To determine the angle that the stream makes with the road in a right triangle, you can use trigonometric relationships. Here’s a detailed, step-by-step solution:
1. Identify the sides of the right triangle:
- One leg of the triangle is the stream, which measures 20 yards.
- The other leg of the triangle is the road, which measures 27 yards.
- The hypotenuse is the fence, but we don't need the hypotenuse to solve for the angle.
2. Determine which trigonometric function to use:
- Since we are given the lengths of the two legs (opposite and adjacent sides of the right angle), we use the tangent function ([tex]\(\tan\)[/tex]) which is defined as the ratio of the opposite side to the adjacent side:
[tex]\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\][/tex]
3. Set up the equation:
- For our scenario:
[tex]\[
\tan(\theta) = \frac{27}{20}
\][/tex]
4. Solve for [tex]\(\theta\)[/tex]:
- To find the angle [tex]\(\theta\)[/tex], take the arctangent (inverse tangent) of both sides:
[tex]\[
\theta = \arctan\left(\frac{27}{20}\right)
\][/tex]
5. Calculate the angle:
- Use a calculator to find the arctangent:
[tex]\[
\theta \approx \arctan(1.35)
\][/tex]
- Plugging the value into a calculator (ensure the calculator is in degree mode):
[tex]\[
\theta \approx 53.72^\circ
\][/tex]
6. Round the angle to the nearest degree:
- The angle rounded to the nearest degree is [tex]\(54^\circ\)[/tex].
Therefore, the angle that the stream makes with the road is [tex]\(54\)[/tex] degrees.
