You can put this solution on YOUR website!
The radius of the circle is the distance, , from center (h,k) to each of the lines.
The distance from a point (h,k) to a line can be calculated as
, so
which simplifies to
and
With those absolute values, there are too many possibilities, so we should figure out the sign of the expressions inside the absolute values.
It's easy to sketch the triangle and figure out.
With the line equations in that form, it's easy to calculate intercepts:
For example, wit ,
--> --> and
--> --> -->
So, goes through (5/3,0) and (0,5).
similarly,
goes through (1,0) and (0,1/3),
and goes through (-7,0) and (0,7/3).
Sketching, we can find what points are inside the triangle,
including point (h,k), the center of the circle.
The triangle is the space below the lines that go through (0,5) and (-7,0)
(the first and third lines), but above the line for (the with intercepts so close to the origin.
Each of the expressions inside the absolute values would be zero if (h,k) is on the corresponding line, positive if (h,k) is on one side of the line, and negative if it is on the other side. Since none of the lines goes through the origin, we can use the origin as a test point to figure out which side is which.

is true for the part of the plane below ,
where we find the origin, (0,0), the triangle, and (inside the triangle)
the center of the circle (h,k).
So, and

is true for the origin and the part of the plane below ,
but for the other side, where the triangle and (h,k) are.
So, and
is true for the origin and the part of the plane below ,
where we find the origin, (0,0), the triangle, and point (h,k).
So, and

Now that we know the sign of each expression, we can re-write
as

and that is a system of equations that we can solve to get
, and
--> -->

Then we can write the equation of the circle as

You can put this solution on YOUR website!
An angle measuring 100 degrees would be supplementary to any angle measuring 80 degrees, but it cannot be called supplementary by itself.
"Complementary" and "supplementary" are terms that describe the relationship between two angles.
They are comparative words like "larger."
Those words are not applied to an angle by itself, unless it is understood what other angle it is being compared to.

Two angles are called supplementary (to each other) if they add up to 180 degrees.

Two angles are called complementary (to each other) if they add up to a right angle (90 degrees).
Unless you are older than I think, and your teacher has talked about negative angles, there is no angle that can be complementary with a 100 degree angle.

You can put this solution on YOUR website!
With no other restrictions, there should be an infinity number of (x,y) pairs that are solutions.
If x and y must be non-negative integers, (3,0) is the only solution I see.

You can put this solution on YOUR website!
I would say the units for log(x) deserve a new name, because you cannot add the value of log(x) to lengths in meters, just as you cannot add any quantity in any other unit to lengths in meters. You could call it log-meter.
However, I have never heard of a name for such a unit.
The only logs that "sort of" got a name are
1 unit of pH = -log(1 mole of hydrogen ion per liter)
and decibels.
They are not exactly logs, but they involve logs.

You can put this solution on YOUR website!
I would get rid of denominators first by multiplying both sides of the inequality times 2:
-->
Next, I would subtract 6 from both sides:
-->
Finally, I would divide both sides by 3:
-->

You can put this solution on YOUR website!
I say that the area is and here is the triangle:
The small triangle, with sides measuring 6, 8, and 10 units is a right triangle, because the squares of the lengths of the shorter sides add up to the square of the length of the longer side:

A large triangle with sides measuring 15, 8, and 17 units is also a right triangle, because the squares of the lengths of the shorter sides add up to the square of the length of the longer side:

Our triangle with sides measuring 9, 10, and 17 units is the large right triangle minus the small right triangle, and its area is the difference between the areas of the two right triangles.
Area of the large right triangle= square units
Area of the small right triangle= square units
Area of the triangle in the problem= square units

When you only have the lengths of the sides, unless it is a right triangle, the problem is tricky. If you have not been taught enough, you need a combination of luck, ingenuity and triangle construction to figure out what I show in the drawing an explanation above. To construct a triangle with sides measuring 9, 10, and 17 cm, you would draw one of the sides to exact measure, and the use a compass to mark arcs centered at the ends of the first side, with radii matching the other two side lengths. Where the arcs meet is the third vertex of the triangle. Such a construction would help you figure out what I showed above.

