Problem 1 - College Algebra with Modeling and Visualization

Question

Find the area and perimeter of the rectangle with length \(L\) and width \(W\). \(L=15\) feet, \(W=7\) feet

Step-by-Step Solution

Answer

The area is 105 sq ft and the perimeter is 44 ft.

Step 1: Identify the Formulas

To find the area of a rectangle, use the formula: \[ \text{Area} = L \times W \]To find the perimeter of a rectangle, use the formula: \[ \text{Perimeter} = 2L + 2W \]

Step 2: Calculate the Area

Substitute the given values into the area formula:\[ \text{Area} = 15 \text{ feet} \times 7 \text{ feet} \]Calculate the product:\[ \text{Area} = 105 \text{ square feet} \]

Step 3: Calculate the Perimeter

Substitute the given values into the perimeter formula:\[ \text{Perimeter} = 2(15 \text{ feet}) + 2(7 \text{ feet}) \]Calculate the expression:\[ \text{Perimeter} = 30 \text{ feet} + 14 \text{ feet} = 44 \text{ feet} \]

Key Concepts

Understanding Perimeter

The perimeter is the total distance around the outside of a shape. For rectangles, this involves adding up all the sides. Since rectangles have opposite sides of equal length, calculating the perimeter is straightforward.
In a rectangle, you have two lengths and two widths. Therefore, the formula for perimeter becomes:

• \[ \text{Perimeter} = 2L + 2W \]

This formula effectively adds the two lengths and two widths to get the total boundary measurement of the rectangle. This is useful in real-world applications, such as fencing a garden or any task involving boundaries.
To use this in our exercise, we simply substitute the given values for length and width to find the perimeter.

Explaining Area Calculation

The area measures the surface size that a shape covers. For rectangles, it's like counting all the little square inches or feet that fit inside it. The calculation is simple because you only need to multiply the length by the width.

• \[ \text{Area} = L \times W \]

This formula gives the total number of square units needed to completely fill the inside of the rectangle.
To put it into practice, imagine you're trying to cover a floor with tiles, and each tile is one square foot. Knowing the area helps determine how many tiles you need.

Rectangles and Their Characteristics

A rectangle is a four-sided shape with opposite sides that are equal and parallel. It's a type of polygon known as a quadrilateral, and it has four right angles. These properties make rectangles quite useful for various real-life applications.

• The opposite sides are equal: This is why the perimeter calculation uses two lengths and two widths.
• All angles are right angles: Each angle measures 90 degrees, making it a great shape for creating order and symmetry.
• It's easy to compute both area and perimeter: With simple multiplication and addition, you can quickly solve real-world problems involving space and boundaries.

Rectangles often appear in daily life, from doors and screens to fields and rooms, making understanding their properties useful for both academic and practical purposes.