Problem 18 - Elementary Algebra

Question

For Problems 11-32, use the geometric formulas given in this section to help solve the problems. (Objective 3 ) A lawn is in the shape of a triangle with one side 130 feet long and the altitude to that side 60 feet long. Will one bag of fertilizer that covers 4000 square feet be enough to fertilize the lawn?

Step-by-Step Solution

Answer

Yes, one bag of fertilizer is enough.

Step 1: Understanding the Problem

The problem requires finding the area of a triangular lawn to determine if a bag of fertilizer can cover it. We need to use the geometric formula for the area of a triangle.

Step 2: Formula for Area of a Triangle

The formula for the area of a triangle is \( A = \frac{1}{2} \times \text{base} \times \text{height} \). In this context, the base is the given side of the triangle, and the altitude is its height with respect to this base.

Step 3: Substituting the Values

Substitute the given values into the formula: base = 130 feet and height = 60 feet. The formula becomes \( A = \frac{1}{2} \times 130 \times 60 \).

Step 4: Calculation of Area

Calculate the area: \( A = \frac{1}{2} \times 130 \times 60 = 3900 \) square feet.

Step 5: Compare with Fertilizer Coverage

Compare the area of the lawn (3900 square feet) with the coverage of the fertilizer bag (4000 square feet).

Key Concepts

Area of a Triangle

Calculating the area of a triangle is a fundamental concept in geometry, often applied in real-world problems. To find the area of a triangle, we use the formula: \[ A = \frac{1}{2} \times \text{base} \times \text{height} \].This formula requires two key dimensions:

• Base: One of the sides of the triangle, chosen to work with the corresponding height.
• Height (Altitude): The perpendicular distance from the base to the opposite vertex.

By using these values, the formula calculates the region covered by the triangle in square units. This simple multiplication, divided by two, provides the area, which is essential in practical scenarios, such as determining how much material is needed to cover a lawn shaped like a triangle.

Problem Solving in Geometry

Problem-solving in geometry involves more than memorizing formulas. It requires understanding relationships and visualizing shapes. For our example, identifying that a lawn is triangular guides us to use the appropriate geometric principles and formulas. Effective problem-solving often follows these steps:

• Identify the Shape: Recognize the triangle and its dimensions, such as base and height.
• Apply the Correct Formula: Choose relevant geometric formulas; here, it's for the area of a triangle.
• Plug in the Values: Substitute given measurements into the formula carefully.
• Perform Calculations: Carry out the mathematical operations to find the solution.

These steps emphasize logical thinking and careful execution of geometric principles to solve problems efficiently.

Mathematical Calculation

Mathematical calculations often involve substituting numerical values into formulas and executing arithmetic operations.Let's take our triangle problem as an example. Here, the given values are a base of 130 feet and a height of 60 feet. When we substitute these values into the area formula, \[ A = \frac{1}{2} \times 130 \times 60 \],we perform the following steps:

• Multiply the base by the height: \( 130 \times 60 = 7800 \).
• Divide by 2 to apply the formula completely: \( \frac{7800}{2} = 3900 \) square feet.

This results in the area calculation of 3900 square feet, allowing us to compare it with the fertilizer coverage of 4000 square feet. Calculations must be precise, as they lay the foundation for actionable decision-making in practical applications.