You can put this solution on YOUR website!
your equation is 6y = 18
solving for y, you get y = 18/6 = 3
this is true for all values of x.
a graph of this equation is shown below:

your input / output table would look like this:
x / y
-5 3
-4 3
-3 3
-2 3
-1 3
0 3
1 3
2 3
3 3
4 3
5 3
-----

You can put this solution on YOUR website!
tan (15) = .267949192 by the calculator.
tan (15) = 0.267949192431123 by excel spreadsheet function.
-----
I don't know if it just stops there or if excel doesn't have the capability to extend the fraction any further.
-----
For it to be exact, it would have to be the result of the ratio of two integers.
----
I would be at a total loss to determine what integers would result in that fraction.
-----
my guess is that it's probably irrational and so the fraction would go on endlessly without any repeating patterns to indicate that even if it's an endless fraction, it was the result of the ratio of two integers.
-----
but that's just a guess.
I really don't know.
-----

You can put this solution on YOUR website!
amount of aspirin in the bloodstream is 1/2 of the original amount every 20 minutes.
-----
formula is:
y = 200*(.5)^(.05x)
-----
graph of this equation looks like this:

-----
this equations looks like it models it pretty well.
y is the amount of aspirin left in the bloodstream.
x is the number of minutes.
the horizontal lines are at 200,100,50,25 to show you the amount of aspirin left in the bloodstream after 0,20,40,60 minutes.
-----
It does not appear that the bloodstream will ever be completely free of aspirin based on this formula.
.5^(.05x) will never be equal to 0.
It will become very small, but never zero.
the calculator might show 0, but that's because the calculator can only go so far in the number of decimal places it can show, even with scientific notation.
-----
My calculator shows a result of 0 when 200 is multiplied by .5^333
It shows 2.285974783 * 10^-98 when 200 is multiplied by .5^332
Since .05*x = 333 when the calculator give a result of 0, this means that x = 20*333 = 6660 minutes
That's 4.625 days for the body to be completely free of aspirin enough so that my calculator can't determine how much is still left.
-----
best thing to do with this is determine a point at which the level of aspirin in the blood is less than a certain amount.
-----
that can be calculated.
-----
assume that amount is 1 milligram.
-----
formula would be 1 = 200*(.5)^(.05x)
divide both sides of this formula by 200 to get:
1/200 = (.5)^(.05x)
take the log of both sides of this equation to get:
log(1/200) = (.05x)*log(.5)
divide both sides of this equation by log(.5) to get:
log(1/200)/log(.5) = (.05x)
solve for (.05x) to get:
(.05x = 7.64385619
solve for x to get:
x = 152.8771238 minutes
-----
this means that it will take 152.8771238 minutes for the aspirin in the bloodstream to be less than 1 milligrams.
-----
the following graph should show that about as clear as can get with the tools available.

I added the horizontal line at y = 1 to show you the cutoff point better. It's at somewhere around x = 153 which is just about right.

You can put this solution on YOUR website!
perfect score is 10
11 scores with:
mean of 8
mode of 7
median of 8
-----
total of 11 scores with a median of 8 means that a maximum of 5 scores are below 8 and 5 scores are above 8. this will occur when there is only 1 number 8 in the distribution. we will assume only one number 8 in the distribution because that would allow for more number 10's if possible. we need at least one though, because we have 11 numbers and that means that one of the numbers has to be the median in order to get 5 on each side plus the 1 in the middle equal to 11 total.
-----
mode of 7 means that 7 is the score that has occurred with the most frequency.
-----
integer scores means that there are no fractions to worry about.
-----
we need:
1 number 8
a minimum of 2 number 7's
5 numbers above 8
5 numbers below 8
-----
let's start off with 5 number 10's.
that would be the maximum number of 10's without any other constraints except the fact that the median is 8 meaning the number of units above 8 had to be 5 at most.
-----
5 number 10's means that we would have to have 6 number 7's because 7 is the mode which is the number that occurs most frequently. we can't have 6 number 7's because the total of numbers below 8 has to be 5 and 6 is greater than 5.
-----
so 5 number 10's is no good.
-----
can we have 4 number 10's?
-----
this means we must have at least 5 number 7's and number 7 would then be the most frequent number.
if we have 4 number 10's, we would need 1 more number above 8 to make 5 numbers above 8. we'll pick the lowest number which is 9.
so far we have:
4 number 10's equals a total of 40
1 number 9 equals a total of 9
1 number 8 equals a total of 8
5 number 7's equals a total of 35
-----
we have a total of 11 numbers with a grand total of 92
-----
we have exceeded the grand total of 88 so 4 number 10's is not possible given all the constraints.
-----
can we have 3 number 10's?
-----
3 number 10's means we have to have a minimum of 4 number 7's to keep number 7 the most frequently used number in the set.
we also need 2 more numbers above 8 which means we have to choose 9 because 9 is the smallest number above 8.
so far we have:
3 number 10's equals a total of 30
2 number 9's equals a total of 18
1 number 8 equals a total of 8
4 number 7's equals a total of 28
-----
this give us a total of 10 numbers with a grand total of 84
-----
we need 1 more number that equals 4.
adding 1 number 4 give us a total of 11 numbers with a grand total of 88.
-----
the maximum number of 10's we can have is 3.
since 10 is a perfect score, then:
the maximum number of perfect scores we can have is 3.
-----

