Problem 14 - Precalculus Student Solutions Manual 5th

Question

Solve each problem. World Largest Ice Cream Cake The world's largest ice cream cake, made at the Baxy ice cream factory in Beijing, China, on January \(16,2006,\) had length \(5.9 \mathrm{ft}\) greater than its width. Its perimeter was 51 ft. What were the length and width of this 8 -ton cake?

Step-by-Step Solution

Answer

Width: 9.8 ft, Length: 15.7 ft.

Step 1: Define Variables

Let the width of the cake be denoted by \(w\) feet. According to the given information, the length of the cake is 5.9 feet greater than its width, so the length can be represented as \(w + 5.9\) feet.

Step 2: Set Up the Perimeter Equation

The formula for the perimeter of a rectangle is given by: \[ P = 2(\text{length} + \text{width}) \] Here, the perimeter is provided as 51 feet. Substituting the known values: \[ 51 = 2((w + 5.9) + w) \]

Step 3: Simplify the Equation

Expand and simplify the equation: \[ 51 = 2(2w + 5.9) \] \[ 51 = 4w + 11.8 \]

Step 4: Solve for w (Width)

Isolate \(w\) by subtracting 11.8 from both sides of the equation: \[ 51 - 11.8 = 4w \] \[ 39.2 = 4w \] Divide both sides by 4 to find the value of \(w\): \[ w = 9.8 \text{ feet} \]

Step 5: Find the Length

Use the value of \(w\) to find the length: \[ \text{Length} = w + 5.9 = 9.8 + 5.9 = 15.7 \text{ feet} \]

Step 6: Verify the Solution

Check the values by substituting both the length and the width back into the perimeter equation to ensure correctness: \[ P = 2(\text{length} + \text{width}) \] \[ P = 2(15.7 + 9.8) = 2 \times 25.5 = 51 \text{ feet} \] The values satisfy the perimeter equation.

Key Concepts

rectangle perimeter

Understanding the concept of the perimeter of a rectangle is crucial for solving problems involving dimensions. The perimeter of a rectangle is the total distance around the border of the rectangle. It can be calculated using the formula: div> \[ P = 2(\text{length} + \text{width}) \] Here,

• P stands for perimeter
• The length refers to one of the longer sides
• The width stands for one of the shorter sides

In our problem, we are given the perimeter (51 feet), and we need to find the length and width of a large ice cream cake. Using the perimeter formula, we can set up an equation that will help solve for the unknown dimensions.

variable definition

To solve an equation efficiently, it's important to define variables clearly. The variable is a symbol that represents an unknown value that we need to find. In this problem,

• Let w denote the width of the cake in feet.
• According to the problem, the length of the cake is 5.9 feet greater than its width.
• We can, therefore, express the length as \[ l = w + 5.9 \]

Now, we have two expressions: one for the width and one for the length, both in terms of the same variable w.

algebraic manipulation

Algebraic manipulation involves rearranging and simplifying equations to isolate the variable. It's a skill that helps to solve for unknowns methodically. To find the dimensions of the cake, our key steps include:

- Set up the perimeter equation:
\[ 51 = 2((w + 5.9) + w) \]
- Expand and simplify:
\[ 51 = 2(2w + 5.9) \]
\[ 51 = 4w + 11.8 \]
- Isolate the variable w by subtracting 11.8 from both sides:
\[ 51 - 11.8 = 4w \]
- Further simplify:
\[ 39.2 = 4w \] and finally, divide both sides by 4:
\[ w = 9.8 \]

equation solving

Once all the algebraic steps have been completed, we solve the final equation to find the value of the variable. In this scenario: - The width w of the cake is found to be 9.8 feet. - Next, use this value to determine the length:
\[ \text{Length} = w + 5.9 \]
\[ \text{Length} = 9.8 + 5.9 = 15.7 \text{ feet} \] With both dimensions found, it's essential to verify the result by plugging the values back into the original perimeter formula: \[ P = 2(15.7 + 9.8) \] \[ P = 2 \times 25.5 = 51 \text{ feet} \] This confirms our solution is accurate. Understanding step-by-step equation solving can help effectively tackle similar problems in the future.