Question

the volume of an open - top box is found 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. noah graphs v(x) as shown.

1. volume cannot be negative. can / can not
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 15 cubic inches? x = 1

example 3
mai 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: (10 - 2x)
width: (14 - 2x) v(x)=(10 - 2x)(14 - 2x)x
height: (x)

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

v(3)=( - 2(3))(14 - )( )
v(3)=______ cm³

Explanation:

Step1: Recall volume formula

The volume of a rectangular - box is \(V = l\times w\times h\). For the open - top box made by cutting out squares of side length \(x\) from a \(10\times14\) cardboard, the length \(l=(10 - 2x)\), the width \(w=(14 - 2x)\), and the height \(h = x\). So \(V(x)=(10 - 2x)(14 - 2x)x\).

Step2: Substitute \(x = 3\) into the volume formula

When \(x = 3\), we have \(V(3)=(10-2\times3)(14 - 2\times3)\times3\). First, calculate \(10-2\times3=10 - 6 = 4\), and \(14-2\times3=14 - 6 = 8\). Then \(V(3)=4\times8\times3\).

Step3: Calculate the product

\(4\times8\times3=96\).

Answer:

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

\(V(3)=96\ cm^{3}\)