For the period 1997-2003, the number of eggs y (in billions) produced in the United States can be modeled by the function y=-0.27x^2+3.3x+77 where x is the number of years since 1997 a. Write and solve an equation that you can use to approximate the year(s) in which 80 billion eggs were produced. B. Graph the function on a graphing calculator. Use the trace feature to fin the year when 80 billion eggs were produced. Use the graph to check your answer from part (a).

Answer:Part a) The approximate years were 1998 and 2008Part b) The graph in the attached figureStep-by-step explanation:Part a) we have[tex]y=-0.27x^{2}+3.3x+77[/tex]For [tex]y=80\ billion\ eggs[/tex] Solve the quadratic equation[tex]80=-0.27x^{2}+3.3x+77[/tex][tex]0.27x^{2}-3.3x+3=0[/tex]we know that The formula to solve a quadratic equation of the form [tex]ax^{2} +bx+c=0[/tex] is equal to [tex]x=\frac{-b(+/-)\sqrt{b^{2}-4ac}} {2a}[/tex] in this problem we have [tex]0.27x^{2}-3.3x+3=0[/tex]so [tex]a=0.27\\b=-3.3\\c=3[/tex] substitute in the formula [tex]x=\frac{3.3(+/-)\sqrt{-3.3^{2}-4(0.27)(3)}} {2(0.27)}[/tex] [tex]x=\frac{3.3(+/-)\sqrt{7.65}} {0.54}[/tex] [tex]x=\frac{3.3(+)\sqrt{7.65}} {0.54}=11.23\ years[/tex] [tex]x=\frac{3.3(-)\sqrt{7.65}} {0.54}=0.99\ years[/tex] thereforeThe approximate years are1997+11=20081997+1=1998Part b) Using a graphing toolwe have[tex]y=-0.27x^{2}+3.3x+77[/tex][tex]y=80[/tex]The solution of the system of equations is the intersection point both graphsThe intersection point are (0.99,80) and (11.23,80) see the attached figurethereforeThe solution part a) is correct