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From the image you provided, it appears that you have two problems:

### 1. Problem (b)
You are asked to find the time for which the instantaneous rate of change in height is \( 0 \, \text{m/s} \), using information from part (a).

### 2. Problem (4)
The movement of a glacier is modeled by the function:
\[
d(t) = 0.01t^2 + 0.5t
\]
where \( d(t) \) represents the distance in meters that a stake on the glacier has moved relative to a fixed position, and \( t \) is the number of days after the first measurement. You are required to estimate the rate at which the glacier is moving after 20 days.

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### Let's start with **problem 4**:

To find the rate at which the glacier is moving after 20 days, we need to compute the derivative of the function \( d(t) \), which will give us the rate of change of the glacier’s position with respect to time (i.e., the velocity).

1. The given function for the glacier’s movement is:
   \[
   d(t) = 0.01t^2 + 0.5t
   \]
   
2. The instantaneous rate of change (velocity) at time \( t \) is given by the derivative \( d'(t) \):
   \[
   d'(t) = \frac{d}{dt} \left( 0.01t^2 + 0.5t \right)
   \]
   Applying the power rule of differentiation:
   \[
   d'(t) = 0.02t + 0.5
   \]

3. Now, substitute \( t = 20 \) to find the velocity after 20 days:
   \[
   d'(20) = 0.02(20) + 0.5 = 0.4 + 0.5 = 0.9 \, \text{m/day}
   \]

So, the glacier is moving at a rate of **0.9 meters per day** after 20 days.

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Would you like to proceed with **problem (b)** or explore the rate of change further? Feel free to ask for details!

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### Relative Questions:
1. What does the derivative of a function represent in real-world applications?
2. How does the rate of change differ from the total change in this context?
3. What units does the derivative have in this problem, and why is it important?
4. What would the velocity of the glacier be at \( t = 0 \)? What does that represent?
5. How would the rate of change of the glacier differ after more days, say at \( t = 50 \)?

### Tip:
Always double-check units when interpreting the derivative, as it gives insight into the physical meaning of your result (e.g., meters per day in this case).

The movement of a certain glacier can be modeled by d(t) = 0.01t^2 + 0.5t, where d is the distance, in meters, that a stake on the glacier has moved, relative to a fixed position, t days after the first measurement was made. Estimate the rate at which the glacier is moving after 20 days.

Learn how to estimate the rate of glacier movement after 20 days using differentiation. This problem explores key calculus concepts like the instantaneous rate of change, quadratic functions, and the application of the power rule. Suitable for students in Grades 11-12.

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