If the cost of, C(x), for manufacturing x units of a certain product is given by C(x)=x2-70x+4400, find the number of units manufactured at a cost of $7400?
To find the number of units manufactured at a cost of $7400, we need to set the cost equation equal to $7400 and solve for x.
C(x) = 7400
Substituting the given cost equation, we have:
x^2 - 70x + 4400 = 7400
Now, let's rearrange the equation to bring it in standard quadratic form:
x^2 - 70x + 4400 - 7400 = 0
x^2 - 70x - 3000 = 0
To solve this quadratic equation, we can either factorize it or use the quadratic formula. Let's use the quadratic formula:
x = (-(-70) ± √((-70)^2 - 4(1)(-3000))) / (2(1))
Simplifying this expression gives us:
x = (70 ± √(4900 + 12000)) / 2
x = (70 ± √(16900)) / 2
x = (70 ± 130) / 2
Now, we have two possible values of x:
x₁ = (70 + 130) / 2 = 200 / 2 = 100
x₂ = (70 - 130) / 2 = -60 / 2 = -30
Since the number of units cannot be negative, the only valid solution is x = 100.
Therefore, the number of units manufactured at a cost of $7400 is 100 units.
To find the number of units manufactured at a cost of $7400, we need to set up the cost equation and solve for the value of x.
The cost equation is given as C(x) = x^2 - 70x + 4400.
We can set up the equation to represent the cost of manufacturing x units as follows:
C(x) = 7400
Substituting the cost equation into the equation above, we get:
x^2 - 70x + 4400 = 7400
Rearranging the equation to bring all terms to one side, we have:
x^2 - 70x + 4400 - 7400 = 0
Simplifying further:
x^2 - 70x - 3000 = 0
To solve this quadratic equation, we can either factor it or use the quadratic formula. In this case, factoring is not straightforward, so we will use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
For our equation, a = 1, b = -70, and c = -3000. Plugging in these values, we get:
x = (-(-70) ± √((-70)^2 - 4(1)(-3000))) / (2(1))
Simplifying further:
x = (70 ± √(4900 + 12000)) / 2
x = (70 ± √(16900)) / 2
x = (70 ± 130) / 2
This gives us two possible values for x:
x1 = (70 + 130) / 2 = 200 / 2 = 100
x2 = (70 - 130) / 2 = -60 / 2 = -30
Since we cannot have a negative value for the number of units manufactured, the only valid solution is x = 100.
Therefore, the number of units manufactured at a cost of $7400 is 100 units.