To find the area of one triangular shelf, we can use the formula for the area of a triangle given two sides and the included angle:
\[ \text{Area} = \frac{1}{2}ab \sin(C) \]
where:
- \( a \) and \( b \) are the lengths of the two sides,
- \( C \) is the included angle in degrees,
- \( \sin \) is the sine function.
In this case:
- \( a = 22 \) inches,
- \( b = 35 \) inches,
- \( C = 87^\circ \).
First, we need to find \( \sin(87^\circ) \). This can be approximated or calculated:
\[ \sin(87^\circ) \approx 0.998 \]
Now, we can calculate the area of one triangle:
\[ \text{Area} = \frac{1}{2} \times 22 \times 35 \times \sin(87^\circ) \] \[ = \frac{1}{2} \times 22 \times 35 \times 0.998 \] \[ = 11 \times 35 \times 0.998 \] \[ = 385 \times 0.998 \approx 384.93 \text{ square inches} \]
Now, we want to find the area for all three shelves:
\[ \text{Total Area} = 3 \times \text{Area of one triangle} = 3 \times 384.93 \approx 1154.79 \text{ square inches} \]
Rounding this to the nearest square inch gives:
\[ \text{Total Area} \approx 1155 \text{ square inches} \]
Now, let's cross-check the options provided:
- A. 660 inches^2
- B. 1,647 inches^2
- C. 2,038 inches^2
- D. 1,153 inches^2
Since \( 1155 \) square inches is closest to option D (1,153 inches²). Hence the answer is:
D. 1,153 inches².