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 x° angle, and tan x° = seven fifths. Car A drives on First Street for 10 miles to arrive at Oak Street. How far will car B have to travel on Main Street to get to Oak Street? Round your answer to the nearest hundredth of a mile.

Question

7.14 miles
14 miles
20 miles
28.14 miles

Answer 1

We are given that \( \tan x^\circ = \frac{7}{5} \).

In a right triangle where the angle \( x^\circ \) is one of the angles, the relationship \( \tan x = \frac{\text{opposite}}{\text{adjacent}} \) holds, where:

• the opposite side to angle \( x^\circ \) corresponds to the segment along First Street (10 miles),
• the adjacent side corresponds to the segment along Main Street, which we need to find.

Using the definition of tangent, we can write:

\[ \tan x = \frac{\text{opposite}}{\text{adjacent}} = \frac{10}{d} \]

where \( d \) is the distance car B travels on Main Street.

Setting the two expressions for \( \tan x \) equal gives:

\[ \frac{10}{d} = \frac{7}{5} \]

Now, we cross-multiply to solve for \( d \):

\[ 10 \cdot 5 = 7 \cdot d \] \[ 50 = 7d \]

Divide by 7:

\[ d = \frac{50}{7} \approx 7.14 \]

Finally, rounding to the nearest hundredth:

\[ d \approx 7.14 \text{ miles} \]

Thus, car B will have to travel 7.14 miles on Main Street to reach Oak Street.

So, the answer is 7.14 miles.