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Chapter 2: Problem 4

Johannes bicycles from his dorm to the pizza shop that is 3.00 mi east. Darren's apartment is located \(1.50 \mathrm{mi}\) west of Johannes's dorm. If Darren is able to meet Johannes at the pizza shop by bicycling in a straight line, what is the distance and direction he must travel?

Short Answer

Expert verified
Answer: The straight-line distance between Darren's apartment and the pizza shop is 4.50 miles, and Darren must travel due East (0 degrees East) to reach it in a straight line.

Step by step solution

01

Draw a Diagram

Draw a diagram with Darren's apartment on the left, Johannes's dorm in the middle, and the pizza shop on the right with the distances labeled. This will help visualize the information more clearly while deriving the solution.
02

Use the Pythagorean Theorem

Use the Pythagorean theorem to find the distance between Darren's apartment and the pizza shop. Let's denote the unknown distance as d. \[d^2 = (1.50 \mathrm{mi} + 3.00 \mathrm{mi})^2 + (0)^2\] \[d^2 = (4.50 \mathrm{mi})^2\]
03

Calculate the distance (d)

Calculate the distance (d) by taking the square root of both sides of the equation: \[d = \sqrt{ (4.50\, \mathrm{mi})^2}\]
04

Distance Answer

Evaluate the distance: \[d = 4.50\, \mathrm{mi}\]
05

Calculate the Angle

To find the angle, we can use the tangent function, as we have the length of the opposite and adjacent sides to the angle: \[\tan(\theta) = \frac{0}{4.50\, \mathrm{mi} -1.50\, \mathrm{mi}}\] Since the numerator is zero, the tangent will be zero, indicating that the angle is 0 degrees.
06

Direction Answer

The direction in which Darren must bicycle to reach the pizza shop is exactly due East, i.e., 0 degrees East or in a straight line parallel to the eastward direction taken by Johannes. In conclusion, the distance between Darren's apartment and the pizza shop is \(4.50\, \mathrm{mi}\), and Darren must travel due East (0 degrees East) to reach it in a straight line.

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