You can put this solution on YOUR website!
n = 7
m = 8
range = 1 to 10
high = h
low = l
n = 5
m = 7.8
least possible score to remove
-----
T1 = 7*8 = 56
T2 = 5*7.8 = 39
R = 56-39 = 17
-----
H + L = 17
L = 17 - H
-----
highest score is 10
L = 17 - 10 = 7
lowest score had to be 7
-----
the average times the number of items equals the total score
-----
7 * 8 = 56 total score
-----
5 * 7.8 = 39 total score
-----
difference between 7 and 5 is 2
difference between 56 and 39 is 17
2 papers had to have a total score of 17
lowest score plus highest score had to add up to 17.
lowest score could not be lower than 7 because that would make the highest score greater than 10 which is not possible.
-----

You can put this solution on YOUR website!
relationship is expressed by F = 9/5*c + 32
-----
this equation is in the slope-intercept form of y = mx + b where:
m is the slope and b is the y-intercept.
-----
your m is 9/5
your b is 32
-----
graph of your equation is shown below:

-----
click on the following hyperlink to see a picture of this graph with a vertical line at 100 degrees centigrade.
x is degrees in centigrade.
y is degrees in fahrenheit
water freezes at 0 degrees centigrade and 32 degrees fahrenheit. There is a horizontal line at y = 32 degrees fahrenheit to show you that point.
water boils at 100 degrees centigrade and 212 degrees fahrenheit. There is a horizontal line at y = 212 degrees fahrenheit to show you that point.
this is given certain conditions, such as being at sea level.
-----
Picture of Centigrade to Fahrenheit Graph with Vertical Line at 100 degrees Centigrade
-----

You can put this solution on YOUR website!
2*ln(x-5) = 3.65
-----
your equation is:
2*ln(x-5) = 3.65
divide both sides of this equation by 2 to get:
ln(x-5) = 1.825
by the basic laws of logarithms, this can happen if and only if:
e^1.825 = (x-5)
e^1.825 = 6.202795019 (use your calculator to get this)
this means that:
6.202795019 = x-5
add 5 to both sides of this equation to get:
x = 11.202795019
-----
the basic law of logarithms states:
if and only if
-----
with natural logarithms, the base is
your x was actually (x-5)
your y was 3.65 / 2 = 1.825
-----

You can put this solution on YOUR website!
A class has a boy-girl ratio of 5:3. Three more girls join the class, changing the ratio of 10:7. How many students are now in the class?
By trying and just plugging in numbers I got 30 and 18 plus 3 more, so now the total is 51 - but I'm not sure how to set it up as a problem!
-----
let b = number of boys.
let g = number of girls.
-----
b/g = 5/3
this means that b = 5g/3
-----
add 3 girls and the ratio becomes 10/7
this means that:
b/(g+3) = 10/7
this means that:
b = (10g + 30)/7
-----
since both these equations equal to b, then they must equal to each other.
you get:
5g/3 = (10g + 30)/7
multiply both sides by 7 and you get:
35g/3 = 10g + 30
multiply both sides by 3 to get:
35g = 30g + 90
subtract 30g from both sides to get:
5g = 90
divide both sides by 5 to get:
g = 18
-----
original ratio led to bg = 5g/3
5*18/3 = 5*6 = 30
original class size is 30 boys and 18 girls.
-----
add 3 girls and the class size is 30 boys and 21 girls.
boy to girl ratio is now 30/21 which equals 10/7
-----

You can put this solution on YOUR website!
Triangle has none
Rectangle has 1
Pentagon has 2
hexagon has 3
-----
Looks like it's n - 3 where n is the number of vertices.
-----
Stands to reason because one vertex has two adjacent vertices that it can't connect to. The rest it can.
-----
Octagon would be 5
-----
Click on the following hyperlink:
Number of Diagonals of a Polygon
-----
Look at the polygon from one of the vertices only.
Click on more and less to change the number of sides of the polygon.
You'll see that the number of diagonals from one vertex to the other vertices is n-3.
-----
This website goes even further to tell you the total number of diagonals from all vertices.
-----
The formula is n * (n-3) / 2
-----
Example:
hexagon is 6 sides
6 * 3 = 18/2 = 9 total diagonals.
There's 3 from each vertex to all the other vertices.
There's 6 vertices in total.
That's where the n * (n-3) comes from.
Each diagonal touches 2 vertices, so the number of diagonals has to be cut in half.
That's where the / 2 comes from.
-----

