A rectangular garden is 9m by 11m.A pathway of uniform width will be constructed around a garden,forming a larger triangle with area 120m². How wide in meters should the Pathway be ?
Answers
Pathway should be 0.5 m wide if constructed around a rectangular garden 9 * 11 such that area of Larger rectangular = 120 m²
Step-by-step explanation:
A rectangular garden is 9m by 11m
Let say pathway of uniform width = w will be constructed around a garden,forming a larger rectangle with area 120m²
then sides of new rectangle = 9 + 2w & 11 + 2w
Area (9 + 2w)(11 + 2w) = 120
=> 4w² + 40w + 99 = 120
=> 4w² + 40w - 21 = 0
=> 4w² - 2w + 42w - 21 =0
=> 2w(2w - 1) + 21(2w - 1) =0
=> (2w + 21)((2w - 1) = 0
w = 1/2 ( -ve values ignored)
Pathway would be 0.5 m wide
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Answer:
The width of the uniform path around the garden is 0.5 m
Step-by-step explanation:
Let x be the width of the path.
Now we have two rectangles.
1) The outer one which is the largest.
2) The inner one which is the smallest.
Since we have the dimensions of the inner rectangle, we can get the dimensions of the outer rectangle as follows :
Length = 11m + 2x
Width = 9m + 2x
Now, area of a rectangle is given by:
Area = length × width
Substituting we have :
(11 + 2x)(9 + 2x) = 120
99 + 22x + 18x + 4x² = 120
99 + 40x + 4x² = 120
This is a quadratic equation.
4x² + 40x + 99 - 120 = 0
4x² + 40x - 21 = 0
The roots of this quadratic equation are:
42 and - 2
Doing the substitution we have:
4x² - 2x + 42x - 21 = 0
2x(2x - 1) + 21(2x - 1) = 0
(2x + 21)(2x - 1) = 0
2x + 21 = 0
2x - 1 = 0
The value of x is:
x = - 21/2 or ½
Since measurement can't take a negative value, we take ½
So the width of the pathway is 0.5m.