The post office is at the corner of First Street and Main Street, which forms a right angle. First Street intersects with Oak Street to the north, and Main Street intersects with Oak Street to the east. The intersection of First Street and Oak Street forms an [tex]\( x^\circ \)[/tex] angle, and [tex]\(\tan x = \frac{7}{5}\)[/tex]. Car A drives on Main Street for 21 miles to arrive at Oak Street. How far will car B have to travel on First Street to get to Oak Street? Round your answer to the nearest tenth of a mile.

A. 15 miles
B. 20 miles
C. 25.4 miles
D. 29.4 miles



Answer :

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.