3. the volume of an open - top box is modeled by the function\n\n v(x)=x(20 - 2x)(10 - 2x). \n\nanother…

3. the volume of an open - top box is modeled by the function\n\n v(x)=x(20 - 2x)(10 - 2x). \n\nanother function is given by ( y = 100 ). the two graphs are shown below.\n\n(a) what is a reasonable domain for ( x )?\n\n(b) approximately which value of ( x ) gives the box the greatest volume?\n\n(c) what do the points of intersection of these two graphs represent?

3. the volume of an open - top box is modeled by the function\n\n v(x)=x(20 - 2x)(10 - 2x). \n\nanother function is given by ( y = 100 ). the two graphs are shown below.\n\n(a) what is a reasonable domain for ( x )?\n\n(b) approximately which value of ( x ) gives the box the greatest volume?\n\n(c) what do the points of intersection of these two graphs represent?

Answer

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.