If you have been taught about Heron's formula involving the semi-perimeter, you could use that.

If you were taught the law of cosines, you could use it to find the cosine (and then sine) of one angle, and then use the formula for area involving the lengths of two sides and the sine of the angle in between.

USING LAW OF COSINES:
units = measure of shortest side
units = measure of medium length side
units = measure of longest side
=measure of the largest angle, opposite longest.
The law of cosines says that
so -->
Solving for :

so
The area of a triangle can be calculated from the lengths of 2 sides, a and b, and the angle between them, C, as
so

USING HERON'S FORMULA:
The semi-perimeter, , is defined as half of the perimeter.
If the side lengths are , , and , then ,
and the area of the triangle is

and

You can put this solution on YOUR website!
probably the question was to write an expression for that in algebra (not in words) and to simplify that expression.
= the first of two consecutive even integers.
= the next of the two consecutive even integers.
The sum of their squares is

You can put this solution on YOUR website!
The rational expression does not exist when
which happens for and for
(You can find that by factoring , or by solving using the quadratic formula).
Those are the only restrictions on the variable .
For any value of other than and the expression is defined.

You can put this solution on YOUR website!
has just two terms: and
The numeric coefficients have as a common factor:
and
The variable appears as in one term and as in the other,
so is a factor common to both terms.
The variable appears as in one term and as in the other,
so is a factor common to both terms.
The variable appears as in both terms,
so is a factor common to both terms.
Taking out common factors,
and
, so

You can put this solution on YOUR website!
is the sum of a geometric sequence.
I always rediscover the formula for a sum, because it's just as easy as looking it up a cumbersome formula and applying it.


--> -->

You can put this solution on YOUR website!
Considering each wrapping a perfect circle, the first 100 wrappings would be 100 circles of radius 0.2 cm, diameter 0.4 cm.
(I'm counting the radius as going from the very center of the nail to the surface where the nail touches the wire, not the slightly larger diameter to the very center of the wire).
That would account for
per wrapping or for the first 100 wrappings.
The next 100 wrappings are wrapped around nail plus one layer of wire, with a radius of and a radius of
The total length of wire for that second layer would be
.
The third layer would be wrapped around the nail plus two layers of wire with a new, larger diameter of and would total of wire.
The length of wire for each layer keeps increasing, and the total length of wire for the 7 layers, meaning 700 wrappings, on diameters measuring, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, and 1.0 cm is

That is .

You can put this solution on YOUR website!
The legs must be longer than 3, but can be as long as you want.

You can put this solution on YOUR website!
At the point where 2 circles are (mutually) tangent their tangents are the same line, perpendicular to the radius of each circle.
The radii of the 2 circles form a continuous segment from one circle center to the other, and is one of the sides of the triangle in the problem.
So, the lengths of the sides of your triangle are:

and


It looks like
if the legs (shorter sides) of the right triangle measure and
If the longest side is the one measuring ,
then I will not draw the whole largest circle

In the first case,
--> --> --> -->
If you are good at factoring, you factor the quadratic into
and find the positive solution you need as
If you are not that good at factoring, the quadratic formula will give you the same answer.

In the second case,
--> --> --> -->

It could have a few more answers, but I have to leave for my paying job now.
The problem most likely assumes that each circle is outside the other two.
(Circles can also be internally tangent, with one circle inside the other and just touching at one point. That would complicate the problem and give you extra solutions, but few people would think of internally tangent circles when reading or writing a problem).

You can put this solution on YOUR website!
When I want to make an expressions sit on top of another, I use the special codes from this website. I write the algebraically correct expression
((8y^3-27)/(64y^3-1))*((16y^2+4y+1)/(4y^2-9))=
and wrap it between three {'s and 3 }'s.
It's much easier to write it with just paper and pencil,
but when talking to or through a machine, you have to use precise language.

Your problem must be simplifying that rational expression and that requires factoring.

At this point you simplify by dividing both numerator and denominator by the same expressions,
which we do by crossing out matching top and bottom expressions, saying that "they cancel each other."
=

NOTES:
1) You have to remember that there are special products that result in a difference of squares, and in difference or sum of cubes.
where and can be any mathematical expression.
Also, and
, where the quadratic polynomials
and cannot be factored further.

