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