Let's solve the problem step-by-step:
1. Understand the given information:
- The intersection of First Street and Main Street forms a right angle.
- Car A drives north on Main Street for 21 miles to arrive at Oak Street.
- The angle [tex]$x$[/tex] formed at the intersection of First Street and Oak Street has a tangent value of [tex]$\frac{7}{5}$[/tex].
2. Identify the goal:
- We need to find the distance car B will travel on First Street to reach Oak Street, rounding to the nearest tenth of a mile.
3. Apply the tangent function:
- In a right triangle, the tangent of an angle is the ratio of the length of the opposite side to the adjacent side.
- Here, [tex]$\tan(x) = \frac{\text{opposite}}{\text{adjacent}} = \frac{7}{5}$[/tex].
- Since car A's travel on Main Street (21 miles) lies along the opposite side relative to angle [tex]$x$[/tex], we can use the given tangent ratio and the known length of the opposite side to find the length of the adjacent side.
4. Calculate the distance for car B:
- Let's denote the distance car B will travel on First Street as [tex]\( \text{distance}_B \)[/tex].
- From the tangent ratio:
[tex]\[
\text{distance}_B = \text{distance}_A \times \tan(x)
\][/tex]
where [tex]\(\text{distance}_A\)[/tex] is 21 miles.
Given that [tex]$\tan(x) = \frac{7}{5}$[/tex]:
[tex]\[
\text{distance}_B = 21 \times \frac{7}{5}
\][/tex]
5. Simplify the calculation:
[tex]\[
\text{distance}_B = 21 \times \frac{7}{5} = 21 \times 1.4 = 29.4 \text{ miles}
\][/tex]
6. Round the result:
- Our calculation gives exactly 29.4 miles, so no further rounding is needed.
7. Conclusion:
- Car B will have to travel 29.4 miles on First Street to get to Oak Street.
So, the correct answer is 29.4 miles.
