Question

Erica is participating in a road race. The first part of the race is on a 5.2-mile-long straight road oriented at an angle of 25∘ north of east. The road then turns due north for another 5.0 mi to the finish line.
In miles, what is the straight-line distance from the starting point to the end of the race?

Answer 1

Answer:

8.6 miles

Explanation:

We need to calculate the components of the total displacement along the east-west and north-south directions first.

In the first part, Erica moves 5.2 miles at 25∘ north of east. So the components of this displacement along the two directions are:

East: $d_{1x} = 5.2 cos 25^{\circ}=4.7 mi$

North: $d_{1y} = 5.2 sin 25^{\circ}=2.2 mi$

In the second part, Erica moves 5.0 miles north. So, the components of this displacement are:

East: $d_{2x}=0$

North: $d_{2y} = 5.0 mi$

So the components of the total displacement are

East: $d_x = d_{1x}+d_{2x}=4.7 + 0 = 4.7 mi$

North: $d_y = d_{1y}+d_{2y}= 2.2 + 5.0 = 7.2 mi$

Therefore the magnitude of the displacement, which is the straight-line distance from the starting point to the end of the race, is

$d=\sqrt{d_x^2 +d_y^2}=\sqrt{4.7^2+7.2^2}=8.6 mi$