Exponential Growth

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Answer Key & Explanations

Exponential Growth · Grade 9 · Worksheet 1

1. Compare f(x)=10^x and g(x)=x^10 for x=11 Answer: f(11) Solution: Calculate f(11) = 10^11 = 100,000,000,000 Calculate g(11) = 11^10 = 25,937,424,601 Compare the values: 100,000,000,000 > 25,937,424,601 Since f(11) > g(11), the exponential function f(x) is larger at x=11 The answer is f(11).
   Full step-by-step solution

   Step 1: Calculate f(11) = 10^11 = 100,000,000,000 Step 2: Calculate g(11) = 11^10 = 25,937,424,601 Step 3: Compare the values: 100,000,000,000 > 25,937,424,601 Step 4: Since f(11) > g(11), the exponential function f(x) is larger at x=11 The answer is f(11).

2. A research team is studying the spread of a new social media trend. On day 1, 50 people shared the trend. The number of people sharing grows exponentially, tripling every 2 days. Using the exponential growth model N(t) = N₀ × a^t, where N₀ is the initial number, t is time in days, and a is the daily growth factor, determine how many people will be sharing the trend after 8 days. Answer: 4050 Solution: Initial number N₀ = 50 people Tripling time = 2 days Time elapsed t = 8 days If the trend triples every 2 days, then: a^2 = 3 Therefore: a = 3^(1/2) = sqrt(3) N(t) = N₀ × a^t N(8) = 50 × (sqrt(3))^8 (sqrt(3))^8 = (3^(1/2))^8 = 3^(8/2) = 3^4 = 81 N(8) = 50 × 81 = 4050 After 8 days, 4050 people…
   Full step-by-step solution

   Step 1: Identify the given values Initial number N₀ = 50 people Tripling time = 2 days Time elapsed t = 8 days Step 2: Find the daily growth factor a If the trend triples every 2 days, then: a^2 = 3 Therefore: a = 3^(1/2) = sqrt(3) Step 3: Apply the exponential growth formula N(t) = N₀ × a^t N(8) = 50 × (sqrt(3))^8 Step 4: Simplify the exponent (sqrt(3))^8 = (3^(1/2))^8 = 3^(8/2) = 3^4 = 81 Step 5: Calculate the final answer N(8) = 50 × 81 = 4050 After 8 days, 4050 people will be sharing the trend.

3. 2^3 × 3^2 - √81 = ? Answer: 63 Solution: Evaluate the exponents: 2^3 = 8 and 3^2 = 9 Evaluate the square root: √81 = 9 Perform multiplication: 8 × 9 = 72 Perform subtraction: 72 - 9 = 63 The answer is 63.
   Full step-by-step solution

   Step 1: Evaluate the exponents: 2^3 = 8 and 3^2 = 9 Step 2: Evaluate the square root: √81 = 9 Step 3: Perform multiplication: 8 × 9 = 72 Step 4: Perform subtraction: 72 - 9 = 63 The answer is 63.

4. Dr. Chen is studying the spread of a viral social media post. The post initially reaches 50 people, and the number of people who see it increases by 15% each hour. Write an exponential function in the form V(t) = V₀ × (1 + r)^t that models the number of people who have seen the post after t hours, where V₀ is the initial number of people and r is the hourly growth rate. Then determine how many people will have seen the post after 6 hours. Answer: 116 Solution: Identify the initial value V₀ = 50 people Convert the percentage growth rate to decimal form: r = 15% = 0.15 Write the exponential function: V(t) = 50 × (1 + 0.15)^t = 50 × (1.15)^t Calculate V(6) = 50 × (1.15)^6 Compute 1.15^6 = 1.15 × 1.15 × 1.15 × 1.15 × 1.15 × 1.15 = 1.3225 × 1.15 = 1.520875…
   Full step-by-step solution

   Step 1: Identify the initial value V₀ = 50 people Step 2: Convert the percentage growth rate to decimal form: r = 15% = 0.15 Step 3: Write the exponential function: V(t) = 50 × (1 + 0.15)^t = 50 × (1.15)^t Step 4: Calculate V(6) = 50 × (1.15)^6 Step 5: Compute 1.15^6 = 1.15 × 1.15 × 1.15 × 1.15 × 1.15 × 1.15 = 1.3225 × 1.15 = 1.520875 × 1.15 = 1.74900625 × 1.15 = 2.0113571875 × 1.15 = 2.313060765625 Step 6: Multiply by initial population: 50 × 2.313060765625 = 115.65303828125 Step 7: Round to the nearest whole person: 116 people The answer is 116.

