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Something like this:
.
1. Column I: F is the midpoint of segment EH ... Column II: Given
2. Column I: Segement EF congruent to Segment FH ... Column II: Definition of midpoint
3. Column I: F is the midpoint of segment IG ... Column II: Given
4. Column I: Segement IF congruent to Segment FG ... Column II: Definition of midpoint
5. Column I: Angle EFI congruent to Angle GFH ... Column II: Vertical Angles are congruent.
6. Column I: Triangle EFI congruent to Triangle HFG ... Column II: Side-Angle-Side congruent to Side-Angle-Side
7. q.e.d.
.
If you have a question about vertical angles, Wikipedia has a good definition showing what
they are and how they are congruent.
.
You can probably understand this problem better if you sketch it and mark the congruent
parts as you identify them in the proof. It will help you to visualize what is going on.
.
You may have to put this "proof" in terms that are acceptable to your teacher, but this "proof"
contains the basic elements you need for the problem.
.
Hope this helps you with two column proofs.
.

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The slope of a line is defined as the change in the value of y divided by the change in the
value of x. To have an undefined slope requires that the change in the value of x be zero. This
generates a division by zero in computing the slope, and since division by zero is not allowed in
algebra, this means the slope is undefined.
.
The equation for such a graph is of the form x = k where k is a constant that shows the
value of x no matter where you are on the vertical line.
.
For this problem one point on the line is given as (-2, 0) which says that at one point on the
line, the value of x equals -2. Since the line is vertical, whatever value you set y at, the
corresponding value of x equals -2.
.
As examples of this, the following points are all on the line: (-2, -100), (-2, -10), (-2,0),
(-2, 5), (-2, 45), and (-2, 200). You can tell if a point is on this line just by seeing if
that point has -2 as its value of x.
.
The line you are looking for is defined by the equation x = -2 and that is the answer to
this problem. The line is defined by the equation x = -2. This line is vertical and has an
undefined slope.
.
Hope this somewhat helps you to understand what is going on with this problem.
.

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Probably varies depending on the model of the calculator. On the TI-89 it goes this way:
.
Suppose you are given a 3 by 3 matrix to inverse. And suppose that matrix is:
.
+1 0 0
-1 1 0
+1 1 1
.
The plus signs are just added for typing purposes in this answer. The minus sign is needed,
but the plus signs are just inserted to keep the columns of numbers aligned in the typed
version.
.
On the entry line ... enter exactly as below:
.
([1,0,0;-1,1,0;1,1,1])^-1
.
A couple of notes about this entry:
.
Note that there is a difference between the symbols "(" and "[" and their counterparts in
the reverse direction. Do not interchange them or use a parenthesis in place of a bracket.
.
Note that the symbol representing the end of a line in the matrix is a semicolon, not
a colon.
.
Note that the minus sign at the beginning of the second line is from the key that looks
like (-). It is not the subtraction function key that has just a minus sign on it. The same
thing applies for the minus sign in the exponent ... use the (-) key not the - key to
enter the exponent -1.
.
When you type the line exactly as shown above, you are raising the matrix to the minus 1 power.
.
You will get an answer that shows the inversed matrix is:
.
+1 +0 0
+1 +1 0
-2 -1 1
.
As before, the plus signs are added here just to keep the number columns aligned. They
do not show in the calculator's answer but the columns are aligned.
.
Hope this helps you with your calculator. If it does not, read your instruction book on how
to enter a matrix. Then try raising it to the -1 power to see if that procedure works.
Use the problem and the answer above as a check to see if the calculator gives the answer
above.
.
Hope this helps you out ...

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The volume of a sphere (V) can be calculated using the equation:
.

.
in which R represents the radius of the sphere. If you multiply out the constants:
.

.
you get a product 4.18879025 and substituting this product into the volume equation simplifies
it to:
.

.
Using this form of the equation, all you have to do for this problem is enter the constant 4.18879025
into your calculator's memory. Then for each of the given spheres, cube its radius and
multiply that by the constant which you can recall from memory.
.
For example. The first pair ... one sphere has a radius of 1 ft. Cube that to get 1 cu ft
and multiply that by 4.18879025 to get an answer of 4.18879025 cu ft.
.
The first pair ... second sphere has a radius of 2 ft. Cube that and you get 8 cu ft. Then
multiply that by 4.18879025 and you get the volume to be 33.51032164 cu ft.
.
It may be apparent to you now that when you doubled the radius the volume went up 8 times.
That is because the cube of R is and the cube of 2R is .
The ratio of these two is .
.
You can now apply this same technique to the remaining 2 pairs of spheres. When you do, you
should get the following answers:
.
Pair 2. The sphere with the radius of 3 ft has a volume of cu ft
and the sphere with the radius of 6 ft has a volume of cu ft
.
Pair 3. The sphere with the radius of 4 ft has a volume of cu ft
and the sphere with the radius of 8 ft has a volume of cu ft
.
In each pair, if you take divide the answer for the volume of the first sphere into the answer
for the volume of the second sphere you should get an answer of 8.
.
Hope this helps you to understand the problem and how you can work it.
.

