beach shelter canvas image
tahoe.girl
There's a problem on my calculus homework I don't know how to do and would really appreciate some help!
"A canvas shelter for the beach ahs two square sides, a rectangular back, and a rectangular top. Suppose that 96 square feet of canvas are to be used. Find the dimensions of the shelter for which the volume is maximized."
I realize that people usually aren't that into doing other's homework for them, but if someone would at least point me in the right direction or tell me what I need to do I'd really appreciate it.
Answer
Let x be the length and y be the width. and height
Volume= xy^2
Surface area = 2xy+2y^2=96
2xy =96-2y^2
x=(96-2y^2) / 2y
x= 48/y -y
Volume = (48/y-y)y^2
Volume = 48y -y^3
V=48y-y^3
differentiate with respect to y
dV/dy = 48-3y^2=0
3y^2=48
y^2-16
y=4 --- height and width
x = 48 / y - y = 48 / 4 - 4 = 8 -- length
The dimensions are length 8, width 4, height 4
d^2V/dy^2 =-6y = -6(4)=-24 < 0, so V is maximized.
Let x be the length and y be the width. and height
Volume= xy^2
Surface area = 2xy+2y^2=96
2xy =96-2y^2
x=(96-2y^2) / 2y
x= 48/y -y
Volume = (48/y-y)y^2
Volume = 48y -y^3
V=48y-y^3
differentiate with respect to y
dV/dy = 48-3y^2=0
3y^2=48
y^2-16
y=4 --- height and width
x = 48 / y - y = 48 / 4 - 4 = 8 -- length
The dimensions are length 8, width 4, height 4
d^2V/dy^2 =-6y = -6(4)=-24 < 0, so V is maximized.
I really need help only on this optimization problem. Plz help if you can. Thnx!?
sdt3
The canvas wind shelter is to be constructed for use on Padre Island beaches. It is to have a back, 2 square ft of canvas are available, find the length of the shelter for which the space inside is maximized assuming all the canvas is used.
A) length=15 ft
B) length=17ft
C) length=18ft
D) length=16ft
E) none of these
Answer
I think you need to supply more information.
Is the shelter a triangle (like an upside-down V) with one end solid?
Also, 2 square feet total fabric doesn't make sense. 200 maybe?
The very first thing you need to do is compute the area of the cross-section as a function of the length of the side and the height. The largest cross-section comes from a tent with a side angle of 45 degrees (optimise area as a function of angle by taking the derivative).
At this point, you will develop a formula for the volume of the shelter as a function of length, given the constant amount of fabric available using the two formulas:
V = area*length (area is the cross section)
fabric amount = side length*2*length + end panel
I think you need to supply more information.
Is the shelter a triangle (like an upside-down V) with one end solid?
Also, 2 square feet total fabric doesn't make sense. 200 maybe?
The very first thing you need to do is compute the area of the cross-section as a function of the length of the side and the height. The largest cross-section comes from a tent with a side angle of 45 degrees (optimise area as a function of angle by taking the derivative).
At this point, you will develop a formula for the volume of the shelter as a function of length, given the constant amount of fabric available using the two formulas:
V = area*length (area is the cross section)
fabric amount = side length*2*length + end panel
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