Question

in △vwx, m∠x = 23°, v = 25 yards, and w = 47 yards, where v is the length of the side opposite v and w is the length of the side opposite w. find the area of △vwx. write your answer as an integer or as a decimal rounded to the nearest tenth. yd² submit

Explanation:

Step1: Recall the formula for the area of a triangle with two sides and included angle

The formula for the area \( A \) of a triangle when two sides \( a \) and \( b \) and the included angle \( C \) are known is \( A=\frac{1}{2}ab\sin C \). In \( \triangle VWX \), we know sides \( v = 25 \) (opposite \( V \)), \( w=47 \) (opposite \( W \)) and angle \( \angle X=23^{\circ} \). The angle \( \angle X \) is included between sides \( v \) and \( w \) (since \( v \) is opposite \( V \), \( w \) is opposite \( W \), so the sides adjacent to \( \angle X \) are \( v \) and \( w \)). So we can use the formula \( A=\frac{1}{2}vw\sin(\angle X) \).

Step2: Substitute the given values into the formula

Substitute \( v = 25 \), \( w = 47 \), and \( \angle X=23^{\circ} \) into the formula:
\( A=\frac{1}{2}\times25\times47\times\sin(23^{\circ}) \)

First, calculate \( \frac{1}{2}\times25\times47 \):
\( \frac{1}{2}\times25\times47=\frac{25\times47}{2}=\frac{1175}{2} = 587.5 \)

Then, find \( \sin(23^{\circ}) \). Using a calculator, \( \sin(23^{\circ})\approx0.3907 \)

Now, multiply \( 587.5 \) by \( 0.3907 \):
\( 587.5\times0.3907\approx587.5\times0.3907 \)
\( 587.5\times0.3907 = 587.5\times(0.4 - 0.0093)=587.5\times0.4-587.5\times0.0093 \)
\( 587.5\times0.4 = 235 \)
\( 587.5\times0.0093 = 5.46375 \)
\( 235 - 5.46375=229.53625 \approx229.5 \) (rounded to the nearest tenth)

Answer:

\( 229.5 \)