Question

Rate at which risk of down syndrome is changing is approximated by function r(x) = 0.004641x2 − 0.3012x + 4.9 (20 ≤ x ≤ 45) where r(x) measured in percentage of births per year and x is maternal age at delivery. find function f giving risk as percentage of births when maternal age at delivery is x years, given that risk of down syndrome at 30 is 0.14% of births.

Answer 1

`\(C\)` is the constant of integration. To find
`\(C\),` use the information that the risk of Down syndrome at
`\(x = 30\) is \(0.14\%\)` of births:

`\[ f(30) = 0.004641 \cdot (1)/(3)(30)^3 - 0.3012 \cdot (1)/(2)(30)^2 + 4.9 \cdot 30 + C \]`

Step-by-step explanation:

To find the function
`\(f(x)\)` , which gives the risk as a percentage of births when maternal age at delivery is
`\(x\)` years, you need to integrate the given rate function
`\(r(x)\)` with respect to
`\(x\)`. The given rate function is a quadratic function, and integrating it will give you the desired function
`\(f(x)\)`. The integration process involves finding the antiderivative of
`\(r(x)\)`.

We have:
`\(r(x) = 0.004641x^2 - 0.3012x + 4.9\)`

To find
`\(f(x)\):`

`\[ f(x) = \int r(x) \, dx \]`

`\[ f(x) = \int (0.004641x^2 - 0.3012x + 4.9) \, dx \]`

`\[ f(x) = 0.004641 \cdot (1)/(3)x^3 - 0.3012 \cdot (1)/(2)x^2 + 4.9x + C \]`

Here,
`\(C\)` is the constant of integration. To find
`\(C\),` use the information that the risk of Down syndrome at
`\(x = 30\) is \(0.14\%\)` of births:

`\[ f(30) = 0.004641 \cdot (1)/(3)(30)^3 - 0.3012 \cdot (1)/(2)(30)^2 + 4.9 \cdot 30 + C \]`

Given that
`\(f(30) = 0.14\% = 0.0014\)`, solve for
`\(C\).`

`\[ 0.0014 = 0.004641 \cdot (1)/(3)(30)^3 - 0.3012 \cdot (1)/(2)(30)^2 + 4.9 \cdot 30 + C \]`

After solving for
`\(C\),` substitute it back into the expression for
`\(f(x)\):`

`\[ f(x) = 0.004641 \cdot (1)/(3)x^3 - 0.3012 \cdot (1)/(2)x^2 + 4.9x + C \]`

This will be the function
`\(f(x)\)` representing the risk as a percentage of births when maternal age at delivery is
`\(x\)`years.

Answer 2

The rate of change of the risk of down syndrome (in percentage of births per year) is
r(x) = 0.004641x² - 0.3012x + 4.9, 20≤ x ≤ 45
where
x = maternal age at delivery.

The function giving risk as a percentage of births when maternal age is x is the integral of r(x). That is,
f(x) = 0.001547x³ - 0.1506x² +4.9x + c

When x = 30, f = 0.14%. Therefore
0.001547(30³) - 0.1506(30²) + 4.9(30) + c = 0.14
41.769 - 135.54 + 147 + c = 0.14
c = -53.089

Answer:
f(x) = 0.001547x³ - 0.1506x² + 4.9x - 53.089, 20 ≤ x ≤ 45

The function is graphed as shown below.