The air quality index, I, in a large city can be modelled by the equation, 𝑰 = πŸ‘πŸ 𝐬𝐒𝐧[πŸπŸ“(𝒕 βˆ’ πŸ“)] + πŸπŸ– 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?

 Apr 25, 2018

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




cool cool cool

 Apr 25, 2018

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