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Fish tank is rectangular. Adding the rocks results in a rise of 2cm.
So adding the rock displaced enough water to fill a "slice" that is 2cm x 40cm x 70cm
The volume in that slice is

cubic cm
1 Liter is 1000cc, so the rock has a volume of 5.6 liters.

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Get out a pencil, your protractor and a ruler :)
http://homeschoolmath.blogspot.com/2006/04/finding-values-of-sine-without.html

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Sum of the angles in a triangle is 180 degrees

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The sum of any two sides of a triangle must be larger than the length of the remaining side.
1) 20 + 15 < 40. So the side cannot be 20
2) 30 + 15 > 40
15+40 > 30
40 + 30 > 15
So 30 can be an answer
3) 40 + 40 > 15
15 + 40 > 40
So 40 can be an answer
4) 50+40 > 15
40+15 > 50
50 + 15 > 40
So 50 can be an answer

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a) Find A(5).




b) If the height of the sail is x +3 meters, then what is the length of the base of the sail?
Area of a triangle is





If you need a nice place to factor expressions. try this
http://72.3.253.76/webMathematica3/quickmath/page.jsp?s1=algebra&s2=factor&s3=basic

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The triangle ABC has 2 45degree angles. That means the third angle (angle at c) is 90 degrees. Since two angles are equal, the sides opposite those angles are equal. Call their length L. You are told the hypotenuse is 16m
So use pythagorean theorem.

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here is a URL for volume --> http://math.about.com/od/formulas/ss/surfaceareavol.htm
Let's think about this first.
We have a sphere that will sink in the water.
The cylinder is wide enough to contain the sphere. (one could argue that a sphere with radius 10 won't fit into a cylinder with radius 10. But we will let that one go for now)
When we drop the sphere into the cylinder of water, it sinks to the bottom.
How much water is displaced by the sphere we dropped in?
The volume of a sphere is given by
So we know how much water is displaced.
Now that we know the amount of water that 'moved', and we know the container holding that water is a right circular cylinder, all we need to know is the formula for the volume of such a cylinder.

Now we can equate those two volumes and solve for h


We are told r = 10 , so that yields = centimeters

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this problem deals with units conversion. In this case, you are told that 6oz=168granms
You are told how expensive 6oz of yogurt is. You are then asked how much a larger amount of yogurt (in grams) will cost.
There are several wasy to do this. Here is one way
1) Start by converting the cost of 6oz into the cost of that same amount in grams (for other problem yet to come, you will most certainly be asked to do similar converions. BUt, soon enough you will have to use more typical unit conversion ratios. For instance, you'll be told that 1oz = 28 grams rather than 6oz=168)
You are told 6oz= 168g, so you know that 168g of yogurt costs the same as 6oz. So the cost fo 168g of yogurt is 1.50
Now, you can use ratios to find the 'final answer'



This makes sense since 840/168 = 5 and 7.50/1.50 is also 5

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First find the volume in a square pyramid.
http://www.mathsteacher.com.au/year10/ch14_measurement/25_pyramid/21pyramid.htm


cubic feet
Next find the volume in one cube shaped apartment


cubic feet
Now see how times larger in volume the pyramid is compared to the apartment



So the volume of 17183 complete apartments fits in there.

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move any powers to negative exponents in the numerator, down into the denominator.
Move any powers to negative exponents in the denominator, up into the numerator.

associate and consolidate powers

cancel terms as appropriate (in this case there is a b^2 in both numerator and denominator, so they cancel out (b^2/b^2) is equal to 1) )

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http://www.math.com/school/subject3/lessons/S3U2L3GL.html#
Pick one :) I like trapezoid

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vertical asymptotes occur when the denominator goes to 0. So find the values of x that make the denominator zero and you have a possible asymptote.
You factored everything correctly, so the vertical asymptote is at x = 6
You can graph this function here --> http://72.3.253.76/webMathematica3/quickmath/page.jsp?s1=graphs&s2=equations&s3=basic
Or use geogebra or a graphing calulator

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Use this tool --> http://www.algebra.com/algebra/college/linear/solving-linear-system-by-substitution.solver as follows

