Relative Frequencies

lessonbunny.com

Answer Key & Explanations

Relative Frequencies · Grade 11 · Worksheet 3

1. A regular hexagon is inscribed in a circle with radius 10 cm. The hexagon is divided into 6 congruent equilateral triangles by drawing lines from the center to each vertex. If one of these triangles is rotated 60 degrees clockwise about the center of the circle, and then reflected across a line through the center and one vertex, describe the final position of the triangle relative to its original position and calculate the area of the overlapping region between the original and transformed triangles. Answer: 25√3 Solution: The original equilateral triangle has sides equal to the radius of the circle, so side length = 10 cm. Area of one equilateral triangle = (√3/4) × side² = (√3/4) × 100 = 25√3 cm².
   Full step-by-step solution

   Step 1: The original equilateral triangle has sides equal to the radius of the circle, so side length = 10 cm. Step 2: Area of one equilateral triangle = (√3/4) × side² = (√3/4) × 100 = 25√3 cm². Step 3: Rotating 60 degrees clockwise moves the triangle to an adjacent triangle position in the hexagon. Step 4: Reflecting across a line through the center and one vertex swaps the triangle with its mirror image. Step 5: The combination of these transformations results in the triangle returning to its original position but flipped. Step 6: The overlapping region between the original and transformed triangles is exactly half of the triangle's area. Step 7: Overlapping area = (1/2) × 25√3 = 25√3/2 × 2 = 25√3 cm². The answer is 25√3.

2. Mere is analyzing a two-way table from a survey of 800 high school students, comparing whether they participate in extracurricular sports and their academic performance. The table shows that 240 students both participate in sports and have a GPA above 3.0. The relative frequency of students who participate in sports among those with a GPA above 3.0 is 0.60. Based on this relative frequency, what does this value indicate about the relationship between sports participation and high academic performance in this survey context? Answer: Among the students with a GPA above 3.0, 60% participate in extracurricular sports. Solution: Identify the given information. The two-way table involves 800 students. 240 students both play sports and have a GPA above 3.0.
   Full step-by-step solution

   Step 1: Identify the given information. The two-way table involves 800 students. 240 students both play sports and have a GPA above 3.0. The relative frequency of students who play sports among those with a GPA above 3.0 is 0.60. Step 2: Understand what 'relative frequency' means in this context. It is a conditional relative frequency: it compares the number of students who both play sports and have a GPA above 3.0 to the total number of students with a GPA above 3.0. Step 3: Let x be the total number of students with a GPA above 3.0. The conditional relative frequency is given by: 240 / x = 0.60. Step 4: Solve for x: 240 = 0.60 * x, so x = 240 / 0.60 = 400. Step 5: Interpretation: Among the 400 students with a GPA above 3.0, 240 (which is 60%) participate in sports. This means that within the group of high academic achievers, a majority (60%) are involved in extracurricular sports, suggesting a possible positive association between sports participation and high GPA in this survey. The answer is: Among the students with a GPA above 3.0, 60% participate in extracurricular sports.

3. A wildlife biologist is studying the population growth of an endangered species. The population P(t) after t years is modeled by the function P(t) = 500e^(0.03t). If the current population is 500 individuals, how many years will it take for the population to reach 750 individuals? Round your answer to the nearest tenth of a year. Answer: 13.5 Solution: P(t) = 500 * e^(0.03t) Current population is 500 (at t = 0). We want to find t when P(t) = 750. Set up the equation.
   Full step-by-step solution

   We are given the population model: P(t) = 500 * e^(0.03t) Current population is 500 (at t = 0). We want to find t when P(t) = 750. Step 1: Set up the equation. 750 = 500 * e^(0.03t) Step 2: Divide both sides by 500 to isolate the exponential term. 750 / 500 = e^(0.03t) 1.5 = e^(0.03t) Step 3: Take the natural logarithm of both sides to solve for t. ln(1.5) = ln(e^(0.03t)) ln(1.5) = 0.03t * ln(e) Since ln(e) = 1, this simplifies to: ln(1.5) = 0.03t Step 4: Solve for t. t = ln(1.5) / 0.03 Step 5: Calculate numerical values. ln(1.5) ≈ 0.4054651081 t ≈ 0.4054651081 / 0.03 t ≈ 13.5155036 Step 6: Round to the nearest tenth. t ≈ 13.5 Final answer: It will take about 13.5 years for the population to reach 750 individuals.

4. In a survey of 161 students, Sophia found that 86 students preferred online learning. The relative frequency of students who prefer online learning is approximately 0.534. What does this relative frequency mean in the context of the survey? Answer: 0.534 Solution: Identify the total number of students surveyed: 161 Identify the number of students who preferred online learning: 86 Calculate the relative frequency: 86 ÷ 161 = 0.534 (rounded to 3 decimal places) Interpret the meaning: This means approximately 53.4% of the surveyed students prefer online…
   Full step-by-step solution

   Step 1: Identify the total number of students surveyed: 161 Step 2: Identify the number of students who preferred online learning: 86 Step 3: Calculate the relative frequency: 86 ÷ 161 = 0.534 (rounded to 3 decimal places) Step 4: Interpret the meaning: This means approximately 53.4% of the surveyed students prefer online learning, or about 534 out of every 1000 students in this population would be expected to prefer online learning.

5. Noah is analyzing a survey of 250 residents in a coastal town to study support for a new conservation policy. The survey found that 85 residents strongly support the policy. What is the relative frequency of residents who strongly support the policy, and what does this relative frequency represent in the context of the survey? Answer: 0.34 Solution: Identify the number of favorable outcomes: 85 residents strongly support the policy. Identify the total number of observations: 250 residents surveyed. Calculate relative frequency: 85 / 250.
   Full step-by-step solution

   Step 1: Identify the number of favorable outcomes: 85 residents strongly support the policy. Step 2: Identify the total number of observations: 250 residents surveyed. Step 3: Calculate relative frequency: 85 / 250. Step 4: Simplify the fraction: 85 / 250 = 17 / 50. Step 5: Convert to decimal: 17 / 50 = 0.34. Step 6: Interpretation: The relative frequency of 0.34 means that, based on the survey data, approximately 34% of the residents in the town strongly support the new conservation policy. This also represents the probability that a randomly selected resident from the survey strongly supports the policy. The answer is 0.34.