10.5 combinations
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There are 4 volunteers (Jamal, Kay, Li, Max) to be painters and 3 spots are available. The order of the painters does not matter, so this is a combination problem. There are 4C3 = 4!/(4-3)!3! = 4 possible combinations of choosing 3 painters out of the 4 volunteers.
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This document discusses trigonometric functions like sine, cosine, and tangent. It defines the trig functions using ratios of sides of a right triangle. It then provides examples of evaluating the six trig functions for various angles in different right triangles, as well as examples of using trig functions to solve for unknown side lengths of right triangles.
Lesson 67
The document discusses antilogarithms and how they are the inverse operation of logarithms. It states that the antilogarithm of a number is equal to the exponent when that number is the logarithm of the base. It then provides examples of finding the antilogarithm of 4.63 for different bases, including bases 2, 10, and the mathematical constant e, which is approximately equal to 2.71828.
Lesson 66
This document discusses formulas for solving systems of equations, calculating phase shifts in sinusoidal functions, and accounting for changes in period. It provides the general formulas for solving a system of two equations with two unknowns. It also explains that if the period of a sinusoid is not 2π or 360°, the coefficient must be calculated by dividing 360° or 2π by the actual period. Finally, it notes that if given a sinusoid with an amplitude, center, coefficient, phase angle, and period, both the sine and cosine equations can be written.
Lesson 65
This document provides 3 problems involving radicals in trigonometric equations, graphs of logarithms. The first two problems ask the reader to solve trigonometric equations containing radicals given certain conditions. The third problem asks the reader to sketch the graph of the logarithmic function y = log3.5 x.
11.6 geometric sequences
This document discusses geometric sequences and provides examples of calculating terms of geometric sequences given initial terms and common ratios. It also covers finding geometric means between two terms by taking the square root of the product of the terms. Examples include finding the 6th term of the sequence 3, -12, 48, -192, finding the 7th term of a sequence where the 2nd term is 24 and 5th term is 3, and finding geometric means between 4 and 9 by taking the square root of their product.
To know test 15
This document contains a list of mathematical concepts and problems including equations of circles, logarithms, trigonometric functions, permutations, systems of inequalities, areas, factoring complex numbers, and properties of circles, lines, and graphs. It appears to be a study guide or checklist for reviewing multiple topics in precalculus or calculus.
Lesson 63
This document contains 3 problems involving completing the square to write equations of circles in standard form and graphing the circles. Problem 1 has the equation x^2 + y^2 - 6x + 8y + 21 = 0 and asks to complete the square and graph the circle. Problem 2 gives the equation x^2 + y^2 + 4x - 6y + 6 = 0 and asks to do the same. Problem 3 gives the equation 2x^2 + 2y^2 - x + y - 3 = 0 and asks to complete the square and describe the resulting circle.
11.5 arithmetic sequences
The document discusses arithmetic sequences and how to find specific terms (tn) using the common difference (d) between terms. It explains that the general formula for an arithmetic sequence is tn = a1 + (n-1)d, where a1 is the first term and d is the common difference. It then provides examples of using this formula to find the nth term for given arithmetic sequences and finds the arithmetic mean between two numbers.
11.3 sequences
The document contains instructions for finding terms of different sequences. It defines a sequence where the first term is 4 and each subsequent term is half of the previous term. It also defines a Fibonacci sequence where the first two terms are 1 and each term is the sum of the previous two terms. It asks the reader to find the first five terms of these sequences.
Lesson 62
This document discusses solving linear equations for variables x and y. It also presents a real world example where the total cost of producing a product varied linearly with the number produced. When 100 were produced the cost was $700, and when 400 were produced the cost was $2200. An equation is given to model this linear relationship between cost (C) and number produced (N) as C = mN + b.
Lesson 61
This document discusses analyzing data through measures like the mean, standard deviation, and box and whisker plots. It provides examples of calculating the mean and standard deviation for different data sets. It also discusses how the standard deviation indicates how data points are distributed around the mean. Finally, it includes an example of using a graphing calculator to analyze a data set and draw a box and whisker plot.
11.2 multiplying matrices
The document discusses matrix multiplication. It explains that two matrices can be multiplied if the number of columns in the first matrix is equal to the number of rows in the second matrix. It provides an example of multiplying two matrices by multiplying the elements in each row of the first matrix with each column of the second matrix and adding the products. It then provides examples of multiplying specific matrices and determining if the products are defined based on the dimensions of the matrices.
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10.5 combinations
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- 2. 10.5 Combinations.notebook March 28, 2013 Suppose that 3 painters are chosen from a group of 5 student volunteers. How many choices are there for the 3 painters?
- 3. 10.5 Combinations.notebook March 28, 2013 Tell whether the question can be answered using a permutation or a combination. Then find the answer. For summer reading, you are asked to read 2 books from a list of 6 books. How many different pairs of books can you choose to read? Twelve students enter a talent show. Awards are given for first place through fifth place. In how many ways can the students finish first through fifth?
- 4. 10.5 Combinations.notebook March 28, 2013 Consider a standard deck of 52 playing cards. The order in which the cards are dealt for a "hand" does not matter. How many different 5card hands are possible? How many different 5 card hands have all 5 cards of a single suit?
- 5. 10.5 Combinations.notebook March 28, 2013 How many ways can you choose 3 dogs and 2 cats from 5 dogs and 6 cats?