Math Semester Review

How long will it take to triple $1500 compounded continuously at 3.5% interest rate?

A=Pe^rt <--REMEMBER THIS. find what 1500 tripled is: 4500. set up the equation: 4500=1500e^1.035t. to get the rate or r: add 1 to the percent so 1 + .035. Then, you just divide by 1500 to get 3=e^1.035t. next you do ln on both sides to get rid of e: ln3=1.035t. divide by 1.035 on both sides. Then you do ln3 and then divide that by 1.035. This is how many years it takes to triple the money. the answer should be 31.4 years.

how to solve 4sin^2x-1=0

Add one, then divide by four so it is sin^2x = 1/4, then square root both sides. Make sure to remember that the square root of 1/4 has to be plus/negative answers. The square root of 1/4 is 1/2. So the answer is sinx = +/- 1/2. Then, you look at the unit circle to find the angles that have a positive or negative 1/2 as the y value.

solve for all the unknown measures in triangle ABC. Round to the nearest hundredth. m<A=42°, b=14, c=19.

Draw triangle. Find which measurements you need to find. In this case, its m<B and C and side length a. first find side length a by using side length formula. This is in another question. write down the long version of answer as well as rounding to hundredths. Next, find m<B by using angle formula which is also in another question. solve and also write out all numbers before rounding to hundredth. remember the °. Then, use the sinA/a = sinB/b = sinC/c to find angle C. You need to remember to do sin^-1(sinC)=sin^-1(.999955642.) to find the real answer.

arithmetic and geometric sequences: Mr. Lehman's marathon training schedule called for a beginning run of 1.5 miles and an increase of 0.1 miles per day. After 120 days, how many total miles had Mr. Lehman run?

First, you need to plug the numbers into the fn=dn +a(little 0) equation: fn= .1n + 1.4. Then, you will plug 120 into n because you want to know after 120 days how much he is running. when you plug 120 into this equation, the answer should come out to: 13.4. This is the amount of miles he will have run overall.

Set up, but do not solve, an expression for the probability that out of a group of 36 people, at least two people will share a birthday.

Remember to double check the number used, it could be 35 or 36 or something completely different. then the equation is: 1-([365!/329!]/365^36)

You create five-digit zip codes where all digits are unique (using digits 0-9). What is the probability that five-digit zip code contains a four and a seven?

You need to make a fraction, so on the bottom, you put 10 x 9 x 8 x 7 x 6 because that is the amount of choices you have each time you choose a number. Then on the top, you put 5P1 because you need a 4 at least once and you have 5 spots, then you multiply that by 4P1 because you only have 4 spots left for the 7 and then you multiply that by 8P3 because you have 8 numbers left to use and only three spots. The answer doesn't have to be simplified

A school dance committee of eleven people is selected at random from a group of 20 ninth graders, 19 tenth graders, and 21 eleventh graders. What is the probability that the committee has no ninth graders?

You need to make a fraction, so on the bottom, you put the probability of all the students: 60C11. on the top, you add up 19 and 21 to get 40. So the fraction should look like: 40C11/60C11. then, you find what they equal and no simplification needed, you got your answer

equations needed for arithmetic and geometric sequences:

a(little n)=dn + a(little 0) sum= n/2(a1+an) if the problem has a percent increase: sum=a1(1-r^n/1-r) an=a1(1+r)^n-1 a1/1-r

arithmetic and geometric sequences: Mr. Lehman's starting salary is still $32,000 but with a salary increase of 1.5% per year. Determine (a) the salary during the thirty-fifth year and (b) the total compensation through thirty-five years.

a) first, you need to make the equation using the a1(1+r)^n-1 equation: an=32,000(1+.015)^35-1. So you will do 1.015^34 and then multiply that by 32,000 which will give you an answer of $53087.88. b) you need to use the equation: a1(1-r^n/1-r) which will look like: 32,000(1-1.015^35/1-1.015). This will equal: $1,458,946.81

arithmetic and geometric sequences: Mr. Lehman's starting salary is $32,000 with a salary increase of $450 per year. Determine (a) the salary during the thirty-fifth year and (b) the total compensation through thirty-five years.

a) use the an=dn+a(little 0) equation. so the equation is an=450n + 31,550. this is the original equation. To find the salary during the 35th year, plug 35 in for n in the equation. the answer is $47,300. b) to find this, use the sum formula: n/2(a1+an). so the equation is: 35/2(32,000+47,300) which equals $1,387,750. You put 35 in as n because it is the compensation over 35 years.

solve the exponential function and show work: 2^(3+x)-1/4 = 0

add 1/4 so the equation is now: 2^(3+x) = 1/4. put it into log form: log(little 2)1/4=3+x. Subtract 3 to the other side. Don't worry about the 3 for a second and change the problem into a fraction like: log1/4 over log2. Solve that into a number and then subtract 3 from that number. The final answer should be x=-5.

How to solve 3sec^2x-4=0

add four, divide by three so the problem is now sec^2x= 4/3. square root both sides and remember to keep positive negative of the square root of 4/3. the square root is not a nice number, so you change the problem so it is secx = square root of 4/ square root of 3. this simplifies to 2/square root of 3. To find the answers on the unit circle, it needs to be in cosine or sine, so you change it to cos. It is now: cosx = positive negative square root of 3/2. Then you find angles with that as the x value.

In how many ways may 15 people line up to get on a bus?

because it matters who is in line first, it is a permutation: 15P15 = 1.307674368x10^2

Write the exponential equation in logarithmic form: e^x=4

e is the base, 4 is the answer, x is the exponent. Rewritten in log form: log(little e)4=x. But, log(little e) is the same as ln, so the answer is ln4=x.

