Model Comparison

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Answer Key & Explanations

Model Comparison · Grade 11 · Worksheet 3

1. Aroha is comparing three functions: f(x) = 9x + 14, g(x) = 2x² + 11, and h(x) = 4^x. For large values of x, which function grows fastest? Answer: h(x) = 4^x Solution: Identify the function types. f(x) = 9x + 14 is linear (degree 1). g(x) = 2x² + 11 is quadratic (degree 2).
   Full step-by-step solution

   Step 1: Identify the function types. f(x) = 9x + 14 is linear (degree 1). g(x) = 2x² + 11 is quadratic (degree 2). h(x) = 4^x is exponential (base 4 > 1). Step 2: Compare linear and quadratic growth. For large x, the quadratic term 2x² dominates the linear term 9x. So g(x) grows faster than f(x) for sufficiently large x. Step 3: Compare quadratic and exponential growth. For large x, exponential functions with base > 1 eventually outgrow any polynomial function. Since 4^x grows much faster than x² for large x, h(x) grows faster than g(x). Step 4: Conclusion. For large x, the growth rates are ordered: f(x) < g(x) < h(x). Therefore, h(x) = 4^x grows fastest.

2. The graphs of three functions are drawn on the same coordinate plane: f(x) = 5x + 10, g(x) = x^2 + 5, and h(x) = 5(2^x). For x ≥ 0, which function will eventually have the greatest y-value as x increases without bound, and at what approximate x-value does the exponential function overtake the quadratic function? (Use integer or fractional approximations.) Answer: h(x) = 5(2^x); approximately x = 5 Solution: Evaluate the functions at x = 0: f(0)=5(0)+10=10, g(0)=0^2+5=5, h(0)=5(2^0)=5. Step 2: At x = 5: f(5)=5(5)+10=35, g(5)=25+5=30, h(5)=5(32)=160. At x = 4: f(4)=30, g(4)=16+5=21, h(4)=5(16)=80.
   Full step-by-step solution

   Step 1: Evaluate the functions at x = 0: f(0)=5(0)+10=10, g(0)=0^2+5=5, h(0)=5(2^0)=5. Step 2: At x = 5: f(5)=5(5)+10=35, g(5)=25+5=30, h(5)=5(32)=160. At x = 4: f(4)=30, g(4)=16+5=21, h(4)=5(16)=80. At x = 3: f(3)=25, g(3)=9+5=14, h(3)=5(8)=40. At x = 2: f(2)=20, g(2)=4+5=9, h(2)=5(4)=20. The exponential equals the quadratic at x=2 (both 20) and is already larger at x=3 (40 > 14). The exponential overtakes the quadratic between x=2 and x=3, but for the overtaking of the quadratic specifically (when h(x) > g(x) for all larger x), we check x=2: h(2)=20, g(2)=9 (exponential already larger). Actually, at x=1: h(1)=10, g(1)=6; exponential already larger. So the exponential is already above the quadratic at x=1. To find where exponential overtakes linear: at x=5, f(5)=35, h(5)=160 (exponential much larger). The exponential dominates all others for x >= 5. The final answer: the exponential function h(x)=5(2^x) eventually has the greatest y-value, and it overtakes the quadratic function by approximately x = 2 (or more conservatively x = 5 for clear dominance).

3. Charlotte is an environmental scientist studying three different models for the spread of an invasive plant species in a national park. The area covered by the species (in square meters) is predicted by three models, where t represents the number of years since the species was first detected. Model L (linear): A(t) = 850 + 90t. Model Q (quadratic): A(t) = 7t² + 850. Model E (exponential): A(t) = 850(1.12)^t. After 12 years, which model predicts the largest area covered by the invasive species? Answer: E Solution: Calculate the area for Model L (linear) at t = 12. A(12) = 850 + 90(12) = 850 + 1080 = 1930 square meters. Calculate the area for Model Q (quadratic) at t = 12.
   Full step-by-step solution

   Step 1: Calculate the area for Model L (linear) at t = 12. A(12) = 850 + 90(12) = 850 + 1080 = 1930 square meters. Step 2: Calculate the area for Model Q (quadratic) at t = 12. A(12) = 7(12)² + 850 = 7(144) + 850 = 1008 + 850 = 1858 square meters. Step 3: Calculate the area for Model E (exponential) at t = 12. A(12) = 850(1.12)^12. First compute (1.12)^12. Use repeated multiplication: (1.12)^2 = 1.2544, (1.12)^4 = (1.2544)^2 = 1.5735, (1.12)^8 = (1.5735)^2 = 2.4759, then (1.12)^12 = (1.12)^8 * (1.12)^4 = 2.4759 * 1.5735 = 3.8958. Then A(12) = 850 * 3.8958 = 3311.43 square meters. Step 4: Compare the three results. Model L: 1930 square meters. Model Q: 1858 square meters. Model E: 3311 square meters. Model E predicts the largest area. The answer is E.

