Solving Exponential Equations - College Algebra

1. A savings account has $P(t)=2500e^{0.03t}$ (t in years); when does it reach $5000$?
   Answer: $t\approx 23.1$ years This question tests solving exponential equations related to real-world contexts in college algebra. Exponential equations model scenarios where quantities grow or decay at a constant rate over time, using the formula P = $P_0$ * $e^{rt}$ where P is the final amount, $P_0$ the initial amount, r the rate, and t the time. In the given scenario, a savings account starts with $2500 and grows at a rate of 0.03 per year, students are expected to solve for t when it reaches $5000 using the provided values. The correct answer correctly applies the exponential formula to determine the time t ≈ 23.1 years, demonstrating understanding of growth processes. A common distractor arises from misinterpreting the growth rate, such as using 0.06 instead of 0.03, which leads to incorrect results. To help students, emphasize the importance of accurately interpreting the exponential formula and understanding the context of the problem. Practice solving similar equations with varying rates and initial values to build confidence.
2. A radioactive sample has $A(t)=60e^{-0.20t}$ (t in days); when does it reach $30$ grams?
   Answer: $t\approx 3.47$ days This question tests solving exponential equations related to real-world contexts in college algebra. Exponential equations model scenarios where quantities grow or decay at a constant rate over time, using the formula P = $P_0$ * $e^{rt}$ where P is the final amount, $P_0$ the initial amount, r the rate, and t the time. In the given scenario, a radioactive sample starts with 60 grams and decays at a rate of 0.20 per day, students are expected to solve for t when it reaches 30 grams using the provided values. The correct answer correctly applies the exponential formula to determine the time t ≈ 3.47 days, demonstrating understanding of decay processes. A common distractor arises from misinterpreting the decay rate, such as using 0.10 instead of 0.20, which leads to incorrect results. To help students, emphasize the importance of accurately interpreting the exponential formula and understanding the context of the problem. Practice solving similar equations with varying rates and initial values to build confidence.
3. A bacteria culture follows $P(t)=500e^{0.30t}$ (t in hours); when does it reach $4000$?
   Answer: $t\approx 6.93$ hours This question tests solving exponential equations related to real-world contexts in college algebra. Exponential equations model scenarios where quantities grow or decay at a constant rate over time, using the formula P = $P_0$ * $e^{rt}$ where P is the final amount, $P_0$ the initial amount, r the rate, and t the time. In the given scenario, a bacteria culture starts with 500 and grows at a rate of 0.30 per hour, students are expected to solve for t when it reaches 4000 using the provided values. The correct answer correctly applies the exponential formula to determine the time t ≈ 6.93 hours, demonstrating understanding of growth processes. A common distractor arises from misinterpreting the growth rate, such as using 0.15 instead of 0.30, which leads to incorrect results. To help students, emphasize the importance of accurately interpreting the exponential formula and understanding the context of the problem. Practice solving similar equations with varying rates and initial values to build confidence.
4. A bacteria culture follows $P(t)=900e^{0.10t}$ (t in hours); when does it reach $2700$?
   Answer: $t\approx 10.99$ hours This question tests solving exponential equations related to real-world contexts in college algebra. Exponential equations model scenarios where quantities grow or decay at a constant rate over time, using the formula P = $P_0$ * $e^{rt}$ where P is the final amount, $P_0$ the initial amount, r the rate, and t the time. In the given scenario, a bacteria culture starts with 900 and grows at a rate of 0.10 per hour, students are expected to solve for t when it reaches 2700 using the provided values. The correct answer correctly applies the exponential formula to determine the time t ≈ 10.99 hours, demonstrating understanding of growth processes. A common distractor arises from misinterpreting the growth rate, such as using 0.05 instead of 0.10, which leads to incorrect results. To help students, emphasize the importance of accurately interpreting the exponential formula and understanding the context of the problem. Practice solving similar equations with varying rates and initial values to build confidence.
5. A bacteria culture follows $P(t)=800e^{0.12t}$ (t in hours); when does it reach $1600$?
   Answer: $t\approx 5.78$ hours This question tests solving exponential equations related to real-world contexts in college algebra. Exponential equations model scenarios where quantities grow or decay at a constant rate over time, using the formula P = $P_0$ * $e^{rt}$ where P is the final amount, $P_0$ the initial amount, r the rate, and t the time. In the given scenario, a bacteria culture starts with 800 and grows at a rate of 0.12 per hour, students are expected to solve for t when it reaches 1600 using the provided values. The correct answer correctly applies the exponential formula to determine the time t ≈ 5.78 hours, demonstrating understanding of growth processes. A common distractor arises from misinterpreting the growth rate, such as using 0.06 instead of 0.12, which leads to incorrect results. To help students, emphasize the importance of accurately interpreting the exponential formula and understanding the context of the problem. Practice solving similar equations with varying rates and initial values to build confidence.