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Start by letting D represent the cost of the desk and T represent the tax paid.
.
The first sentence tells you that the cost of the desk plus the tax equals $131.44. In
equation form this is:
.
D + T = 131.44
.
The second sentence tells you that the tax T equals (1/16) times the cost of the desk less 0.31 dollars.
In equation form this is
.
T = (1/16)D - 0.31
.
Substitute the right side of this equation for T into the first equation for the total price
and the first equation becomes:
.
D + (1/16)D - 0.31 = 131.44
.
Get rid of the -0.31 on the left side by adding 0.31 to both sides to get:
.
D + (1/16)D = 131.75
.
Since D is equivalent to (16/16)D you can substitute that for D and you get:
.
(16/16)D + (1/16)D = 131.75
.
The sum of the two terms on the left side is:
.
(17/16)D = 131.76
.
You can solve this equation for D by either dividing both sides by (17/16) or by multiplying
both sides by (16/17). In either case you get:
.
D = (16/17)*131.75 = (16*131.75)/17
.
Calculator time. 16 times 131.75 = 2108 and dividing that product by 17 gives and answer of:
.
D = $124.
.
That's the answer. The cost of the desk was $124.00
.
Hope this helps you to understand the problem.

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You have two unknowns and therefore you need at least two independent equations to be able to
solve for the two unknowns.
.
One of the ways to solve for two equations with two unknowns is to make one of the equations have
a term in it that is equal in size but has the opposite sign as the corresponding term in the
other equation. Then add the two equations together in vertical columns and one of the unknown
terms drops out. That leaves you with a new equation that has only one variable, and you can
solve for that variable. Once you get one of the variables, you can go back to one of the original
equations, substitute for that variable, and solve for the second variable. Sounds complicated,
but it's not too bad ... let's just do the problem step by step.
.
Given the two equations:
.
4x + 5y = -14
8x + 10y = -20
.
Let's try to get rid of the variable x. We can do that by multiplying the top equation
(both sides ... all terms) by -2. This changes the top equation to:
.
-8x - 10y = +28
.
and when we put this over the original bottom equation the pair of equations is then:
.
-8x - 10y = +28
8x + 10y = -20
.
Notice now what happens when we add these two equations vertically in columns. The -8x
in the top equation cancels the +8x in the bottom equation ... so there is no x term in
the resulting equation.
.
But this problem also has a trick in it. Notice that when you add the terms in the y column,
the -10y and the +10y also cancels out. So we can't solve this problem as we normally would.
.
Since both the x-terms and the y-terms cancel out and the two terms on the right side are different ...
these two equations are for parallel lines. On a graph they will never cross and therefore,
there is no point of the form (x,y) which will satisfy both equations.
.
Not a good problem to start with if you are having trouble understanding the concept of
solving two equations with two unknowns. Just remember the process above and post a new
example that might have a common solution.
.
Sorry for the confusion above, but you've seen an example of a set of two linear equations in which
both variables disappear if you try variable elimination. And if that ever happens to you
again you now know what it means.
.

