Question 536710: i am building a rectangular table top. the perimeter is 22 ft. the area is 24feet squared. what are the dimensions of the tabletop?
Can you help by showing your work?
Thank you!
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! Let's let L represent the Length and W represent the Width.
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To find the perimeter we add all the distances of the four sides or edges of the table top. Therefore, we add:
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L + W + L + W = Perimeter
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Combining the common terms this becomes:
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2L + 2W = Perimeter
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The problem tells you that the perimeter is 22 ft. So we can substitute that value into the equation and we get:
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2L + 2W = 22
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and we can simplify this a little by dividing both sides (all terms) by 2 to get:
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L + W = 11
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(We wouldn't have to do this division and it would all work out in the end, but this makes it a little easier.)
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The area of a rectangle is found by multiplying the length times the width. In equation form this is:
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L*W = Area
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The problem tells you that the area is 24 square feet. Substitute that into the area formula and you get:
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L*W = 24
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Let's solve this area equation for one of the variables. We could solve for either one, but let's choose to solve for L. We do so by dividing both sides of the area equation by W to get:
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L = 24/W
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Now let's go back to the perimeter equation (L + W = 11) and substitute 24/W for L to get:
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24/W + W = 11
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Get rid of the W in the denominator of the first term by multiplying both sides (all terms) by W to get:
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24 + W^2 = 11W
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Get this into standard quadratic form by subtracting 11W from both sides to get (with a little rearrangement of terms):
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W^2 - 11W + 24 = 0
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Notice that this quadratic equation factors into:
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(W - 3)*(W - 8) = 0
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Also notice that this equation will work out if either of the factors equals zero, because if either factor equals zero the left side has a multiplication by zero and will therefore equal the zero on the right side.
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So either W - 3 = 0 and by adding +3 to both sides this says W = 3
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or
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W - 8 = 0 and by adding +8 to both sides this says W = 8
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We can ignore W = 8. Why? Because if W = 8 then the length is smaller than W since L + W equals 11 and L + 8 = 11. This would make L = 3 which certainly is smaller than W.
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So we have W = 3 ft. and since L + W equals 11, L must be 8 ft.
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Check: does 3 + 8 + 3 + 8 = the 22 ft perimeter. Yes it does.
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and
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Does 8 * 3 = the 24 sq ft area? Yes it does. So our answer of Width = 3 ft and Length = 8 ft is correct.
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Hope this helps you and makes you thing a little bit about solving quadratic equations if you can factor them.
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