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(Refer to Exercise 87.) Use the following system of equations to determine the forces or weights W1 and W2 exerted on each rafter for the truss shown in the figure.
Exercise 87.
The simplest type of roof truss is a triangle. The truss shown in the figure is used to frame roofs of small buildings. If a 100-pound force is applied at the peak of the truss, then the forces or weights W1 and W2 exerted parallel to each rafter of the truss are determined by the following linear system of equations.
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Refer to Exercise 97. If the bacteria are not cultured in a medium with sufficient nutrients, competition will ensue and growth will slow. According to Verhulst’s model, the number of bacteria Nj at time 40(j – 1) in minutes can be determined by the sequence
where K is a constant and j ≥ 1.
(a) If N1 = 230 and K = 5000, make a table of Nj for j = 1, 2, 3, . . . , 20. Round values in the table to the nearest integer.
(b) Graph the sequence Nj for j = 1, 2, 3, . . . , 20. Use the window [0, 20] by [0, 6000].
(c) Describe the growth of these bacteria when there are limited nutrients.
(d) Make a conjecture about why K is called the saturation constant. Test the conjecture by changing the value of K in the given formula.
Exercise 97.
(a) Write Nj+1 in terms of Nj for j ≥ 1.
(b) Determine the number of bacteria after 2 hr if N1 = 230.
(c) Graph the sequence Nj for j = 1, 2, 3, . . . , 7, where N1 = 230. Use the window [0, 10] by [0, 15,000].
(d) Describe the growth of these bacteria when there are unlimited nutrients.
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If certain bacteria are cultured in a medium with sufficient nutrients, they will double in size and then divide every 40 minutes. Let N1 be the initial number of bacteria cells, N2 the number after 40 minutes, N3 the number after 80 minutes, and Nj the number after 40(j – 1) minutes.
(a) Write Nj+1 in terms of Nj for j ≥ 1.
(b) Determine the number of bacteria after 2 hr if N1 = 230.
(c) Graph the sequence Nj for j = 1, 2, 3, . . . , 7, where N1 = 230. Use the window [0, 10] by [0, 15,000].
(d) Describe the growth of these bacteria when there are unlimited nutrients.
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One of the most famous sequences in mathematics is the Fibonacci sequence, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, . . . . (Also see Exercise 33.) Male honeybees hatch from eggs that have not been fertilized, so a male bee has only one parent, a female. On the other hand, female honeybees hatch from fertilized eggs, so a female has two parents, one male and one female. The number of ancestors in consecutive generations of bees follows the Fibonacci sequence. Draw a tree showing the number of ancestors of a male bee in each generation following the description given above.
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Write out in full and verify the statements S1, S2, S3, S4, and S5 for the following. Then use mathematical induction to prove that each statement is true for every positive integer n.
2 + 4 + 6 + · · · + 2n = n(n + 1)
Let Sn represent the given statement, and use mathematical induction to prove that Sn is true for every positive integer n. Follow these steps. (a) Verify S1.
(b) Write Sk .
(c) Write Sk+1. (d) Assume that Sk is true and use algebra to change Sk to Sk+1. (e) Write a conclusion based on Steps (a) – (d).
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The factorial of a positive integer n can be computed as a product.
n! = 1 · 2 · 3 · g · n
Calculators and computers can evaluate factorials very quickly. Before the days of modern technology, mathematicians developed Stirling’s formula for approximating large factorials. The formula involves the irrational numbers p and e.
n! ≈ √2πn · nn · e-n
As an example, the exact value of 5! is 120, and Stirling’s formula gives the approximation as 118.019168 with a graphing calculator. This is “off” by less than 2, an error of only 1.65%.
Repeat Exercises 59 and 60 for n = 13. What seems to happen as n gets larger?
Exercises 59.
Use a calculator to find the exact value of 10! and its approximation, using Stirling’s formula.
Exercises 60.
Subtract the smaller value from the larger value in Exercise 59. Divide it by 10! and convert to a percent. What is the percent error to three decimal places?
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Use the fundamental principle of counting or permutations to solve the problem.
For many years, the state of California used 3 letters followed by 3 digits on its automobile license plates.
(a) How many different license plates are possible with this arrangement?
(b) When the state ran out of new plates, the order was reversed to 3 digits followed by 3 letters. How many additional plates were then possible?
(c) When the plates described in part (b) were also used up, the state then issued plates with 1 letter followed by 3 digits and then 3 letters. How many plates does this scheme provide?
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What will happen when an infectious disease is introduced into a family? Suppose a family has I infected members and S members who are not infected but are susceptible to contracting the disease. The probability P of exactly k people not contracting the disease during a 1-week period can be calculated by the formula
where q = (1 – p)I, and p is the probability that a susceptible person contracts the disease from an infected person. For example, if p = 0.5, then there is a 50% chance that a susceptible person exposed to 1 infected person for 1 week will contract the disease. Give all answers to the nearest thousandth.
(a) Compute the probability P of 3 family members not becoming infected within 1 week if there are currently 2 infected and 4 susceptible members. Assume that p = 0.1. To use the formula, first determine the values of k, I, S, and q.
(b) A highly infectious disease can have p = 0.5. Repeat part (a) with this value of p.
(c) Determine the probability that everyone will become sick in a large family if, initially, I = 1, S = 9, and p = 0.5.
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