The Fibonacci sequence is a series of numbers in which each term is found by adding the two previous numbers together. 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144... (0+1)=1, (1+1)=2, (1+2)=3, and so on. The Fibonacci sequence has been used or mentioned in movies called Pi, 21, and the film versions of A Wrinkle in Time and The Da Vinci Code. It is an interesting sequence and in 8th grade I found the first 50 or so terms.
On the 45-45-90 triangle, the sides that are not the hypotenuse are equal lengths. This makes sense as the angles are equal. The equal lengths are why dividing a by b results in one and dividing either of them by c equals the same number, 0.7071067812. The length of c is the length of one of the sides times 1.414213562. On this triangle, the 30-60-90, the short a side is 1/2 of the hypotenuse c. The length of b is always the same ratio when compared to a or c.
When the government pays for the interest on a loan while you are in school, it is a government subsidized loan. You have to start paying as soon as you graduate, but the government pays while you are in school. In an unsubsidized loan, interest starts accumulating when you take the loan. Currently, government loans have an interest rate of 4.66%.
Bank loans are unsubsidized loans and also tend to have a higher interest rate. You can find the total amount you will have to pay using the equation A=P(1+r/n)^nt. A=amount P=principal r=rate of interest n=number of times per year interest is compounded t=time in years An example would be if you took out a $5,000 each of the fours years of college. This gives a total of $20,000. One plan would be to pay it back over 20 years. When the numbers are plugged in, the equation is A=20000(1+0.0466/1)^1*20. This is A=20000(1+0.0466)^20. When solved for A, the answer is $49,733. When paid once a month, the cost is $207.22 because there are 240 months in 20 years and 49,733/240=207.22. a= where bottom = 0 but does not cancel
hole= where bottom = 0 and cancels out A vertical asymptote has a graph where there are two lines that move toward each other but never cross a certain x. This is what makes them vertical. It is like a half-life, it keeps halving, becoming smaller, never quite hitting its zero, the x. A hole is the same expect that there is a second x, one where there is a hole, a gap, a skip in the graph. In terms of the equations, both asymptotes and holes are equations in which the bottom is equal to 0. The difference is, asymptotes are simply where the bottom equals zero while holes are where the bottom equals 0 and the top can cancel with part of the bottom. (̶x̶+̶1̶0̶)̶ / (x-10)(̶x̶+̶1̶0̶)̶ In this case, x=-10 is the hole and x=10 is the asymptote. Fractions tend to scare students away from math. This could be because they are harder to visualize than integers. Integers, one knows which number is larger or smaller. With fractions, the number with the larger denominator has smaller sections (1/8 < 1/2). That is straightforward enough. However, there is then the fact that the numerator can change (5/8 > 1/2). This changes whether it is more or less. This makes fractions a little harder to visualize in the beginning. Additionally, fractions can have many names for the same amount (16/32 = 4/8 = 1/2). Another factor is that there are two numbers, already making it look worse. The best way to deal with fractions when starting out is just to find a common denominator and remember that while doing so, multiplication/division done to the bottom must be done to the top and vice versa. In Mr. Kelly's method of reducing radicals, you take the smallest factor (other than 1) of the number you are working with. You take down what you get when you divide by that factor Dividing 500 by 2 gives 250, dividing that by 2 gives 125. Next, you divide by 5 because that is the smallest factor of 125. This gives 25, which is then divided by 5, giving 5. When 5 is divided by 5, you get 1. When you hit 1, you're done. Because it is a square root, you find pairs. If it was a cube root, you would find groups of three. This gives pairs of 2s and 5s with one 5 remaining. The pairs go outside the radical as 2*5, and the 5 goes inside. This gives 10 √5.
(2[_500 2)[_250 (5[_125 5)[_25 5[_5 1 This works because the pairs are being squared. That means they are square roots that do not need to remain as radicals. (2*5)^2*5=500 This answer is better than a calculator answer because it is more accurate than the calculator. the calculator gives a decimal, and approximation. The calculator answer would be better in instances where you need to see which answers are greater or lesser than. The decimal would also be better to use in science because equipment can only measure to a certain accuracy. (x^y)*(x^z)=x^yz is sometimes true. Most of the time, it is incorrect because when multiplied the exponents should add together, giving an answer of x^y+z. However, if both of the exponents were to be 2, the final answer would be both x^2+2 and x^2*2. This means the equation is true sometimes.
x^3+3x^2-54x=0
To factor, check if there is anything that is in all of the terms. In this case, there is an x in each term. If you divide by x, you then have x(x^2+3x-54)=0. You can put an x out in front of the rest of the equation. Next, you factor what is inside the parentheses. To do this, you need to know the factors of -54. These include (6,-9), (-6,9), (3,-18), (-3,18), (2,-27), (-2,27). When added together, the factors need to equal positive 3. The only factors that equal positive 3 when added together are -6 and 9. If we forget the outside x, the equation looks like (x+9)(x-6). This gives a final factoring of x(x+9)(x-6)=0. Checking gives x(x^2-6x+9x-54)= x^3+3x^2-54x=0. To solve, you use the factored equation. Because you are multiplying for 0, one term of the equation must equal 0. This means the first x=0. From x+9=0, you subtract 9 meaning x=-9. From x-6=0, you add 6 meaning x=6. Factoring is one method of solving quadratic equations. When an equation has been factored, you are able to plug in 0 to solve for x because the factoring involves multiplying. There are three ways to solve systems of two equations with two variables.
y=3x-12 3x+3(3x-12)=12 3x+9x-36=12 12x-36=12 12x=48 x=4 y=3(4)-12 y=12-12 y=0
Ex. 2x+4y=4 and -2x+2y=8 (2x=4y=4)+(-2x+2y=8) 6y=12 y=2 -2x+2(2)=8 -2x+4=8 -2x=4 x=-2 |
AuthorI am a runner and I enjoy math. Archives
June 2015
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