Using the exponential function graph, it is found that:
1. 8.53 micrograms of insulin broke down in the second minute.
2. 0.1216 = 12.16% of the initial amount is present in the last minute.
What is an exponential function?
A decaying exponential function is modeled by:
[tex]A(t) = A(0)(1 - r)^t[/tex]
In which:
- A(0) is the initial value.
- r is the decay rate, as a decimal.
In this problem:
- The patient who is diabetic receives 100 micrograms of insulin, hence [tex]A(0) = 100[/tex].
- From the graph, each minute, the amount is 90% of the previous minute amount, hence [tex]1 - r = 0.9[/tex].
Hence, the equation is:
[tex]A(t) = 100(0.9)^t[/tex]
Item 1:
This amount is [tex]A^{\prime}(2)[/tex], hence:
[tex]A^{\prime}(t) = 100\ln{0.9}(0.9)^t[/tex]
[tex]A^{\prime}(2) = 100\ln{0.9}(0.9)^2 = -8.53[/tex]
8.53 micrograms of insulin broke down in the second minute.
Item 2:
This is the amount at the 20th minute, hence:
[tex]A(20) = 100(0.9)^{20} = 12.16[/tex]
12.16 micrograms out of 100 micrograms remain, hence 0.1216 = 12.16% of the initial amount is present in the last minute.
You can learn more about exponential functions at https://brainly.com/question/25537936
