SOLUTION: John increased the area of his garden by 120 ft^2. The original garden was 12 ft. by 14 ft., and he increased the length and the width by the same amount. Find the exact dimensions

Algebra ->  Functions -> SOLUTION: John increased the area of his garden by 120 ft^2. The original garden was 12 ft. by 14 ft., and he increased the length and the width by the same amount. Find the exact dimensions      Log On


   

Question 148258: John increased the area of his garden by 120 ft^2. The original garden was 12 ft. by 14 ft., and he increased the length and the width by the same amount. Find the exact dimensions of the new garden and approximate the dimensions in feet and inches.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Let x=amount he increased the dimensions of his garden, A%5Bo%5D=area of original garden and A%5Bn%5D=area of new garden
Since the original garden is 12 ft by 14 ft, this means that the area of the original garden is

A%5Bo%5D=12%2A14=168

So A%5Bo%5D=168 which means that the area of the original garden is 168 ft^2


Because the area of the new garden is 120 ft^2 larger than the original garden, this means that A%5Bn%5D=A%5Bo%5D%2B120


A%5Bn%5D=168%2B120 Plug in A%5Bo%5D=168


A%5Bn%5D=288 Add


So the area of the new garden is 288 ft^2


Now since he increased the dimensions by some unknown amount, this means that the area of the new garden is equal to:

A%5Bn%5D=%2812%2Bx%29%2814%2Bx%29


288=%2812%2Bx%29%2814%2Bx%29 Plug in A%5Bn%5D=288


288=168%2B26x%2Bx%5E2 FOIL


0=168%2B26x%2Bx%5E2-288 Subtract 288 from both sides.


0=x%5E2%2B26x-120 Combine and rearrange the terms.


Let's use the quadratic formula to solve for x


x+=+%28-b+%2B-+sqrt%28+b%5E2-4ac+%29%29%2F%282a%29 Start with the quadratic formula


x+=+%28-%2826%29+%2B-+sqrt%28+%2826%29%5E2-4%281%29%28-120%29+%29%29%2F%282%281%29%29 Plug in a=1, b=26, and c=-120


x+=+%28-26+%2B-+sqrt%28+676-4%281%29%28-120%29+%29%29%2F%282%281%29%29 Square 26 to get 676.


x+=+%28-26+%2B-+sqrt%28+676--480+%29%29%2F%282%281%29%29 Multiply 4%281%29%28-120%29 to get -480


x+=+%28-26+%2B-+sqrt%28+676%2B480+%29%29%2F%282%281%29%29 Rewrite sqrt%28676--480%29 as sqrt%28676%2B480%29


x+=+%28-26+%2B-+sqrt%28+1156+%29%29%2F%282%281%29%29 Add 676 to 480 to get 1156


x+=+%28-26+%2B-+sqrt%28+1156+%29%29%2F%282%29 Multiply 2 and 1 to get 2.


x+=+%28-26+%2B-+34%29%2F%282%29 Take the square root of 1156 to get 34.


x+=+%28-26+%2B+34%29%2F%282%29 or x+=+%28-26+-+34%29%2F%282%29 Break up the expression.


x+=+%288%29%2F%282%29 or x+=++%28-60%29%2F%282%29 Combine like terms.


x+=+4 or x+=+-30 Simplify.


So the possible answers are x+=+4 or x+=+-30

However, since a negative length is not possible, this means that he increased his garden by 4 feet.


Now simply add 4 to each dimension 12 and 14 to get:

12+4=16 by 14+4=18

So the dimensions of the new garden are

16 ft by 18 ft


note: the approximate answers are the same as the exact answers since there are no square roots, fractions, decimals, etc. in the answer