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The answer to this problem is choice b. But how do you find that answer to be correct.
.
Recognize two things. First, if two lines are parallel, they have the same slope.
.
Second, if you have an equation of the form y = mx + b, then m (the multiplier of x) is the slope
of the line represented by that equation.
.
You are given the line y = 4x - 2. From this you can tell that the slope of this line is +4
because +4 is the multiplier of x. [You can also tell that the graph crosses the y-axis at the
point where y = -2, but you don't need to know that for this problem.]
.
Now all you have to do is rearrange each of the equations in the four choices to find their slopes.
When you find the equation with the multiplier of x equal to +4 then you will know that
it represents the line that is parallel to the given equation.
.
Do answer a.
.
Given 12x - 4y = 8
.
Subtract 12x from both sides to get:
.
-4y = -12x + 8
.
Divide both sides (all terms) by -4 to solve for y and you have:
.
y = 3x - 2
.
In this equation the multiplier of x is +3 not +4. So it does not represent a parallel line.
.
Do answer c.
.
Given 5y - 25x = -10
.
Add +25x to both sides to get rid of the -25x on the left side and you have:
.
5y = 25x - 10
.
Solve for y by dividing both sides (all terms) by +5 and you have:
.
y = 5x - 2
.
The slope of the line represented by this equation is 5 and therefore it cannot be parallel either.
.
Do answer d.
.
Given -x + 4y = -8
.
Get rid of the -x on the left side by adding x to both sides to make the equation become:
.
4y = +x - 8
.
Solve for y by dividing both sides (all terms) by 4 and you have:
.
y = (1/4)x - 2
.
This cannot be parallel because its slope is +(1/4)
.
Finally do answer b
.
Given 4y - 16x = 10
.
Get rid of the -16x on the left side by adding +16x to both sides to get:
.
4y = 16x + 10
.
Solve for y by dividing both sides (all terms) by +4 and the equation becomes:
.
y = 4x + 10/4
.
The slope (the multiplier of x) is +4, the same as the slope of the equation you were originally
given. Therefore, this line is parallel to the given line. Answer b is the correct answer.
.
Hope this helps you to see how to do the problem.
.

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Since you know that the function produces a linear graph, you can represent this function in
slope intercept form as:
.
f(x) = mx + b
.
where m is the slope of the graph and b is the point where the graph crosses the y-axis.
.
The problem tells you that f(2) = -3. And this tells you that when x = 2, then f(x) = -3.
Substituting these values into the slope intercept equation results in:
.
-3 = m(2) + b
.
Which, when you rearrange the order of multiplication in the first term on the right side becomes:
.
-3 = 2m + b <=== call this equation 1
.
Similarly, the problem also tells you that f(-2) = 5. And this tells you that when x = -2, then
f(x) = 5. Substituting these values into the slope intercept equation results in:
.
5 = m(-2) + b
.
And again, rearranging the order of multiplication in the first term on the right side makes
this equation become:
.
5 = -2m + b <=== call this equation 2
.
So now we have the set of equation 1 and equation 2 as being:
.
-3 = 2m + b and
5 = -2m + b
.
You can eliminate the variable m by adding the two equations together vertically. Note
that the 2m and the -2m cancel each other when added. Therefore, the sum of the two equations
results in:
.
+2 = 2b
.
Solve for b by dividing both sides of this equation by 2 to get 1 = b.
.
Now that you know what b is, you can return to either equation, substitute that value for b
and solve for m. Let's return to equation 1:
.
-3 = 2m + b
.
Substituting +1 for b results in this equation becoming:
.
-3 = 2m + 1
.
Subtract 1 from both sides to get rid of the 1 on the right side and the equation reduces to:
.
-4 = 2m
.
Divide both sides of this equation by 2 to solve for m and you get:
.
m = -4/2 = -2
.
But in the slope intercept form, m (which is the multiplier of x) equals the slope of
the line. So the answer to this problem is that the slope is -2.
.
Check ... now that you know the values of m and b you can write the slope intercept form
for this equation as:
.
f(x) = mx + b = -2x + 1
.
So you have just:
.
f(x) = -2x + 1
.
To find f(2) substitute 2 for every x and you get:
.
f(2) = -2(2) + 1 = -4 + 1 = -3 .... that checks.
.
and to find f(-2) substitute -2 for every x and you get:
.
f(-2) = -2(-2) + 1 = +4 + 1 = +5 ... and that also checks.
.
We found the correct equation f(x) = -2x + 1 because everything checks out as specified
in the problem. Therefore, the slope of -2 (which is the multiplier of x) is correct.
.
Hope this helps you to understand the problem.
.

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"a number" suggests that you have an unknown number. Represent it by x.
.
The sum of "a number" and 7, can therefore be represented by (x + 7). Then 3 times that is:
3(x + 7).
.
"Yields the same result as" is just a fancy way of saying "equals" ... So up to this point
you have:
.
3(x + 7) =
.
Next, twice the same number is 2x and that decreased by 4 is 2x - 4. So the total algebraic
statement becomes:
.
3(x + 7) = 2x - 4
.

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The slope is defined as the vertical distance divided by the horizontal distance.
.
For this problem you are told that the vertical rise is 7.25 feet, and you are also told
that the horizontal distance over which this rise takes place is 13.71 feet. So for this
problem the slope is:
.

