Math homework help please- volume?

Hi Troublemaker

SA =2a^2 + 4ah =253.5

where 'a' is the side of base and 'h' the height

4ah = 253.5 - 2a^2

h = (253.5 -2a^2)/4a

Volume V = a^2h

V = a^2 ((253.5 -2a^2)/4a

V = a(253.5 -2a^2)/4

for Max Volume

find V' (i differentiate w.r.t a) and equate to 0 to get value of a

V' = (253.5 - 6a^2)/4 = 0

6a^2 = 253.5

a^2 = 253.5/6

a = 6.5 cms

and

h = (253.5 -2a^2)/4a

h = (253.5 - 84.5)/4*6.25

h = 169/25

h = 6.76 cms

Largest volume cuboid will be of size

6.5cm x 6.5cm x 6.76cm and

V = 287.3 cubic cms

Let the length of one side of the base be x. Then the area of that base is x*x, or x^2. So is the area of the "top" (the side opposite the base).

Now you have four other sides to your solid. Each of them has x as one dimension and y as the other. So the area of each of these sides is x*y, and you have 4 of them, so you have 4xy.

So the total surface area of your solid is 2x^2 + 4xy, or 2(x^2 + 2xy)

We know that the surface area is 253.5

So 2(x^2 + 2xy) = 253.5

Divide each side by 2

x^2 + 2xy = 126.75

You need to find the values of x and y that maximize the volume (which is length * width * height, or x*x*y, or (x^2)*y)