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