5. A biologist is studying a population of bacteria that doubles every 3 hours. She starts with an initial population of 500 bacteria. Write an exponential function in the form P(t) = P₀ × 2^(t/k) that models the population after t hours, where P₀ is the initial population and k is the doubling time. Answer: P(t) = 500 × 2^(t/3) Solution: Identify the given values from the problem. - Initial population \( P_0 = 500 \) - Doubling time = 3 hours - General exponential growth formula (for doubling): \( P(t) = P_0 \times 2^{(t/k)} \) where \( k \) = doubling time in hours.
   Full step-by-step solution

   Let's go step by step. --- **Step 1: Identify the given values from the problem.** - Initial population \( P_0 = 500 \) - Doubling time = 3 hours - General exponential growth formula (for doubling): \( P(t) = P_0 \times 2^{(t/k)} \) where \( k \) = doubling time in hours. --- **Step 2: Understand the meaning of \( k \).** If \( t = k \), then \( P(k) = P_0 \times 2^{(k/k)} = P_0 \times 2^1 = 2P_0 \). So indeed \( k \) is the time it takes for the population to double. --- **Step 3: Substitute the given numbers into the formula.** We have \( P_0 = 500 \) and \( k = 3 \). So: \( P(t) = 500 \times 2^{(t/3)} \) --- **Step 4: Verify with an example to check the reasoning.** After \( t = 3 \) hours: \( P(3) = 500 \times 2^{(3/3)} = 500 \times 2^1 = 1000 \) → doubled, correct. After \( t = 6 \) hours: \( P(6) = 500 \times 2^{(6/3)} = 500 \times 2^2 = 2000 \) → quadrupled, correct. --- **Final Answer:** P(t) = 500 × 2^(t/3)

6. Isabella is comparing the growth of two functions to understand which one will eventually dominate. The first function is exponential: f(x) = 2^x, and the second is a cubic polynomial: g(x) = x^3. She tests values for x = 1, 2, 3, and 4, and notices that g(x) is larger for these small values. Determine the smallest positive integer value of x for which the exponential function f(x) = 2^x first exceeds the cubic function g(x) = x^3, proving that exponential growth eventually overtakes polynomial growth. Answer: 10 Solution: We need the smallest positive integer x such that f(x) = 2^x is greater than g(x) = x^3. Test values systematically starting from x = 5. For x = 5: f(5) = 2^5 = 32, g(5) = 5^3 = 125.
   Full step-by-step solution

   Step 1: Understand the problem. We need the smallest positive integer x such that f(x) = 2^x is greater than g(x) = x^3. Step 2: Test values systematically starting from x = 5. For x = 5: f(5) = 2^5 = 32, g(5) = 5^3 = 125. 32 < 125, so f is still smaller. For x = 6: f(6) = 2^6 = 64, g(6) = 6^3 = 216. 64 < 216. For x = 7: f(7) = 2^7 = 128, g(7) = 7^3 = 343. 128 < 343. For x = 8: f(8) = 2^8 = 256, g(8) = 8^3 = 512. 256 < 512. For x = 9: f(9) = 2^9 = 512, g(9) = 9^3 = 729. 512 < 729. For x = 10: f(10) = 2^10 = 1024, g(10) = 10^3 = 1000. 1024 > 1000. Step 3: At x = 10, f(10) = 1024 and g(10) = 1000, so f(x) first exceeds g(x) at x = 10. The answer is 10.

7. 2^(x+1) = 32, x = ? Answer: 4 Solution: Write 32 as a power of 2: 32 = 2^5 The equation becomes: 2^(x+1) = 2^5 Since the bases are equal, set the exponents equal: x + 1 = 5 Solve for x: x = 5 - 1 x = 4 The answer is 4.
   Full step-by-step solution

   Step 1: Write 32 as a power of 2: 32 = 2^5 Step 2: The equation becomes: 2^(x+1) = 2^5 Step 3: Since the bases are equal, set the exponents equal: x + 1 = 5 Step 4: Solve for x: x = 5 - 1 Step 5: x = 4 The answer is 4.