2) Factoring is a skill essential to math survival, and it requires practice. You thought you were done with polynomials, but factoring keeps coming back to haunt you, all the way through calculus and beyond. I found it the hard way when I was your age.

3) Parentheses are sometimes needed.
, but if you key into your calculator
10-1/9-1= , your calculator, who knows and obeys the rules of order of operations, correctly interprets it as
=8.88888...

You can put this solution on YOUR website!
is the simplest example I can think of.
works too.
That kind of function/inequality is what is probably expected.
We could get fancy, as in
, but there's no need for that.

You can put this solution on YOUR website!
I have a solution (two solutions actually), but there has to be a better way to get to the solution, and the results are so complicated that I suspect a typo in the problem.
The midpoint of AB is (-3,3).
The slope of the line through A and B is

The perpendicular bisector of AB is
--> -->
The center of a circle passing through A and B has to be on that line, so its coordinates would be (h,k) with

The equation of the circle is
where is the radius.
I can even substitute and get as equation for our circle

Substituting the coordinates of B, I get an expression for
--> --> -->
Now I can write the equation for our circle as

Simple. I just have to find .
I know the circle is tangent to the line
-->
so that line and the circle have just 1 intersection point.
Substituting into the equation for the circle,

I can find that intersection point and more.




If that quadratic equation must have just one solution, the discriminant must be zero, so



The best I can do with that unwieldy equation is divide everything by 4 to get

Applying the quadratic formula



That can be simplified to

Substituting into , we get
with and
with
The approximate values give us points (-1.95,0.91) and (-72.49,141.98) for centers.
Exact values for are
and
and approximate values are and corresponding to the centers above, respectively.
and
The equation of the circles are
for the small circle, and
for the large circle.
Asking for the equation of circle in general form is cruel.

You can put this solution on YOUR website!
Take out as a common factor:
-->
Either
or -->

You can put this solution on YOUR website!
If the thickness of the pipe wall is inch=inch,
we have to add to the inner diameter twice that thickness to get the outside diameter, which is
inch=inch=inch

For the silk problem, we need to get all those numbers as fractions of a yard,
rather than mixed numbers with mixed units.
6 inches = (1/2) foot and 1 foot = 1/3 yard, so
6 inches = (1/2) of (1/3) of a yard or of a yard
3yards 6 inches = 3 yards + 1/6 yard =
Let's add all those yards of fabric:

Now we need a common denominator,
and is the least common denominator we could use

It was a total of 12 yards.


Now we need a common denominator,
and is the least common denominator we could use

I get by dividing 133 by 8. The quotient is 16 whole units and the remainder, , divided by is .

A piece of metal 3 7/8 inches long is to be cut from a longer piece of 5 1/16 inches. what will be the length of the remaining piece?


add 7 5/8 and 5 5/8.

then from their sum subtract 11 5/16.


Now we need a common denominator,
and is the least common denominator we could use


15* 4 2/3 Evaluate.

You can put this solution on YOUR website!
The area will be the area of the rectangular part, plus the area of the semicircular part.

For the second part:
If the radius is ft, the diameter, which is also the width of the rectangle, is
ft.
That would be the base of the downwind.
and if the height of the rectangular part is twice the base, then that height is
feet=feet.
Then the area of the rectangular part is
The area of half of a circle with radius is

The total area is
= approximately .

For the first part, a rectangle 3 feet by 6 feet would have an area of 72 square feet, and a semicircle with a 6 feet diameter (3 feet radius) would have an area of
=approximately ,
for a total area of approximately .
That is what you found, aside from the rounding difference.
(I used a more accurate version of , but I would get 86.13 too, If I used 3.14 por ).

You can put this solution on YOUR website!
A fifth grader could explain it, but not using systems of linear equations. That requires unnecessary complication.

= number of weekday bleachers seats tickets
= number of weekend lawn seats tickets

Total number of ticket you are willing to purchase is
because
"You want a total of 7 tickets."

weekday bleachers seats tickets will cost $
weekend lawn seats tickets will cost $
because both cost $11 per ticket.

