Problem 91 - Precalculus: Functions and Graphs

Question

The table gives the numbers of minutes of daylight occurring at various latitudes in the northern hemisphere at the summer and winter solstices. $$\begin{array}{|c|c|c|}\hline \text { Latitude } & \text { Summer } & \text { Winter } \\\\\hline 0^{\circ} & 720 & 720 \\\\\hline 10^{\circ} & 755 & 685 \\\\\hline 20^{\circ} & 792 & 648 \\\\\hline 30^{\circ} & 836 & 604 \\\\\hline 40^{\circ} & 892 & 548 \\\\\hline 50^{\circ} & 978 & 462 \\\\\hline 60^{\circ} & 1107 & 333 \\\\\hline\end{array}$$ (a) Which of the following equations more accurately predicts the length of day at the summer solstice at latitude \(L ?\) (1) \(D_{1}=6.096 L+685.7\) (2) \(D_{2}=0.00178 L^{3}-0.072 L^{2}+4.37 L+719\) (b) Approximate the length of daylight at \(35^{\circ}\) at the summer solstice.

Step-by-Step Solution

Answer

Equation 2 is more accurate; estimated daylight at 35° is 860 minutes.

Step 1: Calculate Predicted Summer Daylight with Equation 1

Substitute \(L = 35\) into the equation \(D_1 = 6.096L + 685.7\).\[D_1 = 6.096 \times 35 + 685.7 = 213.36 + 685.7 = 899.06\]

Step 2: Calculate Predicted Summer Daylight with Equation 2

Substitute \(L = 35\) into the equation \(D_2 = 0.00178L^3 - 0.072L^2 + 4.37L + 719\).\[D_2 = 0.00178 \times 35^3 - 0.072 \times 35^2 + 4.37 \times 35 + 719\]Calculate each term: - \(35^3 = 42875\) - \(0.00178 \times 42875 = 76.285\)- \(35^2 = 1225\) - \(0.072 \times 1225 = 88.2\)- \(4.37 \times 35 = 152.95\)Add these values together: \[D_2 = 76.285 - 88.2 + 152.95 + 719 = 860.035\]

Step 3: Determine More Accurate Equation

To determine which of the equations predicts the daylight hours more accurately, compare each prediction to the closest given data point in the table. - For 30°, the table gives 836 minutes and Equation 1 predicts 899.06. - Equation 2 prediction for 30° needs minor adjustment for illustration but predicts 860.035. Compare both values with actual values at 30° from the table (836 minutes). Equation 2 seems closer.

Step 4: Conclusion for Part (a)

Based on proximity of the prediction to the actual data at 30°, Equation 2 \(D_2 = 0.00178L^3 - 0.072L^2 + 4.37L + 719\) more accurately predicts the length of day at the summer solstice at latitude \(L\).

Step 5: Approximate Daylight at 35° Latitude

From Step 2, Equation 2 at 35° latitude provides a daylight prediction of approximately 860 minutes. This is the predicted length of daylight at 35° latitude during the summer solstice.

Key Concepts

Latitude

Latitude is essentially a geographical coordinate that specifies the north-south position of a point on the Earth's surface. It is measured in degrees from the equator, which is at 0°. As you move towards the North or South Pole from the equator, the latitude increases up to 90° N (North Pole) or 90° S (South Pole).

Latitude has a significant effect on the amount of daylight received at different times of the year, particularly noticeable during the solstices. This geographical marker includes:

• Near the equator (0° latitude), daylight duration remains almost constant throughout the year.
• Moving northward, the difference between summer and winter daylight becomes more pronounced.
• A high latitude, such as within the Arctic Circle, experiences very long daylight in summer and very short daylight or even polar night in winter.

Understanding latitude helps us predict daylight duration more accurately and sheds light on its influence on seasonal variations.

Summer Solstice

The summer solstice occurs around June 21st and marks the longest day of the year in the northern hemisphere. It happens when the Earth's axial tilt is closest to the Sun at 23.5° relative to its orbit, meaning the North Pole is tilted towards the Sun. During this time:

• The Sun appears at its highest point in the sky at noon, providing the greatest amount of daylight.
• Daylight duration increases as one moves higher in latitude from the equator, but eventually hits a maximum.
• Around the Arctic Circle (66.5°), the sun doesn't set for days, experiencing a phenomenon known as the "Midnight Sun."

Understanding the dynamics of the summer solstice can assist in predicting daylight duration using mathematical equations, as it impacts different latitudes uniquely.

Winter Solstice

Around December 21st, the winter solstice marks the shortest day and longest night of the year in the northern hemisphere. Opposite to the summer solstice, it occurs when the Earth's axial tilt is farthest from the Sun. During the winter solstice:

• The Sun reaches its lowest point in the sky at noon in the northern hemisphere.
• Daylight hours are at their minimum with longer and colder nights.
• In high latitudes, like the Arctic Circle, regions may experience no daylight at all, known as "Polar Night."

The winter solstice demonstrates the inverse of summer solstice conditions and is crucial in understanding how seasonal daylight variations are calculated across latitudes.

Polynomial Approximation

Polynomial approximation is a method used in mathematics to estimate a function using polynomial functions. In the context of predicting daylight duration based on latitude, more complex polynomial equations often provide better approximations than simpler linear equations. Here are key points to remember:

• Polynomials can be used to model relationships where data points show a more curved or complex trend.
• The degree of a polynomial is determined by the highest power of the variable; higher degree polynomials can model data with more twists and turns.
• Polynomial interpolation can accurately estimate daylight for latitudes not directly listed in the given data, by considering multiple terms that describe the underlying trend more comprehensively.

The polynomial equation, such as the cubic one given in the problem, may yield more accurate predictions, particularly when daylight variation is complex, as observed near the poles or in seasonal changes.