You can put this solution on YOUR website!
mean is 76.4 and standard deviation is 6.1
You need a Z-table or Z-table calculator to show you what the values are.
Here's a Z-table calculator.
Z-Table Calculator
-----
With a mean of 76.4 and a Standard Deviation of 6.1, only .1147% of the students scored above 95.
-----
95 is more than 3 standard deviations above the mean (95-76.4)/6.1 = 3.049180328
-----
You were right about the standard deviations above the mean. I's approximately 3.05.
-----
You need the table or the calculator to show you what the percentage is.
-----
Click on the hyperlink to go to the Table Calculator.
-----
In the top graph, enter a mean of 76.4 and a Standard Deviation of 6.1
-----
Select Above and enter 95.
-----
It will show you Shaded area: 0.001147 which is 0.1147%.
-----
Go to the bottom graph.
Enter 0 for mean and 1 for Standard Deviation.
Enter .001147 in the box for Shaded Area.
Select Above.
It will tell you the Standard Deviations above the mean.
-----
The answer was 3.0493 which is close to what you calculated.
The 3.0493 is more accurate.
Actually 3.0491803428 is more accurate. There probably some rounding going on. The showed .001147 which was probably rounded.
-----
Go back to the top graph and enter 0 for the mean and 1 for the standard deviation.
-----
Select Above and enter 3.0493.
You'll get .001147 as your answer.
-----
Whether you used a mean of 76.4 and a standard deviation of 6.1, you will get the same answer as if you entered a man of 0 and a standard deviation of 1.
-----
This is not surprising since the Z-Table is calculated from the mean and standard deviation.
-----
To find the percentage of students that scored below 85%, do the following:
Go to the top graph.
Enter a mean of 76.4
Enter a Standard Deviation of 6.1
Select Below and enter 85
-----
Answer is .920706 which means that 92% of the students scored below 85.
-----
Go to the bottom graph.
Enter a mean of 0 and a standard deviation of 1.
Enter .920706 in the shaded area box.
Select Below.
-----
It tells you that 85 is 1.4098 standard deviations above the mean.
-----
Go back to the top graph.
Enter a mean of 0 and a standard deviation of 1.
Select Below.
Enter 1.4098 (It's above the mean so it's still positive.
-----
It will tell you that the Shaded area is .920706 which is the same answer you got when you entered a mean of 76.4 and a standard deviation of 6.1
-----
The Z table converts mean and standard deviation to mean of 0 and standard deviation of 1.
-----
The formula is Value minus mean divided by standard deviation.
-----
mean is 76.4
standard deviation is 6.1
When Value is 76.4, Z-Table Value is 0 (76.4-76.4)/6.1 = 0
-----
When Value is 95, Z-Table Value is (95-76.4)/6.1 = 3.04918.....
When Value is 85, Z-Table Value is (85-76.4)/6.1 = 1.409836...
-----
Percentages are taken from the table.
They usually tell you the percent to the left or to the right of your value.
If the table tells you to the right, you need to get 1 minus what the table tell you to get to the left.
-----
Calculator that I linked you to does it all for you.
You just tell it what you want.
-----

You can put this solution on YOUR website!
5/3 can't be part of the domain.
If you divide by (3x-5) then when x = (5/3) you will be dividing by 0 because 3*(5/3) - 5 = 5-5 = 0.
That makes x = (5/3) not part of the domain.
-----
Your equation will be something divided by (x-5/3).
-----
Now we need to restrict your domain to values of x >= -4
-----
We can do that with square root of something that makes the something inside the square root negative when x < -4.
-----
Equation of should do it.
-----
When x < -4, expression within the square root sign is negative. This forces the domain to be >= -4.
-----
When x = 5/3, denominator is 0 causing value of the equation to be undefined. This forces the domain to exclude x = 5/3.
-----
Graph of this equation is shown below:

-----
Table of Values from x = -10 to x = 10 are shown below:
-----
x_______________y
-10____________#NUM!
-9____________#NUM!
-8____________#NUM!
-7____________#NUM!
-6____________#NUM!
-5____________#NUM!
-4____________0
-3____________-0.071428571
-2____________-0.128564869
-1____________-0.216506351
0____________-0.4
1____________-1.118033989
(5/3)________#DIV/0!
2____________2.449489743
3____________0.661437828
4____________0.404061018
5____________0.3
6____________0.243252128
7____________0.207289049
8____________0.182321138
9____________0.163888694
10____________0.149666295

You can put this solution on YOUR website!
Formula for present value of a series of payments is:
PRESENT VALUE OF A PAYMENT

PV = present value
PMT = payment per time period
i = interest rate per time period
n = number of time periods
-----
6.55% per year compounded monthly is:
6.55/12 = .554166667% per month which equals a rate of:
.00554166667 per month.
Number of month is 2 years * 12 months = 24 months
payment is $500 per month.
-----
PV = what we want to find.
PMT = $500 per month
i = .00554166667 per month
n = 24 months
Formula becomes:

which equals: $11,224.28901
-----
The loan was $11,224.28901
The lender received $12,778.06 by the end of the loan (Future Value of a Series of Payments).
Principal was $11,224.29
Interest was $1,553.77

You can put this solution on YOUR website!


This becomes:

This becomes:

This becomes:
x = 9.006468342
-----
to prove this is true, substitute in original equation to get:

This becomes:
200 = 200 confirming the answer of x = 9.006468342 is good.
-----
It would take 9.006468342 years to double your money at 8% compounded yearly.
-----

