Exponential Form

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Answer Key & Explanations

Exponential Form · Grade 9 · Worksheet 1

1. Aroha is analyzing a population of rare ferns in a conservation area. The graph below (described in words) shows the number of ferns over time. At time t = 0 (the year 2020), there are 27 ferns. The population triples every 3 years. The graph passes through the points (0, 27), (3, 81), and (6, 243). Write an exponential function of the form f(t) = a * b^t that models the number of ferns after t years. Answer: f(t) = 27 * (3^(1/3))^t or f(t) = 27 * (cube root of 3)^t Solution: Identify the initial value a. At t = 0, f(0) = 27, so a = 27. Step 2: The population triples every 3 years, so f(3) = 3 * 27 = 81.
   Full step-by-step solution

   Step 1: Identify the initial value a. At t = 0, f(0) = 27, so a = 27. Step 2: The population triples every 3 years, so f(3) = 3 * 27 = 81. Step 3: In the form f(t) = 27 * b^t, at t = 3 we have 27 * b^3 = 81. Step 4: Divide both sides by 27: b^3 = 81 / 27 = 3. Step 5: Take the cube root of both sides: b = cube root of 3 (or 3^(1/3)). Step 6: Therefore, the function is f(t) = 27 * (3^(1/3))^t. The answer is f(t) = 27 * (cube root of 3)^t.

2. Isabella is tracking the spread of a rumor in her school. She creates a graph where the x-axis represents time in hours (t) and the y-axis represents the number of students who have heard the rumor. The graph passes through the points (0, 12) and (3, 324). If the rumor spreads according to an exponential function of the form f(t) = a · b^t, what is the value of b (the growth factor per hour)? Answer: 3 Solution: Use the point (0, 12). Since f(0) = a * b^0 = a, we get a = 12. Write the function as f(t) = 12 * b^t.
   Full step-by-step solution

   Step 1: Use the point (0, 12). Since f(0) = a * b^0 = a, we get a = 12. Step 2: Write the function as f(t) = 12 * b^t. Step 3: Use the point (3, 324): 324 = 12 * b^3. Step 4: Divide both sides by 12: 324 / 12 = 27 = b^3. Step 5: Take the cube root of both sides: b = cube root of 27 = 3. The answer is 3.

3. A pharmaceutical company is testing a new medication that decays exponentially in the bloodstream. A patient takes a 400 mg dose, and after 6 hours, only 50 mg remains active. Write an exponential function in the form M(t) = ab^t that models the amount of active medication in milligrams after t hours. Answer: M(t) = 400*(1/2)^(t/2) Solution: Exponential decay functions model situations where a quantity decreases by a constant factor over equal time intervals. The general form is y = a(b)^t, where a is the initial amount and b is the decay factor per time unit.
   Full step-by-step solution

   Exponential decay functions model situations where a quantity decreases by a constant factor over equal time intervals. The general form is y = a(b)^t, where a is the initial amount and b is the decay factor per time unit. To find the decay factor, we can use the relationship between amounts at different times.

4. Sophia is a botanist studying the spread of a rare orchid in a protected forest. The current orchid population is 240 plants. Due to favorable conditions, the population triples every 7 years. Write an exponential function in the form P(t) = a·b^t that models the orchid population after t years. Answer: P(t) = 240 · (3^(t/7)) Solution: Identify the initial population. The current population is 240 plants, so a = 240. Determine the growth factor.
   Full step-by-step solution

   Step 1: Identify the initial population. The current population is 240 plants, so a = 240. Step 2: Determine the growth factor. The population triples every 7 years, so the base b is 3. Step 3: Express the time in units of the given period. Since the population triples every 7 years, the number of 7-year periods in t years is t/7. Step 4: Write the exponential function in the form P(t) = a·b^t. Here, the exponent must reflect the number of periods, so P(t) = 240 · (3^(t/7)). The exponential function is P(t) = 240 · (3^(t/7)).

5. 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 50 mg of the drug remains active. Write an exponential function in the form D(t) = ab^t that models the amount of active drug in the bloodstream after t hours. Answer: D(t)=800*(1/2)^t Solution: Identify the initial amount a = 800 mg Use the point (4, 50) to find b: 50 = 800 * b^4 Divide both sides by 800: b^4 = 50/800 = 1/16 Take the fourth root of both sides: b = (1/16)^(1/4) = 1/2 Write the final function: D(t) = 800 * (1/2)^t The answer is D(t)=800*(1/2)^t.
   Full step-by-step solution

   Step 1: Identify the initial amount a = 800 mg Step 2: Use the point (4, 50) to find b: 50 = 800 * b^4 Step 3: Divide both sides by 800: b^4 = 50/800 = 1/16 Step 4: Take the fourth root of both sides: b = (1/16)^(1/4) = 1/2 Step 5: Write the final function: D(t) = 800 * (1/2)^t The answer is D(t)=800*(1/2)^t.

6. A right triangle is drawn on a coordinate plane with vertices at (0,0), (6,0), and (6,8). The triangle is reflected across the line y = x. What is the area of the new triangle formed after reflection? Answer: 24 Solution: Find the area of the original triangle. The original triangle has vertices at (0,0), (6,0), and (6,8). The base is 6 units (from (0,0) to (6,0)).
   Full step-by-step solution

   Step 1: Find the area of the original triangle. The original triangle has vertices at (0,0), (6,0), and (6,8). The base is 6 units (from (0,0) to (6,0)). The height is 8 units (from (6,0) to (6,8)). Area = (1/2) × base × height = (1/2) × 6 × 8 = 24 square units. Step 2: Understand reflection across y = x. When reflecting across the line y = x, the x and y coordinates swap. Point (0,0) becomes (0,0) Point (6,0) becomes (0,6) Point (6,8) becomes (8,6) Step 3: Find the area of the reflected triangle. The reflected triangle has vertices at (0,0), (0,6), and (8,6). This forms a right triangle with legs of length 6 and 8. Area = (1/2) × 6 × 8 = 24 square units. Step 4: Verify the answer. Reflection is a rigid transformation that preserves area. Therefore, the area remains 24 square units. The answer is 24.

7. Lena is studying the decay of a radioactive isotope used in medical imaging. The initial mass of the isotope is 800 milligrams, 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) = 800(0.85)^t Solution: Exponential decay functions model situations where a quantity decreases by a fixed percentage over equal time intervals. The initial value 'a' represents the starting amount, while the base 'b' represents the decay factor (1 minus the decay rate as a decimal).
   Full step-by-step solution

   Exponential decay functions model situations where a quantity decreases by a fixed percentage over equal time intervals. The initial value 'a' represents the starting amount, while the base 'b' represents the decay factor (1 minus the decay rate as a decimal). For example, if a different substance decayed by 20% each period, the decay factor would be 0.80.