Exponential Growth

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Exponential Growth · Grade 9 · Worksheet 3

1. A research scientist is studying the spread of a new virus in a city with a population of 50,000 people. The number of infected people follows an exponential growth model I(t) = 200 × e^(0.15t), where t is time in weeks since tracking began. How many people will be infected after 4 weeks? Round your answer to the nearest whole number. Answer: 364 Solution: Write down the given exponential function: I(t) = 200 × e^(0.15t) Substitute t = 4 into the function: I(4) = 200 × e^(0.15 × 4) Calculate the exponent: 0.15 × 4 = 0.6 Calculate e^0.6 using a calculator: e^0.6 ≈ 1.8221 Multiply: 200 × 1.8221 = 364.42 Round to the nearest whole number: 364 The…
   Full step-by-step solution

   Step 1: Write down the given exponential function: I(t) = 200 × e^(0.15t) Step 2: Substitute t = 4 into the function: I(4) = 200 × e^(0.15 × 4) Step 3: Calculate the exponent: 0.15 × 4 = 0.6 Step 4: Calculate e^0.6 using a calculator: e^0.6 ≈ 1.8221 Step 5: Multiply: 200 × 1.8221 = 364.42 Step 6: Round to the nearest whole number: 364 The answer is 364 infected people after 4 weeks.

2. A right triangle is drawn on a coordinate plane with vertices at (0,0), (6,0), and (6,8). A circle is circumscribed around the triangle such that all three vertices lie on the circle's circumference. What is the area of this circumscribed circle? (Use π = 3.14) Answer: 78.5 Solution: A = (0, 0) B = (6, 0) C = (6, 8) Points A and B are on the x-axis, so side AB is horizontal. Points B and C have the same x-coordinate, so side BC is vertical. Find the hypotenuse.
   Full step-by-step solution

   Step 1: Understand the triangle and circle. We have a right triangle with vertices at: A = (0, 0) B = (6, 0) C = (6, 8) Points A and B are on the x-axis, so side AB is horizontal. Points B and C have the same x-coordinate, so side BC is vertical. Therefore, angle B is the right angle (90 degrees). Step 2: Find the hypotenuse. The legs are: AB = 6 - 0 = 6 BC = 8 - 0 = 8 Using the Pythagorean theorem: AC^2 = AB^2 + BC^2 = 6^2 + 8^2 = 36 + 64 = 100 AC = sqrt(100) = 10 So the hypotenuse AC = 10. Step 3: Important property of a circumscribed circle around a right triangle. For any right triangle, the hypotenuse is the diameter of the circumscribed circle (Thales' theorem). Since angle B = 90 degrees, the side AC is the diameter of the circle. Step 4: Find the radius. Diameter d = AC = 10 Radius r = d / 2 = 10 / 2 = 5 Step 5: Calculate the area of the circle. Area = π * r^2 Using π = 3.14: Area = 3.14 * (5^2) = 3.14 * 25 Step 6: Final multiplication. 3.14 * 25 = 78.5 Therefore, the area of the circumscribed circle is 78.5.

3. Tane is studying the growth of two functions to demonstrate a key mathematical principle. He defines an exponential function f(x) = 11^x and a cubic function g(x) = x^9. For large positive integer values of x, which function will eventually produce larger outputs? Evaluate both functions at x = 12 and x = 18, and use these comparisons to explain why exponential growth eventually exceeds polynomial growth. Answer: f(x) = 11^x grows faster Solution: Evaluate f(x) = 11^x at x = 12. First compute powers: 11^2 = 121, 11^4 = (11^2)^2 = 121^2 = 14641, 11^8 = (11^4)^2 = 14641^2 = 214,358,881. Then 11^12 = 11^8 * 11^4 = 214,358,881 * 14,641 = 3,138,428,376,721.
   Full step-by-step solution

