Part (a)

The cost of the candles is $0.75 per candle times the number and candles = 0.75c

The cost of soap is $1.25 per bar of soap times the number of bars of soap = 1.25s

the total cost is, Cost = 0.75c+ 1.25s

The sale value of the candles is $1.75 per candle times the number of candles = 1.75c

The sale value of soap is $3.25 per bar of soap times the number of bars of soap = 3.25s

the total sale value, Sale = 1.75 c+ 3.25 S

The profit that the students should make is ≥ $200

Therefore the Profit is , Profit = Sale - Cost ≥ 200

Substituting sale and profit in the inequality you get

1.75 c + 3.25 s - ( 0.75 c + 1.25 s ) ≥ 200

solving 1.75 c + 3.25 s - 0.75 c -1.25 s ≥ 200

and c + 2 s ≥ 200

part (b)

By rearranging this inequality you get c ≥ 200 -2 s or c ≥ -2 s + 200

I f we consider the equation c = -2 s+ 200 , the slope of this line is -2 and the y intercept is 200 that is the point (0,200) on the Y axis, the x intercept would be 100 that is the point (100,0) on the x axis . therfore the graph would be a straight line passing through the points of (0,200) and (100,0) with a slope of -2. However the inequality calls for all the c values above or greater than the line. With the line being in the first quadrant , the shaded area that represent the inequality is all the values above and to the rght of that line. With constraints of 80 s and 140 c from the supplier, a wedge may be cut from the shaded area that represent the boundary constraints of the inequality.

Part (c)

the shaded area of the wedge bounded by the constraints represent all possible combinations of c and s that will achieve the goal of providing ≥ $ 200 in profit.