Let's solve the problem step-by-step, using the provided information.
1. Identify key features of the quadratic function:
- The roots (or x-intercepts) of the quadratic function are [tex]\( x = 0 \)[/tex] and [tex]\( x = 126 \)[/tex].
- The vertex of the quadratic function is given as [tex]\( (63, 1500) \)[/tex].
2. Interpret the given information:
- The vertex is the highest point on the parabola of a quadratic function, which means it represents the maximum profit.
- The roots represent the points where the profit is zero.
3. Determine the maximum profit and the number of products produced at that point:
- From the vertex [tex]\((63, 1500)\)[/tex], we can conclude that:
- The maximum profit is 1500 dollars.
- This maximum profit is achieved when 63 items are produced.
4. Interpret the significance of the roots:
- The first root, [tex]\( x = 0 \)[/tex], tells us that the profit will be zero when 0 products are produced.
- This means if the company does not produce any products, the profit is zero, which makes sense practically.
- The second root, [tex]\( x = 126 \)[/tex], tells us that once 126 items are produced, the company is no longer making any profit.
- This could indicate that beyond this production point, costs might outweigh the revenue, leading to zero or negative profit.
Putting this all together, we get the following conclusions:
1. The maximum profit of 1500 dollars is reached when 63 items are produced.
2. The first root tells us that the profit will be zero when 0 products are produced.
3. The second root indicates that once 126 items are produced, the company is no longer making any profit.
