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Optimization of a Function (Answers)

Module by: Debbie Trahan. E-mail the author

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Summary: This is the answer 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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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.

a

At=360−15t A t 360 15 t

Bt=45+10t B t 45 10 t

Figure 1
Figure 1 (optans1c.jpg)

b

A10=210 A 10 210

B10=145 B 10 145

The distance between Ship A and Ship B after 10 hours is 210 2 +145 2 =65125≈255.196 km 210 2 145 2 65125 255.196 km

c

A16=120 km A 16 120 km

B16=205 km B 16 205 km

Ship B is 205 km from Shipwreck Island when Ship A is 120 km.

d

Ship A and Ship B are 171 km from Shipwreck Island after 12.6 hours.

e

A24=0 A 24 0

B24=285 B 24 285

Ship B is 285 km from Shipwreck Island when Ship A reaches Shipwreck Island.

f

dt=360−15t2+45+10t2 d t 360 15 t 2 45 10 t 2

g

Figure 2:
Figure 2 (optans3.png)
Figure 3
Figure 3 (optans2.png)

At approximately 15.230 hours, the distance between Ship A and Ship B is the least. The distance at this time is approximately 237.134 km.

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