The maximum occupancy of a concert hall is 1,200 people. the hall is hosting a concert, and 175 people enter as soon as the doors open in the morning. the number of people coming into the hall then increases at a rate of 30% per hour. if t represents the number of hours since the doors open, which inequality can be used to determine the number of hours after which the amount of people in the concert hall will exceed the occupancy limit?

Hence, after 8 hours the amount of people in the concert hall will exceed the occupancy limit.

Step-by-step explanation:

Let represents the number of people in the hall at 't' hours after the doors are open.

Also it is given that  175 people enter as soon as the doors open in the morning.

i.e. at t=0

also number of people coming into the hall then increases at a rate of 30% per hour.

i.e. the number of people at t hours after opening of door is given by the recurrence relation as:

also:

Also it is given that:

The maximum occupancy of a concert hall is 1,200 people

i.e.

Hence we are asked to determine the number of hours after which the amount of people in the concert hall will exceed the occupancy limit i.e

1200" class="latex-formula" id="TexFormula6" src="https://tex.z-dn.net/?f=P_t%3E1200" title="P_t>1200">

Taking logarithmic function on both side we have:

\log (\dfrac{1200}{175})t>\dfrac{\log(\dfrac{1200}{175})}{\log 1.3}" class="latex-formula" id="TexFormula8" src="https://tex.z-dn.net/?f=t%20%5Clog1.3%3E%5Clog%20%28%5Cdfrac%7B1200%7D%7B175%7D%29%5C%5C%5C%5Ct%3E%5Cdfrac%7B%5Clog%28%5Cdfrac%7B1200%7D%7B175%7D%29%7D%7B%5Clog%201.3%7D" title="t \log1.3>\log (\dfrac{1200}{175})t>\dfrac{\log(\dfrac{1200}{175})}{\log 1.3}">

on solving we have:

Hence, after 8 hours the amount of people in the concert hall will exceed the occupancy limit

( since :

at t=7 which is approximately equal to 1098

while at t=8 which is approximately equal to 1428 )

step-by-step explanation:

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