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 angle [tex]$x^{\circ}$[/tex], and [tex]$\tan x^{\circ} = \frac{7}{5}$[/tex]. Car A drives on Main Street for 21 miles to get to 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 :

To determine how far Car B has to travel on First Street to reach Oak Street, we need to analyze the given information and use trigonometric relationships.

### Given Information:
1. Car A drives on Main Street for a distance of 21 miles to reach Oak Street.
2. The angle [tex]\( x^\circ \)[/tex] at the intersection of First Street and Oak Street satisfies [tex]\(\tan(x) = \frac{7}{5}\)[/tex].

### Steps to Solve the Problem:

1. Understanding the Tangent Function:
The tangent of an angle in a right triangle is defined as the ratio of the length of the opposite side to the length of the adjacent side. Mathematically,
[tex]\[ \tan(x) = \frac{\text{opposite}}{\text{adjacent}} \][/tex]
Here, the opposite side corresponds to the distance that Car B travels on First Street, and the adjacent side corresponds to the distance that Car A travels on Main Street.

2. Assigning Variables:
Let's denote:
- The distance Car A drives as [tex]\(d_A = 21\)[/tex] miles (Main Street).
- The distance Car B drives as [tex]\(d_B\)[/tex] miles (First Street).

From the given problem, [tex]\(\tan(x) = \frac{d_B}{d_A} = \frac{7}{5}\)[/tex].

3. Setting Up the Equation:
Given [tex]\(\tan(x) = \frac{7}{5}\)[/tex], we can write:
[tex]\[ \frac{d_B}{21} = \frac{7}{5} \][/tex]
To find [tex]\(d_B\)[/tex], solve the equation for [tex]\(d_B\)[/tex].

4. Solving the Equation:
Multiply both sides of the equation by 21:
[tex]\[ d_B = 21 \times \frac{7}{5} \][/tex]
Performing the multiplication:
[tex]\[ d_B = 21 \times 1.4 = 29.4 \][/tex]

5. Rounding the Answer:
Since we are asked to round the answer to the nearest tenth of a mile, we note that our calculated answer already is in the correct form, 29.4 miles.

### Conclusion:
Car B has to travel approximately 29.4 miles on First Street to reach Oak Street. Thus, the distance Car B must drive is 29.4 miles, which matches one of the given choices.