x: 4 units of a and 5 units of b per hour\ny: 3 units of a and 10 units of b per hour\ncost to run machine…

Explanation:

Step1: Find intersection points

First, rewrite the inequalities as equations: $4x + 3y=90$ and $5x + 10y=150$. From $4x + 3y=90$, we get $y = 30-\frac{4}{3}x$. Substitute into $5x + 10y=150$: $5x+10(30 - \frac{4}{3}x)=150$. $5x + 300-\frac{40}{3}x=150$. $15x+900 - 40x = 450$. $- 25x=-450$, so $x = 18$. Then $y=30-\frac{4}{3}\times18=30 - 24 = 6$.

Inter - section of $4x + 3y=90$ and $5x + 10y=150$ is $(18,6)$.

Intersection of $4x + 3y=90$ and $x = 0$: When $x = 0$, $y = 30$. So the point is $(0,30)$.

Intersection of $5x + 10y=150$ and $x = 0$: When $x = 0$, $y = 15$. So the point is $(0,15)$.

Intersection of $5x + 10y=150$ and $y = 0$: When $y = 0$, $x = 30$. So the point is $(30,0)$.

Intersection of $4x + 3y=90$ and $y = 0$: When $y = 0$, $x=\frac{90}{4}=22.5$. So the point is $(22.5,0)$.

Step2: Evaluate objective function

The objective function is $C = 22x+25y$. For point $(0,30)$: $C=22\times0 + 25\times30=750$. For point $(0,15)$: $C=22\times0+25\times15 = 375$. For point $(18,6)$: $C=22\times18+25\times6=396 + 150=546$. For point $(30,0)$: $C=22\times30+25\times0 = 660$. For point $(22.5,0)$: $C=22\times22.5+25\times0=495$.

Answer:

The minimum cost is $$375$ and occurs at $(0,15)$