Answer :
To determine the number of bacteria present at different times for this certain product, we'll be using the function [tex]\( f(t) = 500e^{0.1t} \)[/tex], where [tex]\( t \)[/tex] is the time in days. Let's find the number of bacteria at [tex]\( t = 2 \)[/tex] days, [tex]\( t = 3 \)[/tex] days, and [tex]\( t = 7 \)[/tex] days (which is equivalent to 1 week).
Let's go through each time point step-by-step:
### (a) 2 days
1. Plug [tex]\( t = 2 \)[/tex] into the function [tex]\( f(t) = 500e^{0.1t} \)[/tex].
2. Compute the expression [tex]\( 500e^{0.1 \times 2} \)[/tex].
After computation:
- The result is approximately 611 million bacteria.
- Rounding to the nearest integer, we have 611 million bacteria at 2 days.
### (b) 3 days
1. Plug [tex]\( t = 3 \)[/tex] into the function [tex]\( f(t) = 500e^{0.1t} \)[/tex].
2. Compute the expression [tex]\( 500e^{0.1 \times 3} \)[/tex].
After computation:
- The result is approximately 675 million bacteria.
- Rounding to the nearest integer, we have 675 million bacteria at 3 days.
### (c) 1 week (7 days)
1. Plug [tex]\( t = 7 \)[/tex] into the function [tex]\( f(t) = 500e^{0.1t} \)[/tex].
2. Compute the expression [tex]\( 500e^{0.1 \times 7} \)[/tex].
After computation:
- The result is approximately 1007 million bacteria.
- Rounding to the nearest integer, we have 1007 million bacteria at 7 days.
### Summary:
- At 2 days, the approximate number of bacteria is 611 million.
- At 3 days, the approximate number of bacteria is 675 million.
- At 1 week (7 days), the approximate number of bacteria is 1007 million.
These approximations are rounded to the nearest integer as required.
Let's go through each time point step-by-step:
### (a) 2 days
1. Plug [tex]\( t = 2 \)[/tex] into the function [tex]\( f(t) = 500e^{0.1t} \)[/tex].
2. Compute the expression [tex]\( 500e^{0.1 \times 2} \)[/tex].
After computation:
- The result is approximately 611 million bacteria.
- Rounding to the nearest integer, we have 611 million bacteria at 2 days.
### (b) 3 days
1. Plug [tex]\( t = 3 \)[/tex] into the function [tex]\( f(t) = 500e^{0.1t} \)[/tex].
2. Compute the expression [tex]\( 500e^{0.1 \times 3} \)[/tex].
After computation:
- The result is approximately 675 million bacteria.
- Rounding to the nearest integer, we have 675 million bacteria at 3 days.
### (c) 1 week (7 days)
1. Plug [tex]\( t = 7 \)[/tex] into the function [tex]\( f(t) = 500e^{0.1t} \)[/tex].
2. Compute the expression [tex]\( 500e^{0.1 \times 7} \)[/tex].
After computation:
- The result is approximately 1007 million bacteria.
- Rounding to the nearest integer, we have 1007 million bacteria at 7 days.
### Summary:
- At 2 days, the approximate number of bacteria is 611 million.
- At 3 days, the approximate number of bacteria is 675 million.
- At 1 week (7 days), the approximate number of bacteria is 1007 million.
These approximations are rounded to the nearest integer as required.