.
Using a calculator to divide 13.71 into 7.25 you get an answer of 0.528811086 and this slope rounds
off to the nearest hundredth as 0.53
.
The answer has no dimension associated with it. It is just a decimal value.
.
The answer you are looking for is just slope = 0.53
.
Hope this helps you to understand the problem.
.
And as added information, roofers and architects don't talk about what the decimal value
of a slope is. They generally talk in terms of how much the roof rises in a horizontal
distance of 12 inches. For this case the rise is .53 times 12 inches which is 6.36 inches.
So a roofer would say that the pitch (or slope) of the roof is 6.36 in 12.
.
Just for info only ...

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The first 5 prime numbers (divisible only by themselves and 1 and not counting the number 1) are
2, 3, 5, 7, and 11
.
So the most extra credit a student can receive on a test is 11 points. Since there are 5 cards
that you can draw and only one of those cards contains the number 11, the probability of
drawing that care is 1 out of 5 which is a probability of or or 20%.
.
Hope this helps you to understand a little more about probability and how it works.
.

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The answer is false. The graph of is shaped like the letter "V" as shown by this graph.
.

.
On the left side of the y-axis the slope of the graph is -1 ... a negative slope because the graph is
going down as you go to the right along the x-axis.
.
You need to convince yourself that the graph shown above is the graph of the absolute value
of x. Do this by selecting negative values and positive values of x and seeing if they give
you values of y that match the graph.
.
Hope this helps you to understand the problem and how to view it so you can come up with the
answer to the question ... and to recognize that the answer is false.
.

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The answer to this problem is found by first determining how many eggs are hidden. The number
of eggs that are hidden is the total of 12 and 16 which is a total of 28 eggs. Then you have
12 chances of finding a yellow egg as your first egg because there are 12 yellow eggs that are
hidden. So the probability of finding a yellow egg first is the ratio of the number of
yellow eggs to the total number of eggs. In fractional form this is:
.

.
and since 4 is a factor of both the numerator and the denominator of this fraction you can reduce
the fraction by dividing the numerator and denominator by 4 to get:
.

.
So D is the correct answer.
.
You can immediately eliminate answers B and C because the correct answer must have 28
as the original denominator. Therefore, the denominator of the correct answer must be
a factor of 28 which are 28, 14, 7, 4, 2 and 1. Therefore, the correct answer must have
one of these numbers as its denominator. Answers A and D do, but answers B and C do not.
Therefore, they can be eliminated.
.
Hope this helps you to understand the problem a little better.
.

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Let's assume that each of the persons in the line has a number ... including Jack and Jill.
Therefore, the pool of numbers is 100. Also assume that the persons in the line have the
number that represents their position in the line. (Jack's number would be 28 and Jill's 68.)
.
How many of those 100 numbers are between 28 and 68? Start with 29 and end with 67. There are 39
numbers between 28 and 68. So if you draw a number at random from a pool of 100 numbers you
have 39 chances that the number you pick is in the range of 29 through 67. Therefore,
the probability of picking one of those numbers is
.
Hope this helps you to understand the problem.
.

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On September 14, 2002 she was 70 yrs old. That means that on September 14, 2003 she was 71.
If she had lived to September 14, 2004 she would have been 72. But she died in March of 2004.
Therefore, she was about 71 and a half when she checked out.
.

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Although it is a small possibility, you could draw all 8 red marbles to begin with. Then you
could draw all 7 blue marbles. This means that you have removed 15 marbles and still don't
have a white one. However, the next three draws would only be white, so you would be done.
.
Therefore, you need to draw 18 of the 20 marbles to be absolutely sure that you had three
marbles of each color.
.
Hope this helps you to understand the problem.
.

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Given to multiply:
.

.
One way to do this is to multiply using the FOIL rule ... multiply the First terms in each set
of parentheses (in this case m times m), then multiply the Outside terms (m times - 4n), then
multiply the Inside terms (-4n times m), and then multiply the Last terms (-4n times -4n).
Then you simplify the results by combining like terms.
.
I prefer to look at it in a little different fashion because the FOIL rule (firsts, outsides
insides, lasts) is really set up for binomials ... two sets of parentheses with two terms in
each set. The approach I use works just as well, but it works no matter how many terms are in
each set of parentheses.
.
The approach I use is to take the first term in the first set of parentheses and use it to
multiply each of the terms in the second set of parentheses. Once that is done, take the second
term in the first set of parentheses and use it to multiply each of the terms in the second set
of parentheses. If there are more terms in the first set of parentheses, just take them one
at a time and use them to multiply all the terms in the second set of parentheses. Once
that is done, simplify the results by combining like terms in all the products.
.
In this problem you had:
.

