5. for a given day, the highest temperature is 85 degrees at 3 pm. the coldest temperature is 55 degrees at…

Explanation:

Step 1: Determine the amplitude (A), vertical shift (K)

The general form of a cosine function for this problem is (y = A\cos\left(B(x - H)\right)+K). The period of a temperature - cycle is (24) hours, so (B=\frac{360^{\circ}}{24}=15^{\circ}). The amplitude (A=\frac{\text{Max}-\text{Min}}{2}=\frac{85 - 55}{2}=15). The vertical shift (K=\frac{\text{Max}+\text{Min}}{2}=\frac{85 + 55}{2}=70). The function has a maximum at (x = 15) (3 PM is (15) hours after midnight), so the phase - shift (H = 15). The function is (y=15\cos\left(15(x - 15)\right)+70).

Step 2: Calculate the temperature at 8 AM ((x = 8))

Substitute (x = 8) into the function: [ \begin{align*} y&=15\cos\left(15(8 - 15)\right)+70\ &=15\cos(-105^{\circ})+70\ &=15\cos(105^{\circ})+70\ &\approx15\times(-0.259)+70\ &=- 3.885+70\ &=66.115\approx66^{\circ} \end{align*} ]

Step 3: Calculate the temperature at 1 PM ((x = 13))

Substitute (x = 13) into the function: [ \begin{align*} y&=15\cos\left(15(13 - 15)\right)+70\ &=15\cos(-30^{\circ})+70\ &=15\times\frac{\sqrt{3}}{2}+70\ &=\frac{15\sqrt{3}}{2}+70\ &\approx12.99+70\ &=82.99\approx83^{\circ} \end{align*} ]

Step 4: Calculate the temperature at midnight ((x = 24))

Substitute (x = 24) into the function: [ \begin{align*} y&=15\cos\left(15(24 - 15)\right)+70\ &=15\cos(135^{\circ})+70\ &=15\times(-\frac{\sqrt{2}}{2})+70\ &=-\frac{15\sqrt{2}}{2}+70\ &\approx - 10.61+70\ &=59.39\approx59^{\circ} \end{align*} ]

Answer:

A. Approximately (66^{\circ}) B. Approximately (83^{\circ}) C. Approximately (59^{\circ})