You can put this solution on YOUR website!
Let S = amount of new solution
Then .6*S = the amount of acid in the new solution.
-----
Amount of Acid in existing 3 gallon solution is equal to .2 * 3 = .6 gallons of acid.
-----
Let X be the amount of acid We need to add to the existing solution.
-----
The new solution is equal to the old solution plus X because we are adding pure acid which is represented by X.
-----
This means that S = X + 3
-----
The amount of acid in the existing solution is .6 gallons
We will be adding X gallons of acid.
The Total amount of acid in the new Solution will be X + .6
That will equal to 60% of the new solution which equals .6*S
-----
This means that (X+.6) = .6*S
-----
Since we know that S = (X+3), we can susbtitute in our equation to get:
(X+.6) = .6*(X+3)
-----
This reduces one equation in two unknown into one equation in one unknown which we can then solve.
-----
We remove parentheses to get:
X + .6 = .6*X + .6*3
We Subtract .6 from both sides of the equation and we simplify to get:
X = .6*X + .6*3 - .6 which becomes:
X = .6*X + 1.2
We subtract .6*X from both sides of this equation to get:
X - .6*X = 1.2 which simplifies to:
.4*X = 1.2
We divide both sides of this equation by .4 to get:
X = 1.2/.4 = 3
-----
Answer is you have to add 3 gallons of acid to 3 gallons of 20% acid to get a solution that is 60% acid.
-----
.6 gallons of acid in the original mixture + 3 gallons of acid = 3.6 gallons in the new mixture.
-----
3 gallons of the original mixture + 3 gallons of of acid = 6 gallons of the new mixture.
-----
3.6 / 6 = .6 * 100% = 60% of acid in the new mixture.

You can put this solution on YOUR website!
The standard form of the quadratic equation is:
ax^2 + bx + c = 0
-----
Your equation is 5x^2 = 4x
Substract 4x from both sides of this equation to get:
5x^2 - 4x = 0
Your a coefficient is 5
Your b coefficient is -4
Your c coefficient is 0
That would be selection d.
-----

You can put this solution on YOUR website!
The statement of your problem has a flaw, or I may be mis-reading it.
-----
the formula states:
Percent remaining = 100 - (42000 divided by N)*T
It should state:
Percent Remaining = 100 Percent - (42000 divided by N) * T
-----
It would then state as follows:
Let R = Percent Remaining
Let N = Number of bytes in the original file.
Let T = Time in Seconds
Let X = Transmission Rate per Second.
-----
Your formula would then be:
R = 100% - (X/N)*T
-----
R would equal 25%
X would equal 42000 bytes per second
N would equal 5000000 bytes
T is what we would want to solve for.
-----
Your formula would become:
25% = 100% - (42000/5000000)*T
-----
We would subtract 25% from both sides of this equation and we would add (42000/5000000)*T to both sides of this equation to get:
(42000/5000000)*T = 100% - 25% which would become:
.0084*T = 75%
We would then divide each side of the equation by .042 to get:
T = 75%/.0084 = 8928.571429% seconds
In order to get seconds, we would then have to divide by 100% to get:
T = 89.28571429 seconds.
-----
This is cumbersome and you have to know that you need to divide % seconds by 100% in order to get seconds at the end.
-----
The formula would have been much better stated as:
.25 = 1 - (42000/5000000)*T
-----
Then that conversion at the end would not have been necessary.
-----
Your general formula would then have been:
R = 1 - (X/N)*T
-----
R would equal Percent Remaining / 100% = 25% / 100% which would equal .25
X would equal 42000 bytes per second
N would equal 5000000 bytes
T is what we would want to solve for and would still be in seconds.
-----
Your Formula would become:
.25 = 1 - (42000/5000000) * T
-----
we would solve for T as before.
-----
T is stated in seconds.
The problem requested minutes.
Divide seconds by 60 seconds per minute to get minutes and your answer becomes:
T = 89.28571429/ 60 = 1.488095238 minutes
-----
To confirm your answer is correct, you would replace your original equation with known values of T and solve to see if it is true.
-----
Your original equation is (after my restatement):
25% = 100% - (42000/5000000)*T
We would replace T with 1.488095238 * 60 = 89.28571429
Formula would become:
25% = 100% - (42000/5000000)*89.28571429 which would become:
25% = 100% - .75 which would become:
25% = 100% - 75% after we converted .75 to 75% and would finally become:
25% = 25% confirming the number of seconds for T was correct.
-----
Again, however, we had the complication of % in the equation which would have been eliminated had we converted Percent to Rate up front.
-----
Your original equation would then have become:
.25 = 1 - (42000/5000000)*89.28571429
which would then have become:
.25 = 1 - .75 which would then have become:
.25 = .25 directly without any conversion being necessary.
-----
I haven't seen Percents in equations for some time now. Most of the time they are converted to rates before the equation is solved. This threw me for a little while but I was finally able to figure it out. Hopefully this analysis will help you to understand better as well.
-----

You can put this solution on YOUR website!

-----
They are right.
Here's why.
Your domain is all real values of x EXCEPT x = -8 and x = 2
-----
(-,8) covers the interval from minus infinity < x < -8.
(2,) covers the interval from 2 < x < infinity.
-----
you are missing the interval from -8 < x < 2 represented by:
(-8,2)
-----
number line would show that as follows:
------------------------(-8)---------------------(2)----------------

