Get detailed and reliable answers to your questions on IDNLearn.com. Discover comprehensive answers to your questions from our community of experienced professionals.

The growth of bacteria makes it necessary to time-date some food products so that they will be sold and consumed before the bacteria count is too high. Suppose for a certain product the number of bacteria present is given by [tex]f(t) = 500e^{0.1t}[/tex], where [tex]t[/tex] is time in days and the value of [tex]f(t)[/tex] is in millions. Find the number of bacteria present at each time.

(a) 2 days
(b) 3 days
(c) 1 week

(a) Approximately ____ million bacteria are present in 2 days.
(Round to the nearest integer as needed.)


Sagot :

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.
Your presence in our community is highly appreciated. Keep sharing your insights and solutions. Together, we can build a rich and valuable knowledge resource for everyone. For clear and precise answers, choose IDNLearn.com. Thanks for stopping by, and come back soon for more valuable insights.