# A rectangular field has a length 10 feet more than it is width. If the area of the field is 264, what are the dimensions of the rectangular field?

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Let L be the length of the field. Let W be the width of the field. We know from your question that

L = W + 10

We also know that W*L = 264

This is because the area of a rectangle is equal to the length times the width.

So now we subsitute in

(W + 10)*W = 264

W^2 + 10W = 264

W^2 + 10W - 264 = 0

Now we have to find the factors. We can factor this out to

(w + 22) (w - 12) = 0

So the width could be -22 or 12. Of course it cannot be negative so

W = 12

and therefore

L = 22.

Let us assume that the length is (L) and the width is (W)

We know that the area of a rectangle is a=L*W

==> LW = 264 ...(1)

But we know that L=W+10

Substitute with (1):

(W+10)W =264

w^2 + 10w = 264

w^2+10w -264=0

Factorize:

(w-12)(w+22)=0

W= 12

L= 12+10 = 22

Let the width of the field be w.

Then as the length of field is 10 feet more than the width, the length will be:

Length = w + 10

Area of a rectangle is given by formula:

Area = Width x Length

Substituting the above values of width and length, and the given area of the field in the above equation we get:

Area = w x (w + 10) = 264

Therefore:

w^2 + 10w = 264

Taking all the terms of the equation on left hand side we get:

w^2 + 10w - 264 = 0

To find the value of w in the above equation we factorise the expression on the left hand side as follows.

w^2 + 10w - 264

= w^2 + 22w - 12w - 264

= w(w + 22) - 12(w + 22)

= (w + 22)(w - 12)

Thus w = -22 , or 12

As width cannot be negative, w = 12

Thus:

Width of the field = 12 feet

Length of the field = Width + 10 = 12 + 10 = 22 feet.

The area of the rectangle is 264 ft^2.

The length is 10 feet more than the breadth. So one way is to examine all the factors, 2*132 , 4*66, 8*33, 8*3*11 or 24*11, 12*22 and find which of the pair of factors has the difference 10. The factors 12 and 22 fulfills thcondition. The other way is assuming l and b as lenth and breadth in feetand then,

l-b = 10 ...........(1)and

lb = 264.............(2)

So (l+b) = sqrt{(l-b)^2+4ab} = sqrt(10^2+4*264} = 34.

l-b = 10...........(1)

l+b = 34..........(3)

(1)+(3) gives: 2l = 10+34 = 44. So, l = 44/2 = 22 .

(3) - (1) gives: 2b = 34-10 = 24. So, b= 24/2 = 12.

Hope this may help.