Bosco

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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Solve x - 6 = sqrt(3x).     

Square both sides               x² – 12x + 36  =  3x     

                                            x2 – 15x + 36  =  0     

                                           (x – 3)(x – 12)   =  0     

                                                          x – 3  =  0     

                                                                x  =  3     

                                                          x – 12  =  0     

                                                                  x  =  12     

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An empty rectangular tank measured 70 cm by 25 cm by 36 cm. A stone volume 4500 cm! is placed in the tank. Then the tank is filled with water flowing from a tap in a rate of 9 liters per minute. How long will it take to fill the tank?     

The tank without the stone will hold 70 • 25 • 36 cm³  =  63,000 cm³.     

The stone takes up                                                            4,500 cm³     

Subtract to find that the remaining volume is                   58,500 cm³     

A liter is 1,000 cm³ so the tank holds (58,500 / 1000) liters  =  58.5 liters     

At 9 liter/min, filling the tank will take (58.5 / 9) minutes        =  6.5 minutes  

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Jeremy is making a trail mix containing raisins and peanuts. Raisins cost $1.50 per pound. Peanuts cost $2.50 per pound. Jeremy spends $10.50 to make 5 pounds of trail mix. Determine the amount of raisins in the trail mix     

(1.50) R + (2.50) P  =  10.50     

R + P  =  5     ——>>   R = 5 – P   substitute this value into the other equation     

(1.50)(5 – P) + (2.50)(P)  =  10.50     

     7.50 – 1.50P + 2.50P  =  10.50     

                                     P  =  3.00     

                               R + P  =  5     

                          R + 2.50  =  5     

                                     R  =  2.00     

    check answer     

                          (1.50) R + (2.50) P = 10.50     

    Does (1.50)(2.00) + (2.50)(3.00) = 10.50 ?     

                      3.00    +    7.50          = 10.50  true     

_.     

Let triangle DEF be equilateral, where the side length is 3.     

A point G is chosen at random inside the triangle.     

Find the probability that the length DG is at most 1.      

Draw an equilateral triangle DEF, side 3.   

Draw a circle, center at D, 1 inch radius.     

Probability  =  area of circle sector inside triangle / area of triangle not in circle sector     

Area of entire circle  =  (3.1416) • 1²  =  3.1416     

Area of sector inside triangle  =  (60 / 360) • 3.1416  =  0.5236     

Area of entire triangle  =  [ sqrt(3) / 4 ] • 3²  =  3.8971     

Area of triangle outside sector  =  3.8971 – 0.5236  =  3.3735      

Probability  =  0.5236 / 3.3735  =  0.1552  as a percent  =  15.52%     

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