You can put this solution on YOUR website!
A common factor is a factor that applies to multiple terms.
Suppose you have number 1 and 3
The only common factor between these two number is 1.
1//1 = 1
3/1 = 3
-----
As a point in fact, though, we do not consider the number 1 as a common factor because all number can be divided by 1.
-----
We look for common factors greater than 1.
-----
Take the numbers 2 and 4
A common factor is 2 because:
2/2 = 1
4/2 = 2
Both number can be divided by 2 and yield an integer result.
-----
Integer result is another requirement, since all numbers can be divided by another number smaller than them and yield an answer if the answer does not have to be an integer.
-----
Common factors for 7 and 5 are?????
There are none becauswe there is no number than can be divided evenly into 7 and 5 at the same time. Matter of fact, there is no number smaller than these number and greater than 1 that can be divided into each of these numbers by itself because these numbers are prime numbers.
-----
Take 9 and 3
3 goes evenly into 9 and 3 goes evenly into 3 so 3 is a common factor of 3 and 9.
-----
Common factors are used to manipulate terms and expressions so they can be solved easier.
-----
We can use them to simplify division.
Take the case of:
5*10*7*3*9 / 5*7*3
-----
We have a common factor of 5 in the numerator that will cancel out a 5 in the denominator. Similarly with 7 and 3.
Out equation reduces to:
10*9 which equals 90.
-----
Using the common factor helped reduce the equation so it could be solved easier.
-----
We can use it to simplify multiplication.
Take the case of 5*7 + 3*7 + 6*7
We have a common factor of 7 that can change this equation to:
7 * (5+3+6) which equals 7 * 14 which equals 98
-----
Using the common factor helped reduce the equation so it coulde be solved easier.
-----
Look at x^ + 2x + 1 / (x+1)
The numerator in this equation can be factored to equals (x+1)^2
This changes our equation to:
(x+1)^2 / (x+1) which becomes ( (x+1) * (x+1) ) / (x+1)
Take out the common factor of (x+1) from the numberator and denominator and you are left with:
x+1
-----
It's true this equation could also have been solved by dividing (x+1) into x^2 + 2x + 1 as follows:
equation of x^2+2x+1 divided by (x+1) = x with a remainder of x+1
remainder of x+1 divided by (x+1) = 1 with no more remainder.
Solution is x+1
-----
We got the same answer but we have to work harder to get it. Recognizing the common factor made solution of the problem easier.
-----

You can put this solution on YOUR website!
x = number of fishing boats
y = number of canoes
-----
profit per fishing boat is $500
profit per canoe is $400
profit equation would be:
p = 500*x + 400*y
-----
your constraints appear to be:
total assembly hours = 100*x + 75*y
total finishing hours = 25*x + 50*y
-----
To keep these straight, it helps to create a table that would look something like:
-----
craft_________________assembly time required__________finishing time required
fishing boat_________________100 hours______________________25 hours
canoe______________________75 hours______________________50 hours
-----
you appear to be missing some constraints like total assembly hours <= ... and total finishing hours <= ...
-----
I will insert some hours and solve for that.
You can then replace whatever I have with whatever you have and solve the problem for yourself.
-----
Without those constraints, the problem is already solved because if he has unlimited time to assemble and finish then he should sell all fishing boats because he gets more profit per fishing boat.
-----
I'll put in some constraints and we'll see how it works out.
-----
Total hours available for assembly are <= 1000
Total hours available for finishing are <= 500
-----
Our constraints become:
100x + 75y <= 1000
25x + 50y <= 500
-----
Summary:
Profit equation is 500*x + 400*y
Constraint equation number 1 is 100x + 75y <= 1000
Constraint equation number 2 is 25x + 50y <= 500
-----
We graph the constraint equations.
-----
We do that by solving for y in both equations.
-----
We solve for equality, not for inequality.
This allows us to graph the equations.
Inequality is handled by looking at the area above or below the graph of the equations.
-----
Constraint equation 1 becomes y = (1000-100x)/75
Constraint equation 2 becomes y = (500-25x)/50
-----
graph of these equations is shown below:

-----
Picture of this graph with the area of possible solutions can be viewed by clicking on the following hyperlink.
-----
Graph of Equations and Area of Possible Solutions
-----
The area of possible solution is bounded by:
The x-axis because the value of y, which is number of canoes, has to be >= 0.
The y-axis because the value of x, which is number of fishing boats, has to be >= 0.
The line y = (1000-100x)/75 because the number of hours to assemble a fishing boat (x) and a canoe (y) has to be <= 1000.
the line y = (500-25x)/50 because the number of hours to finish a fishing boat (x) and a canoe (y) has to be <= 500.
-----
Remember that x equals the number of fishing boats, and y equals the number of canoes.
-----
The equation for total assembly time is 100*x + 75*y <= 1000
We converted this to:
y <= (1000-100*x)/75 in order to graph it, and then we graphed:
y = (1000-100*x)/75
-----
When y equals the value of this equation, y will be ON the line.
When y is smaller than the value of this equation, y will be BELOW the line.
-----
Our area of possible solutions for this equation are thye points ON the line and the points BELOW the line.
-----
Same for the other equation.
-----
The possible solutions for the x-axis are the value of y ON the x-axis and the values of y ABOVE the x-axis.
The possible solutions for the y-axis are the values of x ON the y-axis and the values of x TO THE RIGHT OF the y-axis.
-----
The rules of linear programming states that the maximum/minimum values will be at the intersection of the boundaries of the constraints.
-----
This means we have to look at the points:
(0,10)
(0,0)
(10,0)
(4,8)
-----
We apply these points to our revenue equation of 500*x + 400*y
Point (0,10) yields 500*0 + 400*10 = $4,000.00
Point (0,0) yields 500*0 + 400*0 = $0.00
Point (10,0) yields 500*10 + 400*0 = $5,000.00
Point (4,8) yields 500*4 + 400*8 = $2,000.00 + $3,200 = $5,200
-----
Our maximum profit can be attained when we build and sell 4 fishing boats and 8 canoes.
-----
Why couldn't we get more profit when we build all fishing boats and no canoes?
It would seem that a greater profit on each fishing boat should result in building all fishing boats.
-----
Here's what happened.
-----
When we built 10 fishing boats, we used up all the assembly time available so we had no assembly time left over for the canoes.
That means we had to stop, even though we had some finishing time left for the canoes.
-----
When we built 4 fishing boats, we still have assembly time left over for 8 canoes. 4*100 = 400 subtracted from 1000 = 600 divided by 75 hours for each canoe equals 8 canoes able to be assembled with the time left over. This resulted in a total of 12 craft being assembled and finished and sold (presumption is you are able to sell all that you build). Even though the profit for each canoe was less than the profit for each fishing boat, more craft total yielded more total profit. 4*500 + 8*400 = 5200 while 10*500 = 5000.
-----

