Third flashcards
Formula for Volume of a Triangular Prism
The formula for the volume of a triangular prism is A=12bhl .
Formula for Volume of a Sphere
The formula for the volume of a sphere is V=43πr3 .
Formula for Volume of a Cylinder
The formula for volume of a cylinder is V=πr2h .
Formula for Volume of a Cone
the formula for the volume of a cone is V=13πr2h .
Formula for Area of a Circle
The formula for area of a circle is A=πr2 .
Formula for Surface Area of a Rectangular Prism
The formula for surface area of a rectangular prism is SA=2lh+2lw+2hw .
Surface Area
The surface area of a 3D figure is the sum of the areas of all of its surfaces. Unlike volume, surface area is measured in square units, not cubic units.
Formula for Surface Area of a Cube
The formula for surface area of a cube is SA=6s2 , where s is the side length.
Formula for Surface Area of a Triangular Prism
The formula for surface area of a triangular prism is SA=bh+lb+2lc , where c is the diagonal side of the triangle.
Formula for Surface Area of a Cylinder
The formula for the surface area of a cylinder is: SA=2πr2+2πrh .
Formula for Surface Area of a Sphere
The formula for the surface area of a sphere is SA=4πr2 .
Variables
Variables are symbols, usually letters, that stand for unknown numbers. The most common letter used is x . Algebraic expressions are made up of terms that each contain one or more variables raised to a power. Variables with no power shown are considered to be raised to the power of 1; in terms that show no variable at all, called constant terms, the variable is considered raised to the power of 0. Terms may be multiplied by constants known as coefficients, which are part of the same term, and separate terms are joined by addition or subtraction.
Relations of Ordered Pairs
A relation is any set of ordered pairs relating one variable to another, such as some of the linear and proportional relationships we looked at earlier. A relation pairs numbers in one set, called the domain, with numbers in another set, called the range. A relation is often presented as a set of ordered pairs (x,y) , where x is an input or independent variable and y is an output or dependent variable.
Proportions
A proportion is an equation that says two ratios are equal. Proportions often come up when discussing constant rates, such as the cost of renting equipment or the speed at which a car drives on the highway.
For example, let’s say it costs $75 to rent a bounce house for 3 hours. You know that the rate of the cost per hour is constant, and you want to know how much it will cost to rent it for 2 hours.
We don’t know the cost for 2 hours, so we’ll use a variable x to represent it. We do know that the ratios $75 to 3 hours and $x to 2 hours are in proportion, so they must be equal. That means we can write them as a proportion:
753=x2
When you have two fractions set equal to each other like this, you can cross-multiply, which means you set the product of the numerator of the first fraction and the denominator of the second equal to the product of the denominator of the first fraction and the numerator of the second. Then you should be able to easily solve for x :
75(2)150x===3(x)3x50
It will cost $50 to rent the bounce house for 2 hours.
Slope-Intercept Form
An efficient way to graph an equation is to rewrite it in slope-intercept form and then inspect it. The slope-intercept form of an equation is y=mx+b , where m is the slope of the line, b is the y-intercept, and both m and b are constants. The slope of the line is the rate at which the y-value changes relative to the x-value, and the y-intercept is the point at which the line crosses the y-axis (that is, the y-value when x=0 ).
The Pythagorean Theorem
The Pythagorean theorem states that the square of the length of the hypotenuse of a right triangle equals the sum of the squares of the lengths of the other two sides.
Midpoint Formula
The midpoint formula involves taking the average of the two known x-values to find the x-value that is halfway between them and then taking the average of the two known y-values to find the y-value that is halfway between them. Remember that the average of a set of values is the sum of all the values divided by the number of values, which in this case is 2.
M=(x1+x22,y1+y22)
Linear Inequality
A linear inequality uses the comparison symbols < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to) to show a relationship between two linear expressions. Solving a linear inequality for a variable gives you a range of possible values for that variable known as the solution set rather than a single value, as when solving a linear equation.
Let’s look at an example:
4x+9<2x+15
We can solve this exactly the same way we would solve a linear equation, performing operations on both sides as needed to isolate x :
4x+92xx«<2x+1563
Our solution means that x can be any value that is less than 3, including negative or noninteger values. However, 3 is not a solution because it is not true that 4(3) + 9 < 2(3) + 15.
