Question 492775: Find the dimensions of a rectangle whose perimeter is 26 cm and whose area is 36 cm^2.
I don't understand how to do this problem.
Answer by lmeeks54(111)   (Show Source):
You can put this solution on YOUR website! By examining the problem, we can deduce two equations, one about the rectangle's perimeter and one about its area.
Let's describe our rectangle as having two equal long sides labeled "L" for the length and two equal short sides labeled "W" for the width.
The formula for the rectangle's perimeter is: 2 * L + 2 * W = P (perimeter)
The formula for the rectangle's area is: L * W = A (area)
From our problem, P = 26 and A = 36, so we have two equations with the same variables:
2*L + 2*W = 26
L*W = 36
The easiest method to solve these kinds of problems is to restate one of the equations in terms of one of the variables, then plug that new equality back into the other equation.
For example, L*W = 36 could also be: W = 36/L (divided by L)
Plugging our new identity for W back into the first equation gives us:
2*L + 2*(36/L) = 26, which is also: 2*L + 72/L = 26
We can change both sides of an equation to make the terms easier to understand or work with as long as we change both sides the same. So, multiplying both sides by L let's us get L out of the denominator or 72/L:
L*(2*L) + L*(72/L) = 26*L
Let's clean this up by completing the multiplication by L and putting all terms on one side of the equality, setting the equation = 0. Note: L*72/L = 72 (the L's cancel out):
2*L^2 - 26*L + 72 = 0
We can clean this up further by dividing all terms by 2:
L^2 - 13*L + 36 = 0, which is a quadratic equation that we can factor:
(L - 9) * (L - 4) = 0 We can check our work and see by multiplying out the factors we get:
L^2 - 9*L - 4*L + 36 = 0
So, L = 9 or L = 4 both make our quadratic equation true:
4^2 - 13*4 + 36 = 0, which is: 16 - 52 + 36 = 0 (true)
9^2 - 13*9 + 36 = 0, which is: 81 - 117 + 36 = 0 (also true)
Note: we have two possible answers for L (9 or 4), but in reality, the greater value is what we called length above and the shorter value is what we called width above. But to check our work (ALWAYS check your work), recall we stated W in terms of L:
W = 36/L
W = 36/9
W = 4
And go back to the two original equations for Perimeter (P) and Area (A) using L = 9 and W = 4 to check to see if both work (they do):
2*L + 2*W = P
2*9 + 2*4 = 26
18 + 8 = 26 (true)
L*W = A
9*4 = 36 (true)
We're done.
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