To determine how far Car B has to travel on First Street to reach Oak Street, let's proceed with a detailed solution, step by step.
1. Understanding Given Information:
- Given [tex]\( \tan y = \frac{5}{7} \)[/tex].
- Car A drives 14 miles on Main Street to arrive at Oak Street.
2. Recall the Tangent Function:
- The tangent of an angle in a right triangle is the ratio of the length of the opposite side to the adjacent side:
[tex]\[
\tan y = \frac{\text{opposite}}{\text{adjacent}}
\][/tex]
- For this problem:
[tex]\[
\tan y = \frac{\text{distance on First Street}}{\text{distance on Main Street}}
\][/tex]
3. Relate It to Given Values:
- Distance on Main Street (adjacent side) = 14 miles
- Distance on First Street (opposite side) = [tex]\( D \)[/tex] miles
4. Set Up the Equation Using Tangent:
[tex]\[
\frac{D}{14} = \frac{5}{7}
\][/tex]
5. Solve for [tex]\( D \)[/tex]:
- Cross-multiply to solve for [tex]\( D \)[/tex]:
[tex]\[
D = 14 \times \frac{5}{7}
\][/tex]
- Simplify:
[tex]\[
D = 14 \times 0.7142857 = 10.0
\][/tex]
6. Round to the Nearest Tenth:
- We have already reached a value that is to the nearest tenth, which is 10.0 miles.
7. Conclusion:
- Car B will have to travel [tex]\( 10.0 \)[/tex] miles on First Street to reach Oak Street.
Therefore, the distance that Car B has to travel on First Street to get to Oak Street is 10 miles.
