Exponential Models

lessonbunny.com

Answer Key & Explanations

Exponential Models · Grade 11 · Worksheet 3

1. Emma recorded the temperature of a cooling liquid: Time (min) 0, 5, 10, 15; Temperature (°C) 80, 40, 20, 10. Is an exponential model appropriate? Justify your answer. Answer: A. yes Solution: Check the ratio between consecutive temperature values. From 0 to 5 minutes: 40/80 = 0.5 From 5 to 10 minutes: 20/40 = 0.5 From 10 to 15 minutes: 10/20 = 0.5 Since the ratio is constant (0.5) for equal time intervals, an exponential model is appropriate.
   Full step-by-step solution

   Step 1: Check the ratio between consecutive temperature values. Step 2: From 0 to 5 minutes: 40/80 = 0.5 Step 3: From 5 to 10 minutes: 20/40 = 0.5 Step 4: From 10 to 15 minutes: 10/20 = 0.5 Step 5: Since the ratio is constant (0.5) for equal time intervals, an exponential model is appropriate. The answer is Yes.

2. Matiu analyzed bacterial growth data: Day 0: 15, Day 1: 45, Day 2: 135, Day 3: 405. Is an exponential model appropriate? Justify your answer by calculating the ratio between consecutive y-values. Answer: B. yes Solution: Check the ratio between Day 1 and Day 0: 45 ÷ 15 = 3 Check the ratio between Day 2 and Day 1: 135 ÷ 45 = 3 Check the ratio between Day 3 and Day 2: 405 ÷ 135 = 3 Since the ratio is constant at 3 for each step, and x increases by 1 each time, an exponential model is appropriate.
   Full step-by-step solution

   Step 1: Check the ratio between Day 1 and Day 0: 45 ÷ 15 = 3 Step 2: Check the ratio between Day 2 and Day 1: 135 ÷ 45 = 3 Step 3: Check the ratio between Day 3 and Day 2: 405 ÷ 135 = 3 Step 4: Since the ratio is constant at 3 for each step, and x increases by 1 each time, an exponential model is appropriate. The answer is Yes.

3. A research scientist is studying bacterial growth in a lab culture. The initial population is 500 bacteria, and observations show the population doubles every 3 hours. The scientist models this growth with the function P(t) = P₀ * e^(kt), where t is time in hours. Determine the value of the continuous growth rate k for this bacterial population. Answer: ln(2)/3 Solution: We know the initial population P₀ = 500 bacteria. The population doubles every 3 hours. The model is P(t) = P₀ * e^(k t).
   Full step-by-step solution

   Step 1: Understand the problem and the given information. We know the initial population P₀ = 500 bacteria. The population doubles every 3 hours. The model is P(t) = P₀ * e^(k t). We need to find k. Step 2: Use the doubling time information. If the population doubles in 3 hours, then when t = 3, P(3) = 2 * P₀. So: P(3) = P₀ * e^(k * 3) = 2 * P₀. Step 3: Simplify the equation. Divide both sides by P₀ (P₀ is not zero, so this is fine): e^(3k) = 2. Step 4: Solve for k. Take the natural logarithm (ln) of both sides: ln(e^(3k)) = ln(2). Since ln(e^(3k)) = 3k, we have: 3k = ln(2). Step 5: Isolate k. Divide both sides by 3: k = ln(2) / 3. Step 6: Conclusion. The continuous growth rate is k = ln(2)/3. Notice the initial population 500 was not needed for finding k — only the doubling time mattered.

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 P(t) that models the bacterial population after t hours. Then determine how many bacteria will be present after 10 hours. Answer: P(t) = 500 * 2^(t/3); approximately 3175 bacteria Solution: Exponential growth models follow the form P(t) = P₀ * a^(t/k), where P₀ is the initial amount, a is the growth factor per period, and k is the time period. For doubling scenarios, the growth factor is 2.
   Full step-by-step solution

   Exponential growth models follow the form P(t) = P₀ * a^(t/k), where P₀ is the initial amount, a is the growth factor per period, and k is the time period. For doubling scenarios, the growth factor is 2. The exponent t/k represents how many doubling periods have elapsed.

5. A research biologist is studying bacterial growth in a lab culture. The initial population is 500 bacteria, and observations show the population doubles every 3 hours. The biologist models this growth with the function P(t) = P₀ * e^(kt), where t is time in hours. Determine the continuous growth rate k for this bacterial population. Answer: ln(2)/3 Solution: We are told the population doubles every 3 hours, and the model is P(t) = P₀ * e^(kt). Write the doubling condition in terms of the model. At t = 0, P(0) = P₀ = 500.
   Full step-by-step solution

   We are told the population doubles every 3 hours, and the model is P(t) = P₀ * e^(kt). Step 1: Write the doubling condition in terms of the model. At t = 0, P(0) = P₀ = 500. At t = 3, the population is 2 * P₀. So: P(3) = P₀ * e^(k*3) = 2 * P₀. Step 2: Divide both sides by P₀ (since P₀ is not zero). e^(3k) = 2. Step 3: Take the natural logarithm of both sides. ln(e^(3k)) = ln(2). Step 4: Simplify using ln(e^x) = x. 3k = ln(2). Step 5: Solve for k. k = ln(2) / 3. This is the continuous growth rate per hour.

6. A right triangle is drawn on a coordinate plane with vertices at (0,0), (4,0), and (0,3). The triangle is rotated 90 degrees counterclockwise about the origin. What are the coordinates of the vertex that was originally at (4,0) after this rotation? Answer: (0,4) Solution: We have a right triangle with vertices at (0,0), (4,0), and (0,3). We are rotating the vertex originally at (4,0) by 90 degrees counterclockwise about the origin.
   Full step-by-step solution

   Step 1: Understand the problem. We have a right triangle with vertices at (0,0), (4,0), and (0,3). We are rotating the vertex originally at (4,0) by 90 degrees counterclockwise about the origin. Step 2: Recall the rotation rule for 90° counterclockwise about the origin. For any point (x, y), the new coordinates after a 90° counterclockwise rotation are (-y, x). Step 3: Apply the rule to the point (4,0). Here, x = 4 and y = 0. New x-coordinate = -y = -0 = 0. New y-coordinate = x = 4. Step 4: Write the new coordinates. After rotation, the point (4,0) becomes (0,4). Step 5: Verify with a quick mental check. The point (4,0) is on the positive x-axis. Rotating it 90° counterclockwise moves it to the positive y-axis at the same distance 4 from the origin. So (0,4) makes sense. Final answer: (0,4)

7. A right triangle is positioned on a coordinate plane with vertices at (0,0), (12,0), and (0,5). A circle is circumscribed around this triangle, passing through all three vertices. What is the length of the diameter of this circumscribed circle? Answer: 13 Solution: Identify that we have a right triangle with vertices at (0,0), (12,0), and (0,5). Recognize that for any right triangle, the hypotenuse is the diameter of the circumscribed circle.
   Full step-by-step solution

   Step 1: Identify that we have a right triangle with vertices at (0,0), (12,0), and (0,5). Step 2: Recognize that for any right triangle, the hypotenuse is the diameter of the circumscribed circle. Step 3: Calculate the length of the hypotenuse using the distance formula: sqrt((12-0)^2 + (0-5)^2) = sqrt(144 + 25) = sqrt(169) = 13. Step 4: Since the hypotenuse equals the diameter of the circumscribed circle, the diameter length is 13. The answer is 13.