Question

9. A concert hall is filling with people from noon until show time at 3:00 pm. The table below shows the rate of people entering the concert hall (measured in people per minute) at particular times. Time Noon 1:00 1:30 2:00 2:20 2:40 2:50 2:55 3:00 4 8 11 14 17 21 23 26 25 Rate (people per min) a)Estimate the total number of people attending the show using a midpoint approximation for four partitions: (noon 1:30). [1:30,2:20], [2:20,2:50), and [2:50,3:00) in terms of minutes. b) One possible model for the rate at which people enter the concert hall is given by the function R(t)== 11(10%)e 0,04 (1 +1600e -0.01 2 people per minute, where t is the time in minutes since noon. Write and evaluate (rounded to the nearest person) an expression that represents the total number of people at the concert hall at 3pm, as predicted by this model

this is over integrals

Answer 1

Midpoint Rule: Given, Rate of filling with peoples from noon until show time at 3:00 Pm Noon 1:00 1:30 2:002:20 2:40 2:50 2:55 3:00 Time Rate (people per min 4 8 11 14 17 21 23 26 25 (a) We have to estimate the total number of people attending the show using midpoint approximation for four partitions. Given intervals are in clock time so change it into minutes since noon as follows: [0,90],[90,140] [140,170), [170,180] Length of each interval is 90,50,30,10 Therefore, by the Midpoint Rule:

Number of Peoples = 8x 90+14x50+21x30+26x10 = 720+ 700+ 630+260 = 2310 Hence, the required numbers of peoples are 2310. (6) Possible model for the rate at which people enter is given as follows: R(t)= 11(10º)e-0.011 (1+1600e-0.011 We have to find total number of peoples at 3 pm using given model. From noon to 3 pm it is 3 hours that is 180 minutes so integrate the given model from 0 to 180, we get

180 -0.017 11(109)e-0.011 Number of Peoples = 1 -dt (1+1600e 11x10 180 1600 (-0.01)e-0.011 -dt 1600(-0.01) (1+1600-.1)? -11x10 d 1 1600(-0.01) dt (1+1600e 180 -0.011 180 -0.011 -11x10 1600(-0.01) ((1+1600e = 2160 0 Hence, the required numbers of peoples are 2160).