If the arc length shown in blue is 21.3 inches, then 0 to the nearest hundredth of a
radian is

Step-by-step explanation:

By definition, the Arc lenght can be calculated with the following formula:

Where "s" is the Arc lenght, "r" is the radius and is the central angle measured in radians.

From that equation you can solve for dividing both sides of the equation by the radius "r", then:

According to the information given in the exercise:

And you can identify in the figure that the radius of the circle is:

Therefore, you can substitute values into the equation:

Finally, evaluating, you get the following result:

Rounded to the nearest hundredth of a radian:

f(x) has value 150 at x= 0

and 0 at x= 3

so d(f(x))/dt = slope = -150/3 = -50

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.