Will You Help Me Please? Calculus I Work!!!!?
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A landscape architect plans to enclose a 1600 square foot rectangular region in a botanical garden. She will use shrubs costing $25 per foot along three sides and fencing costing $10 per foot along the north side. Find the minimum total cost. Give your answer correct to the nearest dollar.
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A=L*W=1600 => W=1600/L
(This is because we need one variable for when we differentiate.)
Cost= 2*25*W+25*L+10*L
= 50*W + 35*L
= 50*1600/L+35*L
differentiate to get:
Cost’ = -80000*L^(-2) + 35
We will check when Cost’ = 0 because these points may give max/min.
35 = 80000*L^(-2)
35*L^2 = 80000
L^2 = 2285.71
L= 47.8 (since L>0)
I’m not sure about the endpoints but I’ll say it’s 1and 1600 (because it can’t be zero) Maybe you’re supposed to find the limit as L approaches 0. We had a question like this before at school and our teacher said not to worry about that so I don’t really know. Sub in L= 1 L=47.8 and L=1600, into the equation for Cost (not Cost’ ).
The answer I got is the min is $3347. Hopefully I haven’t made any mistakes.
****** OOOPS *******
Brittle is right, I made a mistake. I assumed three sides at $25 and one at $10, that’s not a rectangle.
*************************
$2190.89
rectangle sides x and y
Cost = C
C = 3*25*x + y*10
C = 75x + 10y
But we know that xy = 1600, therefore, y=1600/x
C(x) = 75x + 16000/x
C’(x) = 75 – 16000/x^2
Minimum C when C’ = 0, therefore
0 = 75 – 16000/x^2
16000/x^2 = 75
x^2 = 16000/75
x = sqrt( 16000/75) = 14.6
Then C= 75 * 14.6+ 16000/14.6 = 2190.89
$34000
400 sq ft (one side) + 400 sq ft (other side)+ 400 sq ft (other side) = 1200 sq foot =east west and south side
1200 x $25 = $30000
400 (north side) x $10 = $4000
$30000 + $4000= $34000
$34000
400×400=1600
1200×25=30000
400×10=4000+30000=34000