You can put this solution on YOUR website!
x = number of lawnk jobs
y = number of landscape jobs
-----
revenue = number of lawn jobs * 35 and number of landscape jobs * 125
r = revenue
r = 35x + 125y
-----
30 minutes per lawn and 90 minutes per landscape job
works at most 10 hours per day
30 minutes equals .5 hours and 90 minutes equals 1.5 hours
z = total hours worked per day = 10
.5*x + 1.5*y <= 10
-----
no more than 3 landscaping jobs per day
y <= 3
-----
your revenue equation is:
r = 35x + 125y
-----
this is the equation you want to maximize
-----
your constraint equations are:
.5*x + 1.5*y <= 10
y <= 3
-----
the first constraint says that he can't work more than 10 hours per day on all jobs.
-----
the second constraint says that he can't work more than 3 landscape jobs and get all his lawn jobs done.
-----
the first thing I noticed was that 3 lawn jobs earns $105 while one landscape job that takes the same time earns $125.
-----
this should indicate that the landscape job gets him more money per hour.
I suspect the answer is that 3 landscape jobs and the rest lawn jobs will maximize his revenue for the day.
-----
let's see how that works.
-----
first you graph your constraints.
-----
the second constraint can be graphed as is.
that equation is y <= 3
the first constraint needs to be modified as follows:
.5x + 1.5y <= 10
subtract .5x from both sides of the equation and divide both sides of the equation by 1.5 to get:
y <= (10 - .5*x)/1.5
-----
you need to graph:
y = 3
y = (10-.5x)/1.5
It is shown below:

-----
if you look at the graph, you'll see the following.
---
the line y = 3 is the maximum number of landscape jobs that can be done.
since landscaping jobs are represented by y, that means that y has to be smaller than or equal to 3. that means the area of the graph on or under the line y = 3 is where your solution will lie.
-----
the line y = (10-.5x)/1.5 is the maximum number of hours that he can spend on either job. that line is the one slanting down to the right. since the number of hours he can spend on either job is less than or equal to 10, the solution must lie on or underneath this line as well.
-----
Click on the following hyperlink to see a picture of this graph. Explanations will follow below the picture.
-----
Picture of Constraint Equations and Area of Possible Solutions
-----
The area beneath the line y = 3 and beneath the line .5x + 1.5y = 10 and above the x-axis is the area of possible solutions.
-----
The rules for linear programming indicate that a maximum / minimum solution will be at the points of intersection of all lines that bound the possible solution.
----
In the picture you can see that:
One of the intersection points is at (0,3)
Another of the intersection points is at (11,3)
Another of the intersection points is at (0,0)
Another of the intersection points is (20,0)
-----
We examine each of the intersection points in the Revenue Equation. The Revenue Equation is y = 35*x + 125*y.
-----
At the point (0,3), this equation becomes:
35*0 + 125*3 = 375
At the point (11,3), this equation becomes:
35*11 + 125*3 = 385 + 375 = 760
At the point (0,0), this equation becomes:
35*0 + 125*0 = 0
At the point (20,0), this equation becomes:
35*20 + 125*0 = 700
-----
Remember x = number of lawn jobs and y = number of landscape jobs.
The maximum revenue is generated when 11 lawn jobs are done and 3 landscape jobs are done.
-----
As I pointed out earlier, this was to be expected since 3 lawn jobs earned less than 1 landscape job so logically doing the maximum number of lawn jobs possible earned the most money.
-----

You can put this solution on YOUR website!
convert them all to decimal.
2.05 = ok as is
25/10 = 2.5
2.0513 = ok as is.
2.059 = ok as is
251/100 = 2.51
2.0515 = ok as is
2.052 = ok as is
2.051 = ok as is.
-----
your numbers are:
2.05
2.5
2.0513
2.059
2.51
2.0515
2.052
2.051
-----
your final order will be:
2.05
2.051
2.0513
2.0515
2.052
2.059
2.5
2.51
-----
look at this order and see what's happening.
2.05 is smaller than 2.051
this is because 2.05 is equivalent to 2.050 which is smaller than 2.051
2.5 is smaller than 2.51
this is because 2.5 is equivalent to 2.50 which is smaller than 2.51
-----

