The air quality index, I, in a large city can be modelled by the equation, 𝑰 = 𝟑𝟐 𝐬𝐢𝐧[𝟏𝟓(𝒕 − 𝟓)] + 𝟐𝟖 where t represents the time, in hours, after midnight. New legislation has been introduced that is expected to decrease the pollution levels in the city so that the index values will decrease to 90% of current values in 10 years.
a) What are the current minimum and maximum values of the index in the city?
b) At what time of the day is the air quality index at a maximum?
c) If an air quality alert is issued for times when the index is above 48, during what time period will an air quality alert be issued?
d) What sinusoidal factors in the equation for air quality index will be affected by the legislation? Explain how they will be affected.
e) What will the new air quality index equation be based on your answers in part d)?
f) How will these changes affect the times when an air quality alert will be issued?
Look at the graph, here : https://www.desmos.com/calculator/jegh0wzpku
a. The max value is 60 ....the min value is -4
b. It is at a max at 11 AM
c. The index is above 48 from 7.579 hours after midnight to 14.421 hours after midnight (this trasnlates to about 7:35AM to about 2:25 PM )
d. The amplitude - the "32" - will be affected ...it will be 90% of its current value = 32*.9 = 28.8
Thus...the index max will be reduced to 28.8 + 28 = 56.8
e. The new equation will be I = 28.8sin [ 15 (t - 5) ] + 28
f. Look at this graph : https://www.desmos.com/calculator/i04rsjwpbt
The new legislation will mean that the index level will be > 48 from about 8PM to about 2PM....thus....it shortens the previous window by about one hour