Model Comparison

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Answer Key & Explanations

Model Comparison · Grade 11 · Worksheet 2

1. Compare f(x)=11x+15, g(x)=x²+13, h(x)=4^x for large x. Which function grows fastest? Answer: h(x)=4^x Solution: Analyze f(x)=11x+15 (linear function). As x increases, f(x) grows by a constant amount of 11 for each unit increase in x. For large x, f(x) is approximately 11x.
   Full step-by-step solution

   Step 1: Analyze f(x)=11x+15 (linear function). As x increases, f(x) grows by a constant amount of 11 for each unit increase in x. For large x, f(x) is approximately 11x. Step 2: Analyze g(x)=x²+13 (quadratic function). As x increases, g(x) grows proportionally to x². For large x, g(x) is approximately x². Step 3: Analyze h(x)=4^x (exponential function). As x increases, h(x) multiplies by 4 for each unit increase in x. For large x, h(x) grows extremely rapidly. Step 4: Compare growth rates. For large x, polynomial functions (linear and quadratic) grow much slower than exponential functions with base greater than 1. Specifically, x² grows faster than 11x, but 4^x eventually surpasses both because exponential growth outpaces polynomial growth for sufficiently large x. Therefore, h(x)=4^x grows fastest for large x.

2. Compare f(x) = 7x + 12, g(x) = 2x² + 7, and h(x) = 7^x for large x. Which function grows fastest? Answer: h(x) = 7^x Solution: Identify the function types. f(x) = 7x + 12 is linear (degree 1). g(x) = 2x² + 7 is quadratic (degree 2).
   Full step-by-step solution

   Step 1: Identify the function types. f(x) = 7x + 12 is linear (degree 1). g(x) = 2x² + 7 is quadratic (degree 2). h(x) = 7^x is exponential (base > 1). Step 2: Compare linear and quadratic growth. For large x, the quadratic term 2x² grows much faster than the linear term 7x. So g(x) > f(x) for sufficiently large x. Step 3: Compare quadratic and exponential growth. For large x, an exponential function with base > 1 (here base 7) grows faster than any polynomial function, including quadratic. This is because exponential growth multiplies by a constant factor each step, while polynomial growth only adds a power of x. Step 4: Conclusion. For large x, h(x) = 7^x grows fastest among the three functions. The answer is h(x) = 7^x.

3. Sophia is an environmental scientist studying the growth of three different algae species in a lake. Species L grows linearly: its population (in thousands) is modeled by L(t) = 21 + 6t, where t is the number of weeks after initial observation. Species Q grows quadratically: Q(t) = 0.5t² + 21. Species E grows exponentially: E(t) = 21(1.26)^t. Sophia wants to know which species will have the largest population after 6 weeks. Determine the population of each species at t = 6 and identify which model predicts the highest population. Answer: E Solution: Calculate the linear model population at t = 6. L(6) = 21 + 6(6) = 21 + 36 = 57 thousand algae. Calculate the quadratic model population at t = 6.
   Full step-by-step solution

   Step 1: Calculate the linear model population at t = 6. L(6) = 21 + 6(6) = 21 + 36 = 57 thousand algae. Step 2: Calculate the quadratic model population at t = 6. Q(6) = 0.5(6)² + 21 = 0.5(36) + 21 = 18 + 21 = 39 thousand algae. Step 3: Calculate the exponential model population at t = 6. First compute 1.26^6. 1.26^2 = 1.5876, 1.26^4 = (1.5876)^2 = 2.5205, 1.26^6 = 1.26^4 * 1.26^2 = 2.5205 * 1.5876 ≈ 4.001. Then E(6) = 21 * 4.001 ≈ 84.021 thousand algae. Step 4: Compare the results. Linear: 57 thousand. Quadratic: 39 thousand. Exponential: approximately 84.021 thousand. The exponential model predicts the highest population after 6 weeks. The answer is E.