Write the sequence as an explicit expression (fn=) or a recursive process (an+1): 5,3,1,-1

find the difference between each number: -2. so the equation is fn=-2n and then you need the zeroth term so you add 2 to 5 so the equation is now fn=-2n+7

Write the sequence as an explicit expression (fn=) or a recursive process (an+1): -2,6,-18,54

find the difference between each number: not even. since it isn't even, you know it is being multiplied by -3. so in the equation, you put the first term: -2 and then use the -3 so the equation is: fn=-2(-3)^n-1

Write the sequence as an explicit expression (fn=) or a recursive process (an+1): 64,32,16,8

find the difference between each number: not even. since it isn't even, you know it is being multiplied by a 1/2. so you put the first term: 64 in the first spot and the equation looks like this: fn=64(1/2)^n-1

Write the sequence as an explicit expression (fn=) or a recursive process (an+1): -2x, -x, 0, x

find the difference between the numbers: 1x. so you start the equation as xn. then you find the zeroth term: -3x. so the equation is fn=xn-3x.

How to solve cos 75° using sum and differences formulas

find two angles that add or subtract to 75°. for example cos(30° + 45°). Then, you use the sum formula which is: cosa x cosb - sina x sinb. so the formula would be cos30° x cos45° - sin30° x sin45°. You then find the fraction version of that on unit circle. in this case it is: square root of 3/2 x square root of 2/2 - 1/2 x square root of 2/2. when multiplied, it is square root of 6/4 - square root of 2/4 and the final answer is square root of 6 - square root of 2 all over 4.

How to solve sin -pi/12 using sum and difference formulas

find two fractions on the unit circle that add or subtract to get to -pi/12. one that works is pi/4 - pi/3. pi/4 = 45° and pi/3 = 60°. so you write it as sin(45°-60°) So the formula for this one is sina x cosb - sinb x cosa. with the numbers, it is sin45° x cos60° - sin60° x cos45°. you need to change to fractions on the unit circle, so you get square root of 2/2 x 1/2 - square root of 3/2 x square root of 2/2. Multiplied together, it is square root of 2/4 - square root of 6/4. The final answer is square root of 2 - square root of 6 all over 4.

arithmetic and geometric sequences: Find the sum of the infinite series: 2, 1, 1/2, 1/4 ...

first make the equation using the an=a1(d)^n-1 so it will look like: an=2(1/2)^n-1. then, you use this to create the sum equation which is a1/1-r. r=d, So you put 2/1-1/2 this equals: 2/-1/2 which simplifies to -4.

In 2000, the population of a town was 1,752. In 2010, the population was 1,643. Use an exponential model to predict when the population will hit 1,350.

first, set a year as 0: t=0=2000. So, now 2010 would be considered 10. First, you need to find the rate by using the two years provided for you. You set up the equation: 1643=1752e^r(10). divide by 1752. then do ln on both sides. then divide by 10 to find r. r should equal -.006423415. Remember to write an equation for all the time: A=1752e^(-.006423415)t. Then, to find when the pop will hit 1,350, you plug 1,350 into the equation above. You do the same thing as when you found the rate, but you divide by the rate instead of the time. then, you should end up with t=40.57863307. You add this to the year you designated as 0 so the year that the pop reaches 1,350 should be 2040.

what are the equations for the patterned looking problems like 5,3,1,-1 and 64,32,16,8

if it is going up or down by subtraction or addition: f(little n)=dn+n(little 0). if it is going up or down by multiplication: f(little n)=a(little 1)(d)^n-1 <-- it is always n-1 no matter what.

Evaluate the function at the indicated value of x: f(x)=log(little 2)x x=16.

plug 16 into the equation: log(little 2)16 = y. then put it into exponential form: 2^y=16. Then just figure out what it takes for 2 to get to 16. y=4 = final answer

Evaluate the function at the indicated value of x: f(x)=log (little a)x x=a^-2

plug a^-2 into the equation: log(little a)a^-2 = y. Put it into exponential form: a^y = a^-2. y=-2 = final answer

Evaluate the function at the indicated value of x: f(x)=lnx x=e^3

plug e^3 into the equation: lne^3 = y. Then, put it into exponential form: e^y=e^3. y=3 = final answer

what are the sum and difference formulas?

sin(a-b) = sina x cosb - sinb x cosa sin(a+b) = sina x cosb + sinb x cosa cos(a+b) = cosa x cosb - sina x sinb cos(a-b) = cosa x cosb + sina x sinb

write the logarithmic equation in exponential form: log(little 3)81=4

the little three is the base, 4 is the exponent and 81 is the answer. Rearrange the equation to 3^4 = 81. This is your final answer.

What are the equations for the law of sines and cosines problems?

to find an angle: cos-1(a^2-b^2-c^2/-2bc) = m<A. to find a side: a^2=b^2+c^2 - 2bc x cosA. to find an angle or side with the right amount of sides and angles known: sinA/a = sinB/b = sinC/c

How do you multiply matrices?

you double check that the two middle numbers match: R x c rxC. then, you go --> on the first matrix and down on the second matrix. then add them together. the outside numbers creates the dimensions of the answer matrix

A school dance committee of eleven people is selected at random from a group of 20 ninth graders, 19 tenth graders, and 21 eleventh graders. What is the probability that the committee has all tenth graders?

you set this up as a fraction. so, you add up all the students which equals 60. so on the bottom of the fraction, you put 60C11 because it is a committee and committee starts with c so it is combination. On the top, you put only the tenth graders: 19C11. Then you find what each of these equals and put them over each other for the final answer. no simplification needed.