   Step 1: Evaluate f(x) = 11^x at x = 12. First compute powers: 11^2 = 121, 11^4 = (11^2)^2 = 121^2 = 14641, 11^8 = (11^4)^2 = 14641^2 = 214,358,881. Then 11^12 = 11^8 * 11^4 = 214,358,881 * 14,641 = 3,138,428,376,721. So f(12) = 3,138,428,376,721. Step 2: Evaluate g(x) = x^9 at x = 12. Compute 12^9: 12^2 = 144, 12^4 = 144^2 = 20,736, 12^8 = 20,736^2 = 429,981,696, then 12^9 = 12^8 * 12 = 429,981,696 * 12 = 5,159,780,352. So g(12) = 5,159,780,352. Step 3: Compare at x = 12: f(12) = 3,138,428,376,721 and g(12) = 5,159,780,352. f(12) / g(12) = 3,138,428,376,721 / 5,159,780,352 = 608.2 (approximately). The exponential is already about 608 times larger at x = 12. Step 4: Evaluate f(x) = 11^x at x = 18. f(18) = 11^18 = 11^12 * 11^6. First compute 11^6 = (11^3)^2. 11^3 = 1331, so 11^6 = 1331^2 = 1,771,561. Then f(18) = 3,138,428,376,721 * 1,771,561 = 5,559,060,000,000,000,000 (approximately 5.55906 x 10^18). Step 5: Evaluate g(x) = x^9 at x = 18. Compute 18^9: 18^2 = 324, 18^4 = 324^2 = 104,976, 18^8 = 104,976^2 = 11,019,960,576, then 18^9 = 11,019,960,576 * 18 = 198,359,290,368. So g(18) = 198,359,290,368. Step 6: Compare at x = 18: f(18) / g(18) = 5.55906 x 10^18 / 1.9836 x 10^11 = 28,024 (approximately). The exponential is now about 28,024 times larger than the cubic. Step 7: Conclusion: At x = 12, f(x) is 608 times larger than g(x). At x = 18, f(x) is 28,024 times larger. The ratio increases dramatically as x grows because the exponential function's base (11) causes repeated multiplication that outpaces the fixed power of the cubic function. Therefore, f(x) = 11^x grows faster and eventually exceeds g(x) = x^9 for large x.

4. On a coordinate plane, Kaia is comparing the growth of two functions: f(x) = 2^x and g(x) = x^3. She graphs both functions for x ≥ 0. For small values of x, the cubic function g(x) is larger. However, Kaia knows that exponential growth eventually overtakes polynomial growth. Determine the smallest integer value of x where f(x) = 2^x first becomes greater than g(x) = x^3. Answer: 10 Solution: Compare f(x) = 2^x and g(x) = x^3 for integer values of x starting from x = 0. For x = 0: f(0) = 2^0 = 1, g(0) = 0^3 = 0. f > g.
   Full step-by-step solution

   Step 1: Compare f(x) = 2^x and g(x) = x^3 for integer values of x starting from x = 0. Step 2: For x = 0: f(0) = 2^0 = 1, g(0) = 0^3 = 0. f > g. Step 3: For x = 1: f(1) = 2^1 = 2, g(1) = 1^3 = 1. f > g. Step 4: For x = 2: f(2) = 2^2 = 4, g(2) = 2^3 = 8. g > f. Step 5: For x = 3: f(3) = 2^3 = 8, g(3) = 3^3 = 27. g > f. Step 6: For x = 4: f(4) = 2^4 = 16, g(4) = 4^3 = 64. g > f. Step 7: For x = 5: f(5) = 2^5 = 32, g(5) = 5^3 = 125. g > f. Step 8: For x = 6: f(6) = 2^6 = 64, g(6) = 6^3 = 216. g > f. Step 9: For x = 7: f(7) = 2^7 = 128, g(7) = 7^3 = 343. g > f. Step 10: For x = 8: f(8) = 2^8 = 256, g(8) = 8^3 = 512. g > f. Step 11: For x = 9: f(9) = 2^9 = 512, g(9) = 9^3 = 729. g > f. Step 12: For x = 10: f(10) = 2^10 = 1024, g(10) = 10^3 = 1000. f > g. Thus, the smallest integer x where the exponential function 2^x exceeds the cubic function x^3 is x = 10. The answer is 10.

