Chapter 10

\(\sum_{k=1}^{100} 100\)

Tossing dice Three dice are tossed. (a) Find the probability that all dice show the same number of dots. (b) Find the probability that the numbers of dots on the dice are all different. (c) Work parts (a) and (b) for \(n\) dice.

Find the rational number represented by the repeating decimal. $$3.2 \overline{394}$$

Without expanding completely, find the indicated term(s) in the expansion of the expression. $$ (\sqrt{c}+\sqrt{d})^{8} ; \text { term that contains } c^{2} $$

(a) What happens if a calculator is used to find \(P(150,50)\) ? Explain. (b) Approximate \(r\) if \(P(150,50)=10^{r}\) by using the following formula from advanced mathematics: $$ \log n ! \approx \frac{n \ln n-n}{\ln 10} $$

(a) Find the number of negative integers greater than \(-500\) that are divisible by 33 . (b) Find their sum.

\(\sum_{k=1}^{1000} 5\)

Find the rational number represented by the repeating decimal. $$1 . \overline{6124}$$

Without expanding completely, find the indicated term(s) in the expansion of the expression. $$ \left(3 x-\frac{1}{4 x}\right)^{6} ; \quad \text { term that does not contain } x $$

Log pile A pile of logs has 24 logs in the bottom layer, 23 in the second layer, 22 in the third, and so on. The top layer contains 10 logs. Find the total number of logs in the pile.

\(\sum_{k=253}^{571} \frac{1}{3}\) $$

Find the rational number represented by the repeating decimal. $$123.61 \overline{83}$$

Without expanding completely, find the indicated term(s) in the expansion of the expression. $$ \left(x y-3 y^{-3}\right)^{8} ; \quad \text { term that does not contain } y $$

Stadium seating The first ten rows of seating in a certain section of a stadium have 30 seats, 32 seats, 34 seats, and so on. The eleventh through the twentieth rows each contain 50 seats. Find the total number of seats in the section.

$\sum_{k=137}^{428} 2.1 $$

Find the geometric mean of 12 and 48 .

Smoking deaths In an average year during 1995-1999, smoking caused 442,398 deaths in the United States. Of these deaths, cardiovascular disease accounted for 148,605 , cancer for 155,761 , and respiratory diseases such as emphysema for 98,007 . (a) Find the probability that a smoking-related death was the result of either cardiovascular disease or cancer. (b) Determine the probability that a smoking-related death was not the result of respiratory diseases.

Approximate (1.2) \({ }^{10}\) by using the first three terms in the expansion of \((1+0.2)^{10}\), and compare your answer with that obtained using a calculator.

Find the geometric mean of 20 and 25 .

Starting work times In a survey about what time people go to work, it was found that \(8.2\) million people go to work between midnight and 6 A.M., \(60.4\) million between 6 A.M. and 9 A.M., and \(18.3\) million between 9 A.M. and midnight. (a) Find the probability that a person goes to work between 6 A.M. and midnight. (b) Determine the probability that a person goes to work between midnight and \(6 \mathrm{~A} . \mathrm{M}\).

Approximate \((0.9)^{4}\) by using the first three terms in the expansion of \((1-0.1)^{4}\), and compare your answer with that obtained using a calculator.

A bicycle rider coasts downhill, traveling 4 feet the first second. In each succeeding second, the rider travels 5 feet farther than in the preceding second. If the rider reaches the bottom of the hill in 11 seconds, find the total distance traveled.

Insert two geometric means between 4 and 500 .

Arsenic exposure and cancer In a certain county, \(2 \%\) of the people have cancer. Of those with cancer, \(70 \%\) have been exposed to high levels of arsenic. Of those without cancer, \(10 \%\) have been exposed. What percentage of the people who have been exposed to high levels of arsenic have cancer? (Hint: Use a tree diagram.)

Simplify the expression using the binomial theorem. $$ \frac{(x+h)^{4}-x^{4}}{h} $$

A contest will have five cash prizes totaling $$\$ 5000$$, and there will be a $$\$ 100$$ difference between successive prizes. Find the first prize.

Insert three geometric means between 2 and 512 .

Computers and defective chips A computer manufacturer buys \(30 \%\) of its chips from supplier A and the rest from supplier B. Two percent of the chips from supplier A are defective, as are \(4 \%\) of the chips from supplier B. Approximately what percentage of the defective chips are from supplier \(B\) ?

Simplify the expression using the binomial theorem. $$ \frac{(x+h)^{5}-x^{5}}{h} $$

A company is to distribute $$\$ 46,000$$ in bonuses to its top ten salespeople. The tenth salesperson on the list will receive $$\$ 1000$$, and the difference in bonus money between successively ranked salespeople is to be constant. Find the bonus for each salesperson.

Using a vacuum pump A vacuum pump removes one-half of the air in a container with each stroke. After 10 strokes, what percentage of the original amount of air remains in the container?

Assuming air resistance is negligible, a small object that is dropped from a hot air balloon falls 16 feet during the first second, 48 feet during the second second, 80 feet during the third second, 112 feet during the fourth second, and so on. Find an expression for the distance the object falls in \(n\) seconds.

Consider the sequence defined recursively by \(a_{1}=5\), \(a_{k+1}=\sqrt{a_{k}}\) for \(k \geq 1\). Describe what happens to the terms of the sequence as \(k\) increases.

