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Question

If 30,000cm2 of material is available to make a box with a square base and an open top, find the largest possible volume of the box.


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Solution

Step 1: Calculation of the relation between height and base.

Suppose the length of one side of the base of the box is xcm and its height is hcm.

The volume of the box is hx2 and the surface area is 4hx+x2.

Equate 4hx+x2 with 30,000cm2 and simplify.

4hx+x2=300004hx=30000-x2h=7500x-x4

Step 2: Calculation of volume in terms of base.

Substitute the expression of h in the expression of volume and simplify.

v=hx2v=7500x-x4x2v=7500x-14x3

Step 3: Calculation of critical points.

Differentiate vx with respect to x.

v'(x)=7500-34x2

Solve v'(x)=0 for x.

v'(x)=07500-34x2=034x2=7500x2=43×7500x2=10,000x=100

Step 4: Calculation of the height:

The box has the largest possible volume for x=100cm.

Calculate the value of h for x=100.

h=7500100-1004h=75-25h=50

Step 5: Calculation of volume.

Calculate the largest possible volume for h=50,x=100

v=hx2v=501002v=500,000

Hence, the largest possible volume is 500,000cm3.


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