Question

find minimum and maximum values of z=30x+20y subject to 25x+40y< or equal to 800. x>or equal to 8. y>or equal to 10. where do i start?

Answer 1

start by graphing your constraints, treat the inequalities as if they are just lines and graph them
\[25x +40y = 800 \rightarrow y = -\frac{5}{8}x+20\]
\[x = 8\]
\[y = 10\]
You should now have 3 lines, one vertical, one horizontal, one diagonal, Look at the "<" or ">" for each one and shade in the bounded area. It should be a small triangular area above the horizontal line and to the right of the vertical line. The max and min values of z will always be at an endpoint or where the lines cross. There should be 3 points, find the coordinates of each point and plug it into "z=30x+20y".
The 3 points should be:
\[(8,10), (8,15), (16,10)\]
z values
\[z = 30(8)+20(10) = 440\]
\[z = 30(8)+20(15) = 540\]
\[z = 30(16)+20(10) = 680\]
max = 680, min = 440