Exponential Form

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Answer Key & Explanations

Exponential Form · Grade 9 · Worksheet 2

1. Liam 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 150% every 2 hours. Write an exponential function in the form V(t) = ab^t that models the number of people who have seen the post after t hours. Answer: V(t) = 50 * 2.5^(t/2) Solution: Identify the initial value a = 50 A 150% increase means the growth factor is 1 + 1.5 = 2.5 Since the growth happens every 2 hours, the exponent needs to be t/2 Combine these into the function V(t) = 50 * 2.5^(t/2) The exponential function is V(t) = 50 * 2.5^(t/2)
   Full step-by-step solution

   Step 1: Identify the initial value a = 50 Step 2: A 150% increase means the growth factor is 1 + 1.5 = 2.5 Step 3: Since the growth happens every 2 hours, the exponent needs to be t/2 Step 4: Combine these into the function V(t) = 50 * 2.5^(t/2) The exponential function is V(t) = 50 * 2.5^(t/2)

2. A pharmaceutical company is testing a new medication that has a half-life of 8 hours in the human body. If a patient takes a 400 mg dose, write an exponential function in the form M(t) = ab^t that models the amount of medication remaining in the patient's system after t hours. Answer: M(t)=400*(1/2)^(t/8) Solution: Identify the initial amount: a = 400 mg The medication halves every 8 hours, so the decay factor per hour is (1/2)^(1/8) After t hours, the amount remaining is 400 × (1/2)^(t/8) Write the function: M(t) = 400 × (1/2)^(t/8) The exponential function is M(t) = 400 × (1/2)^(t/8)
   Full step-by-step solution

   Step 1: Identify the initial amount: a = 400 mg Step 2: The medication halves every 8 hours, so the decay factor per hour is (1/2)^(1/8) Step 3: After t hours, the amount remaining is 400 × (1/2)^(t/8) Step 4: Write the function: M(t) = 400 × (1/2)^(t/8) The exponential function is M(t) = 400 × (1/2)^(t/8)

3. A population starts at 1200 and decreases by 15% each year. Write the exponential function in the form f(t) = a·b^t. Answer: 1200·0.85^t Solution: The initial population is 1200, so a = 1200. A 15% decrease each year means the population retains 85% of its value from the previous year (100% - 15% = 85%). 85% as a decimal is 0.85, so b = 0.85.
   Full step-by-step solution

   Step 1: The initial population is 1200, so a = 1200. Step 2: A 15% decrease each year means the population retains 85% of its value from the previous year (100% - 15% = 85%). Step 3: 85% as a decimal is 0.85, so b = 0.85. Step 4: Write the function using the form f(t) = a·b^t. The exponential function is f(t) = 1200·0.85^t.

4. A biologist is studying bacterial growth in a lab culture. The initial population is 500 bacteria, and the population doubles every 3 hours. Write an exponential function in the form P(t) = ab^t that models the bacterial population after t hours. Answer: P(t) = 500 * 2^(t/3) Solution: Identify the initial population. The problem says the initial population is 500 bacteria. In the exponential model \( P(t) = a b^t \), \( a \) is the initial population when \( t = 0 \).
   Full step-by-step solution

   Let's go step by step. --- **Step 1: Identify the initial population.** The problem says the initial population is 500 bacteria. In the exponential model \( P(t) = a b^t \), \( a \) is the initial population when \( t = 0 \). So: \[ a = 500 \] --- **Step 2: Understand the doubling time.** The population doubles every 3 hours. That means: When \( t = 3 \), \( P(3) = 2 \times 500 = 1000 \). --- **Step 3: Set up the equation using the doubling condition.** We have \( P(t) = 500 \cdot b^t \). At \( t = 3 \), \( P(3) = 500 \cdot b^3 = 1000 \). So: \[ 500 \cdot b^3 = 1000 \] Divide both sides by 500: \[ b^3 = 2 \] --- **Step 4: Solve for \( b \).** \[ b = 2^{1/3} \] So the model becomes: \[ P(t) = 500 \cdot (2^{1/3})^t \] --- **Step 5: Simplify the exponent.** Using exponent rules: \( (2^{1/3})^t = 2^{t/3} \). Thus: \[ P(t) = 500 \cdot 2^{t/3} \] --- **Step 6: Check the doubling behavior.** If \( t = 3 \), \( P(3) = 500 \cdot 2^{3/3} = 500 \cdot 2^1 = 1000 \), which is correct. If \( t = 6 \), \( P(6) = 500 \cdot 2^{6/3} = 500 \cdot 2^2 = 2000 \), which is double 1000. Works. --- **Final answer:** \[ P(t) = 500 \cdot 2^{t/3} \]

