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how many candles and soap do i need to sell?

Candles cost $0.75  each and will be sold for $1.75  and soap costs $1.25 and will be sold for $3.25. The students need to raise at least $200 to cover the trip costs.

A.   write an inequality that relates the number of candles c and the number of bars of soap s to the needed income.

B.  the wholesaler can supply no more than 80 bars of soap and no more than 140 candles. graph the inequality from part a and these constraints using the number of candles for the vertical axis.

C.  what does the shaded area of your graph represent?

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3 Answers

 

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.

 

Hey Sabrina -- looks like candles earn a $2 spread and soap makes a $1 spread ... without constraints, getting $200 means selling 100 candles or 200 soap units.

(A) With candles on the y-axis, your graph looks like a triangle with the longest side dropping from y=100 over to x=200 (soap units).

(B) Constraints form a box boundary: y=140 candles-line and a x=80 soap-line cutting the wedge in half.

(C) You can sell the fatter-half of the wedge: 100 candles, or 80 soaps + 60 candles, or any combo in-between where 1 candle is worth 2 soaps (i.e. 70 candles + 60 soaps, 90 candles + 20 soaps) Regards :)

Okay, I'll answer this in parts.

A.  C + 2S > 200

This is assuming that c is the number of candles that are sold at the fundraiser.  Remember, the profit on C and S is a dollar and two dollars, respectively.  Think about it;  1.75 - .75 and 3.25 - 1.25.  Since the students need at least 200 dollars I used the "greater than or equal to" inequality symbol.

B  I don't know how to draw a graph on this program but I can describe it.  Your vertical axis is C, your horizontal is B.  The inequality will pass through the following three points:  (60, 80) which is the left-upper most point, (100, 0) which is the "S" intercept and (140, -80) which of course shows an unrealistic value for C.  You would then make sure to shade area to the right of this line.  Make sure to use a solid line, as it is an "or equals to..." inequality.  A horizontal line should be drawn from points (140, 80) to (140, -80) and from (140, 80) to (60, 80).  Then there will be a triangle that gets shaded in. 

C.  The shaded are represents the endless array of possibilities that will work.  The portion of the triangle below the "S" axis is unrealistic.  But remember, we are dealing with an inequality; therefore all negative values of C can default to zero and be part of the "or greater than" population.

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