Question

example 1
such that an expression for v(x) that gives the volume of an open - top box made by cutting out corners of a square piece of x length and folding up the sides. the dimensions of the paper are 5 inches by 7 inches. now graph v(x) as shown.

1. volume cannot be negative. true / false
2. the cutouts cannot be more than 2.5 in long.
3. a more appropriate domain for v(x) would be 0 < x < 2.5
4. based on the reasonable domain, which value of x would give a volume of 13 cubic inches? x = 1

example 3
ava wants to make an open - top box by cutting out corners of a piece of cardboard and folding up the sides. the cardboard is 10 centimeters by 14 centimeters. the volume v(x) in cubic centimeters of the box is a function of the side lengths x in centimeters of the square cutouts.

1. write the expressions for v(x).

length: ( - 2x)
width: (14 - ) v(x)=
height: ( )

1. if x (cutout) is 3 centimeters, what is the volume?

v(3)=( - 2(3))(14 - )( )
v(3)=______ cm³
example 2
let f(x) be an equation for the volume of an open - top box made by cutting out corners of a square piece of x length. we want to put v(x) in desmos.

1. what is a reasonable domain? 0 < x < 4
2. based on your domain, if the square cutout is x = 2

example 4
paris wants to make an open - top box by cutting out corners of a piece of paper and folding up the sides. the paper is 8 inches by 10 inches. the volume v(x) in cubic inches of the box is a function of the side lengths x in inches of the square cutouts.

1. write the expressions for v(x).
2. find v(4)

Explanation:

Step1: Determine length, width and height for volume formula

The length of the cardboard is 10 cm, after cutting out squares of side - length \(x\) from each corner, the length of the box \(l = 10 - 2x\). The width of the cardboard is 14 cm, so the width of the box \(w=14 - 2x\), and the height of the box \(h = x\). The volume formula for a rectangular - box is \(V=l\times w\times h\), so \(V(x)=(10 - 2x)(14 - 2x)x\).

Step2: Calculate \(V(3)\)

Substitute \(x = 3\) into the volume formula \(V(x)\).
\[

$$\begin{align*} V(3)&=(10-2\times3)(14 - 2\times3)\times3\\ &=(10 - 6)(14 - 6)\times3\\ &=4\times8\times3\\ &=96 \end{align*}$$

\]

Answer:

1. \(V(x)=(10 - 2x)(14 - 2x)x\)
2. \(V(3)=(10 - 6)(14 - 6)\times3 = 96\)