Let's solve the given questions step by step.

Problem Analysis:

The swimming pool in the diagram has:

• A volume of $8.57 \, \text{m}^3$
• Two half-cylindrical ends with a radius of $1 \, \text{m}$ and an unknown height $h$
• A rectangular prism (middle section) with a length of $2 \, \text{m}$, width equal to the diameter of the cylinder (so $2 \, \text{m}$), and height $h$.

We need to:

1. Calculate the depth $h$ of the swimming pool (5.1).
2. Calculate the surface area of the bottom of the swimming pool to repaint it (5.2).

Step 1: Calculate the Depth of the Swimming Pool (5.1)

Volume Calculation

The total volume of the pool consists of:

• Volume of the rectangular prism: $V_{\text{rectangular prism}} = l \times w \times h$
• Volume of two half-cylinders (which combine to form a full cylinder): $V_{\text{cylinder}} = \pi r^2 h$

Since $l = 2 \, \text{m}$, $w = 2 \, \text{m}$, and $r = 1 \, \text{m}$, the total volume is given by: $8.57 = (2 \times 2 \times h) + (\pi \times 1^2 \times h)$

Simplify the Equation

Substitute and simplify: $8.57 = (4h) + (\pi h)$ $8.57 = h(4 + \pi)$

Now, solve for $h$: $h = \frac{8.57}{4 + \pi}$

Using $\pi \approx 3.1416$: $h = \frac{8.57}{4 + 3.1416} \approx \frac{8.57}{7.1416} \approx 1.2 \, \text{m}$

So, the depth $h$ of the swimming pool is approximately $1.2 \, \text{m}$.

Step 2: Calculate the Surface Area of the Bottom of the Swimming Pool (5.2)

The bottom of the swimming pool consists of:

• The rectangular base of the middle section.
• The circular bases of the two half-cylindrical ends.

Surface Area Calculation

• Area of the rectangular base: $A_{\text{rectangle}} = l \times w = 2 \times 2 = 4 \, \text{m}^2$
• Area of the circular base (for one circle): $A_{\text{circle}} = \pi r^2 = \pi \times 1^2 = \pi \, \text{m}^2$
  • Since there are two half-cylinders, we have one full circle, so total circular area is $\pi \, \text{m}^2$.

Adding these together: $\text{Total surface area of the bottom} = 4 + \pi \approx 4 + 3.1416 = 7.1416 \, \text{m}^2$

Thus, the surface area of the bottom of the swimming pool to repaint is approximately $7.14 \, \text{m}^2$.

Final Answers:

1. Depth of the swimming pool (5.1): $h \approx 1.2 \, \text{m}$
2. Surface area of the bottom of the swimming pool (5.2): $\approx 7.14 \, \text{m}^2$

Would you like further clarification on any steps, or do you need additional details?

Related Questions:

1. How would the depth change if the volume of the pool was different?
2. How is the surface area affected if the radius of the half-cylinders increases?
3. How would the calculation change if the pool had full cylindrical sections on each end rather than half-cylinders?
4. What would be the cost to repaint the bottom if the paint costs $5 per square meter?
5. How would we approach this problem if the pool were three-dimensional?

Tip:

For geometry problems involving volume and surface area, always break down the structure into simpler shapes (like prisms and cylinders) for easier calculations.