The determinant of a 3 × 3 matrix A is defined as follows.

The determinant of a 3 × 3 matrix can also be found using the method of “diagonals.”

Step 1 Rewrite columns 1 and 2 of matrix A to the right of matrix A.

Step 2 Identify the diagonals d₁ through d₆ and multiply their elements.

Step 3 Find the sum of the products from d₁, d₂, and d₃.

Step 4 Subtract the sum of the products from d₄, d₅, and d₆ from that sum:

(d₁ + d) + d₃) – (d₄ + d₅ + d₆).

Verify that this method produces the same results as the previous method given.

(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.

Solve the problem.

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 N_j 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 N_j for j = 1, 2, 3, . . . , 20. Round values in the table to the nearest integer.

(b) Graph the sequence N_j 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 N_j+1 in terms of N_j for j ≥ 1.

(b) Determine the number of bacteria after 2 hr if N₁ = 230.

(c) Graph the sequence N_j 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.

Solve the problem.

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 N_j the number after 40(j – 1) minutes. 

(a) Write N_j+1 in terms of N_j for j ≥ 1.

(b) Determine the number of bacteria after 2 hr if N₁ = 230.

(c) Graph the sequence N_j 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.

Solve the problem.

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.

Exercise 33 

Let Sn represent the given statement, and use mathematical induction to prove that S_n is true for every positive integer n. Follow these steps.

(a) Verify S₁. 

(b) Write S_k. 

(c) Write S_k+1.

(d) Assume that Sk is true and use algebra to change S_k to S_k+1.

(e) Write a conclusion based on Steps (a) – (d).

See the text that illustrates the principle of mathematical induction using an infinite ladder.

2 + 4 + 8 + · · · + 2ⁿ = 2ⁿ⁺¹ – 2

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 S_n represent the given statement, and use mathematical induction to prove that S_n is true for every positive integer n. Follow these steps.
(a) Verify S₁. 

(b) Write S_k . 

(c) Write _Sk+1.
(d) Assume that S_k is true and use algebra to change S_k to S_k+1.
(e) Write a conclusion based on Steps (a) – (d).

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?

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?

Solve the problem.

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.