5. A colony of bacteria doubles in size every 3 hours. If the colony starts with 500 bacteria, how many bacteria will there be after 15 hours? Answer: 16000 Solution: The colony starts with 500 bacteria. It doubles every 3 hours. We want the number after 15 hours.
   Full step-by-step solution

   Let's go step by step. --- **Step 1: Understand the problem** The colony starts with 500 bacteria. It doubles every 3 hours. We want the number after 15 hours. --- **Step 2: Determine how many doubling periods occur in 15 hours** Doubling time = 3 hours. Number of periods = Total time / Doubling time = 15 / 3 = 5 periods. So the colony doubles 5 times. --- **Step 3: Set up the growth formula** Initial amount = 500 After 1 doubling: 500 × 2 After 2 doublings: 500 × 2 × 2 = 500 × 2^2 After n doublings: 500 × 2^n Here n = 5. Final amount = 500 × 2^5 --- **Step 4: Calculate 2^5** 2^5 = 2 × 2 × 2 × 2 × 2 = 32 --- **Step 5: Multiply by initial population** Final amount = 500 × 32 500 × 30 = 15,000 500 × 2 = 1,000 Total = 15,000 + 1,000 = 16,000 --- **Step 6: Conclusion** After 15 hours, there will be 16,000 bacteria. --- **Final answer: 16000**

6. A rare orchid species in a botanical garden is growing exponentially. The number of orchids is modeled by the function O(t) = 120 × 2^(t/4), where t is the time in months. How many orchids will there be after 12 months? Answer: 960 Solution: The function is O(t) = 120 × 2^(t/4) Substitute t = 12 into the function: O(12) = 120 × 2^(12/4) Simplify the exponent: 12/4 = 3 Now we have O(12) = 120 × 2^3 Calculate 2^3 = 8 Multiply: 120 × 8 = 960 The answer is 960 orchids.
   Full step-by-step solution

   Step 1: The function is O(t) = 120 × 2^(t/4) Step 2: Substitute t = 12 into the function: O(12) = 120 × 2^(12/4) Step 3: Simplify the exponent: 12/4 = 3 Step 4: Now we have O(12) = 120 × 2^3 Step 5: Calculate 2^3 = 8 Step 6: Multiply: 120 × 8 = 960 Step 7: The answer is 960 orchids.

7. Aroha is analyzing two different growth models for a school project. The linear model is given by f(x) = 15x + 20, and the exponential model is given by g(x) = 3^x. She wants to determine the smallest positive integer value of x for which the exponential model first produces a larger output than the linear model. What is that value of x? Answer: 4 Solution: Evaluate f(x) = 15x + 20 and g(x) = 3^x for x = 1, 2, 3, 4, ... x = 1: f(1) = 15(1) + 20 = 35, g(1) = 3^1 = 3. g(1) < f(1) x = 2: f(2) = 15(2) + 20 = 50, g(2) = 3^2 = 9.
   Full step-by-step solution

   Step 1: Evaluate f(x) = 15x + 20 and g(x) = 3^x for x = 1, 2, 3, 4, ... x = 1: f(1) = 15(1) + 20 = 35, g(1) = 3^1 = 3. g(1) < f(1) x = 2: f(2) = 15(2) + 20 = 50, g(2) = 3^2 = 9. g(2) < f(2) x = 3: f(3) = 15(3) + 20 = 65, g(3) = 3^3 = 27. g(3) < f(3) x = 4: f(4) = 15(4) + 20 = 80, g(4) = 3^4 = 81. g(4) > f(4) At x = 4, the exponential function g(x) = 81 first exceeds the linear function f(x) = 80. Step 2: Therefore, the smallest positive integer x where exponential exceeds linear is x = 4. Final answer: 4