To solve this problem, let's break it down step by step:
### Step 1: Understand the Given Information
- The post office is at the corner of First Street and Main Street, forming a right angle.
- First Street intersects with Oak Street to the north, and Main Street intersects with Oak Street to the east.
- Main Street and Oak Street form an angle [tex]\( y \)[/tex] where [tex]\( \tan(y) = \frac{5}{7} \)[/tex].
- A car drives on Main Street for 14 miles to arrive at Oak Street.
### Step 2: Solve for the Distance on First Street
We are given the tangent of the angle [tex]\( y \)[/tex], which is the ratio of the length of the opposite side to the adjacent side in a right triangle. Here:
[tex]\[
\tan(y) = \frac{\text{opposite}}{\text{adjacent}} = \frac{5}{7}
\][/tex]
In this scenario:
- The "opposite" side corresponds to the distance traveled on First Street.
- The "adjacent" side corresponds to the distance traveled on Main Street, which is 14 miles.
Let's denote the distance traveled on First Street as [tex]\( d \)[/tex].
Since [tex]\( \tan(y) = \frac{d}{14} \)[/tex], we set up the equation:
[tex]\[
\frac{d}{14} = \frac{5}{7}
\][/tex]
To find [tex]\( d \)[/tex], we solve:
[tex]\[
d = 14 \times \frac{5}{7}
\][/tex]
### Step 3: Calculate the Distance
Now, calculate the distance:
[tex]\[
d = 14 \times \frac{5}{7} = 14 \times 0.7142857142857143 \approx 10.0
\][/tex]
### Step 4: Round the Answer
Since the problem asks us to round the answer to the nearest tenth of a mile, we see that the calculated distance [tex]\( d \)[/tex] is already [tex]\( 10.0 \)[/tex].
### Step 5: Identify the Correct Option
The options are:
- 5 miles
- 7.4 miles
- 10 miles
- 19.6 miles
Given that our rounded distance is [tex]\( 10.0 \)[/tex] miles, the correct option is:
- 10 miles
Thus, car B will have to travel approximately 10 miles on First Street to get to Oak Street.
