Question

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 1

Part a) The approximate years were 1998 and 2008

Part b) The graph in the attached figure

Explanation:

Part a) we have

`y=-0.27x^(2)+3.3x+77`

For
`y=80\ billion\ eggs`

Solve the quadratic equation

`80=-0.27x^(2)+3.3x+77`

`0.27x^(2)-3.3x+3=0`

we know that

The formula to solve a quadratic equation of the form
`ax^(2) +bx+c=0` is equal to

`x=\frac{-b(+/-)\sqrt{b^(2)-4ac}} {2a}`

in this problem we have

`0.27x^(2)-3.3x+3=0`

so

`a=0.27\\b=-3.3\\c=3`

substitute in the formula

`x=\frac{3.3(+/-)\sqrt{-3.3^(2)-4(0.27)(3)}} {2(0.27)}`

`x=\frac{3.3(+/-)√(7.65)} {0.54}`

`x=\frac{3.3(+)√(7.65)} {0.54}=11.23\ years`

`x=\frac{3.3(-)√(7.65)} {0.54}=0.99\ years`

therefore

The approximate years are

1997+11=2008

1997+1=1998

Part b)

Using a graphing tool

we have

`y=-0.27x^(2)+3.3x+77`

`y=80`

The solution of the system of equations is the intersection point both graphs

The intersection point are (0.99,80) and (11.23,80)

see the attached figure

therefore

The solution part a) is correct