An open box is to be made from a rectangular sheet of metal that measures 10 by 8 inches, by cutting out squares of the same size from each corner and bending up the sides. Note The picture:
A) Express the volume of the box as a function of x Simplify.
B) Determine the domain of x f rom physical considerations.
C) use the rational zero theorm and your answer from part b to find any rational value of x that will give the box a volume of 24 cubic inches.
D) Find any value(s) of x that give(s) the box a desired volume of 24 cubic inches.
An open box is to be made from a rectangular sheet of metal that measures 10 by 8 inches, by cutting out squares of the same size from each corner and bending up the sides. Note The picture: Express the volume of the box as a function of x Simplify. Determine the domain of x from physical considerations. use the rational zero theorem and your answer from part b to find any rational value of x that will give the box a volume of 24 cubic inches. Find any value(s) of x that give(s) the box a desired volume of 24 cubic inches.
Solution
A)
V(x) = x(8-2x)(10-2x) = 4x(4-x)(5-x)
B)
x > 0
4-x > 0 -> x < 4
5-x > 0 -> x < 5
So the domain of x is: (0 , 4)
C)
V(x) = 24
-> 4x(4-x)(5-x) = 24
x(4-x)(5-x)= 6
-> x^3 - 9x^2 + 20x - 6 = 0
q|1 -> q = 1
p|(-6) -> p = 2 or 3
-> x = p/q -> x = 2 , -2 or x = 3 , -3
but from (b) : 0<x<4 -> x = 2 or 3
by testing the values:
x = 3
D)
x^3 - 9x^2 + 20x - 6 = (x-3)(x^2 - 6x + 2) = 0
-> roots are:
x = 3
x = 3 - sqrt(7)
x = 3 + sqrt(7)
.
An open box is to be made from a rectangular sheet of metal that measu.docx
1. An open box is to be made from a rectangular sheet of metal that measures 10 by 8 inches, by
cutting out squares of the same size from each corner and bending up the sides. Note The picture:
A) Express the volume of the box as a function of x Simplify.
B) Determine the domain of x f rom physical considerations.
C) use the rational zero theorm and your answer from part b to find any rational value of x that
will give the box a volume of 24 cubic inches.
D) Find any value(s) of x that give(s) the box a desired volume of 24 cubic inches.
An open box is to be made from a rectangular sheet of metal that measures 10 by 8 inches, by
cutting out squares of the same size from each corner and bending up the sides. Note The picture:
Express the volume of the box as a function of x Simplify. Determine the domain of x from
physical considerations. use the rational zero theorem and your answer from part b to find any
rational value of x that will give the box a volume of 24 cubic inches. Find any value(s) of x that
give(s) the box a desired volume of 24 cubic inches.
Solution
A)
V(x) = x(8-2x)(10-2x) = 4x(4-x)(5-x)
B)
x > 0
2. 4-x > 0 -> x < 4
5-x > 0 -> x < 5
So the domain of x is: (0 , 4)
C)
V(x) = 24
-> 4x(4-x)(5-x) = 24
x(4-x)(5-x)= 6
-> x^3 - 9x^2 + 20x - 6 = 0
q|1 -> q = 1
p|(-6) -> p = 2 or 3
-> x = p/q -> x = 2 , -2 or x = 3 , -3
but from (b) : 0<x<4 -> x = 2 or 3
by testing the values:
x = 3
D)
x^3 - 9x^2 + 20x - 6 = (x-3)(x^2 - 6x + 2) = 0
-> roots are:
x = 3
x = 3 - sqrt(7)
x = 3 + sqrt(7)
