Problem 69 - Elementary Algebra

Question

A concrete slab is poured in the shape of a rectangle for a shed measuring 8 feet by 10 feet. Determine the area and perimeter of the slab.

Step-by-Step Solution

Answer

Area: 80 sq ft, Perimeter: 36 ft.

Step 1: Understanding the Formula for Area

The area of a rectangle is calculated by multiplying its length by its width. The formula is given by: \[ \text{Area} = \text{length} \times \text{width} \]

Step 2: Calculate the Area

To find the area of the concrete slab, we substitute the given dimensions (length = 10 feet and width = 8 feet) into the formula:\[ \text{Area} = 10 \times 8 = 80 \text{ square feet} \]

Step 3: Understanding the Formula for Perimeter

The perimeter of a rectangle is the sum of all its sides. The formula is:\[ \text{Perimeter} = 2 \times (\text{length} + \text{width}) \]

Step 4: Calculate the Perimeter

Substitute the given dimensions into the perimeter formula:\[ \text{Perimeter} = 2 \times (10 + 8) = 2 \times 18 = 36 \text{ feet} \]

Key Concepts

Calculating the Area of a Rectangle

When it comes to calculating the area of a rectangle, we use a simple yet essential formula that helps us determine the total surface within the rectangle's boundaries. It involves multiplying two of its dimensions - the length and the width.

Here's a step-by-step look at how the formula works:

• Identify the length of the rectangle. This is typically the longer side. In our example, the length is 10 feet.
• Identify the width of the rectangle. This is typically the shorter side. In our example, the width is 8 feet.
• Multiply these two numbers. The formula to remember is: \[ \text{Area} = \text{length} \times \text{width} \]
• Substituting the given values, we have:\[ \text{Area} = 10 \times 8 = 80 \text{ square feet} \]

Calculating the area is crucial as it tells us how much space an object occupies, like our concrete slab here.

Understanding Perimeter Calculation

The perimeter of a rectangle provides the total distance around its outer edges. This measure is particularly useful for projects requiring fencing, outlining, or any situation needing a boundary.

To determine the perimeter of a rectangle, you follow another key mathematical formula:

• The essence is to add the lengths of all sides. Since opposite sides of a rectangle are equal, this simplifies the calculation.
• Formulate as: \[ \text{Perimeter} = 2 \times (\text{length} + \text{width}) \]
• For our concrete slab, substitute into the formula:\[ \text{Perimeter} = 2 \times (10 + 8) \]
• That equals \[ \text{Perimeter} = 2 \times 18 = 36 \text{ feet} \]

Having the perimeter helps in understanding the lengths of materials needed for construction or decoration around the rectangle.

The Role of Mathematical Formulas

Mathematical formulas are the backbone of problem-solving in geometry. They serve as timeless tools that enable consistent and accurate calculations, easing the complexity of mathematical operations.

Why do formulas matter? Think about this:

• They provide a repeatable method to solve problems, ensuring accuracy in calculating areas, perimeters, volumes, and more.
• In our context, using formulas saves time and provides clarity without the burden of figuring out calculations from scratch each time.
• Once you learn a formula, you can apply it to any rectangle, regardless of its specific dimensions, making these formulas universally applicable.

By using the formulas, you efficiently approach calculations, ensuring your work not only benefits from precision but also saves precious time and cognitive effort—skills invaluable in both academic exercises and practical applications.