3.Suppose C(x) = x^2 -10x+27 represents the costs, in hundreds, to produce x thousand pens.
How many pens should be produced to minimize the cost? What is this minimum cost?
If you know Calculus:
C' (x) = 2x - 10
= 0 for a min of C(x)
solve 2x-10 - 0
then sub that value of x back into C(x) to find the minimum cost
Non-Calculus:
C(x) is just an upward opening parabola
find its vertex and you have both answers
Do this by
1. completing the square, or
2. using ---- the x of the vertex is -b/(2a) = -(-10)/2 = 5
This should be the same as the Calculus method answer,
find C(5) for the actual minimum cost
Oh, I must say, the cost of pens is no laughing matter! But fear not, I'm here to help! To find the number of pens that should be produced to minimize the cost, we need to determine the vertex of the quadratic function C(x) = x^2 - 10x + 27.
The vertex of a quadratic function is found at x = -b/2a, where a, b, and c are the coefficients in the equation. In this case, a = 1, b = -10, and c = 27.
Using the formula, we find that x = -(-10)/(2*1) = 5.
So, to minimize the cost, 5 thousand pens should be produced. Now let's find the minimum cost.
Plugging x = 5 into the cost function, C(5) = 5^2 - 10(5) + 27 = 25 - 50 + 27 = 2.
Therefore, the minimum cost is 2 hundred. Keep in mind that this is a simplified answer since we are given the cost in hundreds.
I hope this brings a smile to your face, even when talking about costs!
To find the number of pens that should be produced to minimize the cost, you need to find the vertex of the quadratic function C(x) = x^2 -10x + 27.
The x-coordinate of the vertex can be found using the formula x = -b / (2a), where a, b, and c are the coefficients of the quadratic equation in the form ax^2 + bx + c.
In this case, a = 1, b = -10, and c = 27. Substituting these values into the formula, we get:
x = -(-10) / (2*1)
x = 10 / 2
x = 5
So, the number of pens that should be produced to minimize the cost is 5 thousand pens.
To find the minimum cost, substitute the value of x = 5 into the C(x) equation:
C(5) = 5^2 - 10*5 + 27
C(5) = 25 - 50 + 27
C(5) = 2
Therefore, the minimum cost is $200.
To find the number of pens that should be produced to minimize the cost, we need to determine the x-value for which C(x) is at its minimum. In this case, C(x) represents the cost function.
To find the minimum of a quadratic function, we can make use of calculus. The minimum occurs at the vertex of the parabola, which is given by the formula x = -b / (2a), where the quadratic function is in the form ax^2 + bx + c.
In the given function C(x) = x^2 - 10x + 27, we can compare it to the standard form by rearranging the terms:
C(x) = 1x^2 - 10x + 27
From this, we can deduce that a = 1, b = -10, and c = 27.
To find the x-value for the minimum cost, we substitute these values into the formula:
x = -(-10) / (2 * 1)
x = 10 / 2
x = 5
Therefore, the number of pens that should be produced to minimize the cost is 5 thousand pens.
To find the minimum cost, we substitute this value of x back into the cost function C(x):
C(5) = 5^2 - 10(5) + 27
C(5) = 25 - 50 + 27
C(5) = 2
Hence, the minimum cost is 2 hundred (or $200).