4. the students in mr. michael’s art class are decorating a booth for harvest day. they have blue cloth that is 60 inches long, gold cloth that is 48 inches long, and white cloth that is 72 inches long. they want to cut all of the cloth into pieces of equal length. a. what is the greatest possible length of the pieces without having any cloth left over? explain your reasoning. b. how many pieces of each color cloth will they have?

Length of blue cloth is 60 inches , length of gold cloth is 48 inches and length of white cloth is 72 inches.

a.  

As, the length of all pieces are equal, so for getting the greatest possible length of the pieces, we need to find GCF(greatest common factor) of 60, 48 and 72.

First we will factor out all three numbers completely......

The common factors are: 2, 2 and 3

Thus the GCF

So, the greatest possible length of the pieces without having any cloth left over will be 12 inches.

b.  

For finding the number of pieces for each color cloth, we will just divide the length of each cloth by 12. So...

Number of pieces for blue cloth

Number of pieces for gold cloth and

Number of pieces for white cloth

1 number

if we use a graphing calculator, there is only one such number that satisfies the conclusion of the mean value theorem, that is, a number such that

where the right-hand side is the average rate of change of f on the interval [0.5,0.75], which is the slope of the secant line.

in the attached graph, the black line is the secant line which has its slope as the average rate of change of f on [0.5,0.75], while the orange line is the only tangent line to the graph of f in the interval that satisfies the mvt conclusion.