Question

. Suppose a basketball player, Riley, has an interesting dynamic with the crowd such that the probability of

making a free throw depends only on how loud the crowd is at the moment Riley shoots the ball. Let P(n) be
the probability Riley makes a free throw when the noise of the crowd is n decibels (dB).
(a) Suppose the louder the crowd, the less likely it is that Riley makes a free throw. What would that mean
about P'(n)?
(b) What are the units of P'(n)? What is the meaning of P'(90) = -0.2?
0.2
(c) Suppose P(n) is not simple, but is in fact given by P(n)= (0.002n+1)3 sin(n) + 0.5. Find the function of
P'(n) using calculus, and show your work.
(d) When the crowd is between 90 and 97 decibels, find the greatest probability of Riley making the free
throw, and determine the corresponding noise level of the crowd. You should explain which values you
needed to check to find your answer. You may use a graph to help you do this, but you must explain how
you found your solution using your knowledge of P'(n) and calculus.

Answer 1

The probability of Riley making a free throw decreases as the noise level increases, which implies a negative derivative. P'(n) indicates the change in this probability per decibel. To maximize the probability of a successful free throw within a specific range, one must consider both endpoints and where the derivative is zero or undefined.

When addressing the question about the basketball player Riley and the impact of crowd noise on free throw success, we need to consider several points:

• Probability and its relationship with crowd noise level.
• Understanding the meaning of the derivative of a probability function.
• Calculating the derivative using calculus.
• Finding the maximum probability within a given noise range.

Part (a)

If louder crowd noise leads to a lower probability of making a free throw, this suggests that P'(n) would be negative; as crowd noise (n) increases, the probability of Riley making the free throw (P(n)) decreases.

Part (b)

The units of P'(n) are the change in probability per decibel (probability/decibel). A value of P'(90) = -0.2 means that for each additional decibel of noise at 90 dB, the probability of making the free throw decreases by 0.2.

Part (c)

Using calculus, we can find P'(n) for the probability function P(n)= (0.002n+1)^3 sin(n) + 0.5. The differentiation would be performed by applying the product rule and the chain rule to the composite and trigonometric functions within P(n).

Part (d)

To determine the greatest probability of making a free throw between 90 and 97 decibels, one must evaluate P(n) and its derivative P'(n) at critical points within the interval, including endpoints and points where P'(n) equals zero or is undefined.