world problem

Question

A rectangular piece of cardboard measuring 12 in by 18 in is to be made into a box with an open top by cutting equal sized squares from each corner and folding up the sides.

A.give the restriction of x

B.Deterimine a function V that gives the volume of the box as a function of x

C. for what values of x will the volume be a maximum? What is this maximum volume?

D. for what values of x will the volume be greater than 80 cubic inches?

Answer 1

A.  x (the side of the square cut from the corner)

 

this must be >0 and <6  or 0<x<6  (if it is greater than six then the box will have not width)

 

B.

 

v = l * w * h

 

l = 18 - 2x

w = 12 -2x

h = x

 

V = (18-2x)*(12-2x)*x

V = 4x³ - 60x² + 216x

 

C.

Find the derivative and set it equal to zero to find the maximum

 

v' = 12x² - 120x + 216

 

0 = 12x² - 120x + 216

 

0 = x² -10x + 18

 

Does not factor nicely so use the quadractic formula

 

x = (10 - Sqrt(28)) /2

 

x = 2.354249  (this will be the value for the maximum volume)

 

 

 try x = 2.2 and x = 2.4 and you will see that the maximum is at the value shown above

 

D.

the volume equals 80 when x equals 5  (you can verify by putting 5 into the equation)

 

You should be able to figure out when it is greater than 80

 

Answer 2

A. x must be > 0, <12

 

B. If you cut x inches square from four corners of 12 x 18 inch sheet

 

Length of box = 18-2x inches

Width of box = 12-2x inches

Height of box = x inches

 

Volume V = (18-2x)(12-2x)(x) = 4x³ - 60x² + 216x

 

C. To find max of this function put dV/dx = 0

 

12x² - 120x + 216 = 0

 

Solve this quadratic equation

 

x = (120 + √(14400 - 10368))/24 = (120 + 63.5)/24 = 2.35 or 7.65

 

max volume = 228 cubic inch or - 68 cubic inch. Since volume cant be -ve so x = 2.35 inch for max volume

 

D.  4x³ - 60x² + 216x > 80

 

=> Factorize

=> x(x-9)(x-6) > 80

 

x = 11 inches

 

:)