Owing to an innovative rural public health program, infant mortality in Senegal, West Africa, is being reduced at a rate of 10% per year. How long will it take for infant mortality to be reduced by 50%?
To determine how long it will take for infant mortality to be reduced by 50% in Senegal, West Africa, we can use the concept of exponential decay. Since infant mortality is being reduced at a rate of 10% per year, this can be interpreted as a yearly decrease of 10% of the remaining mortality.
To calculate the time it takes for a quantity to decrease by a certain percentage using exponential decay, we can use the formula:
t = (ln(0.5)) / (ln(1 - r))
Where:
t is the time it takes for the quantity to reduce to the desired percentage (in this case, 50%)
r is the rate of decay (in this case, 10%)
Let's calculate it step-by-step.
Step 1: Calculate the rate of decay in decimal form.
r = 10% = 0.10
Step 2: Calculate the time it takes for infant mortality to be reduced by 50%.
t = (ln(0.5)) / (ln(1 - 0.10))
Step 3: Simplify the equation.
t = (ln(0.5)) / (ln(0.90))
Step 4: Solve the equation.
t ≈ 7.27
Therefore, it will take approximately 7.27 years for infant mortality to be reduced by 50% in Senegal, West Africa, based on the current rate of reduction of 10% per year.
To determine how long it will take for infant mortality to be reduced by 50% in Senegal, we can use the concept of a decay model.
The decay model can be represented by the equation:
N(t) = N0 * (1 - r)^t
Where:
- N(t) is the population or quantity at time t
- N0 is the initial population or quantity
- r is the decay rate
- t is the time
In this case, the population being reduced is the infant mortality rate, N(t) represents the infant mortality rate at time t, N0 is the initial infant mortality rate, r is the decay rate (10% or 0.1), and we want to find the time it takes for the infant mortality rate to decrease by 50% (N(t) = 50% * N0 = 0.5 * N0).
By substituting the given values into the equation, we have:
0.5 * N0 = N0 * (1 - 0.1)^t
Dividing both sides by N0:
0.5 = (1 - 0.1)^t
Taking the natural logarithm of both sides:
ln(0.5) = ln((1 - 0.1)^t)
Using logarithmic properties (ln(a^b) = b * ln(a)), we can rewrite the equation as:
ln(0.5) = t * ln(1 - 0.1)
Now we can solve for t by dividing both sides by ln(1 - 0.1), and then dividing by ln(0.5):
t = ln(0.5) / ln(1 - 0.1)
Calculating this on a calculator results in:
t ≈ 6.9315 / (-0.1054) ≈ -65.80
Since time cannot be negative in this context, we can discard the negative solution. Therefore, it will take approximately 65.80 years (rounded to the nearest whole year) for the infant mortality rate to be reduced by 50% in Senegal through this innovative rural public health program.
8.42
100%-10%=90% 90/100= .9
Each year there is 90% increase or .9
We are looking for when the mortality rate is down to 50% or .5
The equation is .9^x=.5
Change into Logarithmic form:
log(base .9).5=x
Change of base formula:
log.5/log.9=x
x=6.5788 or 6.57 years