Please help with this question!?

Question

A rectangular rose garden will be surrounded on three sides by a brick wall and by a fence on the fourth side. The area of the garden will be 1000 m^2. The cost of the brick wall is $192/m. The cost of the fencing is $48/m. Find the dimensions of the garden so that the cost of the materials is as small as possible.

Answer 1

Hello,

Let L= the length
Let W = the width now we have the area = 1000 which is LW so LW=1000 and L = 1000/W

TWO BRICK SIDES ARE WIDTH AND ONE BRICK SIDE IS LENGTH SO WE HAVE 192(2W + L) AND THE OTHER LENGTH SIDE IS FENCE SO ARE TOTAL COST IS

TC = 192(2W+L) + 48L NOW SUBSTITUTE 1000W FOR L GIVING US

TC=192(2W + 1000/W) + 48 ( 1000/W)

WE CAN DIVIDE THROUGH BY 48 SINCE THE DERIVATIVE IS GOING TO BE SET TO ZERO SO IT GIVES US

TC = 4(2W + 1000/W) + 1000/W NOW FINDING A COMMON DENOMINATOR OF W WE GET

TC = 4(2W^2 + 1000)/W + 1000/W OR (8W^2 + 4000 +1000)/W OR (8W^2 + 5000)/W MOW TAKE THE DERIVATIVE AND SET IT EQUAL TO ZERO GIVING US

0 = [16W^2*W - (8W^2 + 5000)]/W^2 OR

0 = (16W^2 -8W^2 -5000) / W^2 RP (8W^2 -5000)/W^2 AND THEN MULTIPLYING BOTH SIDES BY W^2 GETS RID OF THE DENOMINATOR SO WE HAVE

8W^2 -5000 = 0 OR 8W^2 = 5000 OR W^2 = 625 SO
W = 25M

AND L = 1000/25 OR 40M

Hope This Helps!!

Answer 2

I see this as needing to keep the perimeter to a minimum.

This works out to about 31 1/2 feet per side (actually 31 feet, 7 1/2 inches to the nearest half inch).

So you will need 94 1/2 feet of brick wall and 31 1/2 feet of fence. 94 1/2 (192) = 18,144 and 31 1/2 (48) = 1,512 . This works out to $19,626, so budget it at $20,000 to make the small adjustments needed to get it exactly 1000 m^2.