Question 251715
standard form of a quadratic equation is:


y = f(x) = ax^2 + bx + c = 0


a = coefficient of x^2 term.
b = coefficient of x term.
c = constant term


x represents head of cattle.


with 0 head of cattle, he gets no yield so the first equation you have is:


y = f(0) = a*0^2 + b*0 + c = 0


this tells him that the constant term of c has to be equal to 0 **********


with 5 head of cattle, he gets 8750 pounds of beef.


this makes the standard form of the quadratic equation equal to:


y = f(5) = a*5^2 + b*5 = 8750


with 10 head of cattle, he gets 15000 pounds of beef.


this makes the standard form of the quadratic equation equal to:


y = f(10) = a*100^2 + b*10 = 15000


he needs to solve for the coefficients of the quadratic equation.


the 2 equations become:


y = f(5) = 25*a + 5*b = 8750 (equation 1)
y = f(10) = 100*a + 10*b = 15000 (equation 2)


he needs to solve these two equations simultaneously to get the value of a and b.


he already has the value of c = 0.


solving these equations simultaneously, he gets;


a = -50
b = 2000


his quadratic equation becomes:


y = f(x) = -50x^2 + 2000x


when x = 5, this equation becomes:


y = f(5) = -50*25 + 2000*5 = -1250 + 1000 = 8750


when x = 10, this equation becomes:


y = f(10) = -50*100 + 2000*10 = -5000 + 20000 = 15000


the values are good.


since the coefficient of the x^2 term is negative, the parabola formed by this equation will point upward and open downward.


it will peak when x = -b/2a.


That is the maximum point of the equation.


x = -b/2a becomes x = -2000/-100 = 20


when x = 20, the number of pounds of beef he produces will be:


y = f(20) = -50*20^2 + 2000*20 = -50*400 + 40000 = -20000 + 40000 = 20000.


his production of beef will peak at 20000 when he has 20 head of cattle.


graph of the his equation is shown below:


the horizontal line is at y = 20000 pounds of beef.


{{{graph(600,600,-10,40,-1000,30000,-50*x^2+2000*x,20000)}}}


if you need help with solving the equations simultaneously, let me know and I'll send you the solution for that.