SOLUTION: 1. Find two consecutive integers such that the sum of there square is 85. 2. A garden area is 30ft long and 20ft wide. A path of uniform width is set around the edge. If the rem

Algebra ->  Quadratic Equations and Parabolas -> SOLUTION: 1. Find two consecutive integers such that the sum of there square is 85. 2. A garden area is 30ft long and 20ft wide. A path of uniform width is set around the edge. If the rem      Log On


   

Question 85444: 1. Find two consecutive integers such that the sum of there square is 85.
2. A garden area is 30ft long and 20ft wide. A path of uniform width is set around the edge. If the remaining garden area id 400ft ^2, what is the width of the path?
Found 2 solutions by scianci, venugopalramana:
Answer by scianci(186) About Me  (Show Source):
You can put this solution on YOUR website!
1. x = first integer
x + 1 = next consecutive integer
x%5E2 + %28x+%2B+1%29%5E2 = 85
x%5E2 + x%5E2 + 2x + 1 = 85
2x%5E2 + 2x + 1 = 85
2x%5E2 + 2x - 84 = 0
x%5E2 + x - 42 = 0
(x + 7)(x - 6) = 0
x + 7 = 0 ; x - 6 = 0
x = -7 ; x = 6
x = -7 , x + 1 = -6 is one pair of solutions
x = 6 , x + 1 = 7 is another pair of solutions
2. Let x = width of the path
Garden area = 20(30) = 600
Remaining Garden area = (30 - 2x)(20 - 2x) = 400
600 - 100x + 4x%5E2 = 400
4x%5E2 - 100x + 600 = 400
4x%5E2 - 100x + 200 = 0
x%5E2 - 25x + 50 = 0
x+=+%2825+%2B-+sqrt%28+%28-25%29%5E2-4%2A1%2A50+%29%29%2F%282%2A1%29+ =
%2825+%2B-+sqrt%28+625-200+%29%29%2F2 =
%2825+%2B-+sqrt%28425%29%29%2F2 =
%2825+%2B-+21.62%29%2F2 =
46.62%2F2 , 3.38%2F2 =
23.31 , 1.69
23.31 is not feasible since the whole width is only 20 ft, so the answer is 1.69 ft.

Answer by venugopalramana(3286) About Me  (Show Source):
You can put this solution on YOUR website!
1. Find two consecutive integers such that the sum of there square is 85.
N^2+(N+1)^2=85
N^2+N^2+2N+1=85
2N^2+2N-84=0
N^2+N-42=0
N^2+7N-6N-42=0
N(N+7)-6(N+7)=0
(N-6)(N+7)=0
HENCE N=6 OR -7
THE 2 INTEGERS ARE 6&7
OR -7 AND -6
2. A garden area is 30ft long and 20ft wide. A path of uniform width is set around the edge. If the remaining garden area id 400ft ^2, what is the width of the path?
GARDEN AREA =400 SQ.FT.
IF THE WIDTH OF PATH IS X FEET , THEN
LENGTH OF GARDEN = 30-X-X=30-2X
WIDTH OF GARDEN =20-X-X=20-2X
AREA OF GARDEN =(30-2X)(20-2X)=400
600-60X-40X+4X^2=400
4X^2-100X+200=0
X^2-25X+50=0
x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
x+=+%2825+%2B-+sqrt%28+25%5E2-4%2A1%2A50+%29%29%2F%282%2A1%29+
x+=+%2825%2B-+sqrt%28+425+%29%29%2F%282%29+=2.2 FT