Problem 15 - Elementary Algebra

Question

For Problems 11-32, use the geometric formulas given in this section to help solve the problems. (Objective 3 ) A dirt path 4 feet wide surrounds a rectangular garden that is 38 feet long and 17 feet wide. Find the area of the dirt path.

Step-by-Step Solution

Answer

The area of the dirt path is 504 square feet.

Step 1: Understand the Problem

We are required to find the area of a dirt path surrounding a rectangular garden. The garden has dimensions of 38 feet (length) and 17 feet (width). The dirt path is 4 feet wide and surrounds the garden completely.

Step 2: Calculate Dimensions with Path

To account for the dirt path around the garden, add twice the width of the path to each dimension of the garden. Thus, the total length including the path is \(38 + 2 imes 4 = 46\) feet, and the total width including the path is \(17 + 2 \times 4 = 25\) feet.

Step 3: Calculate Total Area Including Path

Calculate the area of the entire rectangle that includes both the garden and the dirt path using the formula for the area of a rectangle, \(Area = Length \times Width\). So the total area is \(46 \times 25 = 1150\) square feet.

Step 4: Calculate Garden Area

Find the area of just the garden by using the garden's dimensions. The area of the garden is \(38 \times 17 = 646\) square feet.

Step 5: Final Step: Calculate Area of the Dirt Path

Subtract the area of the garden from the total area (including the path) to find the area of the dirt path: \(1150 - 646 = 504\) square feet.

Key Concepts

Geometric Formulas

Understanding geometric formulas is crucial in solving real-world problems, like calculating the area of a garden and its surrounding path. A formula is a mathematical rule expressed using symbols and numbers. For our exercise, the central formula is for the area of a rectangle. The formula is:\[ \text{Area} = \text{Length} \times \text{Width} \]This formula helps determine the amount of space inside the boundaries of a rectangle. Geometric formulas allow us to calculate areas efficiently, especially when shapes are combined or overlap, as in the case of the garden with its path.
Knowing how and when to apply these formulas simplifies otherwise complex problems by breaking them down into manageable steps.

Rectangular Garden

The rectangular garden is the main focus of our area calculation. It measures 38 feet in length and 17 feet in width. These dimensions are crucial in our calculations.
Understanding the basic properties of a rectangle assists in visualizing the problem. A rectangle has opposite sides of equal length and right angle corners. This straightforward shape makes using formulas like \( \text{Area} = \text{Length} \times \text{Width} \) very accessible.

• Length (L) = 38 feet
• Width (W) = 17 feet

The path surrounding the garden is 4 feet wide on all sides, adding to these dimensions. When considering the path, we increase both the length and width by the path’s width twice, since it surrounds the entire garden:
- Total Length including the path = 38 + 2 × 4 = 46 feet
- Total Width including the path = 17 + 2 × 4 = 25 feet
These adjusted measurements are employed to find the total area.

Step-by-Step Solution

Breaking down problems into a step-by-step solution makes them less daunting. Here’s how we tackled the rectangular garden's path problem:**Step 1: Understand the Problem**
Clearly define what needs solving: the area of a dirt path surrounding a garden.

**Step 2: Calculate Dimensions with Path**
Add twice the width of the path to the garden's length and width.

• Total Length: 38 + 8 = 46 feet
• Total Width: 17 + 8 = 25 feet

**Step 3: Calculate Total Area Including Path**
Use the formula \( \text{Area} = \text{Length} \times \text{Width} \) to determine the total enclosed by the outer rectangle:

• Area = 46 × 25 = 1150 square feet

**Step 4: Calculate Garden Area**
Using garden dimensions, find its area:

• Area = 38 × 17 = 646 square feet

**Step 5: Calculate Area of the Dirt Path**
Subtract the garden's area from the total to find the path's area:

• Path Area = 1150 - 646 = 504 square feet

These steps provide a structured approach to ensure all parts of the problem are addressed.