.
Take the m in the first set of parentheses and multiply it by the m and then by the -4n in
the second set of parentheses and you get a product of:
.

.
Next take the second term in the first set of parentheses ... -4n and use it to multiply each of
the terms in the second set of parentheses. When you do, you get a product of:
.

.
Now combine all the terms. The two products are:
.

.
Note that the terms -4mn and -4mn combine to give -8mn. There are no other common terms.
So, as the answer you have:
.

.
Hope this helps you to understand a way of multiplying two quantities, each with multiple terms.
.

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Given:
.

.
One way to solve proportions such as this is to "cross-multiply". Multiply the numerator on
one side by the denominator on the other side. Do it for both numerators and put the equal sign
between the two products.
.
In this case, multiply the numerator on the left side by the denominator on the right side
of the equal sign. This is 7 times 8 and the result is 56.
.
Then multiply the numerator on the right side by the denominator on the left side. In this problem
that means multiplying 5 times y to get 5y
.
So the two products are 5y and 56. Put the equal sign between them and you have:
.
5y = 56
.
Solve for y by dividing both sides by 5 to get:
.

.
and that's the answer.
.
Hope this helps to give you an understandable approach to solving proportions.
.

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Given the problem:
.
i+i^2+i^3+...+i^49= ???
.
One way you can do this is by "patterns" ...
.
Note that the first four terms of this series is:
.
i^1 = +i
i^2 = -1
i^3 = -i
i^4 = +1
.
When you add these four terms, they combine to equal zero (+1 - 1 + i - i = 0
.
Now look at the next four terms in the series:
.
i^5 = i^4 * i = +1*i = +i
i^6 = i^4 * i * i = 1 * i * i = i^2 = -1
i^7 = i^4 * i^3 = +1 * -i = -i
i^8 = i^4 * i^4 = +1 * +1 = +1
.
This shows you that the sum in this series is +i - 1 - i + 1 and is the same as the series of
the first four terms. Therefore, it also totals to zero.
.
If you do the next 4 terms in the series ... i^9 through i^12 you will find that it also produces
the four terms +i - 1 - i + 1 and therefore also sums to zero.
.
The pattern cycles through each group of 4 terms ... each group of 4 adding to zero. This means
that up through i^48 you have 12 groups of 4 terms ... each group equaling zero. Therefore,
the sum of the series i + i^2 + i^3 + i^4 + i^5 .... + i^45 + i^46 + i^47 + i^48 totals zero.
So the sum of the series [i + i^2 + i^3 + i^4 + i^5 .... i^45 + i^46 + i^47 + i^48] + i^49
is equal to zero + i^49, and since i^49 is the the first term in the next group of 4, it
should have a value of i. Think of it this way also ... i^49 = i^48 * i = (i^4)^12 * i
and i^4 = +1. So (i^4)^12 is (+1)^12 = 1. So (i^48)*i = 1*i = i.
.
Putting it all together .... the sum of this entire series is 0 for the first 48 terms and + i for
the 49th term. Therefore, through the 49 terms of the series the answer to this problem is
just ... i.
.
Hopefully this all makes sense when you carefully work your way through it.
.

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Given to factor:
.

.
Notice first that each of the numbers in the three terms is divisible by 3, so the factor 3
can be removed. This results in:
.

.
So now let's concentrate on factoring the terms inside the parentheses. Note that can
only be factored into . Therefore, if the trinomial in the parentheses can be
factored, the factors will be of the form:
.
(5x ______)*(x _____)
.
Next notice that the last term in the trinomial is +2. It's factors can only be +2 and +1 or
-2 and -1 because they must multiply together to give +2. Since the middle term of the
trinomial is -7 we have to have minuses in the factors of the trinomial. So we assume that
the factors of +2 that we will use are -2 and -1. This leaves two possibilities:
.
(5x - 1)*(x - 2)
.
or
.
(5x - 2)*(x - 1)
.
When you multiply out the (5x - 1)*(x - 2) you get which reduces to
so this combination does not work.
.
When you multiply out the (5x - 2)*(x - 1) you get and this does reduce
to , just as we need.
.
So now we can say that the given:
.

.
factors to:
.

.
None of the expressions inside the parentheses can be factored, so this is the final answer.
.
Hope this helps to see how the given trinomial can be factored using a "logical" analysis and
a little bit of trial.
.

