SAT II question! Please help!?

Question

Hi! This is a sample question from the math SAT II (level 2) test.

It says:

The number of hours of daylight, d, in Hartsville, can be modeled by:

d = (35/3) + (7/3)sin([2pi/365] * t), where t is the number of days after March 21. The day with the greatest number of daylight hours has how many MORE daylight hours than May 1?

How do you solve that?! I don't know how to find the maximum value of a trigonometric function (only a quadratic function). Please help! I'll pick a best answer TODAY! Thank you so much!

Answer 1

Since you mentioned that you only know how to find the maximum value of quadratic functions, and not trig functions, then you presumably have not yet had calculus.

This can be solved without using calculus.

Look at the function for d. Without doing any math, you know that d will be at a maximum when sin ((2π/365)t) is at a maximum.

When dealing with polar coordinates, you know that sin(x) <= 1 for any x. sin(x) is 1 when x = π/2.

So, d is maximized when (2π/365)t = π/2
t = 365/4 = 91.25 days

From there, solve for d with t = 40 (for May 1), and t = 91.25.

Answer 2

Take the derivative with respect to t and set that equal to 0.

(7/3)(2pi/365)*cos[(2pi/365)t] = 0...which will be true when (2pi/365)*t = odd multiples of pi/2. t = 91.25 will make this true and would make d = 14 and falls on approximately 20 June; it would have 49 minutes more daylight than 1 May.

Answer 3

dmax - d(41)
= (7/3)[1 - sin((2pi/365) * 41)]
= 0.82 hours more daylight
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Ideas: There are 41 days from 3/21 to 5/1, and the maximum of sin(x) is 1.