If the width of a rectangle is 7 centimeters less than twice its length and its area is 30 square centimeters, then the dimensions of the rectangle are 6 centimeters by 5 centimeters.
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To find the dimensions of the rectangle, we can set up a system of equations and use substitution to solve it. We are given that the width, call it w, of the rectangle is 7 centimeters less than twice its length, call it l. This gives the following equation:
w = 2l - 7
We are also given that the area is 30 square centimeters. The area of a rectangle, call it A, can be found using the formula A = l*w, or area equals the rectangle's length times its width. Therefore, we have another equation as follows:
30 = l*w
Now that we have two equations in l and w, we can use substitution to solve. To do this, we first plug w = 2l - 7 into 30 = lw.
30 = l(2l - 7) = 2l2 - 7l
Now, we can solve the equation 30 = 2l2 - 7l to find our length.
30 = 2l2 - 7l
Subtract 30 from both sides and interchange sides.
2l2 - 7l - 30 = 0
Factor the left-hand side.
(2l + 5)(l - 6) = 0
Set each factor equal to 0 and solve.
2l + 5 = 0 or l - 6 = 0
2l = -5 or l = 6
l = -5/2 or l = 6
We get that the length is 6 or -5/2. Since we can't have a negative length, the length is 6 centimeters. Now we plug l = 6 into either one of the original equations to solve for w and find our width.
30 = 6w
5 = w
or
w = 2(6) - 7
w = 5
In both cases we get that the width is 5 centimeters, so we have that the dimensions of the rectangle are 6 centimeters by 5 centimeters.
