Q39. - Algebra 2

Question

Dean Stadler has 20 days in which to plant corn and soybeans. The corn can be planted at a rate of 250 acres per day and the soybeans at a rate of 200 acres per day. He has 4500 acres available for planting these two crops.

 39. Draw the graph showing the feasible region and list the coordinates of the

vertices of the feasible region.

Step-by-Step Solution

Answer

The feasible region of the system of equations is shaded in purple color, and the coordinates of the feasible region are 

$(0,0),(0,20),(18,0),(10,10).$

Step 1 – Construct a system of equations for the given situation.

Let and be the number of acres of corn and soybean respectively.

 Given that Dean Stadler wants to plant corn and soybeans in 20 days.

So, .$c+s\le 20$

 The corn can be planted at a rate of 250 acres per day and the soybeans at a rate of 200 acres per day. 

 He has 4500 acres available for planting these two crops.

So, .$250c+200s\le 4500\Rightarrow 5c+4s\le 90$

Also,$c\ge 0,s\ge 0.$

Then, the system of equations is:

$\begin{array}{c}c+s\le 20\\ 5c+4s\le 90\\ c\ge 0\\ s\ge 0\end{array}$

Step 2 – Find the intersecting point of c + s = 20 , 5 c + 4 s = 90

To find the intersecting point, use the elimination method.

 Multiply the equation $c+s=20$ by 4 and subtract the new resultant equation from $5c+4s=90.$

$\begin{array}{l}\underset{\_}{\begin{array}{c}5c+4s=90\\ c+s=20\end{array}}\begin{array}{c}\\ \text{multiply by 4}\end{array}\underset{\_}{\begin{array}{c}5c+4s=90\\ (-)4c+4s=80\end{array}}\\ c+0=10\end{array}$

So, $c=10.$

 Substitute $c=10$ in $c+s=20$ and solve for $s$:

$\begin{array}{c}c+s=20\\ 10+s=20\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}substitute 10 for }c\\ s=10\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}subtract 10 from both sides}\end{array}$

So, the intersecting point is $(10,10).$

Step 3 – Graphical representation of the above system of inequality

Graph the solution region$c+s\le 20$ for and the solution region for .$5c+4s\le 90$

The intersecting region of$c+s\le 20$ and $5c+4s\le 90$is the solution region of the system of inequalities.

The feasible region of the system of equations is shaded in purple color, and the coordinates of the feasible region are $(0,0),(0,20),(18,0),(10,10).$