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Janet, the best I can do with your description is the following:
It looks like you have a three-dimensional solid with one face in the shape of a trapezoid.
The trapezoid has dimensions of:
Base = 25ft.
Left side (perpedicular to the base) = 9ft.
Right side (perpendicular to the base) = 16ft.
Now extend this trapezoid shape 18ft. (We'll call this the "depth" of the solid) away from, but perpendicular to, the 25ft. base.
To find the volume of this solid, first find the area of the trapezoid.
Substitute , , and



sq.ft.
Now find the volume by multiplying the area of the trapezoid (312.5sq.ft.) by the "depth" (18ft.).

cu.ft.
I hope I have interpreted your description correctly.

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First off, "indirect " variation and "inverse" variation mean exactly the same thing.
What is it.
When one quantity decreases as another quantity increases, the two quantities are said to vary inversely or indirectly.
Here's an example:
The air temperature goes down the higher you go.
So as the height increases, the temperature decreases.
We say that the air temperature varies inversely (indirectly) as the height.
You can write this algebraically as: (h = height and T = temperature)
The k is called the constant of variation and its value will depend upon the initial values of h and T that are provided in the problem.
For example, if you are told that the air temperature is 30 degrees farenheit at a height of 10,000 feet, you have enough information to find the value of k, the constant of variation.
Substitute T = 30 and h = 10,000
Multiply both sides by 10,000
so the equation now is:
and this allows you to find the air temperature at any height, h.

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First factor an x.
Now factor the trinomial.

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Remember; Slope = rise over run.
In this case, the rise is given as 7.25 inches and the run (also known as tread) is given as 9 inches.
So the slope is =

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Factor:
Rewrite this as:
Factor the 3 which is a common factor of both terms.

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Notice that when you do as the problem instructs, you form a right triangle whose hypotenuse is the plank (25ft.) and whose base is the distance from the bottom of the plank to the side of the barn (7ft.).
The hypotenuse (labeled c) and the base (labeled b) can be used as b and c in the Pythagorean theorem. The height reached by the top of the plank will be repesented by a:
Substitute c = 25 and b = 7.
Simplify.
Subtract 49 from both sides
Finally, take the square root of both sides.

The plank will reach exactly 24 feet up the barn wall.

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How to factor:

To factor this trinomial, you need to find two binomials which, when multiplied, will give you the original trinomial.
In other words, find and such that
First, you look at the coefficient of , and that's 5.
Now ask what are the factors of 5?
Well, 5 is a prime number, so the factors are 1 and 5.
So you can start with:
Notice the and
Now look at the constant term, that's 24.
You need to find the two integers, m and n, whose product is 24 and whose sum 5*n+1*m = 34.
Let's factor 24.
1X24 = 24
2X12 = 24
3X8 = 24
4X6 = 24
After a little trial & error, you'll find:
5*6+1*4 = 34
So this means that n = 6 and m = 4
Now you can finish by writing:

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Given the base = height + 2 or:

And the area is 12 sq.in.
Start with the formula for the area of a triangle:
Substitute and
Simplify by multiplying by 2.
Simplify.
or
Rearrange into standard quadratic equation form.
Solve by factoring.
Apply the zero product rule.
or which means that...
or Discard the negative solution as height (h) is a positive quantity.

The height is 4 inches and the base is 6 (h+2)inches.

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Find all roots by factoring:
First, factor an x since this is a common factor in all three terms.
Now factor the trinomial.
Now set
Apply te zero product rule.
or or so the roots are:

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Substitute
Simplify.
The bases are equal, so the exponents are equal.

Check:

so...

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Let n be the number
"Two-thirds of a number increased by 10, gives the same result as 25 subtracted from three times the number."
Multiply through by 3 to clear the fraction.
Subtract 2n from both sides.
Add 75 to both sides.
Finally, divide both sides by 7.

The number is 15.

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Not quite sure what you are asking here!
Is it:
??? If so, then rewrite this in exponential form:
---> so...
---> Now substitute and to get...
The bases are equal, so the exponents are equal, and...
Divide both sides by 2.
so finally, we have...

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Let n = the number.
"...the square of a number (n) is decreased by 15, the result is equal to the number increased by 15."
Add 15 to both sides.
Subtract n from both sides.
Subtract 30 from both sides.
Solve this quadratic equation by factoring.

or
As you can see, there are two answers:
-5 and 6

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Graph:

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Let's see what useful information we have!
The product of their ages is 36.
They have a pair of twin daughters.
Mr. & Mrs. Wang are 40 years old.
What are the possible ages of the children?
Let's first find the factors of 36 which will represent the possible ages of the three children:
1*1*36 = 36
1*2*18 = 36
1*3*12 = 36
1*4*9 = 36
1*6*6 = 36
2*2*9 = 36
3*3*4 = 36
Now select from this list the entries that have double (twin) integers.
1*1*36 = 36
1*6*6 = 36
2*2*9 = 36
3*3*4 = 36
The first entry doesn't qualify because if the parents are 40, they would not have a child who is 36.
This leaves the remaining entries:
1, 6, and 6.
2, 2, and 9.
3, 3, and 4.
as the possible ages.

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Please post your specific problems then a tutor can help you.

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You can start with the slope-intercept form:
Find the slope:
Substitute the x- and y-coordinates from the two given points: (-1,-5) and (-4,-9).
Evaluate.


