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Optimization of a Function (Student Page)

Module by: Debbie Trahan. E-mail the author

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Summary: This is the student page for the Optimization of a Function Module. The student will write and graph linear functions then use these functions to write a new function using the Pythagorean Theorem. The students will then find the minimum value of the new function using a graphing calculator.

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Lesson Objective

During this lesson, you will review linear functions and the Pythagorean Theorem. You will use these topics to help you write a new function then find the minimum value of the new function.

Problem

Ship A is 360 kilometers east of Shipwreck Island traveling due west toward Shipwreck Island at a speed of 15 kilometers per hour (km/hr). Ship B is 45 kilometers north of Shipwreck Island traveling due north away from Shipwreck Island at a speed of 10 km/hr. (see figure 1)

Figure 1
Figure 1 (optfig1.png)

a

Let Shipwreck Island be at the origin.

Write a function for Ship A’s distance from Shipwreck Island in terms of the time t t .

Write a function for Ship B’s distance from Shipwreck Island in terms of the time t t .

Graph the functions on the coordinate plane shown in Figure 2.

Figure 2
Figure 2 (optfig2.jpg)

b

Find the distance, in kilometers, between Ship A and Ship B after 10 hours.

c

Find the location of Ship B when Ship A is 120 miles from Shipwreck Island.

d

At what time are the ships the same distance from Shipwreck Island?

e

Find the distance for Ship B when Ship A reaches Shipwreck Island.

f

Write a function for the distance between Ship A and Ship B in terms of the time.

g

Graph the function that you wrote in part f on a graphing calculator. Use a calculator to help you approximate when the distance between Ship A and Ship B is the least. Approximate the distance at this time.

Answers

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