The total cost for
weekday bleachers seats tickets plus
weekend lawn seats tickets is
$

You can spend the total $ to buy
weekday bleachers seats tickets plus weekend lawn seats tickets,
spending .

To decide what to buy, you would solve the system


However, the system is what we call "dependent".
We also call it "undeterminate" because there is no unique solution.
The equations and are equivalent and dependent,
because they are equivalent equations, with exactly the same solutions.

Here is what "equivalent equations" means:
is what we get from by multiplying both sides of the equal sign times ,
so all the solutions of are solutions of .
is what we get from by dividing both sides of the equal sign by ,
so all the solutions of are solutions of .
In other words, if you have one equation, are you are given the other one,
you are not getting any new information.
All you know is that you can buy 7 tickets with the $77 you have.
(A fifth grader could tell you that).
You can buy all of your 7 tickets for weekday bleachers seats,
or you can buy all of your 7 tickets for weekend lawn seats,
or you could buy any combination of the two kinds of seats.

You can put this solution on YOUR website!
a)
When you get your points plotted (even if you did it by hand), with mean daily calories on the x-axis and infant mortality rate on the y-axis,
you would see that there is some scatter, but it seems obvious that the two variables are related.

The greater the mean daily calorie intake, the lower the infant mortality rate.
The scatter plot shows that and more.
It looks like a straight line would be a good model to represent the data, so you would say that there is a linear relationship.
The relationship line drawn to fit the points would slope down (negative slope), so you would say that infant mortality rate is negatively correlated with mean daily calorie intake.
I do not like to eliminate suspected outliers.
You may think that the point (2671,7), representing a low 7 per 1,000 births infant mortality in a place with a mean daily calorie intake of only 2671 calories is an outlier. Or maybe you think (3429,44) is an outlier. Or maybe you think both are.

b)
With your data, Excel gives me , and .
I would say that
the correlation coefficient is -0.902,
it is a strong correlation (because that close to -1 or 1, everyone agrees to call it strong), and
it is significant (because tabulated critical values for 10 data pairs/points are -0.602 and 0.602, and -0.902 is not between those critical values, meaning that it would be very unlikely to get 10 data points that so strongly suggest correlation if the variables were really unrelated).
The significant strong negative correlation does not mean that low calorie intake causes infant mortality, or that infant mortality causes low calorie intake. The two variables are related, but statisticians are supposed to say "correlation does not imply causation". (It is up to politicians and journalists to give that correlation the spin of their choice).

If you have Excel, there are many versions of it, and the menus became harder to find with all the new features added in the last 10 years.
You need to enter your data table first.
(I would have 2 columns with titles x, for mean daily calories,
and y, for infant mortality rate).
To graph with Excel, selecting the data first,
you would go to the Insert menu and select Chart, selecting the chart type "XY (Scatter)" with the choice showing just the points, no lines.
You might find "Chart Wizard" icons to get to Chart a little faster, going through less menus.
The software guides you through the steps to get your points plotted.
You get x and y axes and your points plotted.
(I like to get rid of the gray "fill" painted into the chart,
by right clicking on that gray area, selecting format plot area,
and selecting "none" for border and fill.
I also like to delete the grid lines, and the "Series 1" legend).
Then, somewhere in the Chart menu, I would find "Add Trendline" and use it to add my regression line (linear type).
Then, right-cliking on the line, I would "Format Trendline" to get the desired "Patterns" (dashed, dotted, or solid, thick, or thin, black, or other color).
(I also like to use "Format Trendline" "Options" to
"Display equation on chart", "Display R-squared value on chart", and extend the line "Forward" and "Backward" a bit using "Forecast".
Outside/without the graph, Excel can give you the correlation coefficient, R, using the "CORREL" function,
and/or you can use the "LINEST" function as an array function to get ten linear regression parameters tabulated in two columns.

You can put this solution on YOUR website!
I'm solving
Multiplying everything (both sides of the inequality sign) times as positive number
results in an equivalent inequality, so multiplying times 4 we would get

Changing the sign of the expressions on both sides
(it is exactly the same as multiplying times )
flips everything to the other side of zero,
so to get an equivalent inequality, you also have to flip the inequality sign, so
-->
I could have done both steps in one by multiplying times .
It's the same thing, but harder to visualize.

