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If a family borrows $23,346 for an addition to their home, and the loan is to be paid off in monthly payments over a period of 10 years, how much should each payment be? (Interest has been included in the total amount borrowed.)     

     23,346 dollars / 120 months  =  194.55 dollars/month     

_.     

In terms of pi, what is the area of the circle defined by the equation 2x^2+2y^2+10x-6y-48=x^2+y^2-2x+4y-10     

                                              2x² + 2y² + 10x – 6y – 48  =  x² + y² – 2x + 4y – 10     

Combine like terms                          x² + y² + 12x – 10y  =  – 10     

Rearrange & add parentheses        (x² + 12x    )  +  (y² – 10y    )  =  – 10     

in preparation to complete sqs     

Complete the squares. Anything     (x² + 12x + 36)  +  (y² – 10y + 25)  =  – 10 + 36 + 25     

added to the left side, has to be     

added to the right side also.     

Simplify                                            (x + 6)²  +  (y – 5)²  =  51     

The equation of a circle is                (x – h)²  +  (y – k)²  =  r²    

where x and y locate the     

center and r is the radius.     

So r² of this circle is 51.  Area of a circle is π r².  So the area is 51 π.     

_.     

Find the ordered pair (s,t) that satisfies the system     

s/2 + 5t = 3 + 5t     

3t - 6s = 9 + 3t      

                                         s/2 + 5t  =  3 + 5t     

by observation                         s/2  =  3     

                                                    s  =  6     

                                           3t – 6s  =  9 + 3t     

by observation                        – 6s  =  9     

                                                    s  =  – 1.5     

You can't have s equal to two different amounts     

in the same system, so this system is invalid.     

_.     

Alan fills an aquarium to a depth of     

3/4 meters in 1 1/5 minutes.     

 What is the unit rate in minutes per meter?      

Convert to equivalents     ³/₄ = 0.75  &  1 ¹/₅ = 1.20     because, decimals are a lot     

                                                                                      easier to divide than fractions     

1.20 minute / 0.75 meter  =  1.60 minute / meter     

If you must do it in fractions          ⁶/₅ ^minute/₁  divided by  ³/₄ ^meter/₁     

To divide by a fraction, you     

invert the divisor and multiply         ⁶/₅ ^minute/₁  •  ⁴/₃ ¹/_meter  =  ²⁴/₁₅ ^minute/_meter     

                                                       ²⁴/₁₅ ^minute/_meter  =  1 ³/₅ ^minute/_meter      

_.     

The wall is 42 feet by 10 feet.     
What is the side measure of the largest square tile that i can use to cover the whole wall so there is no overlap or cutting?     

factors of 10     1, 2, 5, 10     

factors of 42     1, 2, 3, 6, 7, 14, 21, 42     

Looks like the largest square tile would be 2 ft by 2 ft.     

_.     

Two points on a circle of radius 1 are chosen at random.     

Find the probability that the distance between the two points is at most 1.5.     

The first point can be anywhere.  The second point has to be within 1.5 of the first point.     

The second point can be on either side of the first, so you have a length of 3.0 to consider.     

The diameter of the circle is 2, so it has 2π circumference, that is, 6.28 all the way around.     

So the likelihood of hitting the 3.0 segment is a probability of . . . 3.0 / 6.28 = 0.478  =  48.8%.     

_.     

A car uses 1 2/5 gallons of gasoline to travel 50 3/4 miles. How far can the car travel, in miles, on 1 gallon of gasoline?     

     1 ²/₅ = 1.40  and 50 ³/₄ = 50.75 . . . . because decimals are easier to divide than fractions     

     50.75 miles / 1.40 gallons  =  36.25 miles per gallon     

                                       that is,  36.25 miles on one gallon     

_.     

Levans writes a positive fraction in which the numerator and denominator are integers, and the numerator is 1 greater than the denominator. He then writes several more fractions. To make each new fraction, he increases both the numerator and the denominator of the previous fraction by 1. He then multiplies all his fractions together. He has 20 fractions, and their product equals 3. What is the value of the first fraction he wrote?     

  x+1     x+2     x+3     x+4             x+17     x+18     x+19     x+20       3     

 –––  •  –––  •  –––  •  –––  • • • •  ––––  •  ––––  •  ––––  •  ––––  =  ––     

   x        x+1     x+2     x+3             x+16     x+17     x+18     x+19        1     

Now, cancellation . . . .     

  x+1     x+2     x+3     x+4             x+17     x+18     x+19     x+20       3     

 –––  •  –––  •  –––  •  –––  • • • •  ––––  •  ––––  •  ––––  •  ––––  =  ––     

   x        x+1     x+2     x+3             x+16     x+17     x+18     x+19        1     

                                                                                  (x + 20) / (x)  =  3     

                                                                                            x + 20  =  3x     

                                                                                                  2x  =  20     

                                                                                                    x  =  10     

                                                                   11     

                           so the first fraction was  –––     

                                                                   10     

_.     

One leg of a right triangle is increased by 70%, and the other leg is decreased by 70%.  By what percent does the area of the triangle decrease?     

