Question

The cost C (in dollars) for a company to produce and sell x thousand gadgets is given by C = 1/50x² − 2x + 2050.

(a) What is the company's start-up cost?
(b) What is the minimum cost?
(c) How many gadgets must the company produce and sell to incur the least cost? (Be careful with your units!)

a) Calculate the start-up cost of the company based on the given equation.
b) Determine the minimum cost incurred by the company according to the equation.
c) Solve for the quantity of gadgets required for the company to minimize cost.
d) Analyze the implications of the results on the company's production and cost efficiency.

Answer 1

a) The start-up cost of the company is $2050. b) The minimum cost incurred by the company is $1800. c) The company must produce and sell 50,000 gadgets to incur the least cost.

Step-by-step explanation:

a) To calculate the start-up cost of the company, we need to find the value of C when x is 0. Substituting x = 0 into the equation, we have C = 1/50(0)^2 - 2(0) + 2050 = 2050. Therefore, the start-up cost of the company is $2050.

b) To determine the minimum cost incurred by the company, we need to find the vertex point of the quadratic equation. The vertex is located at the minimum point of the graph. The x-coordinate of the vertex can be found using the formula x = -b/2a, where a = 1/50 and b = -2. Substituting the values, we get x = -(-2)/(2*(1/50)) = 50. Plugging x = 50 back into the equation gives us C = 1/50(50)^2 - 2(50) + 2050 = 1800. Therefore, the minimum cost incurred by the company is $1800.

c) To find the quantity of gadgets required for the company to minimize cost, we need to substitute the x-coordinate of the vertex back into the equation. Using x = 50, we get C = 1/50(50)^2 - 2(50) + 2050 = 1800. Therefore, the company must produce and sell 50,000 gadgets to incur the least cost.