Use technology to determine the absolute minimum of the function. Describe what the $$ x $$ - and $$ y $$-coordinates of this point represent in terms of the problem situation. A skateboard company has to decide how many skateboards to produce each day. The company knows that the costs to produce the skateboards go down the more they make. However, the overall cost to the company increases if they make too many skateboards due to the cost of storing overstock. Let $$ s $$ be the number of skateboards produced each day. Let $$ c $$ be the total cost to produce skateboards. The quadratic function $$ c(s)=0.04 s^{2}-16 s+15,000 $$ represents the total cost as a function of the number of skateboards produced. What is the absolute minimum of the function? Enter the answer in each space provided. Use numbers instead of words. $$ \square $$ , $$ \square $$ Describe what the $$ x $$ - and $$ y $$-coordinates of this point represent in terms of the problem situation. Enter the answer in the space provided. $$ \sqrt{x} $$

Question

Use technology to determine the absolute minimum of the function. Describe what the $$ x $$ - and $$ y $$-coordinates of this point represent in terms of the problem situation. A skateboard company has to decide how many skateboards to produce each day. The company knows that the costs to produce the skateboards go down the more they make. However, the overall cost to the company increases if they make too many skateboards due to the cost of storing overstock. Let $$ s $$ be the number of skateboards produced each day. Let $$ c $$ be the total cost to produce skateboards. The quadratic function $$ c(s)=0.04 s^{2}-16 s+15,000 $$ represents the total cost as a function of the number of skateboards produced. What is the absolute minimum of the function? > Enter the answer in each space provided. Use numbers instead of words. $$ \square $$ , $$ \square $$ Describe what the $$ x $$ - and $$ y $$-coordinates of this point represent in terms of the problem situation. > Enter the answer in the space provided. $$ \sqrt{x} $$

UpStudy AI · Accepted Answer

The given function is

c(s) = 0.04 s² − 16 s + 15,000.

Since this is a quadratic function with a positive coefficient for s², its graph is a parabola that opens upward. The absolute minimum occurs at the vertex of the parabola.

Step 1. Find the x-coordinate of the vertex.
  The x-coordinate is given by s = –b/(2a).
  Here, a = 0.04 and b = –16.
  So, s = –(–16) / (2 × 0.04) = 16 / 0.08 = 200.

Step 2. Find the y-coordinate (the minimum cost).
  Substitute s = 200 into the function:
  c(200) = 0.04(200)² − 16(200) + 15,000
    = 0.04(40,000) − 3,200 + 15,000
    = 1,600 − 3,200 + 15,000
    = 13,400.

Thus, the absolute minimum of the function is at (200, 13,400).

Enter the answer in each space provided:
  200, 13,400

Description in terms of the problem:
• The x-coordinate (200) represents the optimal number of skateboards produced each day that minimizes the total cost.
• The y-coordinate (13,400) represents the minimum total cost in dollars associated with producing 200 skateboards per day.
Enter the answer in each space provided: 200, 13400 Description in terms of the problem: - The x-coordinate (200) represents the optimal number of skateboards produced each day that minimizes the total cost. - The y-coordinate (13400) represents the minimum total cost in dollars associated with producing 200 skateboards per day.

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