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When you are dealing with a right triangle and you are trying to find lenght of the sides, you will most likely end up using the pythagorean theorem which states: where c is the side opposite the right angle, and is called the hypotenuse.
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Given in your problem: c=5, a=x and b=x+1
so by the pythagorean theorem of a right triangle:
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(expanding the squared term)
(combining terms)
(putting the quadratic equation in standard form}}}
(divide the whole thing by 2, to simplify the equation)
0=(x+4)*(x-3) (factor the expression on the right)
x=-4, x=3, since we are dealing with lengths, the negative value x= -4 is extraneous and can be trown out
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so, one side is a=x=3 and the other is b=x+1=4
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what you have is a special type of right triangle where the sides are in a ratio 3:4:5

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You need to understand the relationship:
d=r*t (distance = rate*time)
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Kim's distance after t hours:
Jason's distance after t hours:
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They are travelling in opposite directions and you need to know the time when they are 210 miles apart, so add their distances
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210/70 =t
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t = 3 hours

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when you have the parabola in the form you provide
the vertex is straight forward, vertex =(h,k)
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So in your problem:
h=1, k=-5, so the vertex = (1,-5)

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Bill swims at a speed of 6.7 mph in still water. The river he's in flows at a at speed of 3.6 mph. How long will it take Bill to swim 1.3 mi upstream?
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You need to understand the relationship:
d=r*t (distance = rate*time)
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Since you need to find time,rework this formula to find time (t=d/r)
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you are given the distance d = 1.3 miles
The speed is the "net speed" he travels during this distance.
Since he is swimming upstream, he is fighting the river flow
Therefore, his net speed is his speed in still water minus the river flow
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net speed: (Bill's speed - river flow)==> r = 6.7mph - 3.6 mph = 3.1 mph
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time to swim 1.3 miles: t = d/r =1.3 miles /3.1 mph = 0.419 hours or 0.4 hour rounded to nearest tenth of an hour

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at layer k there are 3^k parents, where k=1 to 5
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total = k=1 to 5
total =
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note:
In general
total = k=1 to n
total =

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solve 1n(e^3)
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A. two consecutive integers : x and x+1
B. The product of two consecutive integers x*(x+1)
C. 4 less than four times their sum: 4[x+(x+1)]-4
D. x*(x+1)=4[x+(x+1)]-4 (equate the sides)
E. x^2+x=4(2x+1)-4 (simplify the equation)
F. x^2 +x=8x+4-4=8x (simplify the equation)
G. x^2=-x+8x=7x (subtract x from both sides)
H. x=7 (divide both sides by x)
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check solution by substituting x=7 back in original equation
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x*(x+1)=4[x+(x+1)]-4
7*(7+1)=4*(7+(7+1))-4
7*8=4*(15)-4
56=60-4=56

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To test the hypothesis of two variances use the F distribution and a test statistic F=
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Ho:
Ha:
at
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critical value:
test statistic
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reject Ho if test statistic > critical value
since test statistic = 1.853 > critical value=1.806 then reject Ho
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conclusion:
at the level, the evidence suggest that

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(a) bimodal
This is usually a mixture of two different distribution, such as heigths of people, there is the distribution for men and for women, so if we dont separate the genders, there will tend to be a bimodal effect
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(b) approximately rectangular
This is the uniform distribution, where all events have equal probability of occuring. Such as tossing a die, the probability of 1 or 2, 3 etc each has 1/6 chance of occuring
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c) positively skewed
Many distributions are positively skewed (have long tails to the right)
examples are wealth distributons (a few people have the largest amounts of money)
population by country, some countries have most of the populations
country sizxe, a few countries have the biggest square aress,

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Nomimal means a label or category that has no order
Ordinal means a ranking, rating, its not measured on a continuous scale but has order
Interval means it can be measured in a continous scale
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B & D would b e the logical answers

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while the word MATHEMATICS has 11 words, there are really only 8 unique words
M, A and T are duplicated.
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So, your problem becomes number of arrangements of 4 letters from a word of 8 unique letters
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there are 8 choices for the first word
there are then 7 choices for the second word
after that there are 6 choices for the third word
and finally there are 5 choices for the fourth word
8*7*6*5=1680

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unless the video screen is a square, there is not enough information to figure out the length and width. You need to provide either a ratio of L to W or the angle that the diagonal makes to one of the sides.

