The approximate length of the day in Galway, measured in hours from sunrise to sunset, may be calculated using the function $$f(t) = 12.25 + 4.75 imes ext{sin} \left( \frac{2\pi t}{365} \right)$$, where $t$ is the number of days after March 21st and $\frac{2\pi t}{365}$ is expressed in radians - Leaving Cert Mathematics - Question 9 - 2015

We need to solve for $t$ in the equation: $f(t) = 15$

This gives:

$12.25 + 4.75 \text{sin} \left( \frac{2\pi t}{365} \right) = 15$

Rearranging yields: $\text{sin} \left( \frac{2\pi t}{365} \right) = \frac{15 - 12.25}{4.75} = 0.578947$

Now solving: $\frac{2\pi t}{365} \approx 0.6153 \text{ or } 0.9743$

Which gives approximately:

$t \approx 35.87 \implies \text{April 26}$

To find the derivative, we differentiate:

$f'(t) = 0 + 4.75 \times \frac{2\pi}{365} \text{cos} \left( \frac{2\pi t}{365} \right)$

This simplifies to:

$f'(t) = \frac{9.5\pi}{365} \text{cos} \left( \frac{2\pi t}{365} \right)$

The longest day occurs when $\text{cos} \left( \frac{2\pi t}{365} \right) = 1$, which gives:

$t = 12.25 + 4.75 \times 1 = 17 \text{ hours.}$

To find the average length of the day, we compute:

$\frac{1}{b-a} \int_a^b f(t) dt$

Where:

$a = 1, \ b = 184$

The integral becomes:

$\int_1^{184} \left(12.25 + 4.75 \times \text{sin} \left( \frac{2\pi t}{365} \right)\right) dt$

This evaluates to approximately 15 hours and 15 minutes.