6. A bacteria culture follows $P(t)=1000e^{0.22t}$ (t in hours); when does it reach $3000$?
   Answer: $t\approx 4.99$ hours This question tests solving exponential equations related to real-world contexts in college algebra. Exponential equations model scenarios where quantities grow or decay at a constant rate over time, using the formula P = $P_0$ * $e^{rt}$ where P is the final amount, $P_0$ the initial amount, r the rate, and t the time. In the given scenario, a bacteria culture starts with 1000 and grows at a rate of 0.22 per hour, students are expected to solve for t when it reaches 3000 using the provided values. The correct answer correctly applies the exponential formula to determine the time t ≈ 4.99 hours, demonstrating understanding of growth processes. A common distractor arises from misinterpreting the growth rate, such as using 0.11 instead of 0.22, which leads to incorrect results. To help students, emphasize the importance of accurately interpreting the exponential formula and understanding the context of the problem. Practice solving similar equations with varying rates and initial values to build confidence.
7. A bacteria culture grows according to the equation $N(t) = 500 \cdot 3^{t/4}$, where $N(t)$ is the number of bacteria after $t$ hours. After how many hours will the population reach 13,500 bacteria?
   Answer: 12 hours after the initial measurement We need to solve $500 \cdot 3^{t/4} = 13500$. Dividing both sides by 500: $3^{t/4} = 27$. Since $27 = 3^3$, we have $3^{t/4} = 3^3$. Therefore $\frac{t}{4} = 3$, so $t = 12$. Choice B comes from incorrectly setting $3^{t/4} = 3^4$ when $27 = 3^4$ is wrong. Choice C results from the error $\frac{t}{4} = 2.5$ when thinking $27 = 3^{2.5}$. Choice D comes from solving $3^{t/4} = 3^{3.5}$ incorrectly.
8. A bacteria culture follows $P(t)=200e^{0.18t}$ (t in hours); when does it reach $1000$?
   Answer: $t\approx 8.94$ hours This question tests solving exponential equations related to real-world contexts in college algebra. Exponential equations model scenarios where quantities grow or decay at a constant rate over time, using the formula P = $P_0$ * $e^{rt}$ where P is the final amount, $P_0$ the initial amount, r the rate, and t the time. In the given scenario, a bacteria culture starts with 200 and grows at a rate of 0.18 per hour, students are expected to solve for t when it reaches 1000 using the provided values. The correct answer correctly applies the exponential formula to determine the time t ≈ 8.94 hours, demonstrating understanding of growth processes. A common distractor arises from misinterpreting the growth rate, such as using 0.09 instead of 0.18, which leads to incorrect results. To help students, emphasize the importance of accurately interpreting the exponential formula and understanding the context of the problem. Practice solving similar equations with varying rates and initial values to build confidence.
9. A savings account has $P(t)=3000e^{0.06t}$ (t in years); when does it reach $4500$?
   Answer: $t\approx 6.76$ years This question tests solving exponential equations related to real-world contexts in college algebra. Exponential equations model scenarios where quantities grow or decay at a constant rate over time, using the formula P = $P_0$ * $e^{rt}$ where P is the final amount, $P_0$ the initial amount, r the rate, and t the time. In the given scenario, a savings account starts with $3000 and grows at a rate of 0.06 per year, students are expected to solve for t when it reaches $4500 using the provided values. The correct answer correctly applies the exponential formula to determine the time t ≈ 6.76 years, demonstrating understanding of growth processes. A common distractor arises from misinterpreting the growth rate, such as using 0.10 instead of 0.06, which leads to incorrect results. To help students, emphasize the importance of accurately interpreting the exponential formula and understanding the context of the problem. Practice solving similar equations with varying rates and initial values to build confidence.
10. A savings account has $P(t)=10000e^{0.025t}$ (t in years); when does it reach $12000$?
    Answer: $t\approx 7.29$ years This question tests solving exponential equations related to real-world contexts in college algebra. Exponential equations model scenarios where quantities grow or decay at a constant rate over time, using the formula P = $P_0$ * $e^{rt}$ where P is the final amount, $P_0$ the initial amount, r the rate, and t the time. In the given scenario, a savings account starts with $10000 and grows at a rate of 0.025 per year, students are expected to solve for t when it reaches $12000 using the provided values. The correct answer correctly applies the exponential formula to determine the time t ≈ 7.29 years, demonstrating understanding of growth processes. A common distractor arises from misinterpreting the growth rate, such as using 0.05 instead of 0.025, which leads to incorrect results. To help students, emphasize the importance of accurately interpreting the exponential formula and understanding the context of the problem. Practice solving similar equations with varying rates and initial values to build confidence.