Roulette In the American version of roulette, a ball is spun around a wheel and has an equal chance of landing in any one of 38 slots numbered \(0,00,1,2, \ldots, 36\). Shown in the figure is a standard betting layout for roulette, where the color of the oval corresponds to the color of the slot on the wheel. Find the probability that the ball lands (a) in a black slot (b) in a black slot twice in succession

Calculating depreciation The yearly depreciation of a certain machine is \(25 \%\) of its value at the beginning of the year. If the original cost of the machine is \(\$ 20,000\), what is its value after 6 years?

$$ \text { Show that }\left(\begin{array}{l} n \\ 0 \end{array}\right)=\left(\begin{array}{l} n \\ n \end{array}\right) \text { for } n \geq 0 \text {. } $$

If \(f\) is a linear function, show that the sequence with \(n\)th term \(a_{n}=f(n)\) is an arithmetic sequence.

Approximations to \(\pi\) may be obtained from the sequence $$ x_{1}=3, \quad x_{k+1}=x_{k}-\tan x_{k} . $$ Use the TAN key for tan. (a) Find the first five terms of this sequence. (b) What happens to the terms of the sequence when \(x_{1}=6 ?\)

Growth of bacteria A certain culture initially contains 10,000 bacteria and increases by \(20 \%\) every hour. (a) Find a formula for the number \(N(t)\) of bacteria present after \(t\) hours. (b) How many bacteria are in the culture at the end of 10 hours?

The sequence defined recursively by \(x_{k+1}=x_{k} /\left(1+x_{k}\right)\) occurs in genetics in the study of the elimination of a deficient gene from a population. Show that the seauence whose \(n\)th term is \(1 / x\). is arithmetic.

Bode's sequence Bode's sequence, defined by $$ a_{1}=0.4, \quad a_{k}=0.1\left(3 \cdot 2^{k-2}+4\right) \text { for } k \geq 2, $$ can be used to approximate distances of planets from the sun. These distances are measured in astronomical units, with \(1 \mathrm{AU}=93,000,000 \mathrm{mi}\). For example, the third term corresponds to Earth and the fifth term to the minor planet Ceres. Approximate the first five terms of the sequence.

Interest on savings An amount of money \(P\) is deposited in a savings account that pays interest at a rate of \(r\) percent per year compounded quarterly; the principal and accumulated interest are left in the account. Find a formula for the total amount in the account after \(n\) years.

Growth of bacteria The number of bacteria in a certain culture is initially 500 , and the culture doubles in size every day. (a) Find the number of bacteria present after one day, two days, and three days. (b) Find a formula for the number of bacteria present after \(n\) days.

Quality control In a quality control procedure to test for defective light bulbs, two light bulbs are randomly selected from a large sample without replacement. If either light bulb is defective, the entire lot is rejected. Suppose a sample of 200 light bulbs contains 5 defective light bulbs. Find the probability that the sample will be rejected. (Hint: First calculate the probability that neither bulb is defective.)

Rebounding ball A rubber ball is dropped from a height of 60 feet. If it rebounds approximately two-thirds the distance after each fall, use an infinite geometric series to approximate the total distance the ball travels.

Exer. 55-56: Depreciation methods are sometimes used by businesses and individuals to estimate the value of an asset over a life span of \(n\) years. In the sum-of-year's-digits method, for each year \(k=1,2,3, \ldots, n\), the value of an asset is decreased by the fraction \(A_{k}=\frac{n-k+1}{T_{n}}\) of its initial cost, where \(T_{n}=1+2+3+\cdots+n\). (a) If \(n=8\), find \(A_{1}, A_{2}, A_{3}, \ldots, A_{8}\). (b) Show that the sequence in (a) is arithmetic, and find \(S_{8}\). (c) If the initial value of an asset is \(\$ 1000\), how much has been depreciated after 4 years?

The Fibonacci sequence The Fibonacci sequence is defined recursively by $$ a_{1}=1, \quad a_{2}=1, \quad a_{k+1}=a_{k}+a_{k-1} \text { for } k \geq 2 . $$ (a) Find the first ten terms of the sequence. (b) The terms of the sequence \(r_{k}=a_{k+1} / a_{k}\) give progressively better approximations to \(\tau\), the golden ratio. Approximate the first ten terms of this sequence.

Life expectancy A man is 54 years old and a woman is 34 years old. The probability that the man will be alive in 10 years is \(0.74\), whereas the probability that the woman will be alive 10 years from now is \(0.94\). Assume that their life expectancies are unrelated. (a) Find the probability that they will both be alive 10 years from now. (b) Determine the probability that neither one will be alive 10 years from now. (c) Determine the probability that at least one of the two will be alive 10 years from now.

Motion of a pendulum The bob of a pendulum swings through an arc 24 centimeters long on its first swing. If each successive swing is approximately five-sixths the length of the preceding swing, use an infinite geometric series to approximate the total distance the bob travels.

Shooting craps In the game of craps, there are two ways a player can win a pass line bet. The player wins immediately if two dice are rolled and their sum is 7 or 11. If their sum is \(4,5,6,8,9\), or 10 , the player can still win a pass line bet if this same number (called the point) is rolled again before a 7 is rolled. Find the probability that the player wins (a) a pass line bet on the first roll (b) a pass line bet with a 4 on the first roll (c) on any pass line bet