Solutions to Set 7

• Problem #1: 120/17
• Problem #2: 22
• Problem #3: 7
• Problem #4: 147
• Problem #5: 25
• Problem #6: 125
• Problem #7: 8
• Problem #8: 225
• Problem #9: 1/13
• Problem #10: 18

Explanations

• Problem #1: ABC is a right triangle with legs of length 8 and 15. Based on the Pythagorean Theorem, or on knowledge of the Pythagorean triple (8, 15, 17), AC=17.

  Now, notice that BD is an altitude of triangle ABC; if AC is chosen as the base, then BD is the height. On the other hand, if AB is chosen as the base, then BC is the height. This provides us with two ways to calculate the area of triangle ABC, and both ways should give the same value:
  (1/2)(AB)(BC) = (1/2)(AC)(BD)
  (AB)(BC) = (AC)(BD)
  (8)(15) = (17)(BD).

  You can now solve for BD:
  BD = (8)(15)/17
  = 120/17.

  Read more about The Pythagorean Theorem.

• Problem #2: Suppose that Amy, Elizabeth, and Meg had a, e, and m dolls, respectively. Then, based on the information given,
  e = √a
  m = √e

  Square roots are sometimes messy to deal with, so you can square both equations to get rid of the radical signs:
  e² = a
  m² = e

  Using substitution, you have
  (m²)² = a
  m⁴ = a.

  The number of dolls that each girl has must be a positive integer. Since m is a positive integer, a must be 1⁴, 2⁴, 3⁴ etc. The only such "perfect 4th power" greater than 1 and less than 50 is 2⁴=16.

  Therefore,
  a = 16
  e = 4
  m = 2

  The total number of dolls is
  a + e + m
  = 16 + 4 + 2
  = 22 dolls.

  Read more about Translating Word Problems.

• Problem #3: While you could solve this problem algebraically, I personally think that, in this case, it is quicker to make a chart:

  Day #	# of Worms Eaten on That Day	Total # of Worms Eaten
  1	2	2
  2	4	6
  3	6	12
  4	8	20
  5	10	30
  6	12	42
  7	14	56

  Timmy had only eaten 42 worms by the end of the 6th day, but he had eaten 56 by the end of the 7th day. Therefore, he must have eaten the 50th worm on the 7th day.
• Problem #4: Suppose that I drove x miles in Michigan and y miles in Ohio. Then, I drove x/70 hours in Michigan and y/65 hours in Ohio. Based on the given information, the following equations can be written:
  x/70 + y/65 = 3.1
  x + y = 212

  Then, all you need to do is solve the system of equations to find the value of x. You can start by taking the first equation and finding a common denominator for the two fractions:
  x/70 + y/65 = 3.1
  13x/910 + 14y/910 = 3.1
  (13x + 14y)/910 = 3.1
  13x + 14y = 2821.

  Then, take the second equation and multiply the entire equation by 14:
  14x + 14y = 2968.

  After these manipulations, the first equation can be subtracted from the second, and you will get
  x = 147 miles.

• Problem #5: Since both people are running in the same direction, the combined speed is 10-8=2 meters per second. The distance between you and It is decreasing by 2 meters every second. Since there are 50 meters of distance to close, it will take 50/2=25 seconds.
• Problem #6: There are 5 odd digits: 1, 3, 5, 7, 9. Therefore, for each of the digits in a 3-digit number, there are 5 choices. Repeated digits are allowed--for example, 111 and 373 satisfy the conditions--so the number of choices does not decrease with each successive digit. The total number of possibilities, therefore, is
  5 * 5 * 5 = 125.
• Problem #7: Let a, b, and c represent the height, width, and length of the rectangular prism. Then,
  ab = 2
  bc = 4
  ac = 8

  Multiply these three equations together, and you have
  abbcac = 2 * 4 * 8
  a²b²c² = 64
  abc = 8.

  The volume of the rectangular prism is given by the expression abc, so the volume is 8.

• Problem #8: The volume of the water you need is
  70 ft * 40 ft * 5 ft
  = (70 * 40 * 5) cubic ft.

  Since each cubic foot of water is equivalent to 7.5 gallons, the volume of the water in gallons is
  (70 * 40 * 5 * 7.5) gallons.

  Each minute, the pumps will add
  4 * 125 gallons of water.

  Therefore, it will take
  (70 * 40 * 5 * 7.5)/(4 * 125)
  = 225 minutes.

• Problem #9: First, let us find the probability that Lucy is NOT chosen.

  Given 65 girls, the total number of 5-girl groups is (use combinations)
  (65 * 64 * 63 * 62 * 61)/(5 * 4 * 3 * 2 * 1).

  If Lucy is NOT chosen, then the 5 girls must be among the other 64 girls that are not Lucy. The number of ways this can occur is (use combinations again)
  (64 * 63 * 62 * 61 * 60)/(5 * 4 * 3 * 2 * 1).

  So, the probability that Lucy is not chosen is
  [(64 * 63 * 62 * 61 * 60)/(5 * 4 * 3 * 2 * 1)]/[(65 * 64 * 63 * 62 * 61)/(5 * 4 * 3 * 2 * 1)]
  = (64 * 63 * 62 * 61 * 60)/(65 * 64 * 63 * 62 * 61)
  = 60/65
  = 12/13

  Since this is the probability that Lucy is NOT chosen, the probability that she IS chosen is
  1 - 12/13
  = 1/13.

  Read more about Calculating Combinations and about Combining Probabilities.

• Problem #10: The following information is given:
  1 m = 10³ mm
  1 km = 10³ m

  It follows that
  1 km = 10³ * 10³ mm
  1 km = 10⁶ mm.

  Cubing both sides of the equation gives
  1 km³ = (10⁶)³ mm³
  1 km³ = 10¹⁸ mm³.

  Since there are 10¹⁸ cubic millimeters in a cubic kilometer, n=18.