Exponential Form

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Exponential Form · Grade 9 · Worksheet 3

1. A pharmaceutical company is testing a new drug that decays exponentially in the bloodstream. A patient receives an initial dose of 800 mg, and after 4 hours, only 200 mg remain active. Write an exponential function in the form D(t) = ab^t that models the amount of active drug in milligrams after t hours. Answer: D(t) = 800 * (1/2)^(t/2) Solution: Exponential decay functions model situations where a quantity decreases by a constant factor over equal time intervals.
   Full step-by-step solution

   Exponential decay functions model situations where a quantity decreases by a constant factor over equal time intervals. To find the decay factor, you can determine what fraction of the original amount remains after the given time period, then find the hourly decay rate that would produce this result.

2. 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.

3. A radioactive substance decays according to the function A(t) = 800 × (1/2)^(t/12), where A(t) is the amount in grams remaining after t years. How many grams will remain after 36 years? Answer: 100 Solution: Start with the decay function: A(t) = 800 × (1/2)^(t/12) Substitute t = 36 into the function: A(36) = 800 × (1/2)^(36/12) Simplify the exponent: 36/12 = 3, so A(36) = 800 × (1/2)^3 Evaluate (1/2)^3 = 1/8 Multiply: 800 × 1/8 = 800/8 = 100 After 36 years, 100 grams of the radioactive substance…
   Full step-by-step solution

   Step 1: Start with the decay function: A(t) = 800 × (1/2)^(t/12) Step 2: Substitute t = 36 into the function: A(36) = 800 × (1/2)^(36/12) Step 3: Simplify the exponent: 36/12 = 3, so A(36) = 800 × (1/2)^3 Step 4: Evaluate (1/2)^3 = 1/8 Step 5: Multiply: 800 × 1/8 = 800/8 = 100 Step 6: After 36 years, 100 grams of the radioactive substance will remain.

4. Sophia invests $1600 in an account that grows by 6% each year. Write the exponential function f(t) = a·b^t that models this growth. Answer: 1600·1.06^t Solution: Identify the initial amount a = $1600 Convert the growth rate to decimal form: 6% = 0.06 Calculate the growth factor b = 1 + 0.06 = 1.06 Write the function in f(t) = a·b^t form: f(t) = 1600·1.06^t The answer is 1600·1.06^t.
   Full step-by-step solution

   Step 1: Identify the initial amount a = $1600 Step 2: Convert the growth rate to decimal form: 6% = 0.06 Step 3: Calculate the growth factor b = 1 + 0.06 = 1.06 Step 4: Write the function in f(t) = a·b^t form: f(t) = 1600·1.06^t The answer is 1600·1.06^t.

5. A biologist is studying bacterial growth in a lab culture. The initial population is 200 bacteria, and it doubles every 3 hours. Write an exponential function in the form P(t) = a * b^t that models the population P after t hours. Answer: P(t) = 200 * 2^(t/3) Solution: The problem says the initial population is 200 bacteria. In the exponential model \( P(t) = a \cdot b^t \), \( a \) is the initial population when \( t = 0 \). So \( a = 200 \).
   Full step-by-step solution

   Let's go step by step. --- **Step 1: Identify the initial value** The problem says the initial population is 200 bacteria. In the exponential model \( P(t) = a \cdot b^t \), \( a \) is the initial population when \( t = 0 \). So \( a = 200 \). --- **Step 2: Determine the growth factor** The population doubles every 3 hours. If it doubles, the growth factor over 3 hours is 2. But in \( P(t) = a \cdot b^t \), \( b \) is the growth factor **per 1 hour**. Let’s find \( b \): We know: \( P(3) = 200 \cdot b^3 = 400 \) (since it doubles to 400 after 3 hours). So: \( 200 \cdot b^3 = 400 \) Divide both sides by 200: \( b^3 = 2 \) Take cube root: \( b = 2^{1/3} \) So \( P(t) = 200 \cdot (2^{1/3})^t \). --- **Step 3: Simplify the exponent** Using exponent rules: \( (2^{1/3})^t = 2^{t/3} \). So \( P(t) = 200 \cdot 2^{t/3} \). --- **Step 4: Check the reasoning** We can check: after \( t = 3 \) hours: \( P(3) = 200 \cdot 2^{3/3} = 200 \cdot 2^1 = 400 \) — correct, it doubled. After \( t = 6 \) hours: \( P(6) = 200 \cdot 2^{6/3} = 200 \cdot 2^2 = 800 \) — doubled again. --- **Final Answer:** P(t) = 200 * 2^(t/3)

6. A city's population is currently 80,000 people and is growing at a rate of 5% per year. Write an exponential function in the form P(t) = ab^t that models the population after t years. Answer: P(t) = 80000 * 1.05^t Solution: Identify the initial population value, which is 80,000. This becomes the coefficient 'a' in our function. Determine the growth factor 'b'.
   Full step-by-step solution

   Step 1: Identify the initial population value, which is 80,000. This becomes the coefficient 'a' in our function. Step 2: Determine the growth factor 'b'. Since the population grows by 5% each year, we convert 5% to decimal form: 5% = 0.05. Step 3: The growth factor is 1 plus the growth rate: 1 + 0.05 = 1.05. Step 4: Write the function using the initial population and growth factor: P(t) = 80000 * 1.05^t. The complete exponential function is P(t) = 80000 * 1.05^t.

7. An environmental scientist is studying the spread of an invasive plant species in a nature preserve. The initial area covered by the plants is 50 square meters, and observations show the coverage area triples every 2 years. Write an exponential function in the form A(t) = ab^t that models the area covered after t years. Answer: A(t) = 50 * (sqrt(3))^t Solution: Identify the initial value a = 50 (the starting area in square meters) The area triples every 2 years, so when t = 2, A(2) = 50 × 3 = 150 Using the form A(t) = ab^t, we have 50 × b^2 = 150 Solve for b: b^2 = 150/50 = 3 b = sqrt(3) The exponential function is A(t) = 50 × (sqrt(3))^t The answer is…
   Full step-by-step solution

   Step 1: Identify the initial value a = 50 (the starting area in square meters) Step 2: The area triples every 2 years, so when t = 2, A(2) = 50 × 3 = 150 Step 3: Using the form A(t) = ab^t, we have 50 × b^2 = 150 Step 4: Solve for b: b^2 = 150/50 = 3 Step 5: b = sqrt(3) Step 6: The exponential function is A(t) = 50 × (sqrt(3))^t The answer is A(t) = 50 * (sqrt(3))^t.

8. Lena is studying the decay of a radioactive isotope used in medical imaging. The initial mass of the isotope is 80 grams, and it decays by 15% every hour. Write an exponential function in the form M(t) = ab^t that models the remaining mass of the isotope after t hours. Answer: M(t) = 80(0.85)^t Solution: Exponential decay functions model situations where a quantity decreases by a fixed percentage over equal time intervals. The general form is y = a(1 - r)^t, where a is the initial amount, r is the decay rate as a decimal, and t is time.
   Full step-by-step solution

   Exponential decay functions model situations where a quantity decreases by a fixed percentage over equal time intervals. The general form is y = a(1 - r)^t, where a is the initial amount, r is the decay rate as a decimal, and t is time. For example, if a car depreciates by 20% annually, the decay factor would be 0.8, representing the 80% that remains each year.