1) Can someone tell me how the write 4x^3 + y^2 - 4y + 36=0 in standard equation (solve for y and complete the squares).
2) A pencil manufacturer determines that the daily cost c in dollars of producing n dozen pencils is given by c=(n^2/25,000) - (2n/5) +40000.
a) How many dozen pencils hsould be produced per day to minimize the cost?
b)What is the minimum cost? Explain how you got this?
1) To write the given equation in the standard form and solve for y, we need to complete the squares. Here are the steps:
- Start with the equation: 4x^3 + y^2 - 4y + 36 = 0
- Move the constant term to the other side: 4x^3 + y^2 - 4y = -36
- Group the terms that involve y: y^2 - 4y
- Take half of the coefficient of y (-4) and square it: (-4/2)^2 = 4
- Add the square term (4) to both sides of the equation: y^2 - 4y + 4 = -36 + 4
- Complete the square by factoring the left side as a perfect square: (y - 2)^2 = -32
- Rewrite the equation and simplify: (y - 2)^2 = -32
- Solve for y by taking the square root of both sides: y - 2 = ±√(-32)
- Since the value under the radical is negative, there are no real solutions for y. Thus, the equation has no real solutions in standard form.
2) To minimize the cost function, we need to find the value of n that corresponds to the minimum point in the cost equation.
a) To find the number of dozen pencils that should be produced per day to minimize the cost, we need to differentiate the cost function with respect to n and set it equal to zero. Here are the steps:
- Start with the cost function: c = (n^2/25,000) - (2n/5) + 40,000
- Differentiate the cost function with respect to n:
d(c)/d(n) = (2n/25,000) - 2/5
- Set the derivative equal to zero and solve for n:
(2n/25,000) - 2/5 = 0
Simplifying, we get:
(2n/25,000) = 2/5
Cross-multiplying:
2n * 5 = 25,000 * 2
10n = 50,000
n = 5,000 dozen
Therefore, to minimize the cost, 5,000 dozen pencils should be produced per day.
b) To find the minimum cost, substitute the value of n we found in part a into the cost function:
c = (n^2/25,000) - (2n/5) + 40,000
c = (5,000^2/25,000) - (2 * 5,000/5) + 40,000
c = 25,000/25,000 - 10,000/5 + 40,000
c = 1 - 2,000 + 40,000
c = 39,001
Therefore, the minimum cost is $39,001.