Problem 45 - College Algebra with Corequisite Support

Question

For the following exercises, solve for the given variable in the formula. After obtaining a new version of the formula, you will use it to solve a question. If the area model for a triangle is \(A=\frac{1}{2} b h,\) find the area of a triangle with a height of 16 in. and a base of 11 in.

Step-by-Step Solution

Answer

The area of the triangle is 88 square inches.

Step 1: Understanding the formula

The area of a triangle is given by the formula \( A = \frac{1}{2} b h \), where \( A \) is the area, \( b \) is the base length of the triangle, and \( h \) is the height of the triangle. This formula calculates the area by multiplying the base and height, then dividing by 2.

Step 2: Substitute the given values

We know the base \( b = 11 \) inches and the height \( h = 16 \) inches. Substitute these values into the formula: \[ A = \frac{1}{2} \times 11 \times 16 \].

Step 3: Calculate the product of base and height

First, calculate the product of the base length and the height: \( 11 \times 16 = 176 \).

Step 4: Apply the formula to solve for area

Using the calculated product, plug it into the formula: \[ A = \frac{1}{2} \times 176 \]. Now, calculate \( \frac{1}{2} \times 176 = 88 \).

Step 5: Conclusion

The area of the triangle, after substituting the base and height and performing the calculations, is \( 88 \) square inches.

Key Concepts

area model

Understanding the area model is like piecing together a simple puzzle. Imagine a triangle as half of a rectangle.
This is the basic principle behind the triangle's area model.- The area model starts with the formula for the area of a triangle: \( A = \frac{1}{2} b h \).- This formula tells us that the area is half the product of the base \( b \) and the height \( h \).- Think of it as splitting a rectangle in half along a diagonal, which forms a triangle. This is why we divide by two.Using the area model helps us visualize and calculate how much space the triangle occupies. By understanding this model, we can easily calculate the triangle's area by identifying its base and height.

triangle formula

The triangle formula, \( A = \frac{1}{2} b h \), is key to finding the area of a triangle.
This formula is essential knowledge in geometry.- "\( A \)" represents the area of the triangle.- "\( b \)" is the base, one side of the triangle often found along the ground in diagrams.- "\( h \)" is the height, which is the perpendicular distance from the base to the opposite vertex.Understanding this formula allows us to calculate areas efficiently. This particular formula is derived from breaking down triangles into simpler geometric shapes whose areas are more straightforward to determine.

mathematical substitution

Mathematical substitution is the act of plugging known values into an equation to solve for an unknown.
Here, we substituted to find an unknown area.- We know \( b = 11 \) inches and \( h = 16 \) inches.- Substitute these values into the triangle area formula: \( A = \frac{1}{2} \times 11 \times 16 \).By substituting, we convert the general formula into a specific computation. This is a fundamental skill in using formulas to find real-world measurements. It simplifies the complex process of using equations into a straightforward calculation.

geometry computation

Geometry computation involves calculating measurements of geometric shapes using mathematical formulas.
For the area of a triangle, it includes both numerical and algebraic steps.- After substituting the numbers, \( 11 \times 16 = 176 \). This multiplication is the numerical aspect.- Applying \( \frac{1}{2} \times 176 = 88 \) completes the algebraic component.These calculations show how numbers and equations work together to solve geometry problems. Accurate computation ensures we can measure the area of triangles reliably and use this knowledge in various applications, from simple geometry problems to building design and beyond.