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first, your answers, of Jean=32 and Michael 16, have you tested them to see if they are correct? The sum of their ages 3 years ago was 45 years
Let see, 3 years ago Jean=32-3=29, Michael=16-3=13 so their sum is 42, so your answers are incorrect!
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Begin by letting J=Jean's age and M=Michaels age, you have two unknowns J and M
and you need as many equations as unknowns.
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Jean is twice as old as Michael
J=2*M [ this is your first equation ]
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The sum of their ages 3 years ago was 45 years
J-3 is jean's age 3 years ago
M-3 is Michaels age 3 years ago, so....
(J-3) + (M-3)=45 [ this is your second equation]
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simplify the second equation
(J-3) + (M-3)=45
J+M-6=45
J+M=51
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So now you have two equations and 2 unknowns
There are many ways to solve this, substittution, elimination, matrices, graphing, guessing, cramer method, etc......
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Try subtitution since the first equation is already set to one of the variables
J =2*M
substitute this into the second equation
J+M=51
(2*M)+M=51
3*M=51
M=51/3=17
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So if M=17 and J=2*M then J=2*(17) =34
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Check your answers
go back to the original equations
J=2M and (J-3) + (M-3)=45 to see if M=17 and J=34 are valid

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mass of Sun relative to Mercury:
==
(Mass of the sun) /(mass of Mercury) = or 6.021 million times

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Many people will make lots of assumptions to simplify a problem like this and in many cases you will get answers close enough, but you should be aware of the proper way to do things
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1. You are dealing with a test of two independent samples, with unknown population variances.
2. You did not state the underlying distribution that these samples came from, but as long as you are analyzing averages and the samples are larger than 30, you can rely on the Central limit theorem for license to utilize Normal approximations to whatever underlying distributions you started with.
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Now just because you can rely on normal approximations does not mean you can just jump into the Z normal distribution, there is the t normal distribution.
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Since the population variances are not known, this indicates that you should use a "t" distribution. Althought, the samples sufficiently large, both the t distribution and Z distribution will merge and become indistinguishable.
but...... most people believe that this happens at samples even close to 30. In fact, the two distributions do not merge close enough until the samples are in the magnitude of approximately 200 each.
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So in your case you really should stick with the t distribution.
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To do things correctly, there are two basic types of t distribution. One for when the "unknown" population variances can be assummed equal and one where the variances cannot be assumed equal. (There is a F statistic for testing this, and I carried it out and determined that the population variances can be deemed to be equal, but I am not going to cover that here)
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So the particular t test that you use is the one where you can pool the sample standard deviations (since the population variances are deemed to be equal)
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Confidence interval for population mean differences
-/+
and =pooled_variance =
t = critical t value with 0.05 tail probability and 50+35-2 =83 degrees of freedom = 1.663, NOTE: for 90% confidence you will have 5% at the tails
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plugging in your information
=
-/+
3.0 -/+ 5.09
(-2.09, 8.09)is the 90% confidence interval for the population mean differences

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1 thru 4 are correct
#5 is wrong P(A or B) =P(A) + P(B)-P(A and B) = 4/9+1/3-1/9 = 6/9 or 2/3

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this was answered already
see 333044

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assuming there are no duplicated letters, there are n! different ways of ordering a word with n letters
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the word SUNDAY be arranged in 6!=6*5*4*3*2=720 ways
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The are 5! ways of ordering the word SUNDAY if the first letter is rstricted to S, because only the 5 remaining ones are allowed to change.
There are 4! ways of ordering the word SUNDAY if the first letter is restricted to S "and" the last letter is restricted to Y
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Therefore, there are 5!-4! ways that the word SUNDAY can be arranged such that the first letter is S and the last letter "is not" Y
5!-4! = 120-24 =96

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you can apply the binomial method directly which can be tedious and difficult
or "if" the assumptions are met, can apply a normal approximation to the binomial.
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The enable the use of the normal approximation, the following need to be met
1. and and and
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you provided p1 = 0.48 n1 = 100 p2 = 0.052 n2 =100
NOTE: not sure if your p2 is correct. Nevertheless, it looks like the requirements above are met.
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90% confidence interval for difference in the population proportions yields Z=1.645
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Upper confidence interval value:
Lower confidence interval value:
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confidence interval = to
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you can do the math

