The weekly revenue produced by a company manufacturing air conditioning units is seasonal - Leaving Cert Mathematics - Question 8 - 2019

Set the revenue function equal to €26,250:

$$22,500 \, ext{cos} \left( \frac{\pi}{26} t \right) + 37,500 = 26,250.$$

Rearranging gives:

$$22,500 \, ext{cos} \left( \frac{\pi}{26} t \right) = -1,250.$$

Dividing both sides:

$\text{cos} \left( \frac{\pi}{26} t \right) = -\frac{1,250}{22,500} = -\frac{1}{18}.$

Using inverse cosine gives us:

$\frac{\pi}{26} t = \cos^{-1}(-\frac{1}{18}),$

And the other solution:

$\frac{\pi}{26} t = 2\pi - \cos^{-1}(-\frac{1}{18}).$

Solving these provides the two values of $t$ within 52 weeks.

Using the chain rule, the derivative of the revenue function:

$$r'(t) = -22,500 \, ext{sin} \left( \frac{\pi}{26} t \right) \cdot \frac{\pi}{26} = -\frac{11250}{13} \pi \, ext{sin} \left( \frac{\pi}{26} t \right).$$

To determine if revenue is increasing at $t = 30$, evaluate $r'(30)$:

$$r'(30) = -\frac{11250}{13} \pi \, \text{sin} \left( \frac{30\pi}{26} \right).$$

Calculating $\text{sin}\left( \frac{30\pi}{26} \right)$ results in a negative value, leading to:

$$$$

Thus, since $r'(30) > 0$, revenue is increasing at this time.

To find the minimum revenue, set the second derivative:

$r''(t) = 0$

Solving for $t$ gives:

$$$$

This occurs when:

$\frac{\pi}{26} t = n\pi$

where $n$ is an integer. So:

$t = 26n.$

Within the first 52 weeks, the minimum occurs at $t = 26$.