# Exam P Practice Problem 107 – wait time at a busy restaurant

Both 107-A and 107-B use the following probability density function.

$\displaystyle f(x) = \left\{ \begin{array}{ll} \displaystyle \frac{1}{450} \ (30-x) &\ \ \ \ \ \ 0 < x < 30 \\ \text{ } & \text{ } \\ \displaystyle 0 &\ \ \ \ \ \ \text{otherwise} \\ \end{array} \right.$

Problem 107-A

The wait time (in minutes) for a table at a busy restaurant on the weekend is distributed according to the density function $f(x)$ given above.

A customer plans to dine in this restaurant on two different weekends.

Determine the expected value of the longest wait of these two visits to the restaurant.

$\displaystyle \bold ( \bold A \bold ) \ \ \ \ \ \ \ \ \ \ \ \ \bold 1 \bold 0 \bold . \bold 0$

$\displaystyle \bold ( \bold B \bold ) \ \ \ \ \ \ \ \ \ \ \ \ \bold 1 \bold 1 \bold . \bold 0$

$\displaystyle \bold ( \bold C \bold ) \ \ \ \ \ \ \ \ \ \ \ \ \bold 1 \bold 2 \bold . \bold 8$

$\displaystyle \bold ( \bold D \bold ) \ \ \ \ \ \ \ \ \ \ \ \ \bold 1 \bold 3 \bold . \bold 5$

$\displaystyle \bold ( \bold E \bold ) \ \ \ \ \ \ \ \ \ \ \ \ \bold 1 \bold 4 \bold . \bold 0$

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Problem 107-B

The wait time (in minutes) for a table at a busy restaurant on the weekend is distributed according to the density function $f(x)$ given above.

A customer plans to dine in this restaurant on two different weekends.

Determine the expected value of the shortest wait of these two visits to the restaurant.

$\displaystyle \bold ( \bold A \bold ) \ \ \ \ \ \ \ \ \ \ \ \ \bold 4 \bold . \bold 5$

$\displaystyle \bold ( \bold B \bold ) \ \ \ \ \ \ \ \ \ \ \ \ \bold 6 \bold . \bold 0$

$\displaystyle \bold ( \bold C \bold ) \ \ \ \ \ \ \ \ \ \ \ \ \bold 7 \bold . \bold 0$

$\displaystyle \bold ( \bold D \bold ) \ \ \ \ \ \ \ \ \ \ \ \ \bold 8 \bold . \bold 6$

$\displaystyle \bold ( \bold E \bold ) \ \ \ \ \ \ \ \ \ \ \ \ \bold 1 \bold 0 \bold . \bold 0$

$\text{ }$

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