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The fertility decline in many countries can be modeled by a quadratic equation. In a certain country, from the late 1960s to the present, the number of births per woman has declined according to the formula [tex]f(x)=0.00925 x^2-0.3385 x+3.875[/tex], where [tex]x[/tex] is the number of years after 1969.

Step 1 of 2: What was the number of births per woman in 1999? Round your answer to three decimal places and, if necessary, round any intermediate calculations to six decimal places.



Answer :

GinnyAnswer
To determine the number of births per woman in 1999 using the given quadratic formula [tex]\( f(x) = 0.00925x^2 - 0.3385x + 3.875 \)[/tex], where [tex]\( x \)[/tex] is the number of years after 1969, we need to follow these steps:

1. Calculate [tex]\( x \)[/tex]:
The year given is 1999. First, convert the year 1999 to [tex]\( x \)[/tex] in terms of the number of years after 1969.
[tex]\[ x = 1999 - 1969 = 30 \][/tex]

2. Substitute [tex]\( x \)[/tex] into the quadratic formula:
Plug [tex]\( x = 30 \)[/tex] into the quadratic equation [tex]\( f(x) = 0.00925x^2 - 0.3385x + 3.875 \)[/tex]:
[tex]\[ f(30) = 0.00925(30)^2 - 0.3385(30) + 3.875 \][/tex]

3. Perform the intermediate calculations:
- Calculate [tex]\( (30)^2 \)[/tex]:
[tex]\[ 30^2 = 900 \][/tex]
- Calculate [tex]\( 0.00925 \times 900 \)[/tex]:
[tex]\[ 0.00925 \times 900 = 8.325 \][/tex]
- Calculate [tex]\( 0.3385 \times 30 \)[/tex]:
[tex]\[ 0.3385 \times 30 = 10.155 \][/tex]

4. Substitute these values back into the formula:
[tex]\[ f(30) = 8.325 - 10.155 + 3.875 \][/tex]

5. Combine the terms:
[tex]\[ 8.325 - 10.155 = -1.83 \][/tex]
[tex]\[ -1.83 + 3.875 = 2.045 \][/tex]

6. Round the result to three decimal places:
The value 2.045 is already rounded to three decimal places.

Therefore, the number of births per woman in 1999 was [tex]\( \boxed{2.045} \)[/tex].

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