Zero Rule of Exponents
The zero rule of exponents says that any base raised to the power of 0 always equals 1. In other words, x0=1 for any value of x .
Product Rule of Logarithms
The product rule of logarithms says that the logarithm of a product is equal to the sum of the logarithms of the product’s factors:
log(xy)=logx+logy
Zero Rule of Logarithms
The zero rule of logarithms says that the log of 1, no matter what the base, is always 0:
logx1=0 , for all positive values of x .
Quotient Rule of Logarithms
The quotient rule of logarithms says that the logarithm of a quotient is equal to the difference of the logarithms of the dividend and divisor:
log(xy)=logx−logy
Power Rule of Logarithms
The power rule of logarithms says that the logarithm of a base raised to a power is equal to the power times the logarithm of the base:
log(xy)=ylogx
Exponential Functions
An exponential function is any function of the form f(x)=abx , where a and b are constants with a≠0,b>0 , and b≠1 . This is not to be confused with a polynomial function, which may have a variable raised to an exponent.
To evaluate an exponential function for a given value of x , you need to substitute your value for x , simplify the exponent if necessary, and then raise the base to the result. For example, let’s say you are given the function f(x)=(34)−6x and are asked to evaluate f(13) :
f(13)====(34)−6(13)(34)−2(43)2169
Simple Interest
The simplest type of interest is called, fittingly enough, simple interest. With simple interest, you pay a percentage of the principal you borrowed for every year that it takes you to pay back the loan. The formula for simple interest is:
I=Prt
In this equation, I is the total interest owed; P is the principal, or amount borrowed; r is the interest rate, which is a percentage of the principal expressed as a decimal; and t is the time in years it takes to pay back the loan.
Interest
Interest is money that a borrower pays a lender for the privilege of borrowing money in addition to paying back the original amount borrowed. For example, you, as a borrower, pay interest when you take out a loan to buy a house or car or to pay for college. When you deposit money into a savings account, the bank, or borrower in this case, pays you interest.
The amount of interest that you pay depends on the amount you borrow, called the principal; the interest rate, which is typically a percentage of the principal collected per year; and the amount of time it takes you to pay the loan back.
Compound Interest
A more common and slightly more complicated way of calculating interest is called compound interest. When an account earns compound interest, the interest is “compounded” a certain number of times per year. That means the interest earned in that period is added to the principal and the interest earned in the following period is a percentage of that new total rather than a percentage of the original principal. The percentage representing the interest rate stays the same, but the amount we’re taking the percentage of gets bigger, so the amount of interest earned each period gets bigger.
The compound interest formula uses an exponent to represent repeated multiplication by the same percentage every year. The compound interest formula finds the future value of the account after a number of years, which is the principal plus the total interest earned rather than just the interest.
When interest is compounded once a year, the formula for the future value of an account is:
FV=P(1+r)t
In this equation, FV is the future value, P is the principal, r is the rate of interest expressed as a decimal, and t is time in years. By adding r to 1, or 100%, we represent the principal increasing by that amount, and by raising it to the power of t , we represent increasing it by the same percentage again every year.
Installment Buying
Sometimes when you can’t afford to pay the full price for an expensive item such as a home, vehicle, or college education, you need to set up an installment plan. An installment plan is an agreement in which the seller agrees to let you spread your payments out over time in exchange for paying a certain amount of interest. Typically, you will pay the seller a set amount every month for a set number of months, with each payment covering both some of the principal and some of the interest.
The formula for calculating your monthly payment on an installment plan is:
P=(rV1−()−n)
In this equation, P is the monthly payment, r is the monthly rate of interest, V is the initial value of the loan or principal, and n is the number of months in the loan period. Note that, to calculate a monthly payment, both the interest rate r and the period of the loan n must be in terms of months. If those values are given to you in terms of years, quarters, or any other period of time, they must be converted to months.
Buying Bonds
Bonds have less potential for a big payoff than stocks but also have a lot less risk. Bonds are issued by governments or companies that are seeking funding, usually for a specific purpose. When you invest in a bond, you essentially lend money to that government or company with the expectation that you will later be paid back with interest. Bonds generally specify a period of time after which they mature, meaning your investment is paid back with interest. How much interest you will collect depends on overall interest rates and on the success of the entity that issues the bond.
