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.
