In Exercises 57–59, graph the region determined by the constraint...

Question

In Exercises 57–59, graph the region determined by the constraints. Then find the maximum value of the given objective function, subject to the constraints.  This  is a piecewise function. Refer to the textbook.

Accepted Answer

Hello today we're going to be finding the maximum value for the function Z is equal to five X plus Y. So in order to start this, let's go ahead. And first start by graphing our constraints. So let's first go ahead and draw a coordinate axis. And we're gonna go ahead and start by graphene, X is greater than or equal to zero. Now this resembles the form of X is equal to zero. So the graph of X is greater than or equal to zero is going to be the vertical line that passes through the origin. Now keep in mind that we have the greater than or equal to inequality symbol. So we're going to include all the values that lie on the graph. Therefore this is going to be graphed as a solid line. Now this is saying that we want all values of this graph that are greater than or equal to zero. So the total amount or the shaded area of this graph is going to be everything to the right because that's going to be the positive values. Now let's go ahead and start by graphing. Why is greater than or equal to zero? Well this is going to follow the same form where this looks like the graph Y is equal to zero. So this is gonna be the horizontal line that passes through the origin. But keep in mind that we want all the values that are greater than or equal to zero. So we're including all the values that lie on this horizontal line. So this is going to be y greater than equal to zero. And we want all the values that are positive. So all the values that are greater than equal to zero. So that's gonna be the entire region that shaded above this graph here. Now let's go ahead and graph X plus y is less than or equal to one. Now this looks like a linear line but not in standard form. So we need to go ahead and rewrite this as Y is equal to Mx plus B. So in order to do that, I'm just going to go ahead and subtract X to both sides of this inequality and what we are left with is why less than or equal to negative X plus one. So this resembles the line that starts at zero comma one and makes its way down to one comma zero because this is the graph of a line that has a slope of negative one. Now keep in mind that we want all the Y values that are less than negative X plus one. So we're going to graph this solid line and we want all the values that are below the graph. Well keep in mind that we want to go ahead and keep the shaded area that's shared between all the graphs. So right now we have the shape of this triangle and finally we're gonna go ahead and graph x minus y less than or equal to one. Now we're going to do the same thing and write this in slope intercept form. So y is equal to mx plus B. And in order to do that, we're going to subtract X from both sides of the equation that leaves us with negative y less than or equal to negative X plus one. And we need y to be positive. So we need to divide everything by negative one. Doing so is going to give me y is greater than equal to x minus one. So this is going to be the form of the graph that starts at zero comma negative one and has a positive slope of one. So it intercepts at one comma zero. And keep in mind we're including all the values of the graph because we have the greater than or equal to inequality symbol. So we're gonna be drawing a solid line that passes through these two points. Now, since the inequalities opening up towards the Y, we want all the values above this graph. But if we shade above the graph, the shade, the shared area between the four graphs is still going to be this triangle. So this is gonna be the enclosed region for the function Z is equal to five, x plus y. And the three intersecting points among these graphs are going to be zero comma 11 comma zero and zero comma zero. So in order to find the maximum value, we need to go ahead and plug in these points into our given function and see which of these points produces the higher value. So if we start with zero comma one and plug it in to our function, we get Z is equal to five times zero. Plus 15 times zero is just zero. And plus one is just going to give us one. So the 10.0 comma one is going to give us Z is equal to one. Now if we plug in 1:00 we get Z is equal to five times one plus zero while five times one is going to be five and plus zero is going to leave us with five. So the 50.1 comma zero gives us Z is equal to five. And if we plug in +00 we get Z is equal to five times zero plus zero. Well five times zero is just zero and plus zero is still zero. So the 00.0 comma zero gives us the value of Z is equal to zero. Now the greater value is going to be which one of these points produced a higher value. And between the three it's going to be, Z is equal to five. So that's gonna be the maximum value for this given function and the answer to this problem is going to be be. So I hope this video helps you understanding how to find the maximum value of this function and I'll go ahead and see you all in the next video

In Exercises 57–59, graph the region determined by the constraints. Then find the maximum value of the given objective function, subject to the constraints. This is a piecewise function. Refer to the textbook.

Key Concepts

Graphing Constraints

Objective Function

Piecewise Functions

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