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

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.