4. 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

5. Mason is comparing three functions: f(x) = 9x + 14 (linear), g(x) = x² + 11 (quadratic), and h(x) = 4^x (exponential). For large x, which function grows fastest? Answer: h(x) = 4^x Solution: Analyze f(x) = 9x + 14. This is a linear function. As x increases, f(x) increases by 9 for each unit increase in x.
   Full step-by-step solution

   Step 1: Analyze f(x) = 9x + 14. This is a linear function. As x increases, f(x) increases by 9 for each unit increase in x. For large x, f(x) behaves like 9x. Step 2: Analyze g(x) = x² + 11. This is a quadratic function. As x increases, g(x) increases by roughly 2x + 1 for each unit increase in x. For large x, g(x) behaves like x². Step 3: Analyze h(x) = 4^x. This is an exponential function with base 4 > 1. As x increases, h(x) multiplies by 4 for each unit increase in x. For large x, h(x) grows much faster than any polynomial. Step 4: Compare growth rates. For large x, exponential functions always outpace polynomial functions (linear and quadratic). Since 4^x grows faster than x² and 9x for sufficiently large x, h(x) = 4^x grows fastest. The answer is h(x) = 4^x.

6. 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.

7. A right triangle is drawn on a coordinate plane with vertices at (0,0), (6,0), and (0,8). A circle is inscribed in this triangle, tangent to all three sides. What is the area of this inscribed circle? (Use π = 3.14) Answer: 12.56 Solution: Identify the triangle's side lengths. The legs are 6 and 8 units. The hypotenuse can be found using the Pythagorean theorem: sqrt(6^2 + 8^2) = sqrt(36 + 64) = sqrt(100) = 10 units.
   Full step-by-step solution

   Step 1: Identify the triangle's side lengths. The legs are 6 and 8 units. The hypotenuse can be found using the Pythagorean theorem: sqrt(6^2 + 8^2) = sqrt(36 + 64) = sqrt(100) = 10 units. Step 2: For a right triangle, the inradius r is given by r = (a + b - c)/2, where a and b are the legs and c is the hypotenuse. Step 3: Substitute the values: r = (6 + 8 - 10)/2 = (4)/2 = 2 units. Step 4: The area of the inscribed circle is πr^2. Using π = 3.14, area = 3.14 × (2)^2 = 3.14 × 4 = 12.56 square units. The answer is 12.56.

8. Noah is a financial analyst comparing three investment models for a client. Model L predicts linear growth: V(t) = 12000 + 750t dollars, where t is years after investment. Model Q predicts quadratic growth: V(t) = 40t² + 12000 dollars. Model E predicts exponential growth: V(t) = 12000(1.09)^t dollars. After 12 years, which model predicts the highest value, and by approximately how many dollars does it exceed the second highest? Answer: Model E, by approximately $11,114 Solution: Calculate Model L (linear) at t = 12. V(12) = 12000 + 750(12) = 12000 + 9000 = 21000 dollars. Calculate Model Q (quadratic) at t = 12.
   Full step-by-step solution

   Step 1: Calculate Model L (linear) at t = 12. V(12) = 12000 + 750(12) = 12000 + 9000 = 21000 dollars. Step 2: Calculate Model Q (quadratic) at t = 12. V(12) = 40(12)² + 12000 = 40(144) + 12000 = 5760 + 12000 = 17760 dollars. Step 3: Calculate Model E (exponential) at t = 12. First compute (1.09)^12. Using successive multiplication: 1.09^2 = 1.1881 1.09^4 = (1.1881)^2 = 1.4116 1.09^8 = (1.4116)^2 = 1.9926 1.09^12 = 1.09^8 * 1.09^4 = 1.9926 * 1.4116 = 2.8127 (approximately) Then V(12) = 12000 * 2.8127 = 33752.40 dollars. Step 4: Compare the three values. Model L: 21000 dollars Model Q: 17760 dollars Model E: 33752.40 dollars Model E is highest. The second highest is Model L at 21000 dollars. Difference = 33752.40 - 21000 = 12752.40 dollars. Rounding to the nearest dollar: approximately $12,752. However, the problem asks "by approximately how many dollars does it exceed the second highest?" Let's recompute the exponential more precisely: 1.09^12 = e^(12 * ln(1.09)) = e^(12 * 0.0861777) = e^(1.0341324) = 2.8127 (more precisely using calculator: 2.8127). 33752.40 - 21000 = 12752.40, so approximately $12,752. The answer is Model E, by approximately $12,752.