The "properties" are what common sense tells you.
It's what makes sense in real world situations.
Math does not require to memorize much.
You just have know meaning of some symbols and definitions,
and think of what it means,
not by "juggling words", or invoking "voodoo rules", but by using ideas.
With inequalities you can add the same number to both sides and you get an equivalent inequality
You can subtract the same number from both sides with the same result (subtracting is just adding the opposite number).
You can multiply/divide by a positive number both sides and get an equivalent inequality.
However, multiplying/dividing by a negative number requires flipping the inequality sign.

You can put this solution on YOUR website!
The denominator is never zero, so there are no vertical asymptotes and no holes.
(You need zeros in the denominator for those).
The graph is shown in red below.


EXTRA:
It has a slanted asymptote (shown in green) because

and gets smaller and smaller as grows larger and larger.

You can put this solution on YOUR website!
(a)
, and
If then
,
and

If is a root, then substituting into we get
--> --> -->
That means that and
at the same time (for the same real value).
and are the only solutions.

EXTRA:
The imaginary roots are and .
so a factor in the factoring of should be

Dividing we find that

So the other two complex solutions to
are the solutions to
which happen to be

(b)
ONE WAY TO LOOK AT IT:
, , , , , , and so on.
In general, for any integer
, , ,
and
So adding up the terms of the sum in groups of 4 we get
, and so on.
Until when?
Dividing 174 by 4 we get a remainder of 2:
so the last group of 4 would end with
After that we have and , so
+....++....++...+
or if you prefer

ANOTHER WAY:
i+i^2+i^3+i^4+....+i^174 is the sum of a geometric sequence,(or geometric progression, depending on where you are studying math).
From what we may know about geometric sequences and series,
i+i^2+i^3+i^4+....+i^174 =
Since ,

You can put this solution on YOUR website!

First, you figure out if the product will have a minus sign.
If both factors have the same sign, the product is positive,
so no minus sign is needed.

Now, when we multiply fractions,
we multiply the numerators (the top parts) to get the numerator of the result,
and we multiply the denominators (the bottom parts) to get the denominator of the result.

At this point we see if we can simplify,
because it is easier to simplify at this point than afterwards.
so we can write it as that

Then we can cross out the 3's and end up with
because and

You can put this solution on YOUR website!
The side of the k cubes (in cm) must be a divisor of 50, 40, and 30.
The greatest common divisor is 10, so 10cm is the side measure of the largest cubes we can make.
The least number of cubes, correspond to the largest cube size and will be
because we will have 5, 4 and 3 cubes to a side that's 50, 40, and 30cm long respectively.

Alternately, the volume of the cuboid, in , is
and the volume of a cube with 10cm sides, in , is ,
so the number of cubes is

You can put this solution on YOUR website!
NOTE:
If you have studied about angles inscribed in a circle, you would know that a right angle subtends a arc (and the diameter), so the problem did not have to tell you that one side measures 15 cm, because you knew that the diameter was 15 cm.
That was redundant information, given to make the problem easier.

You can put this solution on YOUR website!
If the roots of are and , then
and

(Reasons below)

Then can be easily calculated as:



or or

REASONS:
The sum of the solutions to the quadratic equation equals
and the product equals .
Why?
Because if roots of are and ,
factoring should transform that equation into
--> --> -->
and if that equation must be equivalent to ,
all three coefficients must be the same.
The coefficient of the term in is
--> .
The constant (or independent term) is
-->

You can put this solution on YOUR website!
No side can be longer than the sum of the lengths of the other two.
If two sides are the same length, the other side can be as short as you want, but still must be shorter than the sum of the other two sides.
Whether the two sides' length is or , the other side's length can be .

You can put this solution on YOUR website!
You are probably mixing up results
it is not even a multiple of 3, much less a power of 9

You can put this solution on YOUR website!
It took the cyclist an extra 1 hour for an extra 5 miles.
His speed is miles per hour.