A manufacturing plant earned \$80$80dollar sign, 80 per man-hour of labor when it opened. Each year, the plant earns an additional 5\%5%5, percent per man-hour. Write a function that gives the amount A(t)A(t)A, left parenthesis, t, right parenthesis that the plant earns per man-hour ttt years after it opens.

Step-by-step explanation:

The manufacturing plant earned $80 per man-hour of labor when it opened

The plant earns an additional 5% for every additional man-hour t.

This can be modeled using the function:

where Initial Amount Earned, P=80

Rate of Increase, r=5%=0.05

Therefore, the function that models the amount earned at any time t is:

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.