The number of people expected to have a disease in t years is given by y(t)=A•3^(t/4)
i) if now year(2016) the number of people having disease is 1000, find the value of A?
ii) how many people 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?
so considering t = 0 to correspond with 2016
1000 = A 3^0 , but 3^0 = 1
A = 1000
b) y(5) = 1000 3^(4/3) = .... Use your calculator
c) 100000 = 1000 3^(t/4)
1000 = 3^(t/4
take log of both sides
log 1000 = log 3^(t/4)
3 = (t/4)log 3
t/4 = 3/log3
t = 12/log3 = appr 25.15 years
iv) y = 1000 3^(t/4)
dy/dt = 1000(3^(t/4)(1/4)(ln3)
= 250ln3 (3^(t/4)
plug in t = 0 for now, and t = 10 for the other case
To find the answers to the given questions, we will use the equation y(t) = A * 3^(t/4), where y(t) represents the number of people expected to have the disease at time t, and A is an unknown constant.
i) To find the value of A when the number of people having the disease is 1000 in the year 2016, we substitute the given values into the equation. In 2016, t = 0, and y(0) = 1000. Therefore, we have:
1000 = A * 3^(0/4)
1000 = A * 3^0
1000 = A * 1
A = 1000
So, A is equal to 1000.
ii) To find the number of people expected to have the disease in five years, we need to find y(5). Substituting t = 5 into the equation, we have:
y(5) = 1000 * 3^(5/4)
Using a calculator, we can evaluate this expression to find the answer.
iii) To find when 100,000 people are expected to have the disease, we need to solve the equation y(t) = 100,000. Substituting this into the equation, we have:
100,000 = A * 3^(t/4)
We can solve this equation by taking the logarithm of both sides and isolating the variable t.
iv) To determine how fast the number of people with the disease is expected to grow now and ten years from now, we need to find the derivative of y(t) with respect to t and evaluate it at t = 0 and t = 10.
The derivative of y(t) = A * 3^(t/4) with respect to t is given by y'(t) = (A/4) * ln(3) * 3^(t/4).
To calculate the growth rate now, we evaluate y'(0) by substituting t = 0 into the derivative equation. For the growth rate ten years from now, we evaluate y'(10) by substituting t = 10 into the derivative equation.
I hope this explanation helps you understand how to solve these problems.