Jasmira has a new corner shelving unit that is triangular. The unit has three identical oblique triangle shelves. Jasmira wants to put some shelving paper down but isn't sure how much to buy. If one side of each triangle is 22 inches, an adjoining side is 35 inches, and the angle formed between them is 87 degrees, then how much paper (to the nearest square inch) does she need to cover all 3 shelves?

A. 1,153 inches^2
B. 600 inches^2
C. 1,647 inches^2
D. 2,038 inches^2

1 answer

To determine the area of one triangular shelf and then find the total area for all three shelves, 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 sides,
  • \(C\) is the included angle.

In this case:

  • \(a = 22\) inches,
  • \(b = 35\) inches,
  • \(C = 87^\circ\).

First, we calculate the sine of 87 degrees:

\[ \sin(87^\circ) \approx 0.998 \]

Now, substituting the values into the area formula:

\[ \text{Area} = \frac{1}{2} \times 22 \times 35 \times \sin(87^\circ) \] \[ \text{Area} = \frac{1}{2} \times 22 \times 35 \times 0.998 \] \[ \text{Area} = 0.5 \times 22 \times 35 \times 0.998 \]

Calculating \( 0.5 \times 22 \times 35 \):

\[ 0.5 \times 22 = 11 \] \[ 11 \times 35 = 385 \]

Now multiply by \(0.998\):

\[ \text{Area} \approx 385 \times 0.998 \approx 384.23 \]

So, the area of one triangular shelf is approximately \(384.23\) square inches. For all three shelves, we multiply this area by 3:

\[ \text{Total Area} = 3 \times 384.23 \approx 1152.69 \]

Rounding this to the nearest whole number gives:

\[ \text{Total Area} \approx 1153 \text{ square inches} \]

Therefore, the amount of shelving paper Jasmira needs to buy is approximately:

\[ \boxed{1153} \text{ square inches} \]

The closest answer from the options provided is A. 1,153 inches².