Show that A1= 4x^2 - 260x + 4000?

There is a lawn with a rectangle flower bed in the middle.The lawn measures 80m by 50m and the grass edge on either side of the flower bed is X cm.



Let A1 be the area of the flower bed.

-Show that A1= 4x^2 - 260x + 4000, 0 is greater than or equal to x, x is less than or equal to 25.



Let A2 be the area of the lawn.

-Show that A2= -4x + 260x



Find the value of x for which A1 = A2 ie the area of the flower bed is equal to the area of the lawn
Show that A1= 4x^2 - 260x + 4000?
The area of the lawn including the flower bed = 80 x 50 = 4000 m^2



The width of the grass on either side of the flower bed = x m



So the dimensions of the flower bed are :

Length = 80 - 2x (Subtracting x from both the sides)

Breadth = 50 - 2x



Area of the flower bed, A1 = 4x^2 - 260x + 4000



Now area of the flower bed =%26lt; Area of the lawn

4x^2 - 260x + 4000 =%26lt; 4000

4x^2 - 260x =%26lt; 0

4x (x - 65) =%26lt; 0

Since 4x (x - 65) is -ve, one of 4x and x - 65 =%26lt; 0 and the other is not.



4x =%26lt; 0 or x =%26lt; 0

x - 65 =%26lt; 0 or x =%26lt; 65 (Ypu have written 25 but I think it should be 65)



Area of the lawn excluding the flower bed, A2 = Area of the lawn - Area of the flower bed = 4000 - 4x^2 + 260x - 4000 = -4x^2 + 260x





Now for the 3rd part, put A1 = A2

4x^2 - 260x + 4000 = -4x^2 + 260x

8x^2 - 520x + 4000 = 0

Divide the whole equation by 8 -

x^2 - 65x + 500 = 0

Use the quadratic formula to solve for x -

x = [65 +/- sqrt. (4225 - 2000)] / 2 = (65 +/- 47.17) / 2

x = 88.585, 8.915



We reject 88.585 as the width of the grass cannot be greater than the breadth of the lawn.



So

x = 8.915 m



Hope this helps.



your_guide123@yahoo.com
Reply:So the area of the flower bed is (80-2x)(50-2x) [just draw it and see these are the dimensions] FOIL multiply.



Area = 4000 - 260x + 4x^2. (Your note should say 0 is less than x)



Total area is 4000, so area lawn = 260x - 4x^2 (you left out the exponent)



Set these equal to one another and solve for x.



8x^2 - 520x + 4000 = 0. Divide by 8. x^2 - 65x + 500 = 0



Quadratic equation isn't pretty, but the solution is x = about 9
rashes

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