The cost of producing 20 cakes is $315, while 35 cakes is $495.
Let x = the number of cakes produced
Let y = the cost of preparing those x cakes.
The cost of producing 20 cakes is $315,
When x = 20, y = 315, indicated by the ordered pair (20,315)
while 35 cakes is $495
When x = 35, y = 495, indicated by the ordered pair (35,495)
(a)Find the linear cost equation
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 =
where (x1,y1) = (20,315)
and where (x2,y2) = (35,495)
m = = = 12
Then we use the point-slope (or point-gradient) formula:
y - y1 = m(x - x1)
m = = = 12
y - 315 = 12(x - 20)
y - 315 = 12x - 240
y = 12x + 75
(b)What is the y-intercept and interpret the answer?
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.
(c)What is the gradient and interpret the answer?
The slope or gradient is m=12. That means that the cost
per cake increases by $12 per cake.
(d)Find the cost of producing 55 cakes
We substitute x=55 into
y = 12x + 75
y = 12(55) + 75
y = 660 + 75
y = 735
Answer: $735
(e)How many cakes can be produced with $1000.
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