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 [tex]$x^*$[/tex] angle, and tan [tex]$x^*=\frac{7}{5}$[/tex]. Car A drives on Main Street for 21 miles to arrive at 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.
Answer (2)
Recognize that the streets form a right triangle, where First Street and Main Street are legs.
Use the given tangent value tan x ∗ = 5 7 and the distance car A travels on Main Street (21 miles) to set up a proportion.
Solve the proportion 5 7 = d B 21 for d B , which represents the distance car B travels on First Street.
Find the distance d B = 15 miles. 15 miles
Explanation
Problem Analysis We are given that 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 ∗ = 5 7 . Car A drives on Main Street for 21 miles to arrive at Oak Street. We need to find how far car B has to travel on First Street to get to Oak Street.
Set up the equation Let d A be the distance car A travels on Main Street, and d B be the distance car B travels on First Street. We are given that d A = 21 miles. We are also given that tan x ∗ = 5 7 . Since tan x ∗ = adjacent opposite , in this case, tan x ∗ = d B d A .
Cross-multiply We have the equation 5 7 = d B 21 . To solve for d B , we can cross-multiply: 7 × d B = 5 × 21 .
Solve for d_B Now, we can solve for d B : d B = 7 5 × 21 = 7 105 = 15 .
Final Answer Therefore, car B has to travel 15 miles on First Street to get to Oak Street.
Examples
Imagine you're planning a city layout where streets intersect at right angles. If you know the angle at which one street meets another and the distance a car travels on one street to reach an intersection, you can calculate the distance another car needs to travel on the intersecting street to reach the same point. This is useful for urban planning, navigation, and calculating travel times.
Car B needs to travel 15 miles on First Street to reach Oak Street. This is determined using the tangent of the angle formed at the intersection of the streets and the known distance that car A travels on Main Street. By setting up a proportion based on the tangent ratio, we find d B to be 15 miles.
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