4. Dr. Rodriguez is comparing three different models for predicting city population growth over time. Model A is linear: P(t) = 5,000t + 100,000, Model B is quadratic: P(t) = 200t² + 100,000, and Model C is exponential: P(t) = 100,000(1.04)^t, where t represents years and P(t) represents population. After how many years will the exponential model first predict a higher population than both the linear and quadratic models? Answer: 12 Solution: Step 1: Set up the comparison equations We need to find when C(t) > A(t) and C(t) > B(t) Step 2: Compare exponential and linear models 100,000(1.04)^t > 5,000t + 100,000 Divide both sides by 100,000: (1.04)^t > 0.05t + 1 Test values: t = 10: (1.04)^10 = 1.480, 0.05(10) + 1 = 1.5 → 1.480 < 1.5 t…
   Full step-by-step solution

   Step 1: Set up the comparison equations We need to find when C(t) > A(t) and C(t) > B(t) Step 2: Compare exponential and linear models 100,000(1.04)^t > 5,000t + 100,000 Divide both sides by 100,000: (1.04)^t > 0.05t + 1 Test values: t = 10: (1.04)^10 = 1.480, 0.05(10) + 1 = 1.5 → 1.480 < 1.5 t = 11: (1.04)^11 = 1.539, 0.05(11) + 1 = 1.55 → 1.539 < 1.55 t = 12: (1.04)^12 = 1.601, 0.05(12) + 1 = 1.6 → 1.601 > 1.6 ✓ Step 3: Compare exponential and quadratic models 100,000(1.04)^t > 200t² + 100,000 Divide both sides by 100,000: (1.04)^t > 0.002t² + 1 Test values: t = 10: (1.04)^10 = 1.480, 0.002(100) + 1 = 1.2 → 1.480 > 1.2 ✓ t = 11: (1.04)^11 = 1.539, 0.002(121) + 1 = 1.242 → 1.539 > 1.242 ✓ t = 12: (1.04)^12 = 1.601, 0.002(144) + 1 = 1.288 → 1.601 > 1.288 ✓ Step 4: Verify both conditions are met at t = 12 At t = 12: C(12) > A(12) and C(12) > B(12) At t = 11: C(11) > B(11) but C(11) < A(11) Therefore, the exponential model first exceeds both other models at t = 12 years.

5. Maya is comparing three different business growth models for her startup. Model A grows linearly with revenue R(t) = 5000 + 800t dollars per year. Model B grows quadratically with revenue R(t) = 100t² + 3000 dollars per year. Model C grows exponentially with revenue R(t) = 4000(1.25)^t dollars per year, where t represents years since startup. After 8 years, which model predicts the highest revenue? Answer: C. Exponential Model C Solution: Calculate revenue for Model A after 8 years R(t) = 5000 + 800t R(8) = 5000 + 800(8) = 5000 + 6400 = 11,400 dollars Calculate revenue for Model B after 8 years R(t) = 100t² + 3000 R(8) = 100(8)² + 3000 = 100(64) + 3000 = 6400 + 3000 = 9,400 dollars Calculate revenue for Model C after 8 years R(t)…
   Full step-by-step solution

   Step 1: Calculate revenue for Model A after 8 years R(t) = 5000 + 800t R(8) = 5000 + 800(8) = 5000 + 6400 = 11,400 dollars Step 2: Calculate revenue for Model B after 8 years R(t) = 100t² + 3000 R(8) = 100(8)² + 3000 = 100(64) + 3000 = 6400 + 3000 = 9,400 dollars Step 3: Calculate revenue for Model C after 8 years R(t) = 4000(1.25)^t R(8) = 4000(1.25)^8 = 4000 × 5.9604644775 ≈ 23,842 dollars Step 4: Compare the results Model A: 11,400 dollars Model B: 9,400 dollars Model C: 23,842 dollars Model C has the highest revenue after 8 years. The correct answer is Exponential Model C.

6. Aroha is a marine biologist comparing three models for the growth of a coral reef over time, where t represents years since the start of her study, and R(t) represents the reef area in square meters. Model L is linear: R(t) = 450 + 135t. Model Q is quadratic: R(t) = 7t² + 450. Model E is exponential: R(t) = 450(1.13)^t. After 11 years, which model predicts the largest reef area? Answer: E Solution: Evaluate Model L (linear) at t = 11. R(11) = 450 + 135(11) = 450 + 1485 = 1935 square meters. Evaluate Model Q (quadratic) at t = 11.
   Full step-by-step solution

   Step 1: Evaluate Model L (linear) at t = 11. R(11) = 450 + 135(11) = 450 + 1485 = 1935 square meters. Step 2: Evaluate Model Q (quadratic) at t = 11. R(11) = 7(11)² + 450 = 7(121) + 450 = 847 + 450 = 1297 square meters. Step 3: Evaluate Model E (exponential) at t = 11. R(11) = 450(1.13)^11. First compute (1.13)^11. Calculate step by step: 1.13^2 = 1.2769 1.13^4 = (1.13^2)^2 = 1.2769^2 = 1.6305 1.13^8 = (1.13^4)^2 = 1.6305^2 = 2.6585 1.13^10 = 1.13^8 × 1.13^2 = 2.6585 × 1.2769 = 3.3947 1.13^11 = 1.13^10 × 1.13 = 3.3947 × 1.13 = 3.8360 Then R(11) = 450 × 3.8360 = 1726.2 square meters. Step 4: Compare the results. Model L: 1935 square meters Model Q: 1297 square meters Model E: 1726.2 square meters Model L predicts the largest reef area after 11 years. The answer is E.