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I'm not sure that I understand your last sentence, but I get a different value for his average speed.
.
Use the formula that Distance = Rate times Time
.
Or in equation form this is
.

.
The units must be consistent ... D is in miles, R in miles per hour, and T in hours.
.
Apply this equation three times ... once for each speed.
.
The first application is for the Rate of 45 miles per hour. The Time at this speed is 10 minutes
which is (1/6) of an hour. Substitute these values into the equation and you get:
.

.
So at 45 miles per hour, the distance he travels in ten minutes is 7.5 miles.
.
Use the same process when his speed is 30 miles per hour. He travels at that speed for 5 minutes
which is (1/12) of an hour. Substitute these values into the equation and the result is:
.

.
So at 30 miles per hour, the distance he travels in 5 minutes is 2.5 miles.
.
Finally, at the speed of 15 miles per hour, for 15 minutes (or 1/4 hour) the distance he
travels is:
.

.
And at 30 miles per hour, the distance he travels in 15 minutes is 3.75 miles.
.
Adding up the the distances you get the total distance traveled is:
.
miles
.
And the total time it takes to travel that distance is:
.
minutes = 1/2 hour
.
The average speed is the total distance (13.75 miles) divided by the time (1/2 hour) which
results in an average speed of:
.
miles per hour
.
He covered 13.75 miles in the first 30 minutes. If he rides for the next half hour at an
average speed of 27.5 miles per hour, he will cover 13.75 miles in the second half hour.
So the total distance he will travel is 13.75 miles + 13.75 miles = 27.5 miles, the same distance
he would cover if he just rode at a constant speed of 27.5 miles per hour for 1 hour.
.
Does this answer your question? If not, please repost your problem and maybe some other
tutor will see something that I don't in this problem.
.

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The slope is defined as the vertical distance (V) divided by the horizontal distance (H).
.
This indicates that the slope (S) is given by the equation:
.

.
The problem tells you that V = +8.25 feet and that H = +18.56 ft. Substitute these values into
the equation and you get:
.

.
Do the division on a calculator ... divide 18.56 into 8.25 and you get:
.

.
Rounding this off to the hundredths place results in:
.

.
Answer b in the list of answers is the correct choice.
.
Hope this helps you to understand the problem and how to solve it.
.

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Given:
.

.
To solve for x you can begin by recognizing that you can get rid of the denominator
on the left side by multiplying both sides by 4 as follows:
.

.
On the left side, the 4 in the denominator divides into the 4 in the numerator, a division that
results in 1 times x. You can look at this as just canceling the 4 in the denominator with
the 4 in the numerator as follows:
.

.
After "crossing out" the 4 in the denominator with the corresponding 4 in the numerator, you
are left with just x on the left side. So you have:
.

.
Multiply out the right side. 4 times -6 is -24 and you can substitute -24 for the multiplication
and you have the answer:
.

.
That's the answer ... and how you get it step by step.
.
Hope this helps you to understand the problem and a few of the procedures you can use solve
equations.
.

You can put this solution on YOUR website!
This was previously answered as the posting to problem 117069 (answer 85147). To save you the
trouble of looking that answer up, here it is again:
.
In this forum it's pretty hard to do drawings that support geometric problems such as this one.
.
So, let's just say that for a weight on a ramp that goes up at an angle the
portion of the weight W that acts perpendicular to the ramp is given by the equation:
.

.
and the portion of the weight W that acts parallel to and down the ramp is given by the
equation:
.

.
In this problem, the weight W is the weight of the SUV which is given as 6000 lbs and the
angle of the ramp is 25 degrees.
.
The winch will have to generate enough force to overcome the portion of the weight of the
SUV that acts parallel to and down the ramp. [This assumes the rolling friction of the vehicle
is negligible.] So the winch will have to produce an amount of force equal to:
.

.
Once the vehicle is in motion the winch will have to pull with a force of 2535.71 lbs
parallel to the ramp to keep the SUV rolling up the ramp at a steady rate of movement.
.
Hope this helps you to understand the problem a little better. If you expect to do many of
these problems, you might want to memorize these two "ramp" equations.
.

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Given:
.

.
To solve for x, begin by isolating the term by dividing both sides of this equation
by 3 to get:
.

.
To get this into a more standard arrangement, transpose this equation (switch sides) to
put it into the form:
.

.
Square both sides and you get:
.

.
So the answer to this problem is:
.

.
Hope this helps you to understand the problem and the process you can follow to get the answer.
.

You can put this solution on YOUR website!
Given the function:
.

