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Let's solve the two parts of this problem one by one.

### Part 1: Time to Fully Inflate the Baffles

First, we need to calculate the volume of one baffle and then multiply by four since there are four baffles in the bouncy castle. The baffle is made up of a cylindrical part and a conical part.

#### Volume of the Cylindrical Part
The formula for the volume of a cylinder is:
\[ V_{\text{cylinder}} = \pi r^2 h \]
where:
- \( r \) is the radius of the cylinder (half of the diameter, which is \( 0.55 \) m, so \( r = 0.275 \) m)
- \( h \) is the height of the cylindrical part (\( 2.85 \) m)

#### Volume of the Conical Part
The formula for the volume of a cone is:
\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \]
where:
- \( r \) is the radius (same as the cylinder, \( 0.275 \) m)
- \( h \) is the height of the conical part (\( 0.75 \) m)

Calculating these, we can find the total volume of one baffle:

\[ V_{\text{baffle}} = V_{\text{cylinder}} + V_{\text{cone}} \]

Then, the total volume for all four baffles is:
\[ V_{\text{total}} = 4 \times V_{\text{baffle}} \]

The time to inflate can then be calculated since the air blower pumps air at \( 40 \) m³/h:
\[ \text{Time} = \frac{V_{\text{total}}}{\text{Rate of air blower}} \]

Let's calculate these values:

### Part 2: Angle Calculation
The angle in question is between the top of the cylinder and the slant side of the cone. We can use trigonometric relationships in a right triangle formed by the radius at the top of the cone, the height of the cone, and the slant height.

We'll calculate the slant height using the Pythagorean theorem:
\[ s = \sqrt{r^2 + h^2} \]

The angle \( \theta \) between the top of the cylinder and the slant side can be calculated using the tangent function:
\[ \tan(\theta) = \frac{h}{r} \]

Let's perform these calculations now.

Manaia decides to rent out a bouncy castle, which has four baffles (the corners of the castle). Investigate how long it will take to fully inflate the four baffles of the bouncy castle and find the angle between the top of the cylinder and the slanted side of the triangular baffle when it is fully inflated.

Learn how to calculate the time to fully inflate four bouncy castle baffles using volume formulas for cylinders and cones. Additionally, find the angle between the top of the cylinder and the slanted side of the triangular baffle using trigonometric functions. Suitable for Grades 10-12.

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