Question

The weekly volume (in liters) of milk produced in a farm is given by v(x)=x3+21x2-1480x, where x is the number of cows. Find the number of cows that corresponds to a total production of 1500 liters of milk in a week

Answer 1

Answer:

30 cows

Step-by-step explanation:

The given parameters are;

v(x) = x³ + 21·x² - 1480·x

Where x = The number of cows

V(x) = The weekly volume (in liters) of milk produced

The number of cows that corresponds to a total production of 1500 liters of milk in a week is given as follows;

1,500 = x³ + 21·x² - 1480·x

x³ + 21·x² - 1480·x = 1,500

x³ + 21·x² - 1480·x - 1,500 = 0

x³ + 21·x² - 1480·x - 1,500 = 0

We observe that x = -1 is a solution of the above equation, as follows;

(-1)³ + 21·(-1)² - 1480·(-1) - 1,500 = 0

Therefore, (x + 1) is a factor of x³ + 21·x² - 1480·x - 1,500, which by long division gives;

${}$   x² + 20·x - 1500

(x³ + 21·x² - 1480·x - 1,500)/(x + 1)

    ${}$     x³ + x²

           20·x² - 1480·x - 1500

           20·x² + 20·x        

${}$                        -1500·x - 1500

        ${}$               -1500·x - 1500

Therefore, the other factors is x² + 20·x - 1500, which gives;;

x² + 20·x - 1500 = (x - 30) × (x + 50)

Therefore;

x³ + 21·x² - 1480·x - 1,500 = 0

Which gives;

(x - 30) × (x + 50) × (x + 1) = 0

The solutions are x = 30, or x = -50, or x = -1

Therefore, the only possible solution is the number of cows = x = 30 cows.