Len spent $18 to bowl 4 games. this week he spent $27 to bowl 6 games. if each of these bowling games costs the same amount of money, what is the constant of proportionality between the money spent and the number of games played?

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.