A box with a square base and open top must have a volume of 256000 cm^3. We wish
ID: 2893029 • Letter: A
Question
A box with a square base and open top must have a volume of 256000 cm^3. We wish to find the dimensions of the box that minimize the amount of material used. First, find a formula for the surface area of the box in terms of only x, the length of one side of the square base. Simplify your formula as much as possible. A(x) = Next, find the derivative, A'(x) A' (x) Now, calculate when the derivative equals zero, that is, when A'(x) = 0. A' (x) = 0 when x = We next have to make sure that this value of x gives a minimum value for the surface area. Let's use the second derivative test. Find A"(x) A"(x) = Evaluate A"(x) at the x-value you gave above.Explanation / Answer
Volume of box is given by:
V = x*x*h = 256000 cm^3
x^2*h = 256000
h = 256000/x^2
Now surface area of a open box is given by:
A(x) = x*x + 2*(x*h +x*h)
A(x) = x^2 + 4*x*h
A(x) = x^2 + 1024000/x
d(x^n)/dx = n*x^(n - 1)
Using above formula
A'(x) = 2*x - 1024000/x^2
when A'(x) = 0
2*x - 1024000/x^2 = 0
2x = 1024000/x^2
x^3 = 512000
x = (512000)^(1/3) = 80 cm
Now
A''(x) = 2*1 + 2*1024000/x^3
A''(x) = 2 + 2048000/x^3
At x = 80 cm
A''(80) = 2 + 2048000/80^3 = 6
Since A"(x) > 0, So at this x amount of material used will be minimum.
h = 256000/80^2 = 40 cm
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