Question

Please provide a detailed explanation to this problem?

Answer 1

`color(red)(b_("maximum")=750)`

Explanation:

.

Let's graph these inequalities and take a look at the solution set. To do so, we first turn the inequalities into equations. Then we graph each one. Both are straight lines because they are equations of first degree.

The left edge of the green region is the line whose equation is:

`y=5x`

Our inequality is:

`y <= 5x`

This means we are looking for a region that consist of points whose `y`-coordinates are less then the `y`-coordinates of the points that lie on the left edge line. As such, we shade the region below the line green.

The right edge of the red region is the line whose equation is:

`y=-15x+3000`

Our inequality is:

`y <= -15x+3000`

For the same reason as for the other line, we shade the region below the right edge line red.

As you can see the two regions overlap and create the brown region which is the intersection of the red and green regions. This brown region constitutes the solution set for the system of inequalities.

If a point `(a,b)` lies in the solution set, i.e. somewhere in the brown region, the maximum possible value of `b` would be the maximum value of `y` that exists in the brown region which is where the two edge lines intersect.

Since this `y` value has to be valid in the equations of the two edge lines, we set the two equations equal to each other and solve for `x` which is the value of `a` for the point with maximum `y` value in the solution set.

`5x=-15x+3000`

`20x=3000`

`x=150`

Now, we plug this in for `x` in either equation to solve for `y`:

`y=5(150)=750`

`b_("maximum")=750`