Model Comparison

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Answer Key & Explanations

Model Comparison · Grade 11 · Worksheet 1

1. A right triangle is drawn on a coordinate plane with vertices at (0,0), (4,0), and (4,3). A circle is inscribed in this triangle, tangent to all three sides. What is the radius of this inscribed circle? Answer: 1 Solution: A = (0,0) B = (4,0) C = (4,3) This is a right triangle with the right angle at B = (4,0) because AB is horizontal and BC is vertical.
   Full step-by-step solution

   Let's go step by step. --- **Step 1: Understand the triangle** Vertices: A = (0,0) B = (4,0) C = (4,3) This is a right triangle with the right angle at B = (4,0) because AB is horizontal and BC is vertical. Side lengths: AB = distance from (0,0) to (4,0) = 4 BC = distance from (4,0) to (4,3) = 3 AC = hypotenuse = distance from (0,0) to (4,3) = sqrt(4^2 + 3^2) = sqrt(16+9) = sqrt(25) = 5 So sides: 3, 4, 5. --- **Step 2: Inradius formula for a right triangle** For any right triangle with legs a, b and hypotenuse c, the inradius r is given by: r = (a + b - c) / 2 --- **Step 3: Apply formula** Here a = 3, b = 4, c = 5. r = (3 + 4 - 5) / 2 r = (7 - 5) / 2 r = 2 / 2 r = 1 --- **Step 4: Conclusion** The radius of the inscribed circle is 1. --- **Final answer:** 1

2. Emma is a data analyst comparing three different models for the spread of a virus in a small town. Model L predicts the number of infected people grows linearly: I(t) = 1500 + 120t, where t is days after the first case. Model Q predicts quadratic growth: I(t) = 5t² + 1500. Model E predicts exponential growth: I(t) = 1500(1.08)^t. After 15 days, which model predicts the highest number of infected people? Answer: E Solution: Calculate Model L (linear) at t = 15. I(15) = 1500 + 120(15) = 1500 + 1800 = 3300 infected people. Calculate Model Q (quadratic) at t = 15.
   Full step-by-step solution

   Step 1: Calculate Model L (linear) at t = 15. I(15) = 1500 + 120(15) = 1500 + 1800 = 3300 infected people. Step 2: Calculate Model Q (quadratic) at t = 15. I(15) = 5(15)² + 1500 = 5(225) + 1500 = 1125 + 1500 = 2625 infected people. Step 3: Calculate Model E (exponential) at t = 15. I(15) = 1500(1.08)^15. Compute (1.08)^15 first: 1.08^5 ≈ 1.4693, 1.08^10 = (1.08^5)² ≈ 1.4693² ≈ 2.1589, 1.08^15 = 1.08^10 × 1.08^5 ≈ 2.1589 × 1.4693 ≈ 3.1722. Then I(15) = 1500 × 3.1722 ≈ 4758 infected people. Step 4: Compare the results. Model L: 3300 Model Q: 2625 Model E: 4758 Model E predicts the highest number of infected people after 15 days. The answer is E.

3. Dr. Chen is studying bacterial growth in her lab. She observes that a colony starts with 200 bacteria and doubles every 3 hours. Meanwhile, a chemical reaction she is monitoring produces heat according to the quadratic model H(t) = 2t² + 10, where H is heat in joules and t is time in hours. If she needs to find when the number of bacteria equals the heat output in joules, which type of equation must she solve and what is the general form of this equation? Answer: exponential equation; 200 * 2^(t/3) = 2t² + 10 Solution: The colony starts with 200 bacteria and doubles every 3 hours.
   Full step-by-step solution

   Let's break this down step by step. --- **Step 1: Model the bacterial growth** The colony starts with 200 bacteria and doubles every 3 hours. A general exponential growth formula is: Number of bacteria = initial amount × 2^(t / doubling time) So: B(t) = 200 × 2^(t / 3) --- **Step 2: Write the heat output function** The problem gives: H(t) = 2t² + 10 --- **Step 3: Set up the condition** We want the time t when the number of bacteria equals the heat output in joules: B(t) = H(t) So: 200 × 2^(t / 3) = 2t² + 10 --- **Step 4: Identify the type of equation** On the left side, we have 200 × 2^(t / 3), which is an **exponential function** of t. On the right side, we have 2t² + 10, which is a **quadratic function** of t. An equation with a variable in the exponent (exponential part) and also in a polynomial (quadratic part) is called an **exponential equation** in this context (more specifically, a transcendental equation, but the problem asks for the type — it's exponential because the unknown appears in the exponent). --- **Step 5: Final equation form** The general form of the equation Dr. Chen must solve is: 200 × 2^(t / 3) = 2t² + 10 --- **Final Answer:** exponential equation; 200 * 2^(t/3) = 2t² + 10

