91Ó°ÊÓ

Lauren Johnson signs a 10 -yr contract as a loan officer for a bank, at a salary of \(\$ 84,000\) per year. After 7 yr, the bank offers her early retirement. What is the least amount the bank should offer Lauren, given that the going interest rate is \(7.4 \%,\) compounded continuously?

Total cost from marginal cost. A company determines that its marginal cost, in dollars, for producing \(x\) units of a product is given by $$C^{\prime}(x)=3600 x^{-1.8}, \quad \text { where } x \geq 1$$ Suppose that it were possible for the company to make infinitely many units of this product. What would the total cost be?

In a psychology experiment, the time \(t\) in seconds, that it takes a rat to learn its way through a maze is an exponentially distributed random variable with the probability density function $$f(t)=0.02 e^{-0.02 t}, \quad 0 \leq t<\infty$$. Find the probability that a rat will learn its way through a maze in 150 sec or less.

Assume that birthdays are uniformly distributed throughout the year and that February 29 is omitted from consideration. The probability of at least one shared birthday (month and day only) among \(n\) randomly chosen people is $$P(n)=1-\left(\frac{364 \cdot 363 \cdot 362 \cdots \cdots(366-n)}{365^{n-1}}\right)$$For example, in a group of 10 people the probability of at least one shared birthday is$$P(10)=1-\left(\frac{364 \cdot 363 \cdot 362 \cdots \cdots 356}{365^{9}}\right)=0.117$$ a) Verify the claim made at the start of this section that the probability that two people in a group of 30 have the same birthday is about \(70 \%\) b) How many people are required to make the probability that two of them share a birthday greater than \(50 \% ?\) c) How many people are in your calculus class? What is the probability of at least one shared birthday among you and your classmates? Test your calculation experimentally.

Plutonium- 239 has a decay rate of approximately \(0.003 \%\) per year. Suppose that plutonium 239 is released into the atmosphere for 20 yr at a constant rate of 1 lb per year. How much plutonium- 239 will be present in the atmosphere after 20 yr?

In a normal distribution with \(\mu=-15\) and \(\sigma=0.4\) find the \(x\) -value that corresponds to the a) 46 th percentile b) 92 nd percentile

Explain the uses of integration in the study of probability.