Instructions
Read each section carefully and answer the questions. Show your work in the space provided (or on a separate sheet of paper). You may use a calculator for complex calculations, but challenge yourself to solve as much as you can without one. Good luck!
Part 1: Wobbledog Genetics & Probability
You are a world-class Wobbledog breeder. The strange genetics of these mutating dogs follow the basic rules of probability. Use your pre-algebra skills to predict the traits of your next generation of puppies!
- 1. Dominant Legs: You are breeding two dogs. Dog A is heterozygous for 6 legs (Lg) and Dog B has two recessive genes for 4 legs (gg). The dominant "L" gene results in 6 legs. What is the probability that a puppy from this pair will have 6 legs? Express your answer as a fraction and a percentage.
- 2. Wing Color: In Wobbledogs, blue wings (B) are dominant over yellow wings (b). If you breed two heterozygous dogs (Bb), what is the probability of a puppy having yellow wings?
- 3. Mutation Food: A new type of dog food has interesting side effects. There is a 12% chance it gives a dog a striped pattern, a 20% chance it gives them a curly tail, and a 5% chance for both traits to appear simultaneously. What is the probability that a dog eating this food will develop a striped pattern OR a curly tail?
Part 2: Spike Prime Rover's Journey
Your Lego Education Spike Prime rover must navigate a workshop floor. The floor is a perfect rectangle, 16 meters wide and 12 meters long. Your rover starts in one corner and its target, a dropped component, is in the diagonally opposite corner.
- 1. The Shortest Path: The rover can travel along the edges of the room or take a direct diagonal path. Using the Pythagorean theorem (a² + b² = c²), calculate the length of the direct diagonal path. This will be the rover's shortest route.
- 2. Wheel Rotations: The rover's wheels have a circumference of 25 cm. How many full rotations must each wheel make to travel the entire diagonal path? (Remember to convert your units!)
Part 3: The Algebra of the Core
You've discovered that the stability of a Wobbledog and the efficiency of a Spike Prime robot are governed by a set of three secret variables: x, y, and z. Use your algebra skills to solve the system of equations and find their values.
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1. System of Equations: Solve for x, y, and z.
Equation A:
2x + y + z = 15Equation B:
x - y + 2z = 6Equation C:
x + 2y = 11 -
2. Motor Efficiency: The efficiency rating, E, of your Spike Prime motor over time in seconds, t, is modeled by the quadratic equation
E = -t² + 12t + 13. For what value of t does the motor reach its maximum efficiency? (Hint: Consider the vertex of the parabola). After finding the time t, calculate what that maximum efficiency rating E is.
Answer Key
Part 1: Wobbledog Genetics & Probability
1. Dominant Legs:
Answer: 1/2 or 50%
2. Wing Color:
Answer: 1/4 or 25%
3. Mutation Food:
P(Striped or Curly) = P(Striped) + P(Curly) - P(Both)
P(Striped or Curly) = 0.12 + 0.20 - 0.05 = 0.27
Answer: 27%
Part 2: Spike Prime Rover's Journey
1. The Shortest Path:
16² + 12² = c²
256 + 144 = c²
400 = c²
c = √400 = 20
Answer: 20 meters
2. Wheel Rotations:
Divide the total distance by the wheel circumference: 2000 cm / 25 cm/rotation = 80 rotations.
Answer: 80 rotations
Part 3: The Algebra of the Core
1. System of Equations:
x = 11 - 2y. Substitute this into Equations A and B.Sub into A:
2(11 - 2y) + y + z = 15 -> 22 - 4y + y + z = 15 -> -3y + z = -7 (New Eq. D)Sub into B:
(11 - 2y) - y + 2z = 6 -> 11 - 3y + 2z = 6 -> -3y + 2z = -5 (New Eq. E)Now solve the system of D and E. Subtract D from E:
(-3y + 2z) - (-3y + z) = -5 - (-7) -> z = 2.Substitute z=2 into Eq. D:
-3y + 2 = -7 -> -3y = -9 -> y = 3.Substitute y=3 into Eq. C:
x + 2(3) = 11 -> x + 6 = 11 -> x = 5.Answer: x = 5, y = 3, z = 2
2. Motor Efficiency:
ax² + bx + c occurs at t = -b / (2a).In
E = -t² + 12t + 13, a = -1 and b = 12.Time (t) for max efficiency:
t = -12 / (2 * -1) -> t = -12 / -2 -> t = 6.Now, calculate the maximum efficiency E by plugging t=6 back into the equation:
E = -(6)² + 12(6) + 13 -> E = -36 + 72 + 13 -> E = 36 + 13 = 49.Answer: The motor reaches maximum efficiency at t = 6 seconds. The maximum efficiency is 49.