The number of people expected to have a disease in t years is given by
y(t) = A.3^(t/a)
(i) If now (year 2016) the number of people having the disease is 1000, find the value of A.
(ii) How many people are expected to have the disease in five years?
(iii) When are 100,000 people expected to have the disease?
(iv) How fast is the number of people with the disease expected to grow now and ten years from now?
A = 1000 if t is years after 2016
however without knowing a, can not do.
if a = 4 ??
To solve the given questions, we will use the provided equation: y(t) = A * 3^(t/a)
(i) If now (year 2016) the number of people having the disease is 1000, we can substitute the values into the equation and solve for A.
From the equation, y(t) = 1000, and t = 0 (since it is year 2016), we have:
1000 = A * 3^(0/a) = A * 3^0 = A * 1 = A
Therefore, A = 1000.
(ii) To find the number of people expected to have the disease in five years (t = 5), we substitute the values into the equation:
y(5) = A * 3^(5/a)
Plugging in the value of A as 1000, we get:
y(5) = 1000 * 3^(5/a)
Since the value of 'a' is not given, we cannot simplify the expression any further.
(iii) We need to find the value of 't' when 100,000 people are expected to have the disease. We substitute the values into the equation and solve for 't':
100,000 = A * 3^(t/a)
Plugging in the value of A as 1000, we get:
100,000 = 1000 * 3^(t/a)
Again, we cannot simplify the equation further unless we know the value of 'a'.
(iv) To find how fast the number of people with the disease is expected to grow now (t = 0) and after ten years (t = 10), we can calculate the derivative of y(t) with respect to 't':
dy(t)/dt = (A * 3^(t/a)) * ln(3)/a
At t = 0: (dy(t)/dt) evaluated at t = 0 gives us the growth rate at year 2016.
At t = 10: (dy(t)/dt) evaluated at t = 10 gives us the growth rate ten years from now.
To calculate the exact growth rates, the value of 'a' is needed.
In summary, we were able to find the value of A in question (i) as 1000, but without knowing the value of 'a', we cannot obtain precise answers for questions (ii), (iii), and (iv).