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Maximizing and Minimizing using Derivatives

2) A piece of paper for a poster contains 1000 cm^2. The margins at the top and bottom are 9cm and the side margins are 6 cm. What are the dimensions of the sheet if the printed area is to be a maximum.
Answers 2root3 and 3root3

3) At 9am ship B was 65km due east of ship A. Ship B was then sailing west at 10km/h and A was sailing south at 15km/h. When will they be nearest each other , and how near

Answer 11 am , 15root13

4) A cylindrical container is to hold 64 cm^3. Find the dimensions for minimum surface area when the container
A) is an open cup
B) is a closed can

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Problem #2
Suppose the paper sheet is x cm length and y cm width. Since xy = 1000 cm^2, then y = 1000/x.
From the condition, the printing area is (x-12) cm and (y-18) cm, then the area is
A(x) = (x-12)(y-18) = (x-12)(1000/x - 18) = 1000 - 12000/x - 18x + 216
To maximum A(x), we can set
A'(x) = 12000/x^2 - 18 = 0, then x^2 = 12000/18, then x = sqrt(12000/18) = 20sqrt(15)/3
Then y = 1000/x = 150/sqrt(15) = 10sqrt(15)
So the answer is 20sqrt(15)/3 cm by 10sqrt(15) cm.
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Solution Summary

Derivatives are used to find maximum and minimum values.