Question

if the area of △abc = 134 square feet, m∠c = 41°, and b = 24 feet, then what is the length of side a? round your answer to the nearest whole number.
options:
a) 13 ft
b) 14 ft
c) 15 ft
d) 16 ft

Explanation:

Step1: Recall the area formula for a triangle

The formula for the area of a triangle when two sides and the included angle are known is \( \text{Area} = \frac{1}{2}ab\sin C \), but in this case, we have sides \( b \) and \( a \), and angle \( C \). Wait, actually, the formula is \( \text{Area} = \frac{1}{2}ab\sin C \) where \( a \) and \( b \) are two sides and \( C \) is the included angle between them. Wait, in triangle \( ABC \), side \( b \) is \( AC \), side \( a \) is \( BC \), and angle \( C \) is the included angle between sides \( a \) and \( b \). So the area formula is \( \text{Area} = \frac{1}{2} \times a \times b \times \sin C \).

We know the area is 134 square feet, \( b = 24 \) feet, \( \angle C = 41^\circ \), and we need to find \( a \).

So we can rearrange the formula to solve for \( a \):

\( 134 = \frac{1}{2} \times a \times 24 \times \sin(41^\circ) \)

Step2: Solve for \( a \)

First, simplify the right-hand side:

\( \frac{1}{2} \times 24 = 12 \), so the equation becomes:

\( 134 = 12 \times a \times \sin(41^\circ) \)

We know that \( \sin(41^\circ) \approx \sin(41) \approx 0.6561 \) (using a calculator to find the sine of 41 degrees).

So substitute \( \sin(41^\circ) \approx 0.6561 \):

\( 134 = 12 \times a \times 0.6561 \)

Calculate \( 12 \times 0.6561 \approx 7.8732 \)

So the equation is:

\( 134 = 7.8732 \times a \)

Now, solve for \( a \) by dividing both sides by 7.8732:

\( a = \frac{134}{7.8732} \approx 17 \) (when rounded to the nearest whole number)

Wait, let's check the calculation again.

Wait, \( \sin(41^\circ) \approx 0.656059 \)

So \( \frac{1}{2} \times a \times 24 \times \sin(41^\circ) = 12a \times 0.656059 \approx 7.8727a \)

Then \( 134 = 7.8727a \)

So \( a = \frac{134}{7.8727} \approx 17.02 \), which rounds to 17.

Answer:

17