11. A savings account has $P(t)=1500e^{0.08t}$ (t in years); when does it reach $3000$?
    Answer: $t\approx 8.66$ years This question tests solving exponential equations related to real-world contexts in college algebra. Exponential equations model scenarios where quantities grow or decay at a constant rate over time, using the formula P = $P_0$ * $e^{rt}$ where P is the final amount, $P_0$ the initial amount, r the rate, and t the time. In the given scenario, a savings account starts with $1500 and grows at a rate of 0.08 per year, students are expected to solve for t when it reaches $3000 using the provided values. The correct answer correctly applies the exponential formula to determine the time t ≈ 8.66 years, demonstrating understanding of growth processes. A common distractor arises from misinterpreting the growth rate, such as using 0.04 instead of 0.08, which leads to incorrect results. To help students, emphasize the importance of accurately interpreting the exponential formula and understanding the context of the problem. Practice solving similar equations with varying rates and initial values to build confidence.
12. A savings account has $P(t)=800e^{0.05t}$ (t in years); when does it reach $1000$?
    Answer: $t\approx 4.46$ years This question tests solving exponential equations related to real-world contexts in college algebra. Exponential equations model scenarios where quantities grow or decay at a constant rate over time, using the formula P = $P_0$ * $e^{rt}$ where P is the final amount, $P_0$ the initial amount, r the rate, and t the time. In the given scenario, a savings account starts with $800 and grows at a rate of 0.05 per year, students are expected to solve for t when it reaches $1000 using the provided values. The correct answer correctly applies the exponential formula to determine the time t ≈ 4.46 years, demonstrating understanding of growth processes. A common distractor arises from misinterpreting the growth rate, such as using 0.10 instead of 0.05, which leads to incorrect results. To help students, emphasize the importance of accurately interpreting the exponential formula and understanding the context of the problem. Practice solving similar equations with varying rates and initial values to build confidence.
13. A radioactive substance decays according to $A(t) = A_0 e^{-0.1t}$, where $A(t)$ is the amount remaining after $t$ years. If 75% of the substance has decayed, how many years have elapsed?
    Answer: Approximately 13.86 years, using natural logarithms to solve If 75% has decayed, then 25% remains. So $A(t) = 0.25A_0$. The equation becomes $0.25A_0 = A_0 e^{-0.1t}$. Dividing by $A_0$: $0.25 = e^{-0.1t}$. Taking the natural logarithm of both sides: $\ln(0.25) = -0.1t$. So $t = \frac{\ln(0.25)}{-0.1} = \frac{-\ln(4)}{-0.1} = \frac{\ln(4)}{0.1} = 10\ln(4) \approx 10(1.386) = 13.86$ years. Choice B comes from using $\ln(0.75)$ instead of $\ln(0.25)$. Choice C results from the error $t = \frac{\ln(0.25)}{0.1}$ (missing the negative). Choice D uses an incorrect logarithm value.
14. A bacteria culture follows $P(t)=250e^{0.16t}$ (t in hours); when does it reach $500$?
    Answer: $t\approx 4.33$ hours This question tests solving exponential equations related to real-world contexts in college algebra. Exponential equations model scenarios where quantities grow or decay at a constant rate over time, using the formula P = $P_0$ * $e^{rt}$ where P is the final amount, $P_0$ the initial amount, r the rate, and t the time. In the given scenario, a bacteria culture starts with 250 and grows at a rate of 0.16 per hour, students are expected to solve for t when it reaches 500 using the provided values. The correct answer correctly applies the exponential formula to determine the time t ≈ 4.33 hours, demonstrating understanding of growth processes. A common distractor arises from misinterpreting the growth rate, such as using 0.08 instead of 0.16, which leads to incorrect results. To help students, emphasize the importance of accurately interpreting the exponential formula and understanding the context of the problem. Practice solving similar equations with varying rates and initial values to build confidence.
15. If $4^{x-3} = 2^{x+1}$, what is the value of $x$?
    Answer: $x = 7$ Since $4 = 2^2$, we can rewrite the equation as $(2^2)^{x-3} = 2^{x+1}$. Using the power rule, this becomes $2^{2(x-3)} = 2^{x+1}$, or $2^{2x-6} = 2^{x+1}$. Since the bases are equal, the exponents must be equal: $2x-6 = x+1$. Solving for $x$: $2x - x = 1 + 6$, so $x = 7$. Choice A results from solving $x-3 = x+1$ incorrectly. Choice C comes from the error $2x-6 = x-1$. Choice D results from $2x = x+6$ without properly distributing the exponent.

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