How would you go about solving this problem below?

Question

An open top rectangular box has a volume of 729/2 cubic feet. If the dimensions are calculated so as to minimize the outside surface area (bottom and four sides), then the height of the box will be how many feet?.

possible answers....

1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
or is it none of these

Answer 1

Hi,

The height is 4.5 inches

First the rectangle with the largest area is always a square, so the base will be a square with sides of "x".

To find that, let x = length of a side of the base and h = height of the box. Then its volume is found by V = hx². This would become 729/2 = hx². Solving this for h gives h = 729/(2x²).

To express the total surface area, we want the area of the base plus 4 times the area of a side. This would be

Y = x² + 4(hx)

Now substitute 729/(2x²) for h.

Y = x² + 4(729/(2x²)* x)

This simplifies to

Y = x² + 1458/x

If you graph this and use the minimum command on the TI84, you will find the minimum surface area is when x = 9. To find h, use the formula 729/2 = hx² and substitute 9 for x.

729/2 = h9²
729/2 = h81
4.5 = h, so the height that gives the minimum surface area is 4.5 ft.

I hope that helps!! :-)

Answer 2

All the above problems involve 2 unknowns and one equation. So you cannot get a numerical solution. Best you could get is f(x) or f(y) meaning one variable in terms of the other one. However these are non linear equations so a close form solution is not guaranteed. Lets do the last one: log(y*y + x) = y ----> invert the log on both sides y * y + x = 10 ^ y x = 10 ^ y - y ^ 2 Another approach is to numerically graph y vs. x. At least gives you an idea what f(x) looks like.

Answer 3

V=x*y*z =729/2
S= xy +2(xz+yz)
Minimum S with restraint xyz-729/2=0
Using lagrangian multipliers
F(,x,y,z,k)= xy+2(xz+yz)+k(xyz-729/2)
Fx= y+2z+kyz=0
Fy= x+2z +kxz=0
Fz= 2(x+y) +k xy=0
yx+2zx+k*729/2=0
xy+2zy+k*729/2=0
2z(x-y)=0 so x=y and 4x+kx^2=0 so 4+kx=0 x=-4/k
z(2+kx)=-x and as kx=-4 2z=x and z=x/2
finally
x^3=729 and x =9 and z= 4.5 feet