Question 726707
The cost of producing 20 cakes is $315, while 35 cakes is $495. 
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Let x = the number of cakes produced
Let y = the cost of preparing those x cakes.
</pre>
The cost of producing 20 cakes is $315,
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When x = 20, y = 315, indicated by the ordered pair (20,315) 
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while 35 cakes is $495
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When x = 35, y = 495, indicated by the ordered pair (35,495)
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 (a)Find the linear cost equation
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That is the same as the equation of the line that goes through
the points (20,315) and (35,495)

We use the slope (gradient) formula:

m = {{{(y[2]-y[1])/(x[2]-x[1])}}}
where (x<sub>1</sub>,y<sub>1</sub>) = (20,315)
and where (x<sub>2</sub>,y<sub>2</sub>) = (35,495)

m = {{{(495-315)/(35-20)}}} = {{{180/15}}} = 12

Then we use the point-slope (or point-gradient) formula:

y - y<sub>1</sub> = m(x - x<sub>1</sub>)

m = {{{(495-315)/(35-20)}}} = {{{180/15}}} = 12

y - 315 = 12(x - 20)

y - 315 = 12x - 240

      y = 12x + 75
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 (b)What is the y-intercept and interpret the answer?
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The y intercept is when x = 0

      y = 12x + 75
      y = 12(0) + 75
      y = 0 + 75
      y = 75

So the y-intercept is (0,75).  That means that when
no (x=0) cakes have yet been produced, the fixed costs are 
$75, just for setting up and going into the business
of producing cakes.
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 (c)What is the gradient and interpret the answer?
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The slope or gradient is m=12.  That means that the cost
per cake increases by $12 per cake.
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 (d)Find the cost of producing 55 cakes
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We substitute x=55 into 

      y = 12x + 75
      y = 12(55) + 75
      y = 660 + 75
      y = 735

Answer: $735
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 (e)How many cakes can be produced with $1000.
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We substitute y = 1000 in

      y = 12x + 75

and solve for x:

   1000 = 12x + 75
    925 = 12x 
 77.083 = x, round down to 77.

Answer:  77 cakes can be produced for $1000    

Edwin</pre>