Question

The number of people , d, in thousands applying for medical benefits per week in a particular city can be modules by the equation d(t)=2.5 sin(0.76t+0.3)+3.8 where t is the time in years since January 2000. Based on the equation The Maximum number of people in thousands apply for benefits per year in the city is

Answer 1

4773 peoples.

Explanation:

Given the number of people d, in thousands applying for medical benefits per week in a particular city c modeled by the equation d(t)=2.5 sin(0.76t+0.3)+3.8 where t is the time in years, the maximum number of people tat will apply will occur at d(t)/dt = 0

Differentiating the function given with respect to t, we will have;

`d(t)=2.5 sin(0.76t+0.3)+3.8`

First we need to know that differential of any constant is zero.

`Using\ chain\ rule\\(d(t))/(dt) = 2.5cos(0.76t+0.3) * 0.76 + 0\\ \\(d(t))/(dt) = 1.9cos(0.76t+0.3)`

If
`(d(t))/(dt) =0` then;

`1.9cos(0.76t+0.3) = 0\\\\cos(0.76t+0.3) = 0\\\\0.76t+0.3 = cos^(-1) 0\\\\0.76+3t = 90\\\\3t = 90-0.76\\3t = 89.24\\\\t = 89.24/3\\\\t = 29.75years`

To know the maximum number of people in thousands that apply for benefits per year in the city, we wil substitute the value of t = 29.75 into the modeled equation

`d(t)=2.5 sin(0.76t+0.3)+3.8\\d(29.75) = 2.5 sin(0.76(29.75)+0.3)+3.8\\d(29.75) = 2.5 sin(22.61+0.3)+3.8\\\\d(29.75) = 2.5 sin(22.91)+3.8\\\\d(29.75) = 0.9732+3.8\\d(29.75) = 4.7732\\\\`

Since d is in thousands, the maximum number of people in thousands will be 4.7732*1000 = 4773.2 which is approximately 4773 peoples.