Daylight on day 150 of the year

The cosine is largest when its argument is 0, which happens at $t=172$. So $D$ peaks on day 172, and the days just before it lie on the rising side of the peak. The function is at its midline value a quarter period before, about $172-\frac{365}{4}\approx 80.75$, so day 150 sits between the midline crossing and the maximum.

On that stretch the graph is climbing toward a peak, so it is increasing, and it is concave down, flattening as it nears the top. Increasing and concave down means increasing at a decreasing rate.

The calculator confirms it. Day-to-day changes around day 150: - $D(150)-D(149)\approx 877.663-876.622=1.04$ - $D(151)-D(150)\approx 878.659-877.663=1.00$ - $D(152)-D(151)\approx 879.611-878.659=0.95$

The gains are positive and shrinking. That is choice A.

Each choice

• A (correct). Day 150 is about three weeks before the maximum on day 172, where a sinusoid is rising and leveling off.
• B. Falling while concave down, so dropping faster each day, is how a sinusoid behaves just after a maximum. Placing day 150 after the day 172 peak instead of before it leads here.
• C. The curve is rising, though approaching a peak does not mean speeding up. A rising sinusoid gains faster only between its minimum and its midline; day 150 is past the midline, around day 81, so the daily gains are shrinking.
• D. Falling while concave up, so leveling off, is how a sinusoid behaves just before a minimum. Reading day 172 as the shortest day instead of the longest puts day 150 there; with a positive 160 in front of the cosine, day 172 is the maximum.