Respuesta :
Answer:
Infant mortality will reduce to 33% in 1.63 years
Step-by-step explanation:
Let the population be X
Population after 1 year
[tex]X - 7.3[/tex] % of [tex]X[/tex]
[tex]X -0.073 X\\0.927X[/tex]
As we know
[tex]P = P_0 * e^{rt}[/tex]
Substituting the given values we get -
[tex]\frac{P_0 - 0.33P_0}{P_0} = e^{-0.33 * t}\\0.67 = e^{-0.33 * t}\\[/tex]
Taking log on both sides we get -
[tex]-0.20273 = -0.33 * t\\t = 1.627[/tex]
You can form a general expression for resultant reduction percentage after t years. Then equate it with 33% to get the value of t
The time that will be taken by the program to reduce the infant mortality rate by 33% is approx 3.35 years.
How to find the percentage from the total value?
Suppose the value of which a thing is expressed in percentage is "a'
Suppose the percent that considered thing is of "a" is b%
Then since percent shows per 100 (since cent means 100), thus we will first divide the whole part in 100 parts and then we multiply it with b so that we collect b items per 100 items(that is exactly what b per cent means).
Thus, that thing in number is
[tex]\dfrac{a}{100} \times b[/tex]
How to find a general expression for resultant reduction percentage after t years?
The rate of decrement for infant mortality rate is 7.3% per year.
Let the mortality rate in the start be [tex]r[/tex]
Then after 1 year it becomes
[tex]r - \dfrac{r}{100} \times 7.5 = r(1-\frac{7.5}{100}) = r_1 \text{\: (say)}[/tex]
After 1 more year, we will have:
[tex]r_1 -> r_1(1 - \frac{7.5}{100}) = r(1-\frac{7.5}{100})(1-\frac{7.5}{100}) = r(1-\frac{7.5}{100}) ^2 = r_2 \text{\: (say)}[/tex]
Thus, after t years, we will have:
[tex]r_t = r(1-\frac{7.5}{100})^t[/tex]
Simplifying it, we get:
[tex]r_t = r(1 - 0.075)^t = r(0.925)^t[/tex]
The percent of decrement can be calculated by:
[tex]p = \dfrac{\text{old rate - new rate}}{\text{old rate}} \times 100[/tex]
Thus,
the percent of decrement after t years will be
[tex]p = \dfrac{r - r_t}{r} \times 100 = \dfrac{r(1- (0.925)^t)}{r} \times 100 = 100(1-0.925^t)\\\\p = 100(1-0.925^t)[/tex]
We need the time after which the infant mortality rate would be reduced by 33%, thus, p = 33%
or
[tex]p = 33\\p = 100(1-0.925^t) = 33\\1 -0.925^t = \dfrac{33}{100}\\t = log_{0.925}(1 - \dfrac{33}{100}) \approx 3.3525[/tex]
Thus, after approx 3.35 years, the mortality rate would decrease by 33%
Thus,
The time that will be taken by the program to reduce the infant mortality rate by 33% is approx 3.35 years.
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