A market research company finds that traffic in a local mall over the course of a day could be estimated by the function P(t)=-1800cos(𝝅/𝟔𝒕)+𝟏𝟖𝟎𝟎 where P,is the number of people going to the mall ,and t is the time, in hours ,after the mall opens. The mall opens at 9.30 a.m��i)sketch the graph of function P(t)�ii)When does the mall reach its peak hours and state the number of people.�iii)Estimate the number of people in the mall at 7.30p.m
iv)Determine the time when the number of people in the mall reaches 2570.
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To answer these questions, we will use the given function P(t) = -1800cos(π/8t) + 1800, where P is the number of people going to the mall and t is the time in hours after the mall opens.
i) To sketch the graph of the function P(t), we need to plot some points. Let's consider time intervals of one hour and calculate the corresponding values of P(t).
- For t = 0 (9.30 a.m. when the mall opens): P(0) = -1800cos(π/8 * 0) + 1800 = 0 + 1800 = 1800.
- For t = 1: P(1) = -1800cos(π/8 * 1) + 1800 ≈ 2634.
- For t = 2: P(2) = -1800cos(π/8 * 2) + 1800 ≈ 1800.
- Repeat this process for more values of t to get multiple data points.
Using these data points, you can plot the graph of P(t). The graph will be a periodic function with peaks and valleys.
ii) To determine when the mall reaches its peak hours, we need to find the time when P(t) is at its maximum value. Recall that the cosine function reaches its maximum value of 1 when the angle is 0 (cos(0) = 1).
Therefore, in our function P(t), we need to find when cos(π/8t) = 1 to obtain the peak hours.
Solving cos(π/8t) = 1, we get π/8t = 0, which means t = 0.
So, the mall reaches its peak hours at t = 0 (9.30 a.m.). The number of people at that time is P(0) = 1800.
iii) To estimate the number of people in the mall at 7.30 p.m., we substitute t = 10.5 (since the mall opens at 9.30 a.m. and 7.30 p.m. is 10.5 hours later) into the function P(t).
P(10.5) = -1800cos(π/8 * 10.5) + 1800 ≈ 113.
Therefore, the estimate of the number of people in the mall at 7.30 p.m. is 113.
iv) To determine the time when the number of people in the mall reaches 2570, we need to find t for which P(t) = 2570.
-2570 = -1800cos(π/8t) + 1800
-770 = -1800cos(π/8t)
cos(π/8t) ≈ -770/(-1800)
cos(π/8t) ≈ 0.428
Now, we can use the inverse cosine function to find the corresponding angle (π/8t) that gives cos(π/8t) ≈ 0.428.
π/8t ≈ arccos(0.428)
t ≈ (8/arccos(0.428)) * π ≈ 13.998
Therefore, the time when the number of people in the mall reaches 2570 is approximately 13.998 hours after the mall opens.