Write an inequality that represents the fact that while making each product you can't exceed the amount invested ($5,000) for a business where the products will cost the owner $2.50 and $3.25 to make.

Question

A. Write an inequality that represents the fact that while making each product a business owner can't exceed the amount invested he is willing to invest ($5,000) for a business where the products will cost the owner $2.50 and $3.25 to make. (Be sure to include the cost per item in this inequality.) B. Graph the inequality and label the grapg and shade the appropriate side of the line. C. Choose a point that falls in the shaded region, a point that falls directly on the line, and a point that does not fall in the shaded region, and explain for each point what the x-coordinate and the y-coordinate represent and the significance in terms of cost.

Answer 1

A. 5000/2.5=2000 and 5000/3.25=1538.5. If he only makes $2.50 items he can't exceed 2000, and if he only makes $3.25 items he can't exceed 1538. So the inequality is 2.5A+3.25B<5000 where A represents the number of $2.50 items and B that of $3.25 items.

B. To graph the inequality, we write one item in terms of the other: 3.25B=5000-2.5A. If B is the y axis and A the x axis we get a backward sloping straight line. The y intercept is 5000/3.25=1538.5 and the x intercept is 5000/2.5=2000. Choose a suitable scaling and join the intercepts. The graph only has a positive region so the origin of the axes can be placed very much on the left and towards the bottom of the page. The shaded area is the region bounded by the line and axes.

C. Choose the three points P, Q and R as directed.

P is within the shaded area and represents (A,B) in budget; Q makes use of the whole investment; R is outside budget (exceeds the investment). Although the graph is a straight line, in reality it is a sequence of points because A and B cannot contain fractions. If the graph has been drawn reasonably accurately it should be possible to read off values of A and B. The values A and B must be integers, so they need to be rounded down to the nearest integer.

To work out the cost calculate C=2.5A+3.25B. The quantity Q=A+B. The average unit cost=C/Q.