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Fundamentally, probability is straight foward
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Probability = (Number of a certain type of events)/(total number of events)
The difficult part is sometimes enumarating the number in each part of that fraction.
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In your case is not bad at all to enumarate the events.
probability of obtaining a number which is odd on a fair die, there are a total of 6 events {1,2,3,4,5,6} and there are 3 odd events {1,3,5}
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so P(a number which is odd)=Number of odd events/total events = 3/6=1/2

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difficult to help you with binomial tables through this medium
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some binomial tables are constructed using individual probabilites and other bionamial tables are constructed using cumulative probabilities.
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typically the proportion of interest is labelled across at the top, you would be looking for p=0.5
Then under that column, you would go down looking at for an "n" or sample size value along the right hand side (for 13 in your case). Within that sample size you would see events from 0 to the n or 13 in your case. Find for n=13 and x=2, follow that across and where this meets the column value that would represent either P(X=2) or P(X<=2) depending on how your table is constructed.
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To use the normal approximation to the binomial, keep in mind that there is an assumption that np>10, which is not met in your case, but ignoring this, though you shouldnt, then you would compute the mean and standard deviation, just like you did.
Then you would convert The X variable to a Z variable via
So you need P(X=2),
problem is the binomial (count of successes) is a discrete distribution and the normal distribution is a continuous distribution. soo .. what do you ?
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instead of P(X=2) you actually want to find P(1.5<=x<=2.5)
when you convert x to z you then have

this simplies to
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and from there you need to use standard normal tables, excel, statistical calculator, manual integration or some method to find this probability.

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let x=number of balls whose number is >= 17
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we want
in a sample without replacement
P(X>=1) = 1-P(x=0) (Getting at least one is the same as the complement of getting none)
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P(x=0)=P(all 3 are numbered <17) =
P(X>=1)=1-0.491=0.509
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This could also be solved as

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if you have the standard deviations and you want to order according to such criteria, is that simple.
first data set is the one with smalles std dev, then the next smallest std dev and so on.

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just look at your data!
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your data shows that P(x<=39)=1.0
therefore, P(x=45) or "any" number greater than 39 is 0
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so, no such claim is not supported
If something happened that changed the process and it is not following history, in such case , collect new data to determine the support to such claims, but absence a change in process and based on existing data, the claim is not believable

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If sin x= -4/5 and x is in quadrant III, find the value of tan(x/2
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x=arcSin(-4/5)=233.13 (since it lies in quadrant III)
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tan(233.13/2)=tan(116.57)=-2.0

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my trig test says perform this calculation:
135 degrees 16'45" + 234 degrees 50"
How do I do that?
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you basically have 135degrees 16' 45" + 234 degrees 0' + 50"
the 16' stands for 16 minutes
the 45" stands for 45 seconds
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Note: 1degree=60' and 1'=60"
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you can add common things
so add 135degrees + 234 degrees = 369 degrees
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Now add the minutes
16' + 0'=16'
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Now add the seconds
45"+50"=95"
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so you basically have 369degrees 16' 95" (but wait!!!, this is not in reduced form)
95" can be reduced to 1'35"
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so your answer is 369degrees 17' 35"

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You cannot!
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to solve equations you need 1 equation for every unknown.
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You have one unknown (y), but you need 1 equation to solve it.
what you have provided is not a bonafide equation, its basically an expression duplicated in both sides, Which will just show that the left hand side equals the right hand side.

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in mathematics, there are many ways to solve some problems.
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the domain puts you in the first quadrant, where x=positive, y =positive
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solve for x

(divide both sides by sin(x)
(invert both sides of the equation)

since x=26.565 resides within the domain this is the answer

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Verify the identity
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1. Expand the left hand side of the equation

2. simplify this result

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now you need to utilize the property of Sin(a+b)=sin(a)*cos(b)+sin(b)*cos(a)
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Sin(2x)=sin(x+x)=sin(x)cos(x)+sin(x)*cos(x) = 2*sin(x)*cos(x)
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therefore
==

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cot(t) =1/3 is the same as tan(t)=3
therefore
tan(t)=3 can be solved as t= Arctan(3)=71.565-/+n*180 (note 71.565 is in quadrant I and you want to be in quadrant III)
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since the domain is restricted to π < t < 3π/2
This puts you in quadrant III where both x and y are negative.
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so the answer is
t= Arctan(3)=71.565-/+180=251.565 (or -108.435)

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-120 degrees is going clockwise, so that puts you in quadrant III where both x, and y are negative.
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