.
with the task of finding the domain of t ... the spectrum of values that t may takes.
.
When you are finding the domain of a variable, you look for values of the variable that cause a
problem in finding the value of the function. Ask yourself, "In this problem is there any value
of t that would make it impossible to find the corresponding value of g(t)?" For example,
is there a value for t that would cause a division by zero? [The answer to that question is, "No."]
Or is there a value of t that would result in the square root of a negative number? [Again,
the answer to that question is, "No."]
.
As a matter of fact, there is no real value of t that will cause any difficulty in evaluating g(t).
.
Just to convince yourself of this, you may want to try a spectrum of values for t and see
what you get for g(t). For example, try t = -10,000. That means that g(t) will be:
.

.
The denominator will be very large, meaning that g(t) will be very small. But g(t) will
be a valid real number.
.
How about when t = 0? You will get:
.

.
Any number raised to the zero power will equal +1. So when t = 0, g(t) has the valid real
value of +1.
.
Then try evaluating g(t) when t = +20,000. You will get:
.

.
The answer is that g(t) will be a very large real number, but it can be calculated.
.
This is just a little thought process to help you visualize what values t might assume ... that
is to say, "What is the domain of t?"
.
By now you should be pretty convinced that t can have any real value in the spectrum of
numbers on the number line all the way from minus infinity (-oo) to plus infinity (+oo).
.
In inequality form we can write this as:
.

.
Hope this helps you to understand the problem a little more.
.

You can put this solution on YOUR website!
Given the equation:
.

.
In which P represents the percent (as a whole number) of the inbound e-mail in the U.S. that is spam,
and x represents the number of years after 2001 that will determine what P is.
.
For this problem you are asked to solve for x if P is 90 percent. Begin by substituting
90 for P in the equation to get:
.

.
Get rid of the 12 on the right side by subtracting 12 from both sides. When you do that subtraction,
you reduce the equation to:
.

.
Just to get this into a little more standard form ... with the unknown on the left side ...
transpose (switch sides) this equation to:
.

.
Divide both sides by 50 to reduce the equation to:
.

.
which (leaving out the the step that shows 78 divided by 50} is just:
.

.
This is a logarithmic form of an equation. It can be converted to an equivalent exponential form
by applying a rule that says the logarithmic form:
.

.
is equivalent to the exponential form
.
In this problem b, the base of the logarithm, is e which is the base of the natural logarithms
that are commonly indicated by "ln". By comparing the logarithmic form of this rule to the logarithmic
form derived in our problem, you can see that b = e, y = 1.56, and x = x. Substitute these
values into the exponential form of the rule and you get:
.

.
as being the exponential form that is equivalent to
.
You can use the exponential form to solve for x. Transpose the exponential form to get the
x in the more standard arrangement of having the unknown on the left side and the exponential
form becomes:
.

.
You can now use a scientific calculator to solve for x. Just think of e as being a number that
you are going to raise to the 1.56 power. [The value of e is approximately 2.718281828]
.
So you can use your calculator to raise 2.718281828 to the 1.56 power. Or look on your calculator
for the function key that shows . If you have one, your calculator might work
this way: (1) enter 1.56, then (2) activate the key.
.
Either way you should get an answer of 4.758821245 which rounds off to 4.76. This tells you
that 4.76 years after 2001 the inbound spam in the U.S. can be expected to be 90 percent of
the message traffic.
.
(Assuming that this equation begins on December 31, 2001 ... 4.76 years later
(4 and three-quarters years later) the spam level would reach 90 percent around September 30, 2006
which is nearly 5 years after December 31, 2001 ... actually 4.75 years after December 31, 2001
...
.
Hope this helps you to see your way through this problem and helps you also to learn a little
about working with logarithms ... especially natural logarithms (indicated by ln).

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Use the equation:
.

.
for the two parts of this problem. D represents the distance traveled, R the rate or speed
of the travel, and T the time spent traveling.
.
For the first part of the problem, call the unknown rate of travel S (standing for speed). And the
problem tells you that the bicyclist rode out into the country for a time T equal to 5 hours.
Putting these values into the Distance equation results in:
.

.
On the return trip the Rate is S + 5 because you are told that it is 5 miles per hour faster
than on the ride out into the country. You are also told that the duration of the return
ride is 4 hours. Putting these values into the Distance equation for the return ride results in:
.

.
Do the distributed multiplication on the right side by multiplying the 4 times each of the
two terms in the parentheses. This multiplication results in the Distance equation becoming:
.

