A highway engineer uses the coordinate system below to plan a new highway. Each unit on the coordinate system represents 10 kilometers.

Question

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A. What is the total length of the highway from Las Plumas to Bently? Explain how you found your answer.

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Constructed Response Question
B. A rest area must be located 3/5 of the way from Las Plumas to Bently. What are the coordinates of the rest area? Explain how you found your answer.

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Answer 1

To solve the problems, we need more information about the specific coordinates of Las Plumas and Bently on the coordinate system. However, I can guide you on how you might approach each part of the problem using hypothetical coordinates.

A. Finding the total length of the highway:

Let's say the coordinates for Las Plumas are $(x_1,y_1)$ and for Bently are $(x_2,y_2)$.

1. Calculate the distance between the two points using the distance formula: $$\text{Distance}=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}$$

2. Convert the distance from units on the coordinate system to kilometers: Since each unit represents 10 kilometers, you will multiply the calculated distance by 10 to get the total length in kilometers.

B. Finding the coordinates of the rest area:

To find the rest area located $\frac{3}{5}$ of the way from Las Plumas to Bently, you can follow these steps:

1. Find the directional vector from Las Plumas to Bently: $$\text{Direction Vector}=(x_2-x_1,y_2-y_1)$$

2. Calculate the point $\frac{3}{5}$ of the way: The coordinates of the rest area can be found by taking $\frac{3}{5}$ of the way along the directional vector from Las Plumas: $$\text{Coordinates of Rest Area}=(x_1+\frac{3}{5}(x_2-x_1),y_1+\frac{3}{5}(y_2-y_1))$$

This means that for each coordinate (x and y), you will calculate a position that is three-fifths of the distance from Las Plumas to Bently.

Summary

1. For Part A, use the distance formula and convert units to find the total highway length.
2. For Part B, compute the coordinates that represent three-fifths of the way from Las Plumas to Bently using vector arithmetic.

If you have the specific coordinates, you can substitute them into these formulas to get the numerical answers!