Question

An open box is to be made from a flat piece of material 7 inches long and 5 inches wide by cutting equal squares of length xx from the corners and folding up the sides.

Write the volume VV of the box as a function of xx. Leave it as a product of factors, do not multiply out the factors.
V(x)=

If we write the domain of V(x)V(x) as an open interval in the form (a,b)(a,b),

then what is aa?
a=
and what is bb?
b=

* i need to know V(x), a, b

with explaination please..

Answer 1

(a)V(x)=x(7-2x)(5-2x)

(b)The domain of V(x) is (0,2.5) where a=0, b=2.5

Explanation:

The piece of material is 7 inches long and 5 inches wide.

When you cut a square of length x from each corner

The dimensions of the box to be formed becomes:

Height=x inch

Length=7-2x inch

Width=5-2x inch

(You have 2x because you cut from both the left and right hand side.)

(a)The volume of the box

Volume of a Cuboid=lwh

V(x)=x(7-2x)(5-2x)

(b)The domain of V(x) is/are the range of values at which V(x) is defined.

Since no dimension can be negative,

x>0, 7-2x>0, 5-2x>0

x>0, 7>2x, 5>2x

x>0, x<3.5, x<2.5

The values at which V(x) is defined are at:

x>0 and x<2.5

Combining:

0<x<2.5

The domain of V(x) is (0,2.5) where a=0, b=2.5