt{b^2 - 4ac}}{2a} \] This formula comes from rearranging the quadratic equation to isolate r. Using the given values, the equation becomes:\[ r = \frac{-3 \, \, \sqrt{3^2 - 4 \cdot 1 \cdot (-4)}}{2 \cdot 1} \] Solve this equation step-by-step to get the possible values of r. Since the radius can’t be negative, only positive solutions are valid.

Geometry

Geometry helps us visualize mathematical problems and their solutions. For cylindrical containers, understanding geometry involves knowing how shapes come together.

• The circle: the top and bottom of a cylinder are perfect circles. The formula for their area is: \[ \text{Area} = πr^2 \] where r is the radius.
• The rectangle: when you 'unroll' the side of a cylinder, it becomes a rectangle. This makes it easier to calculate the area. The formula combines base length (circumference of the circles, \[ 2πr \]) and height (h): \[ \text{Area} = 2πrh \]

Putting these areas together gives the total surface area of a cylinder. By working through each geometric shape separately (circles and rectangle) and using the right formulas, you can solve for any unknown dimensions.