5. 2^(3x+1) = 128 = ? Answer: 2 Solution: Write both sides with base 2: 2^(3x+1) = 2^7 Since the bases are equal, set the exponents equal: 3x + 1 = 7 Subtract 1 from both sides: 3x = 6 Divide both sides by 3: x = 2 The answer is 2.
   Full step-by-step solution

   Step 1: Write both sides with base 2: 2^(3x+1) = 2^7 Step 2: Since the bases are equal, set the exponents equal: 3x + 1 = 7 Step 3: Subtract 1 from both sides: 3x = 6 Step 4: Divide both sides by 3: x = 2 The answer is 2.

6. 2^(3x+1) = 16 = ? Answer: 1 Solution: Write 16 as a power of 2: 16 = 2^4 The equation becomes: 2^(3x+1) = 2^4 Since the bases are equal, set the exponents equal: 3x+1 = 4 Subtract 1 from both sides: 3x = 3 Divide both sides by 3: x = 1 The answer is 1.
   Full step-by-step solution

   Step 1: Write 16 as a power of 2: 16 = 2^4 Step 2: The equation becomes: 2^(3x+1) = 2^4 Step 3: Since the bases are equal, set the exponents equal: 3x+1 = 4 Step 4: Subtract 1 from both sides: 3x = 3 Step 5: Divide both sides by 3: x = 1 The answer is 1.

7. Mere's investment of $800 grows at 6% annually. Write the exponential function f(t) = a·b^t. Answer: 800·1.06^t Solution: Identify the initial amount a. The investment starts at $800, so a = 800. Determine the growth factor b.
   Full step-by-step solution

   Step 1: Identify the initial amount a. The investment starts at $800, so a = 800. Step 2: Determine the growth factor b. A 6% annual growth means the amount multiplies by 1 + 0.06 = 1.06 each year, so b = 1.06. Step 3: Write the function in the form f(t) = a·b^t. Step 4: Substitute the values: f(t) = 800·1.06^t. The exponential function is f(t) = 800·1.06^t.

8. A population starts at 200 and doubles every 4 years. Write the exponential function in the form f(t) = a·b^t where t is measured in years. Answer: 200·2^(t/4) Solution: Identify the initial value a. The population starts at 200, so a = 200. Determine the growth factor b.
   Full step-by-step solution

   Step 1: Identify the initial value a. The population starts at 200, so a = 200. Step 2: Determine the growth factor b. The population doubles every 4 years, so the base should be 2. Step 3: Adjust the exponent for the time period. Since doubling occurs every 4 years, the exponent should be t/4. Step 4: Write the function: f(t) = 200·2^(t/4). The exponential function is f(t) = 200·2^(t/4).

9. Olivia's investment starts at $275 and grows by 15% each year. Write the exponential function in the form f(t) = a·b^t. Answer: f(t) = 275·1.15^t Solution: The initial amount a is $275. A 15% growth means the investment multiplies by 1 + 0.15 = 1.15 each year. So b = 1.15.
   Full step-by-step solution

   Step 1: The initial amount a is $275. Step 2: A 15% growth means the investment multiplies by 1 + 0.15 = 1.15 each year. So b = 1.15. Step 3: Substitute a and b into the form f(t) = a·b^t. The function is f(t) = 275·1.15^t.