SOLUTION: Allie orders candles from an online company that offers flat rate shipping. She placed an order for 4 candles for $35. A few months later, she placed an order for 12 candles for $9

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Question 1147822: Allie orders candles from an online company that offers flat rate shipping. She placed an order for 4 candles for $35. A few months later, she placed an order for 12 candles for $95. Write an equation to model the total cost for ordering x candles.
Found 2 solutions by Theo, greenestamps:
Answer by Theo(13342)   (Show Source): You can put this solution on YOUR website!
let x equal the number of candles in each shipment.
if you use straight line equation, then the total cost is equal to y in the following equation:
y = mx + b
x is the number of candles in the shipment.
m is the average change in the cost per candle.
b is the fixed cost, regardless of the number of candles shipped.
you have two (x,y) coordinate points from which to calculate m.
they are (4,35) and (12,95).
the total cost is 35 when 4 candles are shipped.
the total cost if 95 when 12 candles are shipped.
you use these two points to find m.
m = (y2 - y1) / (x2 - x1)
let (x1,y1) = (4,35).
let (x2,y2) = (12,95).
m = (y2 - y1) / (x2 - x1) = (95 - 35) / (12 - 4) = 60/8.
formula becomes y = 60/8 * x + b
to find b, use either point and replace x and y with the values from that point.
i used (4,35).
formula becomes 35 = 60/8 * 4 + b
solve for b to get b = 35 - 60/8 * 4 = 280/8 - 240/8 = (280-240)/8 = 40/8.
formula becomes y = 60/8 * x + 40/8.
when x = 4, formula becomes y = 60/8 * 4 + 40/8 = 35.
when x = 12, formula becomes y = 60/8 * 12 + 40/8 = 95.
formula looks good.
your solution is that the formula to model the cost for x candles is:
y = 60/8 * x + 40/8.
this formula can be graphed as shown below:
$$$
the fixed cost for shipping is 5 dollars.
that's the cost regardless of the number of candles that are shipped.
the variable cost is equal to 60/8 times the number of candles shipped.
when 4 candles are shipped, the fixed cost is 5 and the variable cost is 60/8 * 4 = 240/8 = 30 for a total cost of 5 + 30 = 35.
when 12 candles are shipped, the fixed cost is 5 and the variable cost is 60/8 * 12 = 720/8 = 90 for a total cost of 5 + 90 = 95.
the fixed cost is the cost when no candles are shipped.
per the formula this becomes 60/8 * 0 + 5 = 5.
note that y = 60/8 * x + 40/8 is the same equation as y = 7.5 * x + 5.
the graph of each of those two equations is the same red line.
the variable cost means the same as the incremental cost.
it's the cost for each additional unit sold.

Answer by greenestamps(13203)   (Show Source): You can put this solution on YOUR website!


Informally....

The shipping cost is the same for both orders, so the additional 12-4=8 candles cost an extra $95-$35=$60; the price of each candle is $60/8 = $7.50.

The cost of 4 candles is 4*$7.50 = $30, so the flat shipping cost is $35-$30 = $5.

The cost of an order is $7.50 for each candle plus $5 for shipping:



Formal algebra does exactly the same thing, but using symbols instead of words.
let x be the number of candles; let a be the price per candle and b be the flat shipping cost.  Then

     12a + b = 95    [12 candles plus shipping costs $95]
      4a + b = 35    [4 candles plus shipping costs $35]
    --------------
      8a     = 60    [the difference between the two orders is 8 candles for $60]
       a     = 60/8 = 7.5    [the cost for each candle is $7.50]

     4(7.5) + b = 35    [the cost of 4 candles at $7.50 each, plus shipping, is $35]
     30 + b = 35    [the 4 candles cost $30]
     b = 35-30 = 5    [so the shipping cost is $5]
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