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Let's tackle the given problem step-by-step.

### Problem Statement
The diver’s height \( h(t) \) above the water, in meters, at time \( t \) (in seconds) is modeled by the equation:
\[ h(t) = 10 + 2t - 4.9t^2. \]

#### a) Determine when the diver will enter the water.
The diver enters the water when their height \( h(t) \) becomes zero. To find this time, we need to solve the equation:
\[ 10 + 2t - 4.9t^2 = 0. \]

#### b) Estimate the rate at which the diver’s height above the water is changing as the diver enters the water.
The rate of change of the diver's height is given by the derivative of \( h(t) \). We need to find \( h'(t) \) and then evaluate it at the time when \( h(t) = 0 \).

Let's solve these parts now.For part (a), the diver will enter the water at \( t = 1.65 \) seconds (we only consider the positive time).

### Part (b)
Now, we'll find the rate at which the diver's height is changing at this time by calculating the derivative \( h'(t) \) and then evaluating it at \( t = 1.65 \).For part (b), the rate at which the diver's height is changing as they enter the water is approximately \(-14.14\) meters per second. This negative sign indicates that the diver is descending at this rate when they hit the water.

Do you want more details or have any questions? Here are some related questions:

1. How would the initial velocity affect the time of entry?
2. What if the platform height were different—how would the results change?
3. Can we derive the maximum height the diver reaches before descending?
4. How can we calculate the total time the diver is in the air?
5. What is the acceleration of the diver throughout the dive?

**Tip:** When solving quadratic equations in physics problems, always consider the context to choose the physically meaningful solution.

A diver is on the 10 m platform, preparing to perform a dive. The diver's height above the water, in metres, at time t can be modelled using the equation h(t) = 10 + 2t - 4.9t^2. Determine when the diver will enter the water. Estimate the rate at which the diver's height above the water is changing as the diver enters the water.

Solve a quadratic equation modeling a diver's height above the water. Determine when the diver enters the water and estimate the rate of change at that moment. Includes concepts like the quadratic formula and differentiation.

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