Question 1197304
given function:

B(a) = −0.31x
2 + 16.54x
2 − 151.04

since this is not quite clear, I assume you have

{{{B(a) = -0.31x^2 + 16.54x - 151.04}}}



i. What is the age of unmarried women with the highest birthrate?


if {{{B(a)}}} is the birth rate then the value of "{{{a}}}" at {{{B}}}'{{{(a)=0}}} will tell the age of unmarried women of {{{highest}}}{{{ birth}}}{{{ rate}}}. 


{{{B(a) = -0.31x^2 + 16.54x - 151.04 }}}

derivative is:

{{{ B}}}'{{{(a)=-2*0.31x + 16.54}}}

{{{ B}}}'{{{(a)=-0.62x + 16.54}}}

set it equal to zero

{{{-0.62x + 16.54=0}}}

{{{-0.62x =- 16.54}}}

{{{x =- 16.54/-0.62}}}

{{{x=26.677}}}
 

or, you can find x-coordinate this way: 

{{{B(a) = -0.31x^2 + 16.54x - 151.04 }}} =>{{{a=-0.31}}}, {{{b=16.54}}},{{{c=-151.04}}}

{{{x=-b/2a}}}

{{{x=-16.54/(2*(-0.31))}}}

{{{x=16.54/0.62}}}

{{{x=26.677 }}}



ii. What is the highest birthrate of unmarried women?

if {{{x=26.677}}}, then
 
 {{{B(26.677) =-0.31*26.677^2 + 16.54*26.677 - 151.04 }}}

{{{B(26.677) =-220.61532199+ 441.23758 - 151.04 }}}

{{{B(26.677) =69.582}}}

 

iii. Evaluate and interpret {{{B(40)}}}.


 {{{B(40) =-0.31*40^2 + 16.54*40- 151.04 }}}

{{{B(40) =-496 + 661.6- 151.04 }}}

 {{{B(40) =14.56}}}