A forest has 800 pine trees, but a disease is introduced that kills (1)/(4) of the pine trees in the forest every year. Write a function that gives the number of pine trees remaining P(t) in the forest t years after the disease is introduced. P(t)=◻+=^(+)

Understand Decay Pattern: Step 1: To write the function, we need to understand the pattern of decay of the pine trees. Since one quarter of the trees are killed each year, this means that each year, the forest retains three quarters (or 75%) of its pine trees from the previous year. This is a geometric decay with a decay factor of 0.75 (since 1 - 1/4 = 3/4). Exponential Decay Function: Step 2: The general form of an exponential decay function is P(t) = P_0 * (decay_factor)^t, where P_0 is the initial quantity and decay_factor is the factor by which the quantity decreases each year. In this case, P_0 is the initial number of pine trees, which is 800, and the decay factor is 0.75. Substitute Values: Step 3: Substitute the values into the exponential decay function to get the specific function for this problem. P(t) = 800 * (0.75)^t. This function will give us the number of pine trees remaining in the forest t years after the disease is introduced.

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A forest has 800 pine trees, but a disease is introduced that kills
(1)/(4) of the pine trees in the forest every year.
Write a function that gives the number of pine trees remaining
P(t) in the forest
t years after the disease is introduced.

P(t)=◻+=^(+)

A forest has $800$ pine trees, but a disease is introduced that kills $\frac{1}{4}$ of the pine trees in the forest every year. $\newline$ Write a function that gives the number of pine trees remaining $P(t)$ in the forest $t$ years after the disease is introduced. $\newline$ $P(t)=\square$

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Algebra 2

Interpret the exponential function word problem

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Q. A forest has

800

pine trees, but a disease is introduced that kills

\frac{1}{4}

of the pine trees in the forest every year.

\newline

Write a function that gives the number of pine trees remaining

P(t)

in the forest

t

years after the disease is introduced.

\newline

P(t)=\square

Understand Decay Pattern: Step $1$ : To write the function, we need to understand the pattern of decay of the pine trees. Since one quarter of the trees are killed each year, this means that each year, the forest retains three quarters (or $75\%$ ) of its pine trees from the previous year. This is a geometric decay with a decay factor of $0.75$ (since $1 - \frac{1}{4} = \frac{3}{4}$ ).
Exponential Decay Function: Step $2$ : The general form of an exponential decay function is $P(t) = P_0 \times (\text{decay\_factor})^t$ , where $P_0$ is the initial quantity and decay\_factor is the factor by which the quantity decreases each year. In this case, $P_0$ is the initial number of pine trees, which is $800$ , and the decay factor is $0.75$ .
Substitute Values: Step $3$ : Substitute the values into the exponential decay function to get the specific function for this problem. $P(t) = 800 \times (0.75)^t$ . This function will give us the number of pine trees remaining in the forest $t$ years after the disease is introduced.