Question

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

Answer 1

The largest possible volume of the box is $500.000 cm^3$

Explanation:

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

A Box is a polyhedron with all its faces in rectangular shape. The surface area of the box is

$(area of base) + 4* (area of one side) = s$

$30,000 cm^2 = s^2 + 4hs$

Then we can solve the equation of h in terms of s

$4hs = 30,000- s^2 \\h= \frac{30,000- s^2}{4s}$

By plugging this into the expression for V , we get

$V = \frac{30,000-s^2}{4s} *s^2\\V= 7500s- \frac{1}{4} s^2$

The volume cannot  be negative, therefore V>0 or V=0

The  derivative is $\frac{dV}{ds} = 7500-\frac{3}{4} s^2$

Then find the critical numbers  by setting it to zero:

$7500-\frac{3}{4} s^2 = 0\\ s^2=10.000\\s =+- 100$

The closed interval method is a way to solve a problem within a specific interval of a function. Now we can use the “closed-interval method” to determine the absolute maximum of V

$V (0) = 0\\V (100) = 7500* 100 -\frac{1}{4} * 100^3\\V (100) = 750000 - 250000\\V (100) = 500000$

Therefore the largest possible volume of the box is $500.000 cm^3$

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Answer 2

Answer:464760cm3

Step-by-step explanation:

First we have to find the area of the box, and since it has a square base, the box is a cube with all sides equal

A cube has six sides, but the top side is open, so this box will have 5 sides

We are given the material in cm2 units which is also the unit for area, so we divide the value by the number of sides to get the area of each of the 5 sides

30000/5 = 6000cm2

So finding the volume of a cube is Length x beadth x height, which have the same values, because all sides in a cube are equal

Area = 6000cm2

To find the Length( same as height) we find the root of the Area

Root6000 = 77.46

Length = height = 77.46cm

Volume = Length x Breadth x height = Area x height = 6000 x 77.46 = 464760m3