A box with a square base and open top must have a volume of 78732 cm3. We wish t
ID: 2886536 • Letter: A
Question
A box with a square base and open top must have a volume of 78732 cm3. 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 r, the length of one side of the square base. [Hint: use the volume formula to express the height of the box in terms of a.] Simplify your formula as much as possible. A(z) Preview Next, find the derivative, A'() Preview Now, calculate when the derivative equals zero, that is when A'()0. Hint multiply both sides by A'() 0when We next have to make sure that this value of z gives a minimum value for the surface area. Let's use the second derivative test. Find AT(a). Preview A(z) Evaluate A(r) at the r-value you gave above. NOTE Since your last answer is positive, this means that the graph of A(a) is concave up around that value, so the zero of A (e) must indicate a local minimum Get helpExplanation / Answer
Let the height be y and the side length be x .
Volume = x2 y
78732 = x2y
y = 78732 / x2
And the surface area is
A = x2+ 4 xy
A (x ) = x2 + 4x(78732/x2)
A(x) = x2+ (314928 / x)
A '(x ) = 2x -(314928 / x2)
A'(x) = 0
2x - ( 314928 / x2) = 0
x3= 314928/2
x3 = 157464
x = 1574641/3 = 54
Therefore A '(x) = 0 when x = 54 cm
A "(x) =2 +(629856 / x3)
At x= 54, A "(x) = 2 +(629856/543 ) = 6 cm2
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