You can put this solution on YOUR website!
we don't always use the exponent of 2.
only with some object do we use it.
-----
there are different formulas used depending on the object.
-----
area of a square = s^2
exponent of 2 is used here.
-----
area of rectangle is l*w
exponent of 2 is not used here.
-----
area of circle is pi*r^2
exponent of 2 is used here.
-----
area of triangle is 1/x*b*h
exponent of 2 is not used here.
-----
what is really being used is length times width in some form or other.
the result is in square units.
each object has it's own unique formula to derive length times width.
-----
it's easy to see with the square and the rectangle and the triangle.
it's not so easy to see with the circle, but the derivation of the formula for the circle use the concept of length * width as well.
-----
bottom line is that the exponent of 2 is used in deriving the area of objects only in those cases where the length and the width are the same. The main concept involved in deriving the area of something is length times width.
-----

You can put this solution on YOUR website!
4 year bond.
semi-annual coupon rate of 10.5%
$1,000 bond.
-----
I'm pretty sure this is the way it's done.
-----
A time period is every 6 months.
-----
Your study is for 8 time periods.
-----
Your coupon is 10.5% per year paid semi-annually which means you get half of that every 6 months.
-----
Half of that every 6 months is .105/2 * $1,000 = $52.50 at the end of each time period.
-----
At the end of the 8th time period you will be getting the last of the $52.50 plus you will be getting $1,000.
-----
The rate of return you are looking for is an annual rate of 12% compounded semi-annually.
That means for 8 time periods you will be earning 6% compounded interest each time period.
-----
Present Value of $52.50 payments at the end of each of 8 time periods at 6% compounded per time period equals $326.0141751
-----
Present Value of a future amount of $1,000 for 8 time periods at 6% compounded per time period equals $627.4123713
-----
Add these up together and you get $$953.4265464.
-----
This is what you should pay for the bonds today.
-----
You will earn 12% a year compounded annually.
-----
To confirm this, I looked at it from a cash flow analysis point of view.
-----
The Future Value of $953 for 8 time period at 6% per time period is equal to $1519.617065
-----
If you invest $953 today, you should have $1519.617065 in your account at the end of the 8 time periods.
-----
This is small enough to calculate each year separately, so I'll do it.
-----
You invest $953 at the beginning of time period 0.
End of Time Period 1 = $52.06
End of Time Period 2 = * 1.06 + $52.5 = $108.15
End of Time Period 3 = * 1.06 + $52.5 = $167.14
End of Time Period 4 = * 1.06 + $52.5 = $229.67
End of Time Period 5 = * 1.06 + $52.5 = $295.95
End of Time Period 6 = * 1.06 + $52.5 = $366.20
End of Time Period 7 = * 1.06 + $52.5 = $440.68
End of Time Period 8 = * 1.06 + $52.5 = $519.62 + $1,000 = $1519.62 *****
-----
The cash flow analysis and the Future Value of a Present Amount yield the same result so it's probably accurate.
-----
I think I got the coupon rate right. That looks like the way they do it. the coupon rate is an annual rate based on the face value of the bond. If you get the coupon semi-annually, then you get half of it two times a year.
-----
Formulas I used are shown below:
-----
FUTURE VALUE OF A PRESENT AMOUNT

FV = Future Value
PA = present amount
i = Interest Rate per Time Period
n = Number of Time Periods
-----
PRESENT VALUE OF A FUTURE AMOUNT

PV = Present Value
FA = future amount
i = Interest Rate per Time Period
n = Number of Time Periods
-----
PRESENT VALUE OF A PAYMENT

PV = Present Value
PMT = Payment per time period
i = Interest Rate per Time Period
n = Number of Time Periods
-----

You can put this solution on YOUR website!
You start off with $48 for 4 people.
You end up with $78 for 10 people.
-----
The cost per person for 4 people was $48/4 = $12 per person.
The cost per person for 10 people was $78/10 = $7.8 per person.
-----
The cost per person is going down.
At what rate?
-----
If you take a straight line average, then you would find the average rate of change by dividing the difference in cost per person by the difference in the number of people and you would say that's your rate of change in the cost per person.
-----
Let's see how that works.
-----
Change in cost per person went from $12.00 to $7.8
that's a difference of $4.20
-----
Change in number of people went from 4 to 10
That's a difference of 6
-----
The average rate of change in the cost per person was $-4.20 / 6 = -$.70 per person.
-----
If you have 4 people, the cost is $12.00
If you have 5 people, the cost is $11.30
If you have 6 people, the cost is $10.60
If you have 7 people, the cost is $9.90
If you have 8 people, the cost is $9.20
If you have 9 people, the cost is $8.50
If you have 10 people, the cost if $7.80
etc.
-----
The cost per person is going down by $.70 per additional person.
-----
The equation to express this would be:
y = 12 - .70*(x-4)
The graph of that would be:

The horizontal lines at 12 and 8 are just there to show you the values when x = 4 and when x = 10.
-----
The equation to tell you what the total cost would be can be derived from this.
that equation would be:
y = x*(12 - .70*(x-4))
The graph of that would be:

The horizontal lines at 48 and 78 are just there to show you the values when x = 4 and when x = 10.
While the rate of change in the cost per person was linear (previous graph), the rate of change in the total cost is not.
-----
In general, the rate of change is the change in the value of the dependent variable divided by the change in the value of the independent variable.
-----
y is usually the dependent variable.
x is usually the independent variable.
this translates to a change in the value of y divided by a change in the value of x per unit of x.
-----
We tracked cost per person and identified the rate of change in the cost per person as the number of people changed.
-----

