People are entering a zoo at the rate of 100e^t+75t people per hour where t is the amount of time the zoo has been open on that day measured in hours. If the doors are open at 9:00 AM, how many hundreds of people have entered the zoo at 11:40 AM? (Nearest integer)
To find the total number of people who have entered the zoo at 11:40 AM, we need to calculate the integral of the given rate of people entering the zoo with respect to time, from 9:00 AM to 11:40 AM.
The rate of people entering the zoo is given by the function: 100e^t + 75t
To calculate the integral, we can integrate each term separately. The integral of e^t is e^t, and the integral of t is (1/2)t^2.
∫(100e^t + 75t) dt = ∫100e^t dt + ∫75t dt
= 100 * ∫e^t dt + 75 * ∫t dt
= 100 * e^t + 75 * (1/2)t^2 + C
To find the number of people who have entered the zoo at 11:40 AM, we need to evaluate this expression at t=11.67 (since 11:40 AM is 11 hours and 40 minutes after 9:00 AM).
Substituting t=11.67 into the expression, we get:
100 * e^11.67 + 75 * (1/2)*(11.67)^2 + C
Now, to get the value of C, we need additional information. If we assume that no people had entered the zoo at 9:00 AM (when the doors were opened), then C would be zero.
Using that assumption, we can calculate the total number of people who have entered the zoo at 11:40 AM by evaluating the expression:
100 * e^11.67 + 75 * (1/2)*(11.67)^2
Calculating this expression will give you the total number of people who have entered the zoo at 11:40 AM.
number ' (t) = 100e^t + 75t
number (t) = 100e^t + (75/2)t^2 + c
assuming at opening (t=0) no people were in the zoo,
0 = 100e^0 + (75/2)(0) + c
c = -100
number(t) = 100e^t +(75/2)t^2 - 100
so when t = 2.666.. (at 9:40 am)
number(2.666...) = 100e^2.6667 + 37.5(2.666..) - 100
= appr 1439 hundreds
Looks like way to many people in a zoo