To solve this problem, we're given that Car A drives 14 miles along Main Street to get to Oak Street, and we need to determine how far Car B must travel on First Street to get to Oak Street. Given the tangent of the angle [tex]\(y\)[/tex] as [tex]\(\tan(y^\circ) = \frac{5}{7}\)[/tex], we can derive the required distance step by step.
1. Identify the Angle [tex]\( y \)[/tex]:
- We know that [tex]\(\tan(y^\circ) = \frac{5}{7}\)[/tex]. Using this information, we can find the actual value of the angle [tex]\(y\)[/tex] in degrees using the arctangent function. This results in an angle approximately equal to 35.54 degrees.
2. Use the Tangent Function to Find Distance:
- The tangent of an angle in a right-angled triangle is defined as the ratio of the opposite side to the adjacent side.
- In our case:
[tex]\[
\tan(y) = \frac{\text{distance traveled by Car A}}{\text{distance traveled by Car B}}
\][/tex]
3. Plug in the Known Values:
- We know that [tex]\(\tan(35.54^\circ) = \frac{5}{7}\)[/tex], Car A's distance (adjacent side) = 14 miles.
- Therefore, the distance [tex]\(d\)[/tex] that Car B must travel (opposite side) can be found by rearranging the equation for tangent:
[tex]\[
\frac{5}{7} = \frac{14}{d}
\][/tex]
- Solving for [tex]\(d\)[/tex]:
[tex]\[
d = \frac{7 \times 14}{5} = 19.6 \text{ miles}
\][/tex]
Hence, Car B needs to travel approximately 19.6 miles along First Street to reach Oak Street. Therefore, the correct answer is:
Answer: 19.6 miles
