Answered

IDNLearn.com provides a collaborative environment for finding and sharing answers. Whether it's a simple query or a complex problem, our experts have the answers you need.

Select the correct answer from each drop-down menu.

A deli owner has determined that his revenue, [tex]y[/tex], from selling sandwiches each day is at most [tex]-0.05x^2 + 6x[/tex], where [tex]x[/tex] represents the number of sandwiches he sells. To make a profit, his revenue must be greater than his costs, represented by the expression [tex]1.5x + 45[/tex].

Write a system of inequalities to represent the values of [tex]x[/tex] and [tex]y[/tex] where the deli owner makes a profit. Then complete the statements.

The point [tex](30, 90)[/tex] is [tex]$\square$[/tex] of this system.
The point [tex](60, 160)[/tex] is [tex]$\square$[/tex] of this system.

Sagot :

GinnyAnsweridn
To write a system of inequalities to represent the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] where the deli owner makes a profit, we need to consider the following conditions:

1. The revenue [tex]\( y \)[/tex] from selling [tex]\( x \)[/tex] sandwiches each day is at most [tex]\( -0.05x^2 + 6x \)[/tex].
2. In order to make a profit, the revenue [tex]\( y \)[/tex] must be greater than the costs which are represented by the expression [tex]\( 1.5x + 45 \)[/tex].

Thus, the system of inequalities can be written as:
[tex]\[ \begin{cases} y \leq -0.05x^2 + 6x \\ y > 1.5x + 45 \end{cases} \][/tex]

Now, let’s complete the statements about the points [tex]\( (30, 90) \)[/tex] and [tex]\( (60, 160) \)[/tex]:

1. For the point [tex]\( (30, 90) \)[/tex]:
- Substituting [tex]\( x = 30 \)[/tex] into the revenue function [tex]\( y \)[/tex], we get [tex]\( y = -0.05(30)^2 + 6(30) = 135 \)[/tex].
- Substituting [tex]\( x = 30 \)[/tex] into the cost function [tex]\( y \)[/tex], we get [tex]\( y = 1.5(30) + 45 = 90 \)[/tex].
- Checking the inequalities: The point [tex]\( (30, 90) \)[/tex] does not satisfy the condition [tex]\( y \leq -0.05(30)^2 + 6(30) \)[/tex] since [tex]\( 90 \leq 135 \)[/tex] is true, but it does not satisfy the profit condition [tex]\( y > 1.5(30) + 45 \)[/tex] since [tex]\( 90 > 90 \)[/tex] is false.

Therefore, the point [tex]\( (30, 90) \)[/tex] is not a solution of this system.

2. For the point [tex]\( (60, 160) \)[/tex]:
- Substituting [tex]\( x = 60 \)[/tex] into the revenue function [tex]\( y \)[/tex], we get [tex]\( y = -0.05(60)^2 + 6(60) = 180 \)[/tex].
- Substituting [tex]\( x = 60 \)[/tex] into the cost function [tex]\( y \)[/tex], we get [tex]\( y = 1.5(60) + 45 = 135 \)[/tex].
- Checking the inequalities: The point [tex]\( (60, 160) \)[/tex] does not satisfy the condition [tex]\( y \leq -0.05(60)^2 + 6(60) \)[/tex] since [tex]\( 160 \leq 180 \)[/tex] is true, but it does not satisfy the profit condition [tex]\( y > 1.5(60) + 45 \)[/tex] since [tex]\( 160 > 135 \)[/tex] is false.

Therefore, the point [tex]\( (60, 160) \)[/tex] is not a solution of this system.

Summary:

- The point [tex]\( (30, 90) \)[/tex] is not a solution of this system.
- The point [tex]\( (60, 160) \)[/tex] is not a solution of this system.
We value your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Thank you for choosing IDNLearn.com for your queries. We’re committed to providing accurate answers, so visit us again soon.