.
Now recognize that the two distances ... the one going into the country and the one returning
are equal. So the right sides of the two Distance equations must be equal. In equation form
this is represented as:
.

.
Get rid of the 4S on the right side by subtracting 4S from both sides. Subtracting 4S
from both sides reduces the equation to:
.

.
This tells us that the bicyclist rode at a speed of 20 miles per hour going out into the country,
and, since the rate or speed was 5 miles per hour faster on the return trip, the speed on
the return trip was 25 miles per hour.
.
And as add information (not required by the problem) the distance she rode into the country
can be determined by multiplying her speed of 20 miles per hour on the outbound trip by the 5 hours
she spent on this portion of the trip. 20 times 5 is 100 miles. On the return trip she rode
at a speed of 25 miles per hour for hours, and 25 times 4 also equals 100 miles, just as it is
supposed to since the distances are equal in both directions.
.
Hope this helps you to understand the problem a little better and helps you to understand the
equation that says Distance equals Rate times Time.
.

You can put this solution on YOUR website!
In this forum it's pretty hard to do drawings that support geometric problems such as this one.
.
So, let's just say that for a weight on a ramp that goes up at an angle the
portion of the weight W that acts perpendicular to the ramp is given by the equation:
.

.
and the portion of the weight W that acts parallel to and down the ramp is given by the
equation:
.

.
In this problem, the weight W is the weight of the SUV which is given as 6000 lbs and the
angle of the ramp is 25 degrees.
.
The winch will have to generate enough force to overcome the portion of the weight of the
SUV that acts parallel to and down the ramp. [This assumes the rolling friction of the vehicle
is negligible.] So the winch will have to produce an amount of force equal to:
.

.
Once the vehicle is in motion the winch will have to pull with a force of 2535.71 lbs
parallel to the ramp to keep the SUV rolling up the ramp at a steady rate of movement.
.
Hope this helps you to understand the problem a little better. If you expect to do many of
these problems, you might want to memorize these two "ramp" equations.
.

You can put this solution on YOUR website!
The formula for continuous compounding is:
.

.
in which P is the total future value, C is the initial deposit of money, e is the base of the
natural logarithms and is approximately 2.718281828, r is the annual interest rate expressed
as a decimal, and t is the number of years of the investment.
.
For this problem you are told that the initial deposit of money is $2000. Substituting
this value for C in the equation results in the equation becoming:
.

.
You are also told that the annual interest rate is 6%. In decimal form this is 0.06. Substitute
this decimal for r in the equation to get:
.

.
Now that we are this far, we can substitute the two values of t that you are asked to
find the future value (P) for. In the first case you are asked to find the future value
of the $2000 initial deposit if it is left in the investment for 1 year. Substitute 1 for
t in the equation and the equation becomes:
.

.
Now it's basically a calculator problem. The exponent is 0.06 times 1 so it just equals 0.06
and the problem is then:
.

.
If you use your calculator to raise e (or 2.718281828) to the exponent 0.06 you find that
it is 1.061836547 and the equation reduces to:
.

.
When rounded off the answer is that using continuous compounding at an annual rate of 6%
a $2000 initial investment will be worth $2123.67 at the end of one year.
.
If the $2000 is left on deposit for 2 years, the only thing that changes in the equation is the
value of t. For this calculation, t = 2 and the equation:
.

.
becomes:
.

.
Using your calculator on you should find that it equals 1.127496852 and this reduces
the equation to:
.

.
When rounded off the answer is that using continuous compounding at an annual rate of 6%
a $2000 initial investment will be worth $2254.99 at the end of two years.
.
Hope this helps you to understand how to do continuous compounding for an investment.
.

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Wow! That's a pretty big sphere. A diameter of 20 km is equivalent to a diameter of almost 12.5 miles!
.
Anyhow, the formula for the volume (V) of a sphere is:
.

.
where V is the volume and r is the radius of the sphere.
.
If the diameter of a sphere is 20 km, then the radius of that sphere is half that or the radius
of the given sphere is 10 km.
.
Substituting 10 km for r in the volume equation results in:
.

.
and
.
This says that we can substitute 1000 for in the formula to get:
.

.
Multiplying the 1000 times the results in and dividing the
4000 by 3 results in . Substituting 1333.33333333 for
reduces the equation to:
.

.
and using a scientific calculator we can multiply out 1333.33333333 times to get:
.

.
And since the radius was in kilometers this answer is in cubic kilometers. So the answer to
this problem is:
.
Volume = 4188.790205 cubic kilometers
.
Hope this helps you to understand the problem a little better.
.