6. Two players on a basketball team share playing time. Mark's expected number of pointsscored is SM(x) = 30x – 5x² if he plays the fraction x e [0,1] of the game. John's expectednumber is S(x) = 25x – 5x points if he plays the fraction x of the game.(i) Suppose that Mark plays the entire game. How many points does he expect to score? Howmany points would John expect to score if he played the fraction x = A of the game? Find theaverage scoring rate, S(A)/A, as A approaches zero.(ii) What fraction of the time should Mark play in order to maximize the total points scored byMark and John? Find the expected number of points scored by each player per unit time.

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Asked Jan 31, 2020
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6. Two players on a basketball team share playing time. Mark's expected number of points
scored is SM(x) = 30x – 5x² if he plays the fraction x e [0,1] of the game. John's expected
number is S(x) = 25x – 5x points if he plays the fraction x of the game.
(i) Suppose that Mark plays the entire game. How many points does he expect to score? How
many points would John expect to score if he played the fraction x = A of the game? Find the
average scoring rate, S(A)/A, as A approaches zero.
(ii) What fraction of the time should Mark play in order to maximize the total points scored by
Mark and John? Find the expected number of points scored by each player per unit time.
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6. Two players on a basketball team share playing time. Mark's expected number of points scored is SM(x) = 30x – 5x² if he plays the fraction x e [0,1] of the game. John's expected number is S(x) = 25x – 5x points if he plays the fraction x of the game. (i) Suppose that Mark plays the entire game. How many points does he expect to score? How many points would John expect to score if he played the fraction x = A of the game? Find the average scoring rate, S(A)/A, as A approaches zero. (ii) What fraction of the time should Mark play in order to maximize the total points scored by Mark and John? Find the expected number of points scored by each player per unit time.

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Expert Answer

Step 1

(i) If Mark plays the entire game, then x = 1. Hence, the expected score is:

Economics homework question answer, step 1, image 1

If John played the fraction x = Δ, his expected score is:

Economics homework question answer, step 1, image 2

Step 2

Average scoring rate can be calculated as:

Economics homework question answer, step 2, image 1

As Δ approaches to 0, putting Δ = 0 in above equation,

Economics homework question answer, step 2, image 2

Hence, ASR is 25.

Step 3

(ii) If Mark spends x fraction of time, John would spend (1 – x) fraction of time.

Total score can be calculated as:

Economics homework question answer, step 3, image 1

For finding optimal score, differentiating using first order condition:

Economics homework question answer, step 3, image 2

Hence, Mark’s optimum play time is 75% of time.

So, John plays for 25% of time.

...

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