You can put this solution on YOUR website!
Try these lessons on the web.
-----
jamesbrennan
wtamu
algebralab

You can put this solution on YOUR website!
let x be the number.
Equation states that x - 18 = 3/5 * x
Solve for x to get:
x = 45
The number is 45.
The 2 digits of the number are 1 difference from each other.
-----

You can put this solution on YOUR website!
standard form of a quadratic equation is y = f(x) = ax^2 + bx + c if it opens up or down.
standard form of a quadratic equation is x = f(y) = ay^2 + by + c if it opens left or right.
-----
If your equation is in the form of y = ax^2 + bx + c:
It opens up if a is positive.
It opens down if a is negative.
-----
If your equation is in the form of x = ay^2 + by + c
It opens to the right if a is positive.
It opens to the left if a is negative.
-----
If it's in the form of x = f(y) = ay^2 + by + c (open left or right), you might also see it in the alternate form of:
y = f(x) = +/-
-----
You take the x = f(y) version and solve for y to get the y = f(x) version.
-----
Once again, though, if a is positive it opens to the right and if a is negative it opens to the left.
-----
To see this with real equations, consider:
y = x^2 + 2x + 1
This opens up as shown in the following graph because the a term of 1 is positive:

Now consider:
y = -x^2 + 2x + 1
This opens down as shown in the following graph because the a term of -1 is negative:

-----
Now consider:
x = y^2 + 2y + 1
Before we can graph this, we have to solve for y to get:
y = -1 +/- sqrt(x)
This opens to the right as shown in the following graph because the coefficient of x under the square root sign is positive. It is positive because the a term was positive:

We converted x = f(y) to y = f(x) in order to graph the equation. This is the alternate form i was talking about.
-----
Now consider:
x = -y^2 + 2y + 1
Before we can graph this, we have to solve for y to get:
y = 1 +/- sqrt(-x+2)
This opens to the left as shown in the following graph because the coefficient of x under the square root sign is negative. It is negative because the a term was negative:

We converted x = f(y) to y = f(x) in order to graph the equation. This is the alternate form I was talking about.
-----

You can put this solution on YOUR website!
let T = capacity of the tank when it is full.
-----
(3/4) * T - 21 = (2/5) * T
-----
This equation states that three fourths of the tank capacity minus 21 liters equals two fifths of the tank capacity.
-----
Solve for T:
(3/4) * T - 21 = (2/5) * T
Subtract (2/5) * T and add 21 to both sides of this equation to get:
(3/4) * T - (1/5) * T = 21
Multiply both sides of this equation by 20 to get:
15 * T - 8 * T = 21 * 20
Simplify to get:
7 * T = 21 * 20
Divide both sides of this equation by 7 to get:
T = 21 * 20 / 7
Simplify to get:
T = 60
-----
The capacity of the tank when it is full is 60 liters.
3/4 * this is 45 liters.
Substract 21 from that to get:
24 liters.
24 liters / 60 = 2/5
-----
Capacity of the tank is 60 liters. Remove 21 liters from it when it is only 3/4 full and you are left with 24 liters which is 2/5 of the capacity of the tank.
-----
That's your answer.

You can put this solution on YOUR website!
b^(8/3)*c^5
--------------- divided by a^2*b^(5/3)*d^(1/2) = I am stuck
a^(1/2)
If I understand this correctly, it is:

-----
first rule that you can use is that
you have something in the form of which becomes which becomes
-----
applying this rule, your equation becomes:

-----
In your denominator, you can take advantage of the fact that:
x*(y*z) = (a*y)*z
-----
You can use this to multiply the a factors first.
-----
The denominator of your equation becomes:

-----
Now you take advantage of the fact that
-----
The denominator of your equation becomes:
equals:

-----
You can separate out each factor to see them individually better.
Your equation becomes:

-----
Now you take advantage of the fact that
-----
your equation becomes:
equals:
equals:

-----
Since this equation can't be reduced any further, your final answer should be:

-----
To confirm you calculated this correctly, you substitute real values for your variables in the original equation and then you substitute those same real values in your final equation and you should get the same answer. If you do, you did it right. If you don't, then you go back and find out what you did wrong.
-----
let:
a = 2
b = 3
c = 4
d = 5
-----
your original equation of:
comes out to be:
which equals:
242.8629243
-----
your final equation of:
comes out to be:
which equals:
242.8629243
-----
Since the original equation and the final equation both come out the same then the transformation process was done correctly.
-----
Your answer is the final equation.
-----

You can put this solution on YOUR website!
x invests 9000
y invests 7000
z invests 4000
total investment equals $20,000
-----
profits are 4800 / 3 = 1600 apiece
-----
if they each received profits in the same ratio as their investments,
x would have received 9/20 * 4800 = 2160
y would have received 7/20 * 4800 = 1680
z would have received 4/20 * 4800 = 960
-----
total profits to x, y, and z would be 2160 + 1680 + 960 = 4800
-----
Answer to your question is:
x receives 1600
x would have received 2160
x receives 2160 - 1600 = 560 less than x would have received had the profits been divided based on the ratio of investments.
-----

You can put this solution on YOUR website!
(-5)^2 = (-5)*(-5) = 25
-----
That's +25 because a minus times a minus results in a plus.
-----