Problem 14 - College Algebra with Modeling and Visualization

Question

Find the area of the triangle with base \(b\) and height \(h .\) \(b=24\) inches, \(h=9\) fect

Step-by-Step Solution

Answer

The area of the triangle is 1296 square inches.

Step 1: Convert Units

The height is given in feet, but we need it in inches to match the units with the base. Since 1 foot equals 12 inches, convert the height: \[ h = 9 \text{ feet} \times 12 \text{ inches/foot} = 108 \text{ inches} \]

Step 2: Formula for Area of Triangle

The area \( A \) of a triangle is calculated using the formula:\[ A = \frac{1}{2} \times b \times h \]

Step 3: Substitute Values into Formula

Now substitute the given values for base \( b = 24 \) inches and height \( h = 108 \) inches into the formula:\[ A = \frac{1}{2} \times 24 \times 108 \]

Step 4: Calculate the Area

Perform the multiplication and division:\[ A = \frac{1}{2} \times 24 \times 108 = \frac{1}{2} \times 2592 = 1296 \text{ square inches} \]Thus, the area of the triangle is 1296 square inches.

Key Concepts

Unit Conversion

When solving problems involving measurements, it's crucial to ensure that all units are consistent. In this exercise, the base of the triangle is given in inches, while the height is provided in feet. To compute the area accurately, both measurements need to use the same unit. This is where unit conversion plays a key role.

Unit conversion involves multiplying or dividing a given measurement by a conversion factor. A conversion factor is a ratio that expresses how many of one unit are equal to another unit. For height conversion from feet to inches, we use the conversion factor:

• 1 foot = 12 inches.

So, to convert 9 feet into inches:

• Multiply 9 by 12 to get 108 inches.

Ensuring conversions are accurate eliminates errors in geometric calculations. By using consistent units, the results are reliable and correct.

Mathematical Formula

Formulas are fundamental in mathematics as they provide a systematic way to compute various properties. To find the area of a triangle, we use a specific mathematical formula. This formula enables us to calculate the area knowing only the base and the height of the triangle.

The formula for the area of a triangle is:

• \( A = \frac{1}{2} \times b \times h \)

where:

• \( A \) is the area of the triangle,
• \( b \) is the length of the base,
• \( h \) is the height.

This formula is derived from basic geometric principles and works for any type of triangle as long as the base and the height are perpendicular. Using this formula allows for straightforward, repeatable calculations. Always remember to plug the values into the formula with the correct units for accurate results.

Geometric Calculation

Understanding geometric calculation is critical when working with shapes, such as triangles. Once units are converted and formulas understood, the next step is to perform geometric calculations to determine the area.

To find the area of the given triangle, substitute the known values from the problem into the area formula:

• The base \( b = 24 \) inches.
• The height \( h = 108 \) inches.
• Substitute into the formula: \( A = \frac{1}{2} \times 24 \times 108 \)

By multiplying and dividing as per the formula, you calculate:

• The intermediate product is \( 24 \times 108 = 2592 \).
• The final area is \( \frac{1}{2} \times 2592 = 1296 \) square inches.

This result gives the total surface area of the triangle in square inches, a complete and definitive measurement suitable for any practical applications involving this specific triangle.