BackClose

Select a Subject

Subject Test Math 2

Question 28 of 28

beginning of content:
Tags:
SAT Subject Test

The number of hours of daylight, d, in Hartsville can be modeled by d equals fraction numerator 35 over denominator 3 end fraction plus fraction numerator 7 over denominator 3 end fraction sin function application left parenthesis fraction numerator 2 pi over denominator 365 end fraction t right parenthesis, where t is the number of days after March 21. The day with the greatest number of hours of daylight has how many more daylight hours than May 1? (March and May have 31 days each. April and June have 30 days each.)

Select an Answer

0.8 hr

Correct Answer: 
Yes

1.5 hr

Correct Answer: 
No

2.3 hr

Correct Answer: 
No

3.0 hr

Correct Answer: 
No

4.7 hr

Correct Answer: 
No

The correct answer is A.

To determine how many more daylight hours the day with the greatest number of hours of daylight has than May 1 text end text, find the maximum number of daylight hours possible for any day and then subtract from that the number of daylight hours for May 1.

To find the greatest number of daylight hours possible for any day, notice that the expression fraction numerator 35 over denominator 3 end fraction plus fraction numerator 7 over denominator 3 end fraction sin function application left parenthesis fraction numerator 2 pi over denominator 365 end fraction t right parenthesis is maximized when sin function application left parenthesis fraction numerator 2 pi over denominator 365 end fraction t right parenthesis equals 1, which corresponds to fraction numerator 2 pi over denominator 365 end fraction t equals fraction numerator pi over denominator 2 end fraction, so t equals fraction numerator 365 over denominator 4 end fraction equals 91.25. However, for this problem, t must be a whole number, as it represents a count of days after March 21. From the shape of the graph of the sine function, either t equals 91 or t equals 92 corresponds to the day with the greatest number of hours of daylight, and since fraction numerator 35 over denominator 3 end fraction plus fraction numerator 7 over denominator 3 end fraction sin function application left parenthesis fraction numerator 2 pi over denominator 365 end fraction left parenthesis 91 right parenthesis right parenthesis greater than fraction numerator 35 over denominator 3 end fraction plus fraction numerator 7 over denominator 3 end fraction sin function application left parenthesis fraction numerator 2 pi over denominator 365 end fraction left parenthesis 92 right parenthesis right parenthesis, the expression fraction numerator 35 over denominator 3 end fraction plus fraction numerator 7 over denominator 3 end fraction sin function application left parenthesis fraction numerator 2 pi over denominator 365 end fraction t right parenthesis is maximized when t equals 91 days after March 21. (It is not required to find the day on which the greatest number of hours of daylight occurs, but it is 10 plus 30 plus 31 plus 20 days after March 21,that is, June 20.)

Since May 1 is 10 plus 30 plus 1 equals 41 days after March 21, the number of hours of daylight for May 1 is fraction numerator 35 over denominator 3 end fraction plus fraction numerator 7 over denominator 3 end fraction sin function application left parenthesis fraction numerator 2 pi over denominator 365 end fraction left parenthesis 41 right parenthesis right parenthesis.

Therefore, the day with the greatest number of hours of daylight has left parenthesis fraction numerator 35 over denominator 3 end fraction plus fraction numerator 7 over denominator 3 end fraction sin function application left parenthesis fraction numerator 2 pi over denominator 365 end fraction left parenthesis 91 right parenthesis right parenthesis right parenthesis minus left parenthesis fraction numerator 35 over denominator 3 end fraction plus fraction numerator 7 over denominator 3 end fraction sin function application left parenthesis fraction numerator 2 pi over denominator 365 end fraction left parenthesis 41 right parenthesis right parenthesis right parenthesis almost equal to 0.8 more daylight hours than May 1.

Question Difficulty: 
hard