Malachi and Talia are refreshing a triangular flower bed at their school. They want to know the area, so they know the amount of material they need. They know two adjoining sides measure 26 feet and 32 feet, and the angle formed between those sides measures 98 degrees. What is the area of their triangular flower bed to the nearest square inch?(1 point)

Question

inches2

Answer 1

To find the area of a triangle when you know two sides and the included angle, you can use the formula:

\[ \text{Area} = \frac{1}{2} \cdot a \cdot b \cdot \sin(C) \]

where \( a \) and \( b \) are the lengths of the two sides, and \( C \) is the measure of the included angle.

In this case:

• \( a = 26 \) feet
• \( b = 32 \) feet
• \( C = 98^\circ \)

We can first calculate \( \sin(98^\circ) \).

Using a calculator:

\[ \sin(98^\circ) \approx 0.9945 \]

Now, substitute the values into the area formula:

\[ \text{Area} = \frac{1}{2} \cdot 26 \cdot 32 \cdot 0.9945 \]

Calculating step-by-step:

1. Multiply \( 26 \) and \( 32 \):

\[ 26 \times 32 = 832 \]

2. Now multiply by \( \frac{1}{2} \):

\[ \frac{832}{2} = 416 \]

3. Finally, multiply by \( 0.9945 \):

\[ 416 \cdot 0.9945 \approx 413.352 \]

The area is approximately \( 413.352 \) square feet. To convert this to square inches (since there are 144 square inches in a square foot):

\[ 413.352 \text{ ft}^2 \times 144 \text{ in}^2/\text{ft}^2 = 59420.928 \text{ in}^2 \]

Rounding this to the nearest square inch:

\[ \approx 59421 \text{ in}^2 \]

Thus, the area of the triangular flower bed is approximately \( \boxed{59421} \) square inches.