Question 1172512: : A bakery discovers that if it decreases the price of its birthday cakes by $1, it sells 12 more
cakes each month.
(a) Assuming that monthly sales, M, are related to prices, P, by a linear model, M = aP + b, state
the value of a.
(b) If the bakery sells 240 cakes in a month when the price of the cake is $14, work out the value
of b.
(c) Use this model to estimate monthly sales when the price is $9.
(d) If the bakery can make only 168 cakes in a month, work out the price that it needs to charge
to sell them all
Found 2 solutions by Boreal, math_tutor2020:
Answer by Boreal(15235)   (Show Source):
You can put this solution on YOUR website! The slope is -12, which is a in the equation m=aP+b. Every decrease of cost by $1 is an increase in sales by 12.
-
240=-12(14)+b
$408=b
------
price is $9
M=-12*9+408
M=$300
----
168=-12P+408
-240=-12P
P=$20
Answer by math_tutor2020(3817)  (Show Source):
You can put this solution on YOUR website!
Part (a)
What we'll do is increase M by 12 and reduce P by 1.
Increasing M by 12 means we'll replace M with M+12.
Decreasing P by 1 means we'll replace P with P-1.
After distributing, we'll cancel two pairs of terms.
Let's see what happens when we apply those steps
M = aP + b
M+12 = aP + b ... replace M with M+12
M+12 = a(P-1) + b ... replace P with P-1
aP + b + 12 = a(P-1) + b ... plug in M = aP+b
aP + b + 12 = aP - a + b ... distribute
aP + 12 = aP - a ...... subtract b from both sides
12 = -a ....... subtract aP from both sides
a = -12
This value of 'a' tells us that each time the price goes down 1 dollar, the number of cakes sold goes up by 12.
It also means that each time the price is increased by 1 dollar, the amount of cake sold goes down by 12.
As one variable goes down, the other goes up. This is why the negative is present.
Answer: a = -12
========================================================
Part (b)
We're given
M = 240 cakes sold in a month
P = 14 dollars is the price of the cake
From part (a) we found a = -12
M = aP + b
M = -12P + b ... plug in a = -12
240 = -12P + b ... plug in M = 240
240 = -12(14) + b ... plug in P = 14
Let's solve for b
240 = -12(14) + b
240 = -168 + b
240+168 = b
b = 408
Answer: b = 408
========================================================
Part (c)
We want to find M when P = 9
Earlier we found a = -12 and b = 408
The equation M = aP+b turns into M = -12P+408
Plug in P = 9 and compute
M = -12P+408
M = -12*9+408
M = -108+408
M = 300
We estimate 300 cakes will be sold in a month when the price is $9.
Answer: 300
========================================================
Part (d)
Now we have M = 168 and we want to find its paired value of P
Plug in M = 168 and solve for P
M = -12P+408
168 = -12P+408
168-408 = -12P
-240 = -12P
-12P = -240
P = -240/(-12)
P = 20
The price must be $20 per cake in order to sell 168 cakes per month.
As the tutor @boreal has shown, the blue horizontal line is the line y = 168 (here y = M). Note that it intersects the diagonal line y = -12x+408 at (20,168). This is visual confirmation that we have the pair (P,M) = (20,168).
Answer: $20
|
|
|
