Question

1. the volume of an open - top box is modeled by the function

v(x)=x(20 - 2x)(10 - 2x).

another function is given by ( y = 100 ). the two graphs are shown below.

(a) what is a reasonable domain for ( x )?

(b) approximately which value of ( x ) gives the box the greatest volume?

(c) what do the points of intersection of these two graphs represent?

Explanation:

(a) Domain of \(x\)

Step1: Consider the physical meaning

The length, width and height of the box must be non - negative.
For \(20 - 2x>0\), we have \(x < 10\). For \(10 - 2x>0\), we have \(x<5\). Also, \(x>0\) (since \(x\) represents a dimension).

Step2: Determine the domain

Combining these inequalities \(0 < x<5\).

(b) Value of \(x\) for maximum volume

Step1: Analyze the graph

Looking at the graph of \(y = V(x)\), the vertex (maximum point) of the parabola - like part (for \(0 < x<5\)) occurs approximately at \(x = 2\).

(c) Intersection of the two graphs

Step1: Interpret the intersection

The points of intersection of \(y = V(x)\) and \(y = 100\) represent the values of \(x\) for which the volume of the open - top box is \(100\) cubic inches.

Answer:

(a) \(0 < x<5\)
(b) \(x\approx2\)
(c) The values of \(x\) for which the volume of the open - top box is \(100\) cubic inches.