Problem: A sheet metal worker is planning to make an open-top box by cutting equal squares (x-in. by x-in.) from the corners of a 10-in. by 14-in. piece of copper. A second box is to be made in the same manner from an 8-in. by 10-in. piece of aluminum, but its height is to be one-half that of the first box.
(1)Find a polynomial function for the volume of each box.
1st box dimensions: (10-2x) by (14-2x) by x
Vol = x*(10-2x)*(14-2x)
FOIL
f(x) = x(140 - 48x + 4x^2)
f(x) = 4x^3 - 48x^2 + 140x
2nd box cut out will be .5x in. by .5x in.,(height 1/2 the 1st box), therefore:
the dimensions will be:
(10-x) by (8-x) by .5x
Vol = .5x*(10-x)*(8-x)
FOIL
f(x) = .5x(80 - 18x + x^2)
f(x) = .5x^3 - 9x^2 + 40x
(2) Find the values of x for which the copper box is 72 cubic in. larger than the aluminum box.
Big box vol - small box vol = 72 cu in
(4x^3 - 48x^2 + 140x) - (.5x^3 - 9x^2 + 40x) = 72
Remove brackets
4x^3 - 48x^2 + 140x - .5x^3 + 9x^2 - 40x = 72
Group like terms
4x^3 - .5x^3 - 48x^2 + 9x^2 + 140x - 40x - 72 = 0
3.5x^3 - 39x^2 + 100x - 72 = 0
Solve this by graphing y = 3.5x^3 - 39x^2 + 100x – 72
The integer solution x = 2; & x ~ 1.3; the third solution x ~ 7.8 isn't reasonable
(3)Write the difference between the two volumes (d) as a function of x.
d(x) = (4x^3 - 48x^2 + 140x) - (.5x^3 - 9x^2 + 40x)
which is d(x) = 3.5x^3 - 39x^2 + 100x
(4)Find d for x=1.5
d(x) = 3.5(1.5^3) - 39(1.5^2) + 100(1.5)
d(x) = 3.5(3.375) - 39(2.25) + 150
d(x) = 11.8125 - 87.75 + 150
d(x) = 74.0625
(5)For what value of x is the difference between the two volumes the largest?
I need someone to help me understand how to get the solution for this one step by step. For some reason I wasn't able to figure it out.
Letty
(1)Find a polynomial function for the volume of each box.
1st box dimensions: (10-2x) by (14-2x) by x
Vol = x*(10-2x)*(14-2x)
FOIL
f(x) = x(140 - 48x + 4x^2)
f(x) = 4x^3 - 48x^2 + 140x
2nd box cut out will be .5x in. by .5x in.,(height 1/2 the 1st box), therefore:
the dimensions will be:
(10-x) by (8-x) by .5x
Vol = .5x*(10-x)*(8-x)
FOIL
f(x) = .5x(80 - 18x + x^2)
f(x) = .5x^3 - 9x^2 + 40x
(2) Find the values of x for which the copper box is 72 cubic in. larger than the aluminum box.
Big box vol - small box vol = 72 cu in
(4x^3 - 48x^2 + 140x) - (.5x^3 - 9x^2 + 40x) = 72
Remove brackets
4x^3 - 48x^2 + 140x - .5x^3 + 9x^2 - 40x = 72
Group like terms
4x^3 - .5x^3 - 48x^2 + 9x^2 + 140x - 40x - 72 = 0
3.5x^3 - 39x^2 + 100x - 72 = 0
Solve this by graphing y = 3.5x^3 - 39x^2 + 100x – 72
The integer solution x = 2; & x ~ 1.3; the third solution x ~ 7.8 isn't reasonable
(3)Write the difference between the two volumes (d) as a function of x.
d(x) = (4x^3 - 48x^2 + 140x) - (.5x^3 - 9x^2 + 40x)
which is d(x) = 3.5x^3 - 39x^2 + 100x
(4)Find d for x=1.5
d(x) = 3.5(1.5^3) - 39(1.5^2) + 100(1.5)
d(x) = 3.5(3.375) - 39(2.25) + 150
d(x) = 11.8125 - 87.75 + 150
d(x) = 74.0625
(5)For what value of x is the difference between the two volumes the largest?
I need someone to help me understand how to get the solution for this one step by step. For some reason I wasn't able to figure it out.
Letty
