Planting Crops A farmer has 500 acres of arable land on...

Planting Crops A farmer has 500 acres of arable land on which he wants to plant potatoes and corn. The farmer has $\$ 40,000$ available for planting and $\$ 30,000$ for fertilizer. Planting 1 acre of potatoes costs $\$ 90,$ and planting 1 acre of corn costs $\$ 50 .$ Fertilizer costs $\$ 30$ for 1 acre of potatoes and $\$ 80$ for 1 acre of corn. (a) Find a system of inequalities that describes the number of acres of each crop that the farmer can plant with the available resources. Graph the feasible region. (b) Can the farmer plant 300 acres of potatoes and 180 acres of corn? (c) Can the farmer plant 150 acres of potatoes and 325 acres of corn?

Planting Crops A farmer has 500 acres of arable land on which he wants to plant potatoes and corn. The farmer has $\$ 40,000$ available for planting and $\$ 30,000$ for fertilizer. Planting 1 acre of potatoes costs $\$ 90,$ and planting 1 acre of corn costs $\$ 50 .$ Fertilizer costs $\$ 30$ for 1 acre of potatoes and $\$ 80$ for 1 acre of corn.
(a) Find a system of inequalities that describes the number of acres of each crop that the farmer can plant with the available resources. Graph the feasible region.
(b) Can the farmer plant 300 acres of potatoes and 180 acres of corn?
(c) Can the farmer plant 150 acres of potatoes and 325 acres of corn?

Key Concepts

Linear Programming

Linear programming is a mathematical technique used for optimization, where the goal is to maximize or minimize a linear objective function subject to a set of linear inequalities or equations representing constraints. It is commonly applied in resource allocation problems such as planning and production scheduling.

System of Inequalities

A system of inequalities is a collection of linear constraints that need to be satisfied simultaneously. These inequalities model the limitations in a problem, such as budget, available land, or other resources, and form the basis for determining feasible solutions in linear programming problems.

Feasible Region

The feasible region is the set of all possible points that satisfy the system of inequalities in a linear programming problem. It represents all the viable solutions that meet the constraints, and its boundaries are determined by the equality conditions of the constraints.

Constraints

Constraints represent the limitations or requirements imposed on a problem, such as limited resources or capacity restrictions. In planning problems, these include factors like budget limits, available land, and necessary materials, which must all be taken into account when determining a solution.

Resource Allocation

Resource allocation involves distributing limited resources among various activities or projects in order to maximize or achieve a particular objective. In the context of linear programming, it is about assigning the available resources efficiently while satisfying all the relevant constraints.

Transcript

00:02 Problem 69, chapter 10, section 5, system of inequalities.

00:08 In this question, we are given that we have $40 ,000 in total for planting and $30 ,000 for fertilizers.

00:19 And the cost of planting, potatoes per acre and corn per acre is given.

00:27 So we want to graph the feasible reason and check whether these two conditions can be met.

00:35 First, let us write the system of inequalities which describe this situation.

00:46 So first, let's assume that x is the amount of potatoes that we can plant and y is the amount of corn which we can plant.

01:07 So yeah let us assume we plant x amount of potato and y amount of corn then we have 90 like 90 dollars multiplied by x from this condition plus 50 multiplied by y is less than are equal to 40 ,000 and we have from the second condition 30x plus 80 y is less than equal to 30 ,000 so let us convert these into a form that we are more familiar with the form of y equals mx plus c which is equation of a line so the first one from the first equation we get y is less than equal to 40 ,000 by 50 minus 90x by 50.

03:13 On solving this equation for the we have 8 0 0 minus 9 by 5 x now for the second equation we have y is less than equal to 30 ,000 by 80 minus 30 x divided by 80 divided into okay so again right so we have two equations now and we can plot them so you can do it carefully on a graph paper or you can do it roughly we are just going to do it roughly using our estimates so let's say 800 is somewhere up here and 9 by 5 we know is approximately equal to 2 and minus means a downward slope so this graph would look like sorry let me 9 twice okay so yeah so the first line is the region on this side of the y -axis of this line and second one 375 will be around here minus 3 by 8 x so this slope is greater than 1 3 by 8 is almost 3 times of this so slope looks like this okay so we also have the condition that of course we want our values to be positive and not negative because we are planting something so that's a positive number so this is the region which we this is the region which describes the amount of crops which can be planted within the given budget so now let's look at the next question can we plot can we plant 300 acres of potatoes and 180 acres of corn.

07:26 So potatoes is x...

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