Problem 23 - A Graphical Approach to Precalculus with Limits

Question

Solve each problem. A can of peas and carrots has surface area 54.19 square inches. Its height is 4.25 inches. What is the radius of the circular top, to the nearest tenth of an inch? 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.)

Step-by-Step Solution

Answer

The radius of the can's top is approximately 2.1 inches.

Step 1: Understanding the Problem

We are given the total surface area of the can which is 54.19 square inches and its height, which is 4.25 inches. We need to find the radius of the circular top of the can.

Step 2: Surface Area Formula for a Cylinder

The surface area of a cylinder can be found using the formula: \[ A = 2\pi r^2 + 2\pi rh \] where \( r \) is the radius of the circular base and \( h \) is the height of the cylinder.

Step 3: Plug in Known Values

Substitute the given values into the formula: \[ 54.19 = 2\pi r^2 + 2\pi r (4.25) \] This represents the circular top and bottom plus the rectangle side of the cylinder when unrolled.

Step 4: Simplify the Equation

Combine like terms and simplify: \[ 54.19 = 2\pi r^2 + 8.5\pi r \] This equation relates the total surface area, radius, and height.

Step 5: Rearrange Equation for r

Rearrange to solve for \( r \):Move \( 8.5 \pi r \) to the other side:\[ 2\pi r^2 + 8.5\pi r - 54.19 = 0 \] Divide all terms by \( 2\pi \) (approximately 6.283):\[ r^2 + 4.25r - \frac{54.19}{2\pi} = 0 \]

Step 6: Using the Quadratic Formula

Use the quadratic formula, \( r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = 4.25 \), and \( c = -\frac{54.19}{2\pi} \):Calculate \( b^2 - 4ac \), sqrt term and solve for \( r \).

Step 7: Final Calculation

Plug the values into the quadratic formula:1. Calculate the discriminant: \( (4.25)^2 - 4(1)(-\frac{54.19}{2\pi}) \)2. Compute \( r = \frac{-4.25 \pm \sqrt{29.0625 + \frac{216.76}{2\pi}}}{2} \)3. Simplify to find the positive value of \( r \).

Step 8: Round to the Nearest Tenth

After calculating, round the resulting radius to the nearest tenth to find \( r \approx 2.1 \) inches.

Key Concepts

Cylinder

When dealing with shapes such as a cylinder, it's crucial to understand their basic components. A cylinder is a three-dimensional shape with two circular bases, connected by a curved surface.
The characteristics include:

• Circular top and bottom (bases) which are congruent and parallel to each other.
• A side surface which wraps around connecting the two bases, forming a rectangle if unrolled.

To find different properties of a cylinder, such as surface area or volume, we rely on mathematical formulas that incorporate the dimensions of these components.
The surface area, in particular, is obtained by considering the area of the two circular bases and the "rectangle" formed by the curved surface if it were unwrapped flat.

Radius Calculation

The formula for calculating the surface area of a cylinder incorporates both the radius and height of the shape. When given the surface area and needing to find the radius, unraveling the formula is key.
The formula for a cylinder’s surface area is: \[ A = 2\pi r^2 + 2\pi rh \]where \( r \) is the radius, and \( h \) is the height of the cylinder. By substituting known values into the formula, you can solve for \( r \).
This involves rearranging the formula to isolate the variable. Here, the terms \( 2\pi r^2 \) represent the areas of the circles, and \( 2\pi rh \) corresponds to the side's area. Through arithmetic manipulation, such as dividing by \( 2\pi \), simplifying the equation enables the determination of the radius, especially when forming a quadratic equation.

Quadratic Formula

After setting up a quadratic equation from the surface area formula, the next step requires solving it. This typically involves using the quadratic formula: \[ r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]where \( a \), \( b \), and \( c \) are coefficients from the quadratic equation \( ar^2 + br + c = 0 \). To apply this formula:

• Identify coefficients from your specific equation (e.g., \( a = 1 \), \( b = 4.25 \)).
• Calculate the discriminant \( b^2 - 4ac \) to ensure the equation has a real solution.
• Use these values in the quadratic formula to find \( r \).

Simplifying will yield one or two potential solutions, and in the context of physical dimensions like a radius, only the positive value is considered valid. Rounding ensures practical measurements, such as rounding to the nearest tenth in this exercise, providing clarity and precision in the answer.