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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 Main Street and Oak Street forms a [tex][tex]$y^4$[/tex][/tex] angle, and [tex][tex]$\tan y^{\circ}=\frac{5}{7}$[/tex][/tex]. Car A drives on Main Street for 14 miles 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. 5 miles
B. 7.4 miles
C. 10 miles
D. 19.6 miles


Sagot :

To solve this problem, we're given that Car A drives 14 miles along Main Street to get to Oak Street, and we need to determine how far Car B must travel on First Street to get to Oak Street. Given the tangent of the angle [tex]\(y\)[/tex] as [tex]\(\tan(y^\circ) = \frac{5}{7}\)[/tex], we can derive the required distance step by step.

1. Identify the Angle [tex]\( y \)[/tex]:
- We know that [tex]\(\tan(y^\circ) = \frac{5}{7}\)[/tex]. Using this information, we can find the actual value of the angle [tex]\(y\)[/tex] in degrees using the arctangent function. This results in an angle approximately equal to 35.54 degrees.

2. Use the Tangent Function to Find Distance:
- The tangent of an angle in a right-angled triangle is defined as the ratio of the opposite side to the adjacent side.
- In our case:
[tex]\[ \tan(y) = \frac{\text{distance traveled by Car A}}{\text{distance traveled by Car B}} \][/tex]

3. Plug in the Known Values:
- We know that [tex]\(\tan(35.54^\circ) = \frac{5}{7}\)[/tex], Car A's distance (adjacent side) = 14 miles.
- Therefore, the distance [tex]\(d\)[/tex] that Car B must travel (opposite side) can be found by rearranging the equation for tangent:
[tex]\[ \frac{5}{7} = \frac{14}{d} \][/tex]
- Solving for [tex]\(d\)[/tex]:
[tex]\[ d = \frac{7 \times 14}{5} = 19.6 \text{ miles} \][/tex]

Hence, Car B needs to travel approximately 19.6 miles along First Street to reach Oak Street. Therefore, the correct answer is:

Answer: 19.6 miles