4. Aroha is a data analyst for a tech company. She is comparing three models for the growth of a new social media platform's user base (in thousands). Model L predicts linear growth: U(t) = 15 + 7t, where t is weeks since launch. Model Q predicts quadratic growth: U(t) = 0.5t^2 + 15. Model E predicts exponential growth: U(t) = 15(1.45)^t. After 9 weeks, which model predicts the highest number of users? Answer: E Solution: Evaluate Model L (linear) at t = 9. U(9) = 15 + 7(9) = 15 + 63 = 78 thousand users. Evaluate Model Q (quadratic) at t = 9.
   Full step-by-step solution

   Step 1: Evaluate Model L (linear) at t = 9. U(9) = 15 + 7(9) = 15 + 63 = 78 thousand users. Step 2: Evaluate Model Q (quadratic) at t = 9. U(9) = 0.5(9)^2 + 15 = 0.5(81) + 15 = 40.5 + 15 = 55.5 thousand users. Step 3: Evaluate Model E (exponential) at t = 9. U(9) = 15(1.45)^9. Compute (1.45)^9 step by step: 1.45^2 = 2.1025 1.45^4 = (1.45^2)^2 = 2.1025^2 = 4.4205 1.45^8 = (1.45^4)^2 = 4.4205^2 = 19.5412 1.45^9 = 1.45^8 * 1.45 = 19.5412 * 1.45 = 28.3347 Then U(9) = 15 * 28.3347 = 425.02 thousand users (rounded to two decimals). Step 4: Compare the results. Model L: 78 thousand Model Q: 55.5 thousand Model E: 425.02 thousand Model E predicts the highest number of users after 9 weeks. The answer is E.

5. Compare f(x)=8x+7, g(x)=x²+9, h(x)=5^x for large x. Which function grows fastest? Answer: h(x)=5^x Solution: Analyze f(x)=8x+7 (linear function) As x increases, f(x) grows at a constant rate of 8 per unit increase in x.
   Full step-by-step solution

   Step 1: Analyze f(x)=8x+7 (linear function) As x increases, f(x) grows at a constant rate of 8 per unit increase in x. Step 2: Analyze g(x)=x²+9 (quadratic function) As x increases, g(x) grows proportionally to x squared, which is faster than linear growth. Step 3: Analyze h(x)=5^x (exponential function) As x increases, h(x) grows by multiplying by 5 for each unit increase in x. This is exponential growth. Step 4: Compare growth rates For small x, linear or quadratic might appear faster, but for large x: - Linear: f(x) ~ 8x - Quadratic: g(x) ~ x² - Exponential: h(x) ~ 5^x Since exponential functions (with base > 1) always grow faster than polynomial functions for sufficiently large x, h(x)=5^x grows fastest. The answer is h(x)=5^x.

6. Compare f(x) = 14x + 23, g(x) = 3x² + 11, and h(x) = 4^x for large x. Which function grows fastest? Answer: h(x) = 4^x Solution: Analyze f(x) = 14x + 23. This is a linear function. As x increases, f(x) grows by a constant amount of 14 for each unit increase in x.
   Full step-by-step solution

   Step 1: Analyze f(x) = 14x + 23. This is a linear function. As x increases, f(x) grows by a constant amount of 14 for each unit increase in x. For large x, f(x) is approximately 14x. Step 2: Analyze g(x) = 3x² + 11. This is a quadratic function. As x increases, g(x) grows proportionally to x². For large x, g(x) is approximately 3x². Since x² grows faster than x, g(x) will eventually exceed f(x). Step 3: Analyze h(x) = 4^x. This is an exponential function with base 4 > 1. As x increases, h(x) grows by multiplying by 4 for each unit increase in x. For large x, exponential growth outpaces any polynomial growth. Step 4: Compare growth rates for large x. For example, at x = 10: f(10) = 14(10) + 23 = 163, g(10) = 3(100) + 11 = 311, h(10) = 4^10 = 1,048,576. At x = 20: f(20) = 303, g(20) = 1,211, h(20) = 4^20 ≈ 1.1 × 10^12. The exponential function h(x) = 4^x grows fastest for large x. The answer is h(x) = 4^x.

7. Compare f(x) = 7x + 10, g(x) = x^2 + 8, and h(x) = 4^x for large x. Which function grows fastest? Answer: h(x) = 4^x Solution: Analyze f(x) = 7x + 10. This is a linear function. As x increases by 1, f(x) increases by a constant 7.
   Full step-by-step solution

   Step 1: Analyze f(x) = 7x + 10. This is a linear function. As x increases by 1, f(x) increases by a constant 7. For large x, f(x) behaves like 7x. Step 2: Analyze g(x) = x^2 + 8. This is a quadratic function. As x increases, g(x) grows proportionally to x^2. For large x, g(x) behaves like x^2, which grows faster than 7x. Step 3: Analyze h(x) = 4^x. This is an exponential function with base 4 > 1. As x increases by 1, h(x) multiplies by 4. For large x, h(x) grows much faster than any polynomial function. Step 4: Compare growth rates. For large x, polynomial functions (like linear and quadratic) are eventually outpaced by exponential functions with base > 1. Since 4^x grows faster than x^2 and 7x for sufficiently large x, h(x) = 4^x grows fastest. The answer is h(x) = 4^x.