Solved by pluggable solver: Solving a linear system of equations by subsitution

Lets start with the given system of linear equations Now in order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to choose y. Solve for y for the first equation Subtract from both sides Divide both sides by 7. Which breaks down and reduces to Now we've fully isolated y Since y equals we can substitute the expression into y of the 2nd equation. This will eliminate y so we can solve for x. Replace y with . Since this eliminates y, we can now solve for x. Distribute -4 to Multiply Reduce any fractions Add to both sides Make -13 into a fraction with a denominator of 7 Combine the terms on the right side Make 5 into a fraction with a denominator of 7 Now combine the terms on the left side. Multiply both sides by . This will cancel out and isolate x So when we multiply and (and simplify) we get <---------------------------------One answer Now that we know that , lets substitute that in for x to solve for y Plug in into the 2nd equation Multiply Add to both sides Combine the terms on the right side Multiply both sides by . This will cancel out -4 on the left side. Multiply the terms on the right side Reduce So this is the other answer <---------------------------------Other answer So our solution is and which can also look like (,) Notice if we graph the equations (if you need help with graphing, check out this solver) we get graph of (red) and (green) (hint: you may have to solve for y to graph these) intersecting at the blue circle. and we can see that the two equations intersect at (,). This verifies our answer. ----------------------------------------------------------------------------------------------- Check: Plug in (,) into the system of equations Let and . Now plug those values into the equation Plug in and Multiply Add Reduce. Since this equation is true the solution works. So the solution (,) satisfies Let and . Now plug those values into the equation Plug in and Multiply Add Reduce. Since this equation is true the solution works. So the solution (,) satisfies Since the solution (,) satisfies the system of equations this verifies our answer.

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clear the left side denominator by multipying both sides by 3

expand the right side
collect like terms (subtract 12x from both sides)

isolate x by dividing boths sides by -11

Now verify the answer by pluggin your solution into the original equation


which is true so the naswer is correct

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The following was solved using a 'solver tool'. Since the x coefficients are already 1 and -1, you can ignore that part of the solution. The rest works fine.
If you want to use that tool for other similar problems, look here
http://www.algebra.com/algebra/college/linear/solving-linear-system-by-elimination.solver

Solved by pluggable solver: Solving a System of Linear Equations by Elimination/Addition

Lets start with the given system of linear equations In order to solve for one variable, we must eliminate the other variable. So if we wanted to solve for y, we would have to eliminate x (or vice versa). So lets eliminate x. In order to do that, we need to have both x coefficients that are equal but have opposite signs (for instance 2 and -2 are equal but have opposite signs). This way they will add to zero. So to make the x coefficients equal but opposite, we need to multiply both x coefficients by some number to get them to an equal number. So if we wanted to get -1 and 1 to some equal number, we could try to get them to the LCM. Since the LCM of -1 and 1 is -1, we need to multiply both sides of the top equation by 1 and multiply both sides of the bottom equation by 1 like this: Multiply the top equation (both sides) by 1 Multiply the bottom equation (both sides) by 1 So after multiplying we get this: Notice how -1 and 1 add to zero (ie ) Now add the equations together. In order to add 2 equations, group like terms and combine them Notice the x coefficients add to zero and cancel out. This means we've eliminated x altogether. So after adding and canceling out the x terms we're left with: Divide both sides by to solve for y Reduce Now plug this answer into the top equation to solve for x Plug in Multiply Subtract from both sides Combine the terms on the right side Multiply both sides by . This will cancel out on the left side. Multiply the terms on the right side So our answer is , which also looks like (, ) Notice if we graph the equations (if you need help with graphing, check out this solver) we get graph of (red) (green) (hint: you may have to solve for y to graph these) and the intersection of the lines (blue circle). and we can see that the two equations intersect at (,). This verifies our answer.

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Let the smaller integer be X.
Then the other integer, call it Y, is equal to 3X
Their sum is 108

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Given
1.Approximately how many cases of the infection will there be 16 years from now?


millions

2.In approximately how many years will there be 6 million cases?
square both sides to eliminate the radical

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You are told 1 oz = 28 g , so the conversion between grams and ounces is
You are then asked to convert 26 oz into grams.

Note that when you look at the equation above, the "oz" units cancel jus the same way numbers do. That is, oz/oz = 1 -- so you can 'cancel' it.

