Farmer jim is building a new chicken coop in the shape of a rectangular prism. he needs the height of the coop to be 2 feet shorter than the width, and needs the length of the coop to be 10 feet longer than the width. create an equation to represent the volume of the chicken coop, y, in terms of width, x, in feet.

Let the width of the coop be x. Then the height is x - 2, and the length is x + 10. Then the volume of the coop (a rectangular prism) is:

y = x(x-2)(x+10) = x^3 + 8x^2 - 20x

Based on the given format, this means that:

a = 1

b = 8

c = -20

d = 0

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.