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((pie(9))/6 t)+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.
P(t)=-1800cos((pie(n))/6 t)+1800
Grrrr! The Greek letter is pi
The dessert is pie!
Here is the graph. I expect you can use it to help with the other questions.
http://www.wolframalpha.com/input/?i=-1800cos%289%CF%80%2F6+t%29%2B1800+for+0+%3C%3D+t+%3C%3D+10
To answer these questions, let's go step by step.
i) Sketching the graph of the function P(t):
To sketch the graph of the function P(t) = -1800cos((π/6)t) + 1800, we need to understand the properties of a cosine function.
- The amplitude of the cosine function is the coefficient of cos, so in this case, it is 1800.
- The period of the cosine function is given by T = 2π/B, where B is the coefficient of t inside the cosine function. In this case, B = π/6, so the period is T = 2π/(π/6) = 12 hours.
Based on this information, we can draw the graph as follows:
- The graph will oscillate above and below the x-axis.
- The highest point (peak) will occur at t = 0 hours (when the mall opens), and the lowest point will occur at t = 6 hours (halfway through the period).
- The amplitude of 1800 indicates the maximum number of people visiting the mall, while the negative sign reflects the oscillation above and below the x-axis.
ii) Finding the peak hours and the number of people:
We need to find the time when the mall reaches its peak, which is the highest point on the graph.
To get this value, we can set the derivative of P(t) to zero and solve for t:
P'(t) = -1800(-π/6)sin((π/6)t) = 0
sin((π/6)t) = 0
Since sin(x) = 0 when x = 0, π, 2π, ..., we can solve for t:
(π/6)t = 0, π, 2π, ...
t = 0, 6, 12, ...
Therefore, the mall reaches its peak hours every 12 hours. The number of people can be found by substituting these values into the function P(t):
P(0) = -1800cos((π/6)*0) + 1800 = 0 + 1800 = 1800 people (when the mall opens)
P(6) = -1800cos((π/6)*6) + 1800 = -1800cos(π) + 1800 = -1800(-1) + 1800 = 3600 people (peak hours)
iii) Estimating the number of people at 7.30 p.m. (t = 10.5):
To estimate the number of people at 7.30 p.m., we substitute t = 10.5 into the function P(t):
P(10.5) = -1800cos((π/6)*10.5) + 1800
Use a calculator to evaluate the cosine function, and then calculate P(10.5).
iv) Determining the time when the number of people in the mall reaches 2570:
To find the time when the number of people in the mall reaches 2570, we need to solve the equation P(t) = 2570.
-1800cos((π/6)t) + 1800 = 2570
Solve this equation for t using algebraic or numerical methods.