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Pythagorean Theorem states that given a right triangle, then
converse is : if the sides of a triangle are related by , then it must be a right triangle.
So does
Now just grind out the arithmetic
yepper. So it is a right triangle

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This is a good URL to factor with --> http://72.3.253.76:8080/webMathematica3/quickmath/page.jsp?s1=algebra&s2=factor&s3=basic
Make sure to just enter in 4y2 + 3y - 1

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Let the 3 numbers be given as
I
I+2
I+4
So, the eqaution to solve is

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Sure you can graph it on a graphing calc.
Or go here --> http://www.geocities.com/igraph.online/graphing-calculator.html
At that URl, enter log(5x) and click Draw Graph
On your graphing calc (TI-83 here)
1) Y= button and enter Log(5x) for y1
2) Window button to set Xmin =0 Xmax=10 Xscl=1 Ymin-0 Ymax=10 Yscl=1
3) Graph

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Let the current ages be "x" and "y" (x for father, y for son)
Given



and

Solved by pluggable solver: Solving a linear system of equations by subsitution

Lets start with the given system of linear equations Now in order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to choose y. Solve for y for the first equation Subtract from both sides Divide both sides by -7. Which breaks down and reduces to Now we've fully isolated y Since y equals we can substitute the expression into y of the 2nd equation. This will eliminate y so we can solve for x. Replace y with . Since this eliminates y, we can now solve for x. Distribute -5 to Multiply Reduce any fractions Add to both sides Make 6 into a fraction with a denominator of 7 Combine the terms on the right side Make 2 into a fraction with a denominator of 7 Now combine the terms on the left side. Multiply both sides by . This will cancel out and isolate x So when we multiply and (and simplify) we get <---------------------------------One answer Now that we know that , lets substitute that in for x to solve for y Plug in into the 2nd equation Multiply Subtract from both sides Combine the terms on the right side Multiply both sides by . This will cancel out -5 on the left side. Multiply the terms on the right side Reduce So this is the other answer <---------------------------------Other answer So our solution is and which can also look like (,) Notice if we graph the equations (if you need help with graphing, check out this solver) we get graph of (red) and (green) (hint: you may have to solve for y to graph these) intersecting at the blue circle. and we can see that the two equations intersect at (,). This verifies our answer. ----------------------------------------------------------------------------------------------- Check: Plug in (,) into the system of equations Let and . Now plug those values into the equation Plug in and Multiply Add Reduce. Since this equation is true the solution works. So the solution (,) satisfies Let and . Now plug those values into the equation Plug in and Multiply Add Reduce. Since this equation is true the solution works. So the solution (,) satisfies Since the solution (,) satisfies the system of equations this verifies our answer.

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This is a very poorly worded problem. For the answer to be 1/8, the problem should read :
If 1/2 the area of a garden is vegetable plants and 1/4 the vegetable garden is tomatoes, how much of the area is tomatoes? In that case, the answer would be "1/4 of 1/2" = 1/4 * 1/2 = 1/8
As it reads now, I would say the 1/4 of the area is tomato since "1/4 the total area is tomatoes, how much of the area is tomatoes?"

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Boyles Law says PV is a constant.
So

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Sqrt is undefined for negative numbers. (You need complex numbers for that). So you are being asked "for what values of x is ?"
You know that a product of two numbers is positive when both numbers are postive OR both numbers are negative. The product is negative when one number is postive and the other is negative.
You know the product is 0 when either factor is 0.
So, you can see that the points where the product is 0 are
x = 0 and x = 3
Now break the number line into three intervals.
x<0
0<=x<=3
x>3
Now, pick a value in each interval and see if that product of is positive or negative.
Let's pick x = -2
= . The product is positive, so the result is defined then.
Now pick x = 1
= . The result is negative, so the sqrt(-2) is undefined
Finally, pick x = 5.
= . again the result is positive. So the sqrt is defined.
domain is valid when
x<=0 OR x>=3
It is undefined when

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You are probably used to graphing linear equations in 2 variables. That ends up being a line on a 2 dimensional coordinate system.
When graphing a linear equation in 3 variables, you end up with a plane in a 3 dimensional coordinate system.
See this URL --> http://academic.cuesta.edu/jpandosh/3x3_supp/web/3x3_sys.pdf
The method shown there is a great way to do what you are trying to do.

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The total volume is gallons
Let the concentration of juice in A be given by X
The mixture is formed as follows:




x = 14%