Let's break down the steps required to solve this problem, which involves a hemispherical tank filled with water up to a certain height and the task of calculating the work needed to pump out the water.

Step 1: Understanding the Formula for $r$ and Surface Area $S(h)$

The diagram shows a cross-section of the hemisphere filled with water. The goal is to express $r$ (the radius of the water surface at a height $h$) and then calculate the surface area of the water at height $h$.

From the diagram, we have a right triangle formed by:

• $R$ = 1.9 m (the radius of the hemisphere),
• $r$ (the radius of the water surface at height $h$),
• $h$ (the height of the water in the tank from the bottom up).

Using the Pythagorean theorem, we can find $r$ as: $r = \sqrt{R^2 - (R - h)^2}$

Step 2: Surface Area of the Water

The surface area of the water at height $h$ is the area of a circle with radius $r$, given by: $S(h) = \pi r^2 = \pi (R^2 - (R - h)^2)$

Step 3: Apply the Given Values

Now, let's substitute the values $R = 1.9 \, \text{m}$ and $h = 1.3 \, \text{m}$ into the above equations and compute $r$ and $S(h)$.

I'll do the calculations next.The radius of the water surface at height $h = 1.3 \, \text{m}$ is approximately $r = 1.80 \, \text{m}$.

The surface area of the water at this height is $S(h) \approx 10.21 \, \text{m}^2$.

Would you like further details or have any questions?

Here are 5 related questions to extend this problem:

1. How would the surface area change if the water height was different, say $h = 1.0 \, \text{m}$?
2. How do we calculate the volume of water in the hemispherical tank?
3. What is the total work needed to pump the water out of the tank?
4. How does the density of water influence the amount of work required to pump it?
5. How would the work change if the tank were a full sphere instead of a hemisphere?

Tip: For solving real-world problems involving fluid mechanics, it's useful to understand the relationship between height, volume, and work done due to gravity.