The length of a rectangle is 7cm less than twice it's width. If the area is 30 sq. cm, find the dimensions

The rectangle is 6 centimeters by 5 centimeters.

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.