Problem 23 - A Graphical Approach to College Algebra

Question

Solve each problem. Radius of a Can A can of garbanzo beans has surface. area 54.19 square inches. Its height is 4.25 inches. What is the radius of the circular top? See the figure. (Hint: The surface area consists of the circular top and bottom and a rectangle that represents the side cut open vertically and unrolled.) (IMAGE CAN'T COPY)

Step-by-Step Solution

Answer

The radius of the can is approximately 2 inches.

Step 1: Understand the Surface Area Formula for a Cylinder

The surface area of a cylinder consists of two circular bases and a lateral surface. The formula for the surface area of a cylinder is \( S = 2\pi r^2 + 2\pi rh \), where \( r \) is the radius of the base and \( h \) is the height of the cylinder.

Step 2: Substitute the Given Values

Given that the total surface area, \( S \), is 54.19 square inches, and the height, \( h \), is 4.25 inches, substitute these values into the equation: \( 54.19 = 2\pi r^2 + 2\pi r(4.25) \).

Step 3: Simplify and Solve for \( r \)

Simplify the equation to isolate \( r \): \( 54.19 = 2\pi r^2 + 8.5\pi r \). This is a quadratic equation in the form \( ax^2 + bx + c = 0 \), where \( a = 2\pi \), \( b = 8.5\pi \), and \( c = -54.19 \). Divide through by \( \pi \) to simplify: \( \frac{54.19}{\pi} = 2r^2 + 8.5r \).

Step 4: Solve the Quadratic Equation

To solve for \( r \), use the quadratic formula \( r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 2 \), \( b = 8.5 \), and \( c = -\frac{54.19}{\pi} \). Calculate \( \Delta = b^2 - 4ac \) and substitute these into the quadratic formula to find \( r \).

Step 5: Calculate \( \Delta \) and \( r \)

Calculate \( \Delta \) as follows: \( (8.5)^2 - 4\times2\times(-\frac{54.19}{\pi}) \) and use it to calculate the roots of the equation using the quadratic formula. Simplify to get the positive root since radius cannot be a negative value.

Step 6: Verify the Solution

Confirm that the calculated radius satisfies the original surface area equation. Substituting \( r \) back into the simplified equation should approximately provide the total surface area of 54.19 square inches.

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation of degree two. It typically takes the form \[ ax^2 + bx + c = 0 \]where \( a eq 0 \). The unknown variable \( x \) can have two possible solutions, given that the equation represents a parabola that may intersect the \( x \)-axis at up to two points.
Here's how to understand a quadratic equation in the context of finding the radius of a cylinder:

• The equation is set as \( 2\pi r^2 + 8.5\pi r = 54.19 \), where \( r \) is the radius of the cylinder.
• Identify \( a = 2\pi \), \( b = 8.5\pi \), and \( c = -54.19 \). These coefficients represent the parts of the quadratic equation related to the cylinder's dimensions.
• The quadratic formula is integral in solving these equations: \[ r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

This formula is useful to solve for the variable when direct factoring is not feasible.

Cylinder Geometry

Understanding cylinder geometry is important to solve problems involving surface areas and volumes.A cylinder consists of two circular bases and a curved lateral surface connecting them.
In describing a cylinder:

• Radius \( r \): The distance from the center to the edge of the circular base.
• Height \( h \): The straight-line distance between the bases.

The problem describes a can with these components, aiming to find the radius with a given height and surface area.The challenge is to break the cylinder's surface into recognizable shapes (circles and rectangles) for calculations.This understanding aids in applying surface area formulas correctly by linking physical measurements to mathematical expressions.

Surface Area Formula

The surface area of a cylinder is calculated using:\[ S = 2\pi r^2 + 2\pi rh \]where \( S \) represents total surface area, \( r \) is the radius of the base, and \( h \) is the height.This formula accounts for both the circular areas and the lateral surface area.
Here's the breakdown:

• Circular Bases: The term \( 2\pi r^2 \) calculates the total area of the two circles (top and bottom).
• Lateral Surface: The term \( 2\pi rh \) represents the area of the side of the cylinder, which can be visualized as a rectangle when unwrapped.

Substituting known values into this formula gives a tractable equation to determine unknowns like the radius.