A city's daily high temperature, in degrees Celsius, can be modelled by the function t(d) = -28 cos(2π/365 d) + 10. On days when the temperature is approximately 32 °C or above, th... A city's daily high temperature, in degrees Celsius, can be modelled by the function t(d) = -28 cos(2π/365 d) + 10. On days when the temperature is approximately 32 °C or above, the air conditioners at city hall are turned on. During what days of the year are the air conditioners running at city hall? Read more

Steps to Solve

1. Set up the equation

To find when the temperature reaches or exceeds 32 °C, we set the function equal to 32:

$$ -28 \cos\left(\frac{2\pi}{365} d\right) + 10 \geq 32 $$

2. Rearrange the inequality

First, we need to isolate the cosine term:

$$ -28 \cos\left(\frac{2\pi}{365} d\right) \geq 32 - 10 $$

This simplifies to:

$$ -28 \cos\left(\frac{2\pi}{365} d\right) \geq 22 $$

3. Divide by -28

Since we are dividing by a negative number, we must flip the inequality sign:

$$ \cos\left(\frac{2\pi}{365} d\right) \leq -\frac{22}{28} $$

4. Simplify the fraction

The fraction $-\frac{22}{28}$ simplifies to:

$$ \cos\left(\frac{2\pi}{365} d\right) \leq -\frac{11}{14} $$

5. Find the angles

To solve for $d$, we need to find the angles for which this cosine value occurs within the unit circle. The reference angle is:

$$ \theta = \cos^{-1}(-\frac{11}{14}) $$

Calculating this gives us two angles in the range [0, 2π]:

• $$ \theta_1 = \cos^{-1}(-\frac{11}{14}) $$
• $$ \theta_2 = 2\pi - \theta_1 $$

6. Translate angles back to days

The angles can then be converted back to days using the formula:

$$ d = \frac{365}{2\pi} \theta $$

7. Calculate intervals for $d$

The days when the temperature exceeds 32 °C will be found in two intervals:

• From $d_1$ to $d_2$
• And from $d_3$ to $d_4$, which accounts for periodicity (the cycle repeating over 365 days).