Suppose y , the number of cases of a disease, is reduced by 11% per year.
(a) If there are initially 10,000 cases, express y as a function of t , the number of years elapsed.
(b) How many cases will there be 6 years from now?
(c) How long does it take to reduce the number of cases to 1000?
a) y =10000(0.11)^t
b) y = 10000(0.11)^6 and you solve it
c) 1000 = 10000(0.11)^t and I would divide both sides by 10000 first then either uses exponents or take the log of both sides to solve
it is reduced by .11/yr so multiply by 0.89^t
It is reduced by 11% leaving 89%
so y = 10000(.89)^t
b) y = 10000(.89)^6
c)
10000(.89)^t = 1000
.89^t = .1
t log .89 = log .1
t = -1/-.0506099.. = appr 19.76 years
Thanks math gang! Time for a cup of afternoon tea. I was doing the math in my head and .89 didn't feel just right. LOL! Thanks again for the correction : )
To solve this problem, we need to understand that the number of cases, y, is reduced by 11% per year. We can use this information to create an exponential decay model.
(a) To express y as a function of t, we can use the formula for exponential decay:
y = initial value * (1 - decay rate)^t
In this case, the initial value is 10,000 and the decay rate is 11% or 0.11, so the formula becomes:
y = 10,000 * (1 - 0.11)^t
Simplifying further:
y = 10,000 * 0.89^t
This is the function that represents the number of cases, y, as a function of the number of years elapsed, t.
(b) To find the number of cases 6 years from now, we can substitute t = 6 into the equation:
y = 10,000 * 0.89^6
Calculating this, we get:
y ≈ 6,047.99
Therefore, there will be approximately 6,048 cases 6 years from now.
(c) To find out how long it takes to reduce the number of cases to 1000, we can set y = 1000 and solve for t in the equation:
1000 = 10,000 * 0.89^t
Dividing both sides by 10,000:
0.1 = 0.89^t
To solve for t, we can take the logarithm (base 0.89) of both sides:
log(0.1) = t * log(0.89)
Using logarithm properties, we can isolate t:
t = log(0.1) / log(0.89)
Calculating this expression, we find:
t ≈ 14.7
Therefore, it takes approximately 14.7 years to reduce the number of cases to 1000.