Let's solve this problem step by step.
1. Understanding the relationship given by the tangent:
- We are given that tan [tex]\( y' = \frac{5}{7} \)[/tex].
- The tangent of an angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side.
2. Assign the given values:
- Car A travels along Main Street, which forms the adjacent side of the right triangle. The length of the adjacent side (Main Street) is 14 miles.
- We need to find the length of the opposite side (First Street) for Car B to reach Oak Street.
3. Use the definition of tangent:
[tex]\[
\tan(y') = \frac{\text{opposite}}{\text{adjacent}}
\][/tex]
Since we know that [tex]\(\tan(y') = \frac{5}{7}\)[/tex], the adjacent side is 14 miles, and we denote the opposite side (the distance Car B needs to travel) as [tex]\( x \)[/tex]:
[tex]\[
\frac{5}{7} = \frac{x}{14}
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
We can solve the equation for [tex]\( x \)[/tex] by multiplying both sides by 14:
[tex]\[
x = 14 \times \frac{5}{7}
\][/tex]
5. Calculation:
[tex]\[
x = 14 \times 0.7142857142857143
\][/tex]
[tex]\[
x = 10
\][/tex]
6. Finalize the answer:
The distance Car B has to travel on First Street to get to Oak Street is 10 miles. When we round this to the nearest tenth, it remains 10.0 miles.
So, Car B will have to travel 10 miles on First Street to get to Oak Street. The correct choice from the given options is:
- 10 miles
