Optimization Problem

Luke EmeryCousino High School

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Theme song

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Hangin’ out…down the street

It was 1977 in Point Place, Wisconsin.

The whole gang (Eric, Donna, Kelso, Hyde and Fez) were lounging around in Eric’s basement.

Suddenly, Jackie (Kelso’s girlfriend) barged in the room.

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Hey guys! So, I hope you all remembered my birthday coming up…

Of course, Jackie! How could we forget?

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And Kelso, I just know you probably bought me the most expensive earrings you could find!

Uh, Jackie, DUH. It’s not like I forgot your birthday or anything…

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And Eric, you already picked out my birthday cake I assume? In the proper shape?

Uh…yeah! O-of course. Cylinder shape, right?

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NO!!!! I refuse to eat anything that resembles the shape of a tuna fish can. It must must MUST be a rectangular prism.

Oh, yeah. See, that was my second choice..

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So, what’s the problem?

Jackie demands that she receives a cake with a volume of 40 cubic feet. This cake must also be in the shape of a rectangular prism, with the height of this cake being half of the width.

Oh, and it also has to be strawberry flavored.

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So, what’s the problem?

Eric and Donna need to find the dimensions of a container to bake their cake in.

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So, what’s the problem?

To make this problem a little bit easier on themselves, Eric and Donna sat down a drew a simple diagram illustrating the problem.

Happy birthday!

xy

0.5x

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So, Eric and Donna sat down and started doing some calculations…

Donna figured out that they needed to solve for a variable using the volume given to them by Jackie.

40 = 0.5x2 yY = 40/0.5x2 Y = 80x-2

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So, Eric and Donna sat down and started doing some calculations…

Eric then discovered that they needed to create an equation to find the area of this ridiculous cake using the values that Jackie gave them.

A(x) = 0.5xy + xy + 0.5x2 A(x) = xy + xy + x2

A(x) = 2xy + x2

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Not a thing to do…but do calculus

Eric also said they needed to plug in the Y-value that was found in the first step.

A(x) = 2x(80x-2) + x2

A(x) = x2 + 160x-1

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Not a thing to do…but do calculus

And they also needed to find the derivative of the area function.

A’(x) = 2x – 160x-2

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Not a thing to do…but do calculus

Then, they set that derivative equal to zero and solved for x.

A’(x) = 2x – 160x-2 0 = 2x – 160x-2 X = 4.3089 feet

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Not a thing to do…but do calculus

After that, Donna made sure they used to original volume equation to solve for Y, and using the X-value they just found.

40= 0.5x2y40 = 0.5(4.3089)2yY = 4.3089 feet

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We’re all alright!

Before they started baking, Eric and Donna looked at the diagram they originally started with. Because X and Y both equal 4.3089 feet, the dimensions of the pan to bake Jackie’s ridiculous cake are 4.3089 feet by 4.3089 feet by 2.1544 feet. Happy birthday!

xy

0.5x

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But they’re not done yet…

Eric and Donna now want to find the first derivative of the surface area function.

A’(x) = 2x – 160x-2 And they set it equal to zero in order to solve for x.

0 = 2x – 160x-2 X = 4.3089

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But they’re not done yet…

After that, they plugged in 4 and 5.

A’(4) = 2(4) – 160(4)-2 A’(4) = -2 A’(5) = 2(5) – 160(5)-2 A’(5) = 3.6

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Graphically, you can see that the slope is changing from negative to positive.

And because the graph goes from decreasing to increasing, a minimum has to occur between 4 and 5. The exact value of this minimum is 4.3089.

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Now let’s find the second derivative…

Next, Eric and Donna took the surface area function and found the second derivative.A’’(x) = 2 + 320x-1 And they substituted the x value they found earlierA’’(4.3089) = 2 + 320(4.3089)-1 A’’(4.3089) = 76.2654

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So, when x = 4.3089, the value of the second derivative is positive.

But what does this mean? Well, the graph is concave up and there is a minimum.

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What else can they do with this knowledge?

Being the broke high school students that they are, Eric and Donna want to know the minimum amount of material they’ll need to build the container to bake this ridiculous cake.

Here’s the answer

We know that x = 4.3089

We know that y = 4.3089

Eric and Donna then substituted x and y into the original surface area equation.

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Here’s the answer

Substitution of the surface area formula:

A(x) = 2xy + x2 A(4.3089) = 2(4.3089)(4.3089) + (4.3089)2

Surface area = 55.9661 square feet

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Happy birthday, Jackie!

Oh my gosh guys! This is perfect! Thank you so much!

No problem. Although, it was really hard finding an oven to actually fit this cake.

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The dimensions of the pan were 4.3089 x 4.3089 x 2.1544 feet.

And the overall surface area of the pan was 55.9661 cubic feet.

So, give me a quick recap of your journey to make me my beautiful cake.