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Let's break down the given problem and solve it step-by-step.

### Given Function:
\[
G(x) = -0.2x^2 + 450
\]
where \( G(x) \) is the blood sugar level 1 hour after \( x \) units of insulin are injected.

### Part 1: Finding the Blood Sugar Level for Given Units of Insulin
1. **For \( x = 0 \):**
   \[
   G(0) = -0.2(0)^2 + 450 = 450
   \]

2. **For \( x = 25 \):**
   \[
   G(25) = -0.2(25)^2 + 450
   \]
   Calculate \( 25^2 = 625 \):
   \[
   G(25) = -0.2 \times 625 + 450 = -125 + 450 = 325
   \]

### Part 2: Finding the Rate of Change of Blood Sugar Level
To find the rate of change of blood sugar, we need to compute the derivative of \( G(x) \):

\[
G'(x) = \frac{d}{dx}(-0.2x^2 + 450) = -0.4x
\]

Now, we find the rate of change at the specified values of \( x \):

1. **For \( x = 10 \):**
   \[
   G'(10) = -0.4 \times 10 = -4
   \]

2. **For \( x = 25 \):**
   \[
   G'(25) = -0.4 \times 25 = -10
   \]

### Summary of Answers:
1. Blood sugar levels:
   - For \( x = 0 \): 450
   - For \( x = 25 \): 325

2. Rate of change of blood sugar:
   - For \( x = 10 \): -4
   - For \( x = 25 \): -10

Would you like a more detailed explanation, or do you have any other questions?

### Relative Questions
1. What is the significance of the negative rate of change in the blood sugar level?
2. How would the blood sugar level change if more than 40 units of insulin were injected, given the current model's limitation?
3. How does the quadratic nature of the function \( G(x) \) affect the blood sugar levels at higher doses of insulin?
4. Why is the derivative \( G'(x) = -0.4x \) relevant in determining the rate of change?
5. How would the model change if the coefficient of \( x^2 \) in \( G(x) \) were different?

### Tip
When dealing with quadratic functions, the derivative is linear and helps determine the rate of increase or decrease at any point.

Find the blood sugar level and the rate of change of blood sugar level after injecting the specified units of insulin using the function G(x) = -0.2x^2 + 450, where G(x) is the blood sugar level 1 hour after x units of insulin are injected.

Learn how to solve a quadratic function for blood sugar levels after injecting insulin using the formula G(x) = -0.2x^2 + 450. Also, find the rate of change of blood sugar using derivatives. Suitable for high school students in Grades 9-11.

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