Question

The roots of the quadratic function describing the relationship between number of products produced and overall profit margin are x=0 and 100. The vertex is (50,1000). The maximum profit of $ dollars is reached when items are produced. The first root tells us that the profit will be 0 when 0 products are produced. The second root says once 100 items are made, the company is no longer making any profit. (They do not have production capacity and have to outsource for anything over 50.)

Answer 1

Answer:

I assume that we want to complete the statement:

"The maximum profit of $__ dollars is reached when __ items are produced"

We know that the profit equation is defined between x = 0 and x = 100, which are the two roots of the equation (so the profit is equal to zero for x = 0 and for x = 100).

Then we can assume that the profit will be positive in this range.

Thus, the quadratic equation should have a negative leading coefficient, which would mean that the arms of the graph go downwards.

If this is the case, we know that the maximum will be at the vertex.

Here we know that the vertex is:

(50, 1000)

Where remember, x represents the number of items and y represents the profit.

So, given that the maximum is at the vertex, and we know that the vertex is (50, 1000) we can conclude that the maximum profit is $1000, and this happens when the number of produced items is 50.

Then the complete statement is:

"The maximum profit of $1000 dollars is reached when 50 items are produced"