Sam's Responses in WyzAnt Answers

Dear Louis,

You can figure out the precise distance between these two points by using what you know about right triangles:  that the sum of the squares of the legs equals the square of the hypotenuse.

Here's how.  First, plot both of these points (-2, 5) and (12, -1) on a piece of graph paper.  Then what I want you to do is draw two lines:  the first is a horizontal line going from (-2,5) to (12, 5).  Then, turning the pencil in a different direction, draw a vertical line that goes from (12, 5) to (12, -1).

Voila.  You have a right triangle and you will notice the length of the hypotenuse of this triangle is precisely equal to the length between these two points.  If you count the tick-marks, you'll notice the legs of this right triangle have lengths 14 and 6.  14 squared plus 6 squared is equal to 232.  The answer is, therefore, that the length is the SQUARE ROOT OF 232 or about 15.23. 

The quick method is to think the DIFFERENCE in X is one leg of the triangle and the DIFFERENCE in Y is the other leg of your triangle.   The difference between (-2) and 12 is 14.  The difference between 5 and -1 is 6.  Etc.

Mr. Gets Results

Here's what you do.  They say the AREA is 3 times the square root of 2 + 5.  I am assuming by 2 + 5 you mean 7.  If the number 5 is NOT in the radical, then this whole thing would be 5 + 3sqrt(2) but I interpreted that as 3(sqrt(7)).

Now knowing that the area of this shape is 3sqrt(7) the way you figure out the other dimensions of the shape depends on what kind of shape it is.  If it's a circle, then you can solve for the radius and you will get (3(sqrt(7))/pi.  Then take the square root of THAT number and you have your exact radius to however accurately you want to round it.  God bless calculators.  The answer is about 1.59.  This is the radius of the circle.  Its diameter is about 3.18 and it's circumference is about 9.99.  You can figure all that out with a geometry cheat-sheet.

On the other hand, if it's a square you are looking for, then you just take the square root.  Don't worry about pi.  It's just the square root of 3sqrt(7).  It's about 2.82.  2.82 is the measurement (about) of each of the four sides of the square.  By the way, the perimeter of the square adds up to about 11.27.

If the shape is a rectangle, then you just take ANY pair of numbers you fancy and as long as they have a product of 3squrt(7), you're good.  You could start with 1 and 3sqrt(7) and of course, that's the area you get.  The perimeter of such a rectangle will vary but it will always be greater than 11.27.  I know this because that (11.27) is the perimeter of the square, and a square is the four-sided shape that MAXIMIZES area for a given perimeter length. 

Kudos and keep number crunchin'

Mr. Gets Results

Okay, here's how you figure this out:  you use what they call systems of equations.  We know that the the number of children plus the number of adults is 285.  After all, EVERYBODY is either an adult or a child.  We also know that 2 times the number of children plus 6.8 times the number of adults adds up to 1458. 

Now here's how we use the system.  First, let's assign a variable to children and to adults.  Let's say the number of adults is X and the number of children, Y.  In that case, we know the following:

X + Y = 285     AND WE KNOW:    6.8x + 2y = 1458

Now what we will do is solve for one of these variables.  Either X or Y will do.  I decided to solve for X.  Therefore, what I will do is use the substitution method to get X by itself.  How do I do that?  BY FINDING A WAY TO EXPRESS Y IN TERMS OF X. 

Since X + Y = 285 in one of the equations, I can take this equation, subtract X from each side, and then get Y = 285 - X  

Now that I have solved for Y in the first equation, I can SUBSTITUTE that value of Y in for "Y" in the second equation and BINGO, I have an equation entirely written in terms of X.

6.8x + 2y = 1458 will now turn into....    6.8x + 2(285 - X) = 1458

See what I just did?  I plugged the (285 - X) value of Y in for the Y value of the second equation.  Now I can distribute:  6.8x + 570 - 2x = 1458 is the new value of the equation after I do this.  Then I combine like terms on the left side:

4.8x + 570 = 1458   Now I'll subtract 570 from each side:

4.8x = 888  And now I divide each side by 4.8:

X = 185.

So there you go.  The value of X, the number of adults, is 185.  Since there were 285 people, the number of children must therefore have been 100. 

For best results, make sure your values work in both equations:

X + Y = 285      Now plug and chug to get 185 + 100 = 285.  A true statement

6.8x + 2y = 1458  Now plug and chug to get 6.8(185) + 2(100) = 1458.  Another true statement.  

There were, therefore, 185 adults and 100 children.

Hello.

Kevin S. In Seneca did an excellent job in figuring out that this equation can be factored and set equal to zero.  It can be factored into two binomials:  (4Sin(x) - 3)(Sin(x) - 1) = 0. 

From here, it's a piece of cake.  You just set the factors to zero.  Sin(x) - 1 will be equal to zero when Sin(x) is equal to 1.  This occurs at 90 degrees, or pi/2 depending on whether you are in degree or radian measure.  Now as for making 4Sin(x) - 3 equal to zero; first you do some algebra and get that sin(x) = 3/4.  In other words, the value of Sine in this case must be 0.75.  It is important to note, however, that this is NOT A VALUE of Sine which appears explicitly on the unit circle.  It's not, say, the Sine of 2pi/3 or 45 degrees.  So how do you figure it out?  I have three words for you:  GOD BLESS CALCULATORS. 

If you type Sin(-1)(.75) into any scientific calculator, you will get your answer.  In radian measure, it's about .848.  In degree measure, it's 48.59 degrees.

Mr. Gets Results

Dear Nadia,

The first thing I am going to do in this situation is multiply each SIDE of the equals sign by the common denominator of (x + 6).  That way, that miserable denominator of (X + 6) will be obliterated on both sides and that is a relief.  However, this must be done correctly as each side must be multiplied by that common denominator and the left side of the equals sign CONTAINS TWO TERMS.  To make a long story short, both the fraction and the number 5 must be multiplied by (X + 6) in the first step.

It becomes:    -2x + 5(X + 6) = -x

Which simplifies (when you distribute) to:  -2x + 5x + 30 = -x

Combine like terms to simplify:  3x + 30 = -x

If we add X to each side and subtract 30 FROM EACH SIDE we get the following:

4x = -30  And then we divide each side by 4 to get the answer:  X = -7.5

Best Wishes, Mr. Gets Results