Ms. tang has asked keith and judy to paint small hearts on their classroom walls. keith has painted 33 hearts and is painting at the rate of 10 hearts per hours. judy has painted 25 hearts and is painting at the rate of 14 hearts per hour . in how many hours will they have painted the same numbers of hearts

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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.