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Andrei has a glass tank. First, he wants to put in some marbles, each of which has a volume of 0.04 liters. Then, he wants to fill the tank with water until it's completely full. If he uses 50 marbles, he will have to add 33 liters of water.     

To find the total volume of the tank     

            add the volume of marbles     (50)(0.04 liter)  =    2 liters     

            plus the volume of water                                     33 liters     

                                                           total volume     =  35 liters     

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The local library have a children section with picture books and chapter books the ratio is 2:3. If there is a total of 500 books in the children's section how many chapter books are there in the section     

The ratio is 2 picture books to 3 chapter books.     

This is 2 parts to 3 parts, so the total is 5 parts.     

Since there are 5 parts, each part is 500 / 5 so     

each part is 100 books.  That makes the ratio     

in the values of 200:300.  300 chapter books     

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  The lateral area of a right cylinder which has a base diameter of 12 cm and a height of 8 cm     

  A_Lateral  =  circumference x height     

                =  π d h     

                =  (3.14)(12 cm)(8 cm)     

                =  301.44 cm²     —–>>  Call it 301 cm² to match the choice.    

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An alloy is a mixture of metals. Suppose that a certain alloy is made by mixing 300 grams of an alloy that contains 72% copper with 20 grams of pure copper. How many grams of copper are in the resulting mixture? What percentage of the resulting mixture is copper?     

Which Ingredient   Amount Cu   Amount Not Cu     

    Alloy 1                     216                     84      

    Pure Cu                     20     

    New Alloy                 236                    84     

Amt cu in resulting alloy  =  236     

Pctg cu in resulting alloy  =  (236) / (236+84)  =  0.7375  =  74%     

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Sally used a pencil to trace the circular base of a drinking glass.  She placed four quarters in a line across the diameter of the circle.  Using a ruler, she measured the diameter of each quarter, and calculated the circumference of the glass as 29.78cm.   The quarter could be used as a referent  for a measure of one _______, and, according to Sally’s measurement, each quarter has a diameter of ________cm.      

C  =  π d     

29.78  =  3.14 • d     

d  =  29.78 / 3.14     

d  =  9.48     

4Q  =  d     

4Q  =  9.48     

  Q  =  9.48 / 4     

  Q  =  2.37     <<—— the diameter of one quarter     

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   What is the value of c if $x\cdot(3x+1)     

   In this context, c doesn't stand for an abbreviation of a number  .  

   \cdot is the LaTeX command used to produce a centered dot (⋅)     

   to indicate the multiplication function.     

   The example in the question could be written as x • (3x+1) if you     

   use Alt-0149 on the keypad to insert the dot instead of \cdot.     

   Read it as "x times the quantity three x plus one".     

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It clearly doesn't diverge:

                        A

                F         

             3         

          D                            E

       x                                    3

  B                                            C

AF / AE  = AD / AC    ⇒  AE/AC = AF/ AD

AD/ AE = AB / AC   ⇒    AE / AC  = AD/ AB

AF / AD  = AD / AB

AD^2 = AF * AB

We need to know AF  to solve this rather than EC

The quadratic x^2 - 6x + 60 can be written in the form (x + b)^2 + c, where b and c are constants. What is b + c?     

In the instance of x² – 6x you need (6 / 2)² to complete the square.  So you have x² – 6x + 9     

                                                                                                                                 (x – 3)²     

To take that "9" on up to 60, simply add 51.  The resulting equation becomes (x – 3)² + 51     

Per the terms of the problem, add b + c, that is, add 3 + 51, and the final answer is 54.     

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 Five times a number is divided by 7 more than the original number.     

 If the result is 8, then what was the original number?     

     5X / (X+7)  =  8     

     5X  =  (8)(X+7)     

     5X  =  8X + 56     

  – 3X  =  56     

       X  =  – 56 / 3     

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Let's write the problem like this:  In triangle ABC, with C=90^o, sin A = 4/5.  Find cos B.     

By Pythagoras, that is a 3-4-5 right triangle, thus the third side is 3.     

cosine equals adjacent over hypotenuse   ——>>   cos(B) = 4 / 5     

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  Find cos C.     

  cosine equals adjacent over hypotenuse     

  cos(C)  =  7 / 25     

  cos(C)  =  0.2800     

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What is the coefficient of x^2 when -5x^3 - 5x^2 - 7x + 1 is multiplied by -x^2 + 6x + 1 and the like terms are combined?     

   (–5x³ – 5x² – 7x + 1)(–x² + 6x + 1)  —–>>    – 5x²     

   (–5x³ – 5x² – 7x + 1)(–x² + 6x + 1)  —–>>  – 42x²     

   (–5x³ – 5x² – 7x + 1)(–x² + 6x + 1)  —–>>       – x²     

                combined total of coefficients          – 48     

   It isn't necessary to multply the whole thing out.     

   Just find only the terms that will contain an x².     

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If (2x + 5)(x + 3) = 14, find the sum of the possible values of x.     

       (2x + 5)(x + 3)  =  14     

        2x² + 11x + 15  =  14     

        2x² + 11x + 1  =  0     

        It doesn't seem humanly possible to factor that quadratic, and     

        the quadratic formula involves too much arithmetic, so I used     

        Desmos to plot it, where I found that the two roots are     

                                              x  =  – 5.40754     

                                              x  =  – 0.09246     

                                       total        – 5.50000     

                        so, the sum is      – 5.5     

_.     

If BAC  = 60

Then angle BAD = 30   and angle CAD  = 30

So CAD can't equal 45

arcos (sqrt (.7)) + arcos (sqrt(.3)) = 90°

cos C  =  180 - 90  =  cos (90°) = 0

In triangle LMN, we have angle M = 90, MN = 2, and LM = sqrt(19). What is sin L?     

Use Pythagoras' Theorem to find that the hypotenuse is sqrt(23).     

sine = opposite over hypotenuse, therefore  sin(L) =  2 / sqrt(23)    ( 0.41703 )     

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The sequence 54675, 18225, 6075, 2025, 675, ... is made by repeatedly dividing by 3. How many integers are in this sequence?     

54675, 18225, 6075, 2025, 675, . . . the sequence continues 225, 75, 25, . . .     

Terms following 25 are not integers, so the sequence contains 8 integers.     

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The graph of y = f(x) is shown in red below.  Find f(2).  Assume grid lines are spaced 1  unit apart.     

The edge of the grid is cut off before the curve gets to x = 2 so f(2) can't be read directly.     

However, the curve is labelled y = 3^x – therefore, using that, when x equals 2, then y = 9.     

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  How many numbers can be expressed as the sum of two or more distinct elements of the set {0,1,2,4,8,16}?     

  ₆C₂  =  15     You have to be careful of duplicates, but there aren't any in this total.     

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  A prop for the theater club’s play is constructed as a cone topped with a half-sphere.     

  What is the volume of the prop? Round your answer to the nearest tenth of a cubic inch.     

  Use 3.14 to approximate pi. radius=5in  height=17in  Help pls      

  V_CONE  =  (1/3) π r² h     

               =  (1/3)(3.14)(5²)(17)     

               =  444.83 cu in     

  V_SPHERE  =  (4/3) π r³     

                   =  (4/3)(3.14)(5³)     

                   =  523.33 cu in     

  V_TOTAL  =  444.83 + 523.33     

                =  968.16 cu in     

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For problem 1, the answer is 8*sqrt(5).