Let's break down the problem step-by-step:
1. Given Values:
- The tangent of the angle [tex]\( y \)[/tex] ([tex]\( \tan y \)[/tex]) is given as [tex]\(\frac{5}{7}\)[/tex].
- The distance traveled by Car A on Main Street is 14 miles.
2. Understanding the Tangent Function:
- The tangent of an angle in a right-angled triangle is the ratio of the opposite side to the adjacent side.
- In this context, [tex]\(\tan y = \frac{\text{opposite}}{\text{adjacent}} = \frac{\text{distance on First Street}}{\text{distance on Main Street}}\)[/tex].
- Given [tex]\(\tan y = \frac{5}{7}\)[/tex], the distance on First Street (opposite side) can be calculated by rearranging the formula:
[tex]\[
\text{distance on First Street} = \text{distance on Main Street} \times \tan y
\][/tex]
3. Calculating the Distance on First Street:
- Given [tex]\(\tan y = \frac{5}{7}\)[/tex], and the distance on Main Street is 14 miles.
- Therefore:
[tex]\[
\text{distance on First Street} = 14 \times \frac{7}{5}
\][/tex]
4. Performing the Calculation:
- Simplifying the multiplication:
[tex]\[
\text{distance on First Street} = 14 \times \frac{7}{5} = 14 \div \frac{5}{7} = 14 \times \frac{7}{5} = 14 \times 1.4 = 19.6
\][/tex]
5. Rounding the Result:
- The distance on First Street is approximately 19.6 miles.
6. Conclusion:
- Car B will have to travel approximately 19.6 miles on First Street to get to Oak Street.
Therefore, the correct answer, rounded to the nearest tenth of a mile, is:
[tex]\[
\boxed{19.6 \text{ miles}}
\][/tex]
