Certainly! Let's go through the problem step by step.
We need to find the length of the side [tex]\( y \)[/tex] in a right triangle. The given information includes:
- An angle of [tex]\( 31^\circ \)[/tex]
- The hypotenuse of the triangle is 400 feet.
To find the length of [tex]\( y \)[/tex], we will use the sine function, which relates the angle of a right triangle to the ratio of the length of the opposite side to the hypotenuse. The sine function formula is:
[tex]\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \][/tex]
Here, the angle [tex]\( \theta \)[/tex] is [tex]\( 31^\circ \)[/tex], the hypotenuse is 400 feet, and the opposite side is [tex]\( y \)[/tex]. We can rearrange the formula to solve for [tex]\( y \)[/tex]:
[tex]\[ y = \text{hypotenuse} \times \sin(\theta) \][/tex]
1. Convert the angle from degrees to radians because trigonometric functions in most contexts use radians. The angle [tex]\( 31^\circ \)[/tex] in radians is approximately [tex]\( 0.541 \)[/tex] radians (since [tex]\( 1 \)[/tex] degree [tex]\( \approx 0.01745 \)[/tex] radians).
2. Calculate the sine of the angle [tex]\( \sin(31^\circ) \approx \sin(0.541) \approx 0.5150 \)[/tex].
3. Multiply this sine value by the hypotenuse to get [tex]\( y \)[/tex]:
[tex]\[ y = 400 \times 0.5150 = 206.01522996402167 \][/tex]
4. Round the result to the nearest tenth:
[tex]\[ y \approx 206.0 \text{ feet} \][/tex]
Therefore, the length of [tex]\( y \)[/tex] is approximately [tex]\( 206.0 \)[/tex] feet when rounded to the nearest tenth.
