HOW DO YOU DO THIS??

Question

If 10,800 cm² of material is available to make a box with a square base and an open top, find the largest possible volume (in cm³) of the box.

 ______ cm³

Answer 1

It is not clear what method what method you have studied? However, if you use Lagrange Multipliers, the answer doesn't require a lot of complex algebra to solve.

Let's assume that the dimension of the bottom is "a" and the dimension of the height is "h". Let the surface area = S and the volume = V.

You then are attempting to maximize the volume. Then,

Then V = a²h and S = a² + 4ah and in the language of Lagrange Multipliers, we have

V = a²h + λ(a² + 4ah - 10,800) Differentiating, we have

dV/da = 2ah + λ(2a + 4h) = 0 dV/dh = a² + λ(4a) = 0 dV/dλ = a² + 4ah - 10,800 = 0

Equation 2 says that λ = -a/4 Then using equation 1 & substituting, we find that a = 2h

Looking at the surface area equation, we find that a = 60 and then h = 30

Now, the question remains,,,,, do these values make the volume a maximum?

I won't solve this for you but notice that the equation for the surface area is a parabola!

Put in a values just on each side of 60 (keep area constant) and you find that it is a maximum!

Answer 2

Well suppose the base is b x b, and the height is h

Area of material used = b² + 4bh = 10800

We want to maximise V = b²h

h = (10800 - b²) / 4b

V = b²h = b(10800-b²) /4

dV/db = 2700 - (3b²/4)

We set this to zero giving, b = 60.

The second derivative is -6b/4, <0 when b=60 showing we have a maximum

So b = 60, h = 30, and the maximum volume is 1800 cm³