A cardboard box has a square base, with each edge of the base having length x inches, as shown in the figure. The total length of all 12 edges of the box is 144 in.

Question

a) Express the volume V of the box as a function of x.

b) What is the domain of V? (Use the fact that length and volume must be positive. Enter your answer using interval notation.)

c) graph

d) find the maximum

Thanks!!!

Answer 1

x = the length the base side

y = the height

12 sides = 2(4x) + 4(y) = 8x + 4y = 144

Divide by 4

Total sides = 2x + y = 36

y = 36 – 2x

The height y = 36 – 2x

Volume (V) = yx²

 Replace y with 36 – 2x

V = (36 – 2x)( x²) = 36 x² – 2x³

V = 36 x² – 2x³

Maximum Volume

d/dx(36 x² – 2x³) = 72x - 6x²

= 6x(12 -x) = 0

x = 0  or  x = 12

The side of the base is

x = 12

The max V = 12.12.12 = 12 = 1,728 in³

Answer 2

Total length of all edges = 144 = 8x + 4y where x=edge of base, y=edge of the height

solve for y in terms of x

4y=144-8x

Volume = Area of base times height = x²y = x²(144-8x)

V(x)=x²(144-8x) = -8x³ +144x² cubic inches If you worked this in feet (converting 12 inches to 1 foot) the equation would be V(x)=x²(3-2x)=-2x²+3x² That would make the remaining calculations simpler, but the following retains the inches dimensions.

If x>0, V>0

Maximum V is when the derivative V'(x)=0 or -24x²+288x = 0 divide by -24x to get:

x-12+ 0, x=12 when x=12, V is at its maximum

V(12)=144(144-8(12))= 144(144-96)=144(48)=6912 cubic inches = 4 cubic feet

The domain of V(x) is (0,6912] (inches, cubic inches] or (0,4] for (feet, cubic feet] 48x12x12=6912

The graph of V = -8x³ + 144x² for x>0 to x=12 (0,12}] starts at the origin and increases to its maximum at 6212. If we use feet instead of inches, the graph starts at 0 and goes up to 4 feet maximum, then decreases, going back to zero when y=0 or y= 0 = 144-8x 8x=144, x=18 at (18,0) or (1.5,0) in feet when V=0.

That is the only relevant part of the graph of the cubic equation. (0,0) to maximum V at (1,4) back to (1.5,0)

when x or y = 0 V =0 It goes from the origin up to a peak at (1,4) then back to the x axis at (1.5,0)

when x=12 inches = 1 foot, y=144-8x=144-96=48 inches = 4 feet the base is 12x12, the height=48

Maximums volume is 12x12x48 = 6912 cubic inches or 1x1x4 = 4 cubic feet