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Answer :
To solve this problem, let's carefully break down the steps to determine the number of cells present [tex]\( t \)[/tex] hours after starting the experiment.
1. Identify the initial number of cells:
At the start of the experiment, the initial number of cells is [tex]\( 170 \)[/tex].
2. Determine the decay rate:
The number of cells decreases by [tex]\( 17\% \)[/tex] each hour.
3. Calculate the decay factor:
A decrease of [tex]\( 17\% \)[/tex] means that [tex]\( 83\% \)[/tex] of the cells remain each hour. This can be represented as:
[tex]\[
\text{Decay factor} = 1 - 0.17 = 0.83
\][/tex]
4. Set up the exponential decay formula:
The number of cells after [tex]\( t \)[/tex] hours can be modeled using an exponential decay function. The general form for exponential decay is:
[tex]\[
N(t) = N_0 \cdot ( \text{decay factor} )^t
\][/tex]
Here, [tex]\( N(t) \)[/tex] represents the number of cells at time [tex]\( t \)[/tex], [tex]\( N_0 \)[/tex] is the initial number of cells, and the decay factor is [tex]\( 0.83 \)[/tex].
5. Substitute the known values into the formula:
Given [tex]\( N_0 = 170 \)[/tex] and the decay factor is [tex]\( 0.83 \)[/tex], the formula becomes:
[tex]\[
f(t) = 170 \cdot (0.83)^t
\][/tex]
Now, compare this equation with the provided options:
a. [tex]\( f(t) = 170(0.83)t \)[/tex]
b. [tex]\( f(t) = 170(0.17)^t \)[/tex]
c. [tex]\( f(t) = 170 - 0.17 t \)[/tex]
d. [tex]\( f(t) = 170(0.83)^t \)[/tex]
The correct representation of the number of cells [tex]\( f(t) \)[/tex] present [tex]\( t \)[/tex] hours after starting the experiment is:
[tex]\[
\boxed{f(t) = 170(0.83)^t}
\][/tex]
1. Identify the initial number of cells:
At the start of the experiment, the initial number of cells is [tex]\( 170 \)[/tex].
2. Determine the decay rate:
The number of cells decreases by [tex]\( 17\% \)[/tex] each hour.
3. Calculate the decay factor:
A decrease of [tex]\( 17\% \)[/tex] means that [tex]\( 83\% \)[/tex] of the cells remain each hour. This can be represented as:
[tex]\[
\text{Decay factor} = 1 - 0.17 = 0.83
\][/tex]
4. Set up the exponential decay formula:
The number of cells after [tex]\( t \)[/tex] hours can be modeled using an exponential decay function. The general form for exponential decay is:
[tex]\[
N(t) = N_0 \cdot ( \text{decay factor} )^t
\][/tex]
Here, [tex]\( N(t) \)[/tex] represents the number of cells at time [tex]\( t \)[/tex], [tex]\( N_0 \)[/tex] is the initial number of cells, and the decay factor is [tex]\( 0.83 \)[/tex].
5. Substitute the known values into the formula:
Given [tex]\( N_0 = 170 \)[/tex] and the decay factor is [tex]\( 0.83 \)[/tex], the formula becomes:
[tex]\[
f(t) = 170 \cdot (0.83)^t
\][/tex]
Now, compare this equation with the provided options:
a. [tex]\( f(t) = 170(0.83)t \)[/tex]
b. [tex]\( f(t) = 170(0.17)^t \)[/tex]
c. [tex]\( f(t) = 170 - 0.17 t \)[/tex]
d. [tex]\( f(t) = 170(0.83)^t \)[/tex]
The correct representation of the number of cells [tex]\( f(t) \)[/tex] present [tex]\( t \)[/tex] hours after starting the experiment is:
[tex]\[
\boxed{f(t) = 170(0.83)^t}
\][/tex]
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