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In a typical diabetic, the level of insulin might be given by the following function:

\[
f(t) = \left\{
\begin{array}{ccc}
70t + 115 & \text{if} & 0 \leq t \leq 4 \\
395 & \text{if} & 4 < t \leq 11 \\
-75t + 1220 & \text{if} & 11 < t \leq 14 \\
170 & \text{if} & 14 < t \leq 24 \\
\end{array}
\right.
\]

Here, [tex]f(t)[/tex] is the blood sugar level, in appropriate units, at time [tex]t[/tex] measured in hours from the time of insulin injection. Suppose a patient takes insulin at [tex]5 \text{ AM}[/tex]. Find the blood sugar levels at [tex]10 \text{ AM}[/tex] and at [tex]6 \text{ PM}[/tex].

At [tex]10 \text{ AM}[/tex], the blood sugar level is:
A. 815
B. 395
C. 465
D. 170

At [tex]6 \text{ PM}[/tex], the blood sugar level is:
A. 770
B. 170
C. 395
D. 245

Answer :

To solve the problem of finding the blood sugar levels at 10 AM and 6 PM for a patient who takes insulin at 5 AM, we can follow these steps:

### Step 1: Determine the time elapsed since the insulin injection
- For 10 AM:
- Time since 5 AM is 10 AM - 5 AM = 5 hours.

- For 6 PM:
- Time since 5 AM is 6 PM - 5 AM = 18 hours.

### Step 2: Use the given piecewise function for different intervals to find the blood sugar levels

The function [tex]\( f(t) \)[/tex] is defined as follows:
[tex]\[ f(t)=\begin{cases}
70t + 115 & \text{if } 0 \leq t \leq 4 \\
395 & \text{if } 4 < t \leq 11 \\
-75t + 1220 & \text{if } 11 < t \leq 14 \\
170 & \text{if } 14 < t \leq 24
\end{cases} \][/tex]

#### For 10 AM (t = 5 hours)
- The time, 5 hours, falls in the interval [tex]\( 4 < t \leq 11 \)[/tex].
- According to the piecewise function, if [tex]\( 4 < t \leq 11 \)[/tex], the blood sugar level, [tex]\( f(t) = 395 \)[/tex].

Therefore, the blood sugar level at 10 AM is:
[tex]\[ \boxed{395} \][/tex]

#### For 6 PM (t = 18 hours)
- The time, 18 hours, falls in the interval [tex]\( 14 < t \leq 24 \)[/tex].
- According to the piecewise function, if [tex]\( 14 < t \leq 24 \)[/tex], the blood sugar level, [tex]\( f(t) = 170 \)[/tex].

Therefore, the blood sugar level at 6 PM is:
[tex]\[ \boxed{170} \][/tex]

### Summary
The blood sugar levels at the specified times are:
- At 10 AM, the blood sugar level is [tex]\(\boxed{395}\)[/tex].
- At 6 PM, the blood sugar level is [tex]\(\boxed{170}\)[/tex].

Thanks for taking the time to read In a typical diabetic the level of insulin might be given by the following function f t left begin array ccc 70t 115 text if. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!

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