So now you have:
Substitute the x- and y-coordinate values from any one of the tw given points and solve for b.
Evaluate.
Add to both sides.
The final eaquation is:

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You can start by assigning variables to the unknown quantities, the invested amounts in gold and in silver.
Let G = the number of dollars invested in gold and S = the number of dollars invested in silver.
The sum of these two amounts is given as $3,500.00, so you can write:
1)
Now we must find an equation for the amount of interest earned.
The interest earned on the gold investment is 15%(G) and the interest earned on the silver investment is 12%(S), and the sum of these two amounts is given as $480.00
So, after changing the percents to their decimal equivalents, you can write the equation:
2)
Now let's replace the G in equation 2) with its equivalent: which we get from equation 1) by solving for G.
3) Simplify and solve for S.
3a)
3b) Subtract 525 from both sides.
3c) Finally, divide both sides by -0.03
3d) This is the number of dollars invested in silver.


and this is the number of dollars invested in gold.

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Find the hourly rate of each person.
If Kevin can do the job in 5 hours, then he can do of the job in 1 hour.
If Carrie can do the job in 4 hours, then she can do of the job in 1 hour.
So together, they can do of the job in 1 hour.
This means that it will take them hours to complete the job together.
hours = hours,minutes and seconds.

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I don't know that I can show you a "formula" but I can show you a procedure.
You want your answer to be an equation in the slope-intercept form:
of a line that is perpendicular to the line and which passes through the point (4, 7).
Let's start with the slope:
If two lines are perpendicular, their slopes are the negative reciprocal of each other.
The given line is: . Since this is in the slope-intercept form: you can see that the slope .
The negative reciprocal of 2 is .
So you can start with:
but now you need to find the value of b, the y-intercept.
So we substitute the x- and y-coordinate values of the given point (4, 7) into the equation above and solve for b.
Simplify this:
Add 2 to both sides.
Now that we have the value of b, we can write the final equation.

And just to show you what these look like when graphed:

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Write with no denominators:

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First, think in terms of the amounts of antifreeze in this situation.
Let x = the number of quarts of pure antifreeze to be added.
How much antifreeze is there in the six quarts of 10% antifreeze solution?
6quarts(10%) = 0.6 quarts of antifreeze.
The final antifreeze solution will be 6+x quarts of 30% antifreeze.
So now we can write the necessary equation to solve for x.
x+0.1(6) = 0.3(x+6) Simplify and solve for x.
x+0.6 = 0.3x+1.8 Subtract 0.3x from both sides.
0.7x+0.6 = 1.8 Subtract 0.6 from both sides.
0.7x = 1.2 Finally, divide both sides by 0.7
x = 1.71429
So you'll need to add 1.71429 quarts of pure antifreeze to the 6 quarts of 10% antifreeze solution to obtain 7.71429 (6+1.71429) quarts of 30% antifreze solution.

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Let x = the number of years when the two trees will be the same height.
For the 2ft tree, you can express its height in x years as:
2+1(x)
And for the 6ft. tree...
6+0.5(x)
So you need to find the value of x that would make these equal.
2+x = 6+0.5x Subtract 0.5x from both sides.
2+0.5x = 6 Now subtract 2 from both sides.
0.5x = 4 Finally, divide both sides by 0.5
x = 8
So in 8 years, the two trees will be the same height.

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The height (as a function of time, t) of an object falling from 100 meters is described by:

i) Find the velocity at t = 2
The velocity is the rate of change of distance with respect to time, t.
If we take the first derivative of the height fuction, we will get the eqation for the velocity.
and at t=2, we get:

meters/second. The negative indicates a downward direction.
ii) Find the acceleration at t = 2.
Acceleration, which is the rate of change of velocity with respect to time, t, can be found by taking the second derivative of the height function ():
=meters/second squared.

meters/second squared. A constant.

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The solution(s) to a quadratic equation is(are) x = 9.
Yes, a quadratic equation may have a "single" solution, but this is known as a "double" solution.
To write the equation whose solution(s) is(are) x = 9, you will start by setting x = 9.
To get such a solution you must have found the factors of the quadratic equation to be:
and Now if you multiply these factors together, the result will be the generating quadratic equation:

So the equation is:

Here's the graph of this quadratic equation and you can see why there appears to be a "single" solution:

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Solve
When you are asked to solve a problem such as this, you are being asked to find the value of x that, when added to -5 will give the result of 20.
You can solve this equation for x this way:
To isolate the variable, x, you must add 5 to both sides of the equation. When you do this, you are "undoing" the -5 so that you are left with the variable by itself on the left side of the equation.
Because , we are left with x by itself on the left side:
...and on the right side, you can see that so...

To see if this is really the correct solution for x, we'll substitute into the original equation to see if both sides are equal.
Substitute and, since or we have:

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Factor:

The first term has the factors:
The last term has the factors: or
Since the middle term is: we can get this by , so...

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Multiply & simplify:
Using FOIL, we get:
Simplifying gives us:
or
-

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You could start with the general formula for the height of an object (s) as a function of time (t), s(t), propelled upwards with an initial velocity () from an initial height ():
where g (constant of acceleration due to gravity) = -32 ft/sec^2. So substitute g = 32, , and

Note: The acceleration due to gravity is negative because it is in a downward direction.

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If you look at the tepee from the side you will see an isoceles triangle whose height is 4 feet and whose base is 6 feet.
If you drop a line from the apex perpendicular to the base you will have two right triangles. The height of the right triangle is 4 feet and the base will be just half of the width of the base of the isoceles triangle, or 3 feet.
You need to find the length of the hypotenuse of the right triangle.
Use:
Substitute a = 3 and b = 4

Take the square root of both sides.

The PVC plumbing pipe should be 5 feet long.

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The area of a regular polygon can be expressed in terms of its apothem (r) (the perpendicular distance from the center to a side of the polygon) and the number (n) of sides in the polygon.
A regular hexagon has 6 sides so n = 6 and the apothem (r) is given as
Substitute n = 6 and :
Evaluate.


sq.cm.