Government or municipal bonds carry the lowest risk but also generally promise the smallest returns. One of the safest investments in bonds is investment in Treasury bonds or T-bonds, which are issued and guaranteed by the U.S. federal government. T-bonds carry basically no risk but also generally give only modest returns. They are also often exempt from all or some taxes.
Corporate bonds carry a higher risk because the corporation issuing them might go bankrupt, but they also generally promise larger returns if everything goes well. Corporate bonds that carry an unusually high amount of risk are known as junk bonds, and they often offer a high return in exchange for taking on a high level of risk.
Buying Stocks
Buying a stock in a company is essentially buying a small piece of the that company. The amount of the company that you own is measured in shares, and the number of shares into which a company is divided will vary from one company to another. You buy your shares at a given price per share based on the company’s valuation, or how much investors believe it is worth. Over time, the company’s valuation can go up or down, which will determine the price at which you can sell your shares. The goal in investing in stocks is to “buy low and sell high,” which results in a profit.
Basic Probability
Probability is the branch of mathematics concerned with how likely certain events or outcomes are.
The event whose probability we want to know is called the desired outcome, whether it’s something we actually desire to happen. Assuming that all outcomes are equally likely, the probability (P) of a desired outcome is simply the portion of all possible outcomes that are considered desired, expressed as a fraction, decimal, or percentage. We can write this as a mathematical formula:
P=# of desired outcomes# of total outcomes
Independent Events
Different events are considered independent events if the outcome of one has no effect on the outcome of the other. For example, consider the probability of rolling a 5 on a six-sided number cube with the numbers 1 to 6 and then getting “heads” when flipping a coin. These two events are considered independent because their outcomes do not affect each other.
Fundamental Counting Principle
When we have to keep track of a very large number of possible outcomes to find a probability, we use a similar idea called the fundamental counting principle. The fundamental counting principle essentially says that if there are p ways to perform one action and q ways to perform a second, then there are p×q ways to perform both actions together. This applies no matter how many different actions are being combined.
Combinations
When we need to know the number of different ways that objects can be combined in which order does not matter, we are looking for the number of combinations.
For example, let’s say you have 10 books, and you want to choose 3 to bring with you on vacation. You want to know how many distinct combinations of 3 books can be made from the 10. Since there is no significance to the order in which the 3 books are chosen, any combinations that contain the same 3 books in a different order are considered the same and are NOT counted in the number of distinct combinations.
Since we are choosing 3 books, there will always be 3!=3×2×1=6 different orders in which the same 3 books can be arranged. Therefore, when we want to know the number of combinations rather than permutations, we must divide by this number, leaving us with only 1 combination for each set of 3 books.
The number of possible combinations, C , is written as nCr and read as “n choose r,” where n is the number of objects we are choosing from and r is the number we are choosing. The formula is:
nCr=n!r!(n−r)!
Mutually Exclusive Events
Two events are considered mutually exclusive if it is impossible for both of them to happen.
For example, let’s say you are randomly choosing to turn left or right while taking a walk. The events of choosing to turn in one direction or the other are mutually exclusive because you can’t do both. If you have a 25% chance of choosing to turn left and a 30% chance of choosing to turn right, then the probability of choosing to turn left OR right is simply the sum of the two probabilities: P = 25% + 30% = 55%. It’s OK that these don’t add to 100% because you could also choose to walk straight or turn around.
Mean
The mean, also known as the average, is the sum of all data points divided by the number of data points.
Median
The median is the value that is in the middle when the values are arranged in order from least to greatest.
Mode
The mode is the value or values that occur most often in the data set.
Variance
The variance of a data set is the average squared distance of all points from the mean. We square the distances so that they will all be positive, no matter which side of the mean they are on.
Range
The most basic measure of variation is the range of a data set, which is simply the difference between the greatest and least values in the set. Let’s look at a set of data showing the scores of 10 students on an exam:
{88, 93, 86, 75, 81, 97, 95, 83, 90, 92}
The easiest way to find the range is to start by putting the data points in order from least to greatest:
{75, 81, 83, 86, 88, 90, 92, 93, 95, 97}
Then we can subtract the smallest data point from the largest. In this case, the range is simply 97−75=22 .
