Questions: Survey of Calculus (7) Section 2.5 Activity Question 1, 2.5.1-P8 HW Score: 0%, 0 of 8 points Part 1 of 4 Points 0 of 1 1 ← A chemical substance has a decay rate of 7.6% per day. The rate of change of an amount N of the chemical after t days is given by dN/dt=-0.075 N. a) Let N0 represent the amount of the substance present at t=0. Find the exponential function that models the decay b) Suppose that 600 g of the substance is present at t=0. How much will remain after 6 days? c) What is the rate of change of the amount of the substance after 6 days? d) After how many days will half of the original 600 g of the substance remain? a) N(t)=

Survey of Calculus (7) Section 2.5 Activity Question 1, 2.5.1-P8 HW Score: 0%, 0 of 8 points Part 1 of 4 Points 0 of 1 1 ← A chemical substance has a decay rate of 7.6% per day. The rate of change of an amount N of the chemical after t days is given by dN/dt=-0.075 N. a) Let N0 represent the amount of the substance present at t=0. Find the exponential function that models the decay b) Suppose that 600 g of the substance is present at t=0. How much will remain after 6 days? c) What is the rate of change of the amount of the substance after 6 days? d) After how many days will half of the original 600 g of the substance remain? a) N(t)=
Transcript text: Survey of Calculus (7) Section 2.5 Activity Question 1, 2.5.1-P8 HW Score: $0 \%, 0$ of 8 points Part 1 of 4 Points 0 of 1 $1 \leftarrow$ A chemical substance has a decay rate of $7.6 \%$ per day. The rate of change of an amount N of the chemical after t days is given by $\frac{\mathrm{dN}}{\mathrm{dt}}=-0.075 \mathrm{~N}$. a) Let $\mathrm{N}_{0}$ represent the amount of the substance present at $t=0$. Find the exponential function that models the decay b) Suppose that 600 g of the substance is present at $t=0$. How much will remain after 6 days? c) What is the rate of change of the amount of the substance after 6 days? d) After how many days will half of the original 600 g of the substance remain? a) $N(t)=\square$
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Solution

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Solution Steps

Step 1: Modeling the Decay

The exponential decay function is modeled as $N(t) = N_0 e^{-kt}$, where $N(t)$ is the amount of substance remaining after $t$ days. Given: $k = 0.076$, $N_0 = 600$. The decay function becomes $N(t) = 600 e^{-0.076t}$.

Step 2: Remaining Amount After t Days

Using the decay function, the remaining amount after $t = 6$ days is $N(t) = 600 e^{-0.076*6} = 380.29$.

Step 3: Rate of Change After t Days

The rate of change at $t = 6$ days is given by $\frac{dN}{dt} = -kN_0 e^{-kt} = -28.9$.

Step 4: Time for Half of the Substance to Remain

To find when half of the substance remains, we solve $N(t) = \frac{N_0}{2}$ for $t$, which gives $t = \frac{\ln(2)}{k} = 9.12$ days.

Final Answer:

The remaining amount after 6 days is 380.29. The rate of change after 6 days is -28.9. The time for half of the substance to remain is approximately 9.12 days.

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