Exponential Form

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Answer Key & Explanations

Exponential Form · Grade 9 · Worksheet 3

1. Olivia is studying a population of bacteria in a petri dish. At time t = 0 hours, she observes that there are 40 bacteria. The population grows exponentially, and after 3 hours, the number of bacteria has increased to 320. Write an exponential function of the form f(t) = a * b^t that models the number of bacteria after t hours. Answer: f(t) = 40 * 2^t Solution: The exponential function has the form f(t) = a * b^t. Here, a is the initial amount at t=0. From the problem, a = 40.
   Full step-by-step solution

   Step 1: The exponential function has the form f(t) = a * b^t. Here, a is the initial amount at t=0. From the problem, a = 40. So f(t) = 40 * b^t. Step 2: We know that at t=3, f(3) = 320. Substitute into the equation: 40 * b^3 = 320. Step 3: Divide both sides by 40: b^3 = 320 / 40 = 8. Step 4: Solve for b by taking the cube root of both sides: b = cube root of 8 = 2. Step 5: Therefore, the exponential function is f(t) = 40 * 2^t. The answer is f(t) = 40 * 2^t.

2. Lena is studying the spread of a social media post. When she first shares it, 15 people see it immediately. The number of people who see the post triples every 2 hours. Write an exponential function in the form V(t) = a * b^t that models the number of viewers V after t hours. Answer: V(t) = 15 * (sqrt(3))^t Solution: Identify the initial value 'a'. The problem states 15 people see it immediately, so a = 15. Determine the base 'b'.
   Full step-by-step solution

   Step 1: Identify the initial value 'a'. The problem states 15 people see it immediately, so a = 15. Step 2: Determine the base 'b'. The number of viewers triples every 2 hours. This means that after 2 hours, V(2) = 15 * 3. Step 3: Using the function form V(t) = a * b^t, we can write V(2) = 15 * b^2. Step 4: Set the two expressions for V(2) equal: 15 * b^2 = 15 * 3. Step 5: Divide both sides by 15: b^2 = 3. Step 6: Solve for b: b = sqrt(3). Step 7: Write the final function: V(t) = 15 * (sqrt(3))^t.

3. Isabella is studying the growth of a bacterial colony on a petri dish. She observes that the colony starts as a small circle with area 27 square millimeters. Every 2 hours, the area of the colony triples. Write an exponential function of the form f(t) = a · b^t that models the area of the colony (in square millimeters) after t hours. Answer: f(t) = 27 · (sqrt(3))^t Solution: Identify the initial value. The colony starts at 27 square millimeters, so a = 27. The colony triples every 2 hours.
   Full step-by-step solution

   Step 1: Identify the initial value. The colony starts at 27 square millimeters, so a = 27. Step 2: The colony triples every 2 hours. This means after 2 hours, the area is multiplied by 3. Step 3: Let the function be f(t) = 27 · b^t. We need to find b such that f(2) = 27 · b^2 = 27 · 3. Step 4: Divide both sides by 27: b^2 = 3. Step 5: Solve for b by taking the positive square root (since growth is positive): b = sqrt(3). Step 6: Therefore, the exponential function is f(t) = 27 · (sqrt(3))^t. The answer is f(t) = 27 · (sqrt(3))^t.

4. 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.

5. A pharmaceutical company is testing a new drug that reduces the concentration of a substance in the bloodstream. The initial concentration is 800 mg/L, and it decreases by 25% every 2 hours. Write an exponential function in the form C(t) = ab^t that models the concentration after t hours. Answer: C(t) = 800(0.75)^(t/2) Solution: Exponential decay functions model situations where a quantity decreases by a fixed percentage over regular time intervals. The base represents the decay factor (1 - percentage decrease), and the exponent must account for how often this decay occurs relative to the time unit.
6. 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.

7. 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)

8. 2^(3x) = 64 = ? Answer: 2 Solution: Write 64 as a power of 2: 64 = 2^6 Substitute into the equation: 2^(3x) = 2^6 Since the bases are equal, set the exponents equal: 3x = 6 Solve for x: x = 6 ÷ 3 x = 2 The answer is 2.
   Full step-by-step solution

   Step 1: Write 64 as a power of 2: 64 = 2^6 Step 2: Substitute into the equation: 2^(3x) = 2^6 Step 3: Since the bases are equal, set the exponents equal: 3x = 6 Step 4: Solve for x: x = 6 ÷ 3 Step 5: x = 2 The answer is 2.