Triangle before          A  =  (1/2)(b • h)     

Triangle after             A  =  (1/2)(1.70 b)(0.30 h)     

                                  A  =  (1/2)(b • h)(0.51)     

The area of the triangle decreases by 49%.     

_.    

The functions f and h are defined as follows:     

f(x) = (x + 5)/(x - 7)  and h(x) = (x + 5)(x - 7).     

Explain why the functions f and h are not the same function.     

     At first glance, they look alike, sort of, but look closely at f(x)     

     and you'll see that there's a virgule between (x+5) and (x–7),     

     indicating division, but h(x) does not have a virgule.     

     f(x) is a division operation     

     h(x) is a multiplication operation     

_.     

   is -1.75 the same as -5/4, or is it -7/4     

   – 1.75  =  – ⁷/₄ which is another way of writing – 1 ³/₄     

_.     

Find the length AB.     

        sine = opposite over hypotenuse     

        sin(34)  =  29 / AB     

        AB  =  29 / sin(34)     

        AB  =  29 / 0.5592     

        AB  =  51.86     <<——  so the answer is A.     

_.     

Find the area of the region enclosed by the graph of x^2 + y^2 = 4x - 6y + 6.     

                                                    x² + y²  =  4x – 6y + 6     

                                                    x² + y² – 4x + 6y  =  6     

                                                    x² – 4x + y² + 6y  =  6     

                                                   (x² – 4x    ) + (y² + 6y    )  =  6     

                                                   (x² – 4x + 4) + (y² + 6y + 9)  =  6 + 13     

                                                   (x – 2)² + (y + 3)²  =  19     

The standard equation of a circle is (x – h)² + (y – k)²  =  r²     

which means, in our equation, the radius squared equals 19.     

                                                   Area of a circle is π r²     

                                                   A  =  3.1416 • 19     

                                                   A  =  59.69     

_.     

Find the standard equation of the circle defined by the equation x^2 + y^2 +8x - 12y + 43 = 0     

As you may have deduced from the answer choices,

you are going to have to complete those squares.     

The standard equation of a circle is (x – h)² + (y – k)²  =  r²     

 Starting with                                      x² + y² + 8x – 12y + 43  =  0     

Rearrange and add parentheses      (x² + 8x     ) + (y² – 12y     ) + 43  =  0     

You need to add 16 to the x and 36 to the y.     

To keep the equation in balance you have to     

subtract the same amount that you add.     

                                                          (x² + 8x + 16) + (y² – 12y + 36) + 43 – 52  =  0     

                                                          (x + 4)² + (y – 6)² – 9  =  0     

Normally, you would take that 9 over to the right of the equals sign     

but this is the form in which the choices of answer were presented.     

_.     

The second side of a triangular deck is 6 feet longer than the shortest​ side, and the third side is 6 feet shorter than twice the length of the shortest side. If the perimeter of the deck is 48 ​feet, what are the lengths of the three​ sides?     

        Call "s" the shortest side.     Perimeter is the sum of all three sides.     

        48  =  (s) + (s + 6) + (2s – 6)     

        48  =  4s + 6 – 6     

        4s  =  48     –––>>     s  = 12     

        1st side                    =  12 ft     

        2nd side  =  (12+6)  =  18 ft     

        3rd side   =  (24–6)  =  18 ft     

_.     

Find the value of x in the triangle shown below.     
x=     

     Equal sides show the triangle to be isoceles.     

     So x and the other unlabeled angle are equal.     

     The sum of all angles of every triangle is 180^o.     

     x  =  (1/2)(180^o – 58^o)     

     x  =  (1/2)(122^o)     

     x  =  61^o     

The functions f and h are defined as follows:     

f(x) = (x + 5)/(x - 7)  and h(x) = (x + 5)(x - 7).     

Explain why the functions f and h are not the same function.     

Because     f(x) is (x+5) divided by (x–7)     

whereas     h(x) is (x+5) multiplied by (x–7)     

That's why they're not the same.     

_.     

A tree diagram has outcomes (H, H, H), (H, H, T), (H, T, H), (H, T, T), (T, H, H), (T, H, T), (T, T, H), (T, T, T).What is the probability of one of the coins landing on tails and two of them landing on heads? 1/4 3/8  1/2 3/4     

There are eight possible outcomes,     

of which three are 1 tails & 2 heads.     

So the likelihood is 3 / 8.     

_.     

Let AB be a diameter of a circle, and let C be a point on the circle such that AC=8 and BC=14 The angle bisector of ACB intersects the circle at point M Find CM.     

Angle ACB is subtended by AB, which is a diameter of the circle, therefore is 90^o.     

Pythagoras' Theorem will determine AB which is the hypotenuse of a right triangle.     

AB²  =  AC² + BC²    

AB²  =  8² + 14²  =  260     

AB  =  sqrt(260)     

Since CM bisects the 90^o angle subtended by the diameter, it passes

through the center of the circle, thus becoming a diameter itself.     

And so, being a diameter . . . CM = sqrt(260)     

_.     

          which ratio is the same 2/3 

          1)   8/12     <<—— this is it       2       4        8     

          2)   6/12                                     ––  x  ––  =  ––     

          3) 10/12                                      3        4       12     

          4)   9/12     

_.