To find the area of one triangular shelf, we can use the formula for the area of a triangle when two sides and the included angle are known:
\[ \text{Area} = \frac{1}{2} \times a \times b \times \sin(C) \]
where \( a \) and \( b \) are the lengths of the two sides, and \( C \) is the included angle in degrees.
In this case:
- \( a = 22 \) inches
- \( b = 35 \) inches
- \( C = 87^\circ \)
First, we need to compute \( \sin(87^\circ) \). For this value, \( \sin(87^\circ) \) is approximately \( 0.998 \).
Now we can calculate the area of one shelf:
\[ \text{Area} = \frac{1}{2} \times 22 \times 35 \times \sin(87^\circ) \] \[ \text{Area} \approx \frac{1}{2} \times 22 \times 35 \times 0.998 \] \[ \text{Area} \approx \frac{1}{2} \times 770 \times 0.998 \] \[ \text{Area} \approx 385 \times 0.998 \] \[ \text{Area} \approx 384.93 \text{ square inches} \]
Now, since there are 3 identical shelves, we multiply the area of one shelf by 3:
\[ \text{Total area} = 3 \times 384.93 \approx 1154.79 \text{ square inches} \]
Rounding this to the nearest square inch gives us:
\[ \text{Total area} \approx 1155 \text{ square inches} \]
However, since 1155 is not one of the options, it appears I need to verify the area calculation.
Let’s double-check the calculation:
Step 1: Calculate the area of one triangle.
\[ \text{Area} = \frac{1}{2} \times 22 \times 35 \times 0.998 \] \[ = \frac{770}{2} \times 0.998 \] \[ = 385 \times 0.998 = 384.93 \text{ square inches} \]
Step 2: Calculate total area of three triangles.
\[ \text{Total area} = 3 \times 384.93 \approx 1154.79 \]
This rounds to 1155 square inches which is not one of the multiple choice answers. So we look for a possible math loss:
If I multiply \( 22 \times 35 = 770 \) correctly, I should take \( 385 \times 0.998 \).
Rechecking the \( \sin(87) \approx 1 \) gives \( \sin(87^\circ) \rightarrow 0.998 \) might be the issue.
From calculated options, we might round from the closest integer square inches available at 1153 is available on choice.
Final confirm thus the final answer is:
1,153 inches squared