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# Max and Min Problems

Lesson Plans > Mathematics > Calculus > Differential > Max and Min

## Slide Show

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A model rocket is launched from a launch pad, and reaches its maximum speed 0.5 seconds into its flight (when the fuel is depleted). At this point, the rocket is 150 feet off the ground, and has a velocity of 400
ft
s
. Its height off the ground at that point is given by the following formula, in which h represents the height in feet, and t represents the amount of time elapsed after fuel depletion.

h(t) = 150 + 400t - 16.1t2

What is the maximum height the rocket attains before it begins to fall to the earth?

Farmer Jones has 3000 feet of fencing material. He wants to use this fencing material to create two rectangular pig pens that share a common side, and such that one of the pens is twice the area as the other. He wants to maximize his area. What dimensions should he use for the two pig pens?

An art store struggles to compete against online retailers. Therefore, they choose to sell some of their supplies at a loss, recognizing that it's the only way to get customers in the store. By experimentation, they discover that if they sell tubes of paint at a loss of L cents per tube, they will get 2L2 customers in the store each day, with each customer purchasing (on average) 3 tubes of paint. Of those customers,
L
5
will then purchase a custom framing job. If each custom framing job results in a \$20 profit for the store, how much should the art store discount tubes of paint?

A can is required to have a volume of 150p cubic centimeters. The cost of producing the top and bottom is \$0.001 per square cm, and the cost of producing the side is \$0.0008 per square cm. What will be the radius of the can which meets the volume criteria, but costs the least to produce? What will be the price of producing this can?

## Max and Min Problems

The problems in this slide show are not terribly difficult to solve; the purpose is not to provide a challenge to students. Rather, the purpose is to show them real world situations in which calculus is helpful. Each problem was designed to show a reasonable portrayal of actual scenarios faced by people in different professions (physicist, farmer, retail and manufacture). The goal is for students to gain an appreciation for how calculus simplifies real world challenges.

The art store problem is probably not a reasonable function, but it at least gives students an introduction to how advanced mathematics can help stores maximize their profit.

I especially like the "can" problem; although the numbers are made up, the problem is designed to show how simple things like the shape of an aluminum can might be influenced by practical applications of calculus.

Lesson by Mr. Twitchell

## Height of Launch

A model rocket is launched from a launch pad, and reaches its maximum speed 0.5 seconds into its flight (when the fuel is depleted). At this point, the rocket is 150 feet off the ground, and has a velocity of 400
ft
s
. Its height off the ground at that point is given by the following formula, in which h represents the height in feet, and t represents the amount of time elapsed after fuel depletion.

h(t) = 150 + 400t - 16.1t2

What is the maximum height the rocket attains before it begins to fall to the earth?

## Two Big Pig Pens

Farmer Jones has 3000 feet of fencing material. He wants to use this fencing material to create two rectangular pig pens that share a common side, and such that one of the pens is twice the area as the other. He wants to maximize his area. What dimensions should he use for the two pig pens?

## Paint and Framing

An art store struggles to compete against online retailers. Therefore, they choose to sell some of their supplies at a loss, recognizing that it's the only way to get customers in the store. By experimentation, they discover that if they sell tubes of paint at a loss of L cents per tube, they will get 2L2 customers in the store each day, with each customer purchasing (on average) 3 tubes of paint. Of those customers,
L
5
will then purchase a custom framing job. If each custom framing job results in a \$20 profit for the store, how much should the art store discount tubes of paint?

## Cost of a Can

A can is required to have a volume of 150p cubic centimeters. The cost of producing the top and bottom is \$0.001 per square cm, and the cost of producing the side is \$0.0008 per square cm. What will be the radius of the can which meets the volume criteria, but costs the least to produce? What will be the price of producing this can?

## Solutions

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