What change(s) should Sylvia make to the equation 4800(8)^(t/2) = 600 to find the value of t, given that the deer population halves every 8 years and the half-life decay formula is... What change(s) should Sylvia make to the equation 4800(8)^(t/2) = 600 to find the value of t, given that the deer population halves every 8 years and the half-life decay formula is m(t) = m₀(1/2)^(t/h)?
Understand the Problem
The question describes a scenario of deer population decay and asks what correction should be made to the given equation to correctly model this decay, using the half-life decay formula. We need to identify the error in the current equation and select the statement that proposes the correct adjustment based on the formula m(t) = m₀(1/2)^(t/h), where m(t) is the population after time t, m₀ is the initial population, and h is the half-life.
Answer
8 should be replaced with $\frac{1}{2}$, and $\frac{t}{2}$ should be replaced with $\frac{t}{8}$
Answer for screen readers
B. 8 should be replaced with $\frac{1}{2}$, and $\frac{t}{2}$ should be replaced with $\frac{t}{8}$
Steps to Solve
- Identify the correct half-life decay formula
The half-life decay formula is given as $m(t) = m_0(1/2)^{t/h}$, where $m(t)$ is the population after time $t$, $m_0$ is the initial population, and $h$ is the half-life.
- Compare Sylvia's equation with the correct formula
Sylvia's equation is $4800(8)^{\frac{t}{2}} = 600$. We need to determine what changes should be made to make it consistent with the half-life formula.
- Analyze the base of the exponent
In the correct formula, the base is $\frac{1}{2}$. In Sylvia's equation, the base is $8$. She needs to replace $8$ with $\frac{1}{2}$.
- Analyze the exponent
In the correct formula, the exponent is $\frac{t}{h}$. In Sylvia's equation, the exponent is $\frac{t}{2}$. Since the half-life $h$ is 8 years, she needs to replace $\frac{t}{2}$ with $\frac{t}{8}$.
- Determine the necessary changes
Based on the half-life formula, Sylvia should replace $8$ with $\frac{1}{2}$ and $\frac{t}{2}$ with $\frac{t}{8}$.
B. 8 should be replaced with $\frac{1}{2}$, and $\frac{t}{2}$ should be replaced with $\frac{t}{8}$
More Information
The half-life formula is used to model the decay of a substance or population, where the quantity decreases by half over a constant period.
Tips
A common mistake is to confuse the initial population $m_0$ with the final population $m(t)$. Also, forgetting to change both the base and the exponent according to the half-life formula.
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