Question

Part 1: Read this problem and then solve, explaining how you do each step, as if you were explaining to a younger student. You will want to draw a picture, and or make a table.

Ken and Barbie are going to add a deck with a fancy railing to their dream house. The deck needs to have a total area of 100 square feet. They will only need a railing on three sides of the deck, since it will be connected to the house on one side. The deck costs $12 per square foot and the railing costs $9 per linear foot. What could the length and width of the deck be to keep the cost reasonable? Find the total cost. There is more than one correct answer. Remember to explain!!

Answer 1

• 12' by 8'4" . . . . $1458
• 13'4" by 7'6" . . $1455
• 15' by 6'8" . . . . $1455

Explanation:

The cost of the area of the deck is fixed, because the area is fixed. It will be ...

($12/ft²)×(100 ft²) = $1200

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The cost of the railing is proportional to its length, so it will be minimized by minimizing the length of the railing. If the length of it is x feet parallel to the house, then the length of it perpendicular to the house (for a deck area of 100 ft²) is 100/x.

The total length of the railing is ...

r = 2(100/x) + x

We can minimize this by setting its derivative with respect to x equal to zero:

dr/dx = -200/x² +1 = 0

Multiplying by x² and adding 200, we get ...

x² = 200

x = √200 ≈ 14.142

So, the minimum railing cost will be had when the deck is 14.142 ft by 7.071 ft. That railing cost is about ...

$9 × (200/√200 +√200) ≈ $254.56

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We might imagine that dimensions near these values would have almost the same cost. Here are some other possibilities:

• 13'4" by 7'6" ⇒ $255.00
• 15' by 6'8" ⇒ $255.00
• 12' by 8'4" ⇒ $258.00
• 10' by 10' ⇒ $270.00

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Then the total cost for a couple of possible deck sizes will be $1200 plus the railing cost, or ...

• 12' by 8'4" . . . . $1458
• 13'4" by 7'6" . . $1455
• 15' by 6'8" . . . . $1455

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Note on the solution process

It can be helpful to use a spreadsheet or graphing calculator to do the repetitive computation involved in finding suitable dimensions for the deck.