A wind turbine, used to generate electricity, has three equally spaced blades 65 metres long - Leaving Cert Mathematics - Question a - 2014

The area of the disc traced out by the blades during one full rotation is given by the formula for the area of a circle, which is:

$A = \\pi r^2$

Here, the radius (r) represents the length of the blades, which is 65 metres:

$A = \\pi (65)^2 \\\approx 13273 \, m^2$

Correct to the nearest whole number, the area is approximately 13273 m².

To find the area of the triangle formed by the tips of the blades, we can use the formula for the area of a triangle when the lengths of all three sides are known. Since it is an equilateral triangle:

A = rac{1}{2} imes ext{base} imes ext{height}

Using the sides of length 65 m and the angle of 120° gives:

A \\\approx 5488 \, m^2$$ Thus, the area of the triangle formed is approximately 5488 m².

The total number of rotations can be calculated as follows:

1. Calculate the total operating time in minutes:
   • Operating time in a year = 365 days × 24 hours/day × 60 minutes/hour × 31% = total minutes.
2. Calculate the total rotations:
   • Total rotations = (15 rotations/minute) × (total operating minutes for 25 years).

After performing the calculations, we get:

$15 \times 60 \times 24 \times 365 \times 25 \times 0.31 = 61101000$ $

In scientific notation, it is expressed as:

$6.1101 \times 10^7$.