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# the length of a rectangle is 5m more than three times the width. The area is 232m2. Find the length and width of the rectangle

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Philip P. | Effective and Affordable Math TutorEffective and Affordable Math Tutor
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The length of a rectangle is 5m more than three times the width. The area is 232m2. Find the length and width of the rectangle.

Step 1 - What are the unknowns you're asked to find?

Find the length and width of the rectangle.

Let L= the length of the rectangle
Let W = its width

Step 2 - We have 2 unknowns (L and W), so we need two equations that relate the unknowns

Equation 1. the length of a rectangle is 5m more than three times the width:  L = 5 + 3W
Equation 2.  The area is 232m2:  Area = L*W = 232

Step 3 Substitution

Let's substitute 5+3W (from equation 1) in place of L in equation 2:

L*W = 232                      [Equation 2]
(5+3W)*W = 232            [Substituted 5+3W from equation 1 in place of L]
5W + 3W2 = 232             [We now have an equation with only variable, W, to solve for]

3W2 + 5W - 232 = 0        [A quadratic equation!]

Step 4 - Solve for W.  Use the quadratic formula.

W = -(b/2a) ± (1/2a)√(b2-4ac)   where a=3, b=5, c= -232 from our quadratic equation

W = -(5/6) ± (1/6) √(52-(4)(3)(-232))
W = -(5/6) ± (1/6) √(2809)
W = -(5/6) ± (53/6)

W = 8, -58/6

We can't have a negative width (-58/6), so W = 8 meters

Step 5 - Solve for L
L = 5 + 3W             [Equation 1]
L = 5 + 3(8)            [Put in W=8]
L = 5 + 24
L = 29 meters

Step 6 Check
L*W = 232               [Equation 2]
(29)(8) = 232           [Put in L=29, W=8]
232=232                  CHECK!