This document discusses solving rational equations by finding the least common denominator and making like denominators equal. It contains an example of solving the rational equation (x+2)/(x-3) = (x-1)/x and mentions that questions 1, 6 and 7 involve solving rational equations.
A small community has a workforce of 1800 people, with 1680 employed and 120 unemployed. The document discusses using transition matrices to model how the number of employed and unemployed individuals in the community may change over time. Specifically, it notes that in one year, 10% of currently employed workers will lose their jobs, and 60% of currently unemployed individuals will find jobs. The transition matrix can then be raised to successive powers and multiplied by the initial state vector to model the employment status after 1, 2, 5, and 50 years.
This document discusses solving rational equations by finding the least common denominator and making like denominators equal. It contains an example of solving the rational equation (x+2)/(x-3) = (x-1)/x and mentions that questions 1, 6 and 7 involve solving rational equations.
A small community has a workforce of 1800 people, with 1680 employed and 120 unemployed. The document discusses using transition matrices to model how the number of employed and unemployed individuals in the community may change over time. Specifically, it notes that in one year, 10% of currently employed workers will lose their jobs, and 60% of currently unemployed individuals will find jobs. The transition matrix can then be raised to successive powers and multiplied by the initial state vector to model the employment status after 1, 2, 5, and 50 years.
A rectangular piece of cardboard had its length and width determined by cutting out 3 cm by 3 cm squares from each corner and folding up the sides to form an open box with a volume of 450 cm3. An equation was needed to calculate the original length and width of the cardboard based on this information.
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The document summarizes work done over two summers to restore a truck cab. It describes picking up the cab in Detroit to replace the original, taking out the old cab, sandblasting and priming the cab and frame, doing bodywork like installing a new floor, and finally lifting the restored cab onto the frame.
Dave needs a new computer for university costing $3,294 before taxes. Financing is available at 18.99% compounded monthly over 48, 36, or 24 months. The monthly payments and total paid are calculated for each term. Daniel can also lease the same computer for $106.92/month over 48 months, $132.32/month over 36 months, or $182.88/month over 24 months. The document then provides information about leasing, including what costs make up monthly lease payments and details that would be in a lease agreement.
The document discusses various interest rates and compounding periods for investments, savings accounts, and credit cards, and uses calculations to determine future values, effective interest rates, and the best investment options based on interest rates and compounding frequencies. Formulas like the Rule of 72 are presented for estimating doubling times given interest rates.
The monthly payments and total amount paid will increase as the loan term decreases from 48 months to 36 months to 24 months due to the interest being applied over a shorter period of time.
The document defines key terms and concepts related to circles and their equations. It explains that a circle consists of points equidistant from a fixed center point, and defines the radius as the distance from the center to any point on the circle. It provides the standard equation for a circle with center at the origin, and notes that the standard form includes variables h and k to indicate the center coordinates and r for the radius. It also describes a second form for the circle equation that can be converted to standard form by completing the square.
This document discusses personal finance concepts like the time value of money and compound interest. It provides the basic formulas for calculating future value (FV), present value (PV), interest rate (I%), number of periods (N), principal (P), payments (PMT), periodic interest rate (r), number of compounding periods per year (n), and time (t). The document works through examples of using these formulas to calculate things like how much money you will have after investing a principal amount over a period of time at a given interest rate.
The document contains 6 math problems: 1) Find the value of k that gives equal roots of the quadratic equation f(x)=x^2 + 4x + k. 2) Determine the nature of the roots of the quadratic equation 13x^2 - 15x = 4. 3) Solve the equation x - 3 = 2 for x. 4) Find the quadratic equation with integer coefficients given the roots 3 + 9i and 10. 5) Solve the equation x + 8 = 10x - 81 for x. 6) Solve the equation x/3 = 6 - 2x for x.
This document discusses using the shell method to calculate volumes of solids generated by revolving regions between functions around axes. It provides examples of revolving the function f(x)=x^2 around the x-axis and y-axis, and revolving the region between f(x)=0.5x^2-2x+4 and g(x)=4+4x-x^2 around both the x-axis and y-axis. Instructions are given to use the shell method to find each volume.
Absolute value refers to the distance from zero on the number line. There are two values, -2 and 2, that have an absolute value of 2. Absolute value can never be negative because distance cannot be negative. The document provides examples of using absolute value to solve equations.
This document discusses solving rational equations by finding the least common denominator, combining like terms, and then solving the resulting equation for the variable. It contains an exercise with questions 1, 6, and 7 about solving rational equations.
This document provides instructions and problems for solving vector problems by drawing scale diagrams, adding vectors using the triangle method, and calculating distances and directions from starting points. Specifically, it asks the reader to: 1) draw scale diagrams of vectors for a person walking 13 blocks E15°S and a boat headed 300° at 45 km/h; 2) add the vectors using the triangle method; and 3) solve problems involving distances and directions for a man walking in different directions and a jogger moving north and east over time.
An equation containing a radical is called a radical equation. This document refers to exercises 18 questions 1 through 5 and also questions 8 and 9 which involve solving or working with radical equations. The goal is to extract the key essential information about radical equations from the given document in 3 sentences or less.
The document provides information about vector addition and trigonometric equations. It discusses drawing scale diagrams to represent vectors and their directions and magnitudes. Methods for adding vectors are described, including the triangle method used when vectors are tip-to-tail and the parallelogram method used when vectors are tail-to-tail. An example of each method is worked out to find the resultant vector.
The document contains instructions to find the volumes of solids generated by revolving regions bounded by graphs about axes. It gives the volume as 183.981 when revolving the region between the graphs y = 2x + 4 and y = ex about the x-axis. It also gives the volumes as 7/15 when revolving the region between y = x^2 + 1 and y = x + 1 about the x-axis and 4/5 when revolving the same region about the line y = -1.
The document provides information about vector addition and trigonometric equations. It discusses drawing scale diagrams to represent vectors and their directions and magnitudes. Specific examples are given of adding vectors using the triangle method when vectors are tip-to-tail and the parallelogram method when they are tail-to-tail. Measurements from the scale diagrams along with a protractor can be used to find the resultant vector.
The document discusses vectors and provides examples of identifying quantities as scalar or vector. It also discusses four notations for writing vectors using arrows, bearings, angle-direction-direction, and angle-direction of direction. Examples are given to demonstrate each notation. The document also discusses stating the direction of vectors in five ways and using diagrams of parallelograms to name vectors that are equal, opposite, collinear, or parallel but not equal to other vectors in the diagram.
The document describes a problem where a rectangular piece of cardboard has a length longer than its width. Square pieces are cut from the corners and the sides are folded up to form a box with a volume of 450 cm3. The original length of the cardboard was 16 cm and the width was 11 cm.
This document discusses methods for finding the roots of quadratic equations. It introduces the discriminant formula to determine the type of roots, and explains how to use the quadratic formula to find the exact values of the roots. It also shows how to write a quadratic equation given the sum and product of its roots, or given two integer roots.
The document discusses calculating the volume of solids of revolution using integrals. It provides the formula for finding the volume of a solid rotated about the x-axis between x=a and x=b using a cross-sectional area function A(x). It then works through an example of finding the volume of a right circular cone of height 4 and base radius 1, and confirms the result matches the standard volume formula for a cone.
