The length of a rectangle is 7 cm less than its WIDTH. What are the dimensions of a rectangle if it’s areas 198 cm²?

Answer 1

How about a different approach?

A. Knowing that a square with s = side length = 14 would give nearly the same area (196), and increasing s by x for l, and decreasing s by x for w would give slightly less for A = lw = (s² - x²), you could start a guesstimate with x = 3.5 (half of 7, the given difference between l and w) and have 10.5 X 17.5. This needs to be slightly more, so try 11 X 18 and voila!!

B. Realizing that 3-digit numbers with digits abc will be divisible by 11 when a + c = b, you can start immediately with one of the sides at 11, and dividing 198/11 = 18, and how about that--they differ by 7!

Answer 2

One thing that we should remember from earlier math courses is that for a rectangle, A = L*W.

We don't know the length or width of the rectangle, but we do know that the relationship is that its length is 7 cm less than its width, or L = W - 7.

The key here is to use this relationship and substitute it into the area formula. If we do this, the only variable will be width, and we will be able to solve to find it.

A = L * W Substitute

198 = (W - 7)*W Distribute

198 = W² - 7W Subtract 198 from each side.

W² - 7W - 198 = 0 Factor by finding a pair of numbers that multiplies to -198 and adds to -7.

(W - 18)(W + 11) = 0 Now set each factor = 0

W - 18 = 0 Add 18 to both sides OR W + 11 = 0 Subtract 11 from both sides

W = 18 OR W = -11

Since the negative answer doesn't make sense here, the width must be 18.

Now use the given relationship L = W - 7 = 18 - 7 = 11.

The rectangle has a width of 18 cm and a length of 11 cm.

Hope this helps!

Answer 3

Let w = the rectangle's width. The length is 7 cm less, so the length is w-7.

Area = length x width

198 = (w-7)·w

198 = w² - 7w

0 = w² - 7w - 198

0 = (w-18)(w+11)

w = 18 and -11

Since the width cannot be negative, w = 18. Length = w - 7 = 18 - 7 = 11

Check:

198 = 18 x 11

198 = 198

Check!

Answer 4

Width is x

Length is x-7

Area is L times W

L x W =198

x(x-7) = 198

x^2 - 7x = 198 Subtract 198 from both sides which gives you x^2 - 7x - 198 = 0

The prime factorization of 198 is 2 x 3^2 x 11.

The factors of 198 that differ by 7 are 18 and 11.

Since the numerical coefficient of 7 is negative the problem factored is (x-18)(x+11) = 0

Set both equal to zero. x-18 = 0 x +11 = 0 Solve each.

The 2 possible values are +18 and -11.

Since the distance cannot be negative x = 18.

x-7 becomes 18 - 7 which is 11.

The dimensions are 11 by 18.