Use integration to find the average height of the point A over the first 8 minutes that the wheel is turning - Leaving Cert Mathematics - Question 8(f) - 2022

After setting up the integral, we calculate:

$\int (72 - 60 \cos(\frac{\pi t}{6})) \, dt = 72t - 60 \cdot \frac{6}{\pi} \sin(\frac{\pi t}{6}) + C$

Substituting in the limits from 0 to 8:

$= \left[ 72(8) - \frac{360}{\pi} \sin(\frac{8\pi}{6}) \right] - \left[ 72(0) - \frac{360}{\pi} \sin(0) \right]$

Calculating this gives:

$= 576 - \frac{360}{\pi} (-\sqrt{3}/2) \approx 576 + \frac{180\sqrt{3}}{\pi}$

Now we calculate the numerical value of the integral and compute:

$= 72 - 60 \cdot \cos(\frac{8\pi}{6})$

From the evaluated integral, we find:

$\frac{1}{8} (576 + \frac{180\sqrt{3}}{\pi}) - 72 + 60 \cdot \cos(\frac{8\pi}{6})$

Finally, the average height will be:

$= 72 - 60 \cdot \cos(\frac{4\pi}{3}) = 72 - 60 \cdot \left(-\frac{1}{2}\right) = 72 + 30 = 102$

Thus, the average height is approximately 65.8 m.