A company distributes college logo sweatshirts and sells them for $50 each. The total cost function is linear, and the total cost for 90 sweatshirts is $3564, whereas the total cost for 260 sweatshirts is $5944.(a) Write the equation for the revenue function R(x)
(b) Write the equation for the total cost function
(c) Find the break-even quantity. sweatshirts

The total revenue equation is TR=50Q

The total cost equation TC=2304+14Q

Equilibrium quantity is 64

Step-by-step explanation:

The total cost function can be written TC=a+bQ, where TC is the total cost ,a fixed cost ,b is the variable cost per unit and Q the quantity produced

Revenue function can be given TR=P*Q, where TR is total revenue,P is the price per unit and Q quantity sold

since P is $50

TR=50P

when TC=$3564,

3564=a+90Q equation 1

when TC=$5944

5944=a+260bequation 2

Solving simultaneously,we subtract equation 1 from 2

5944=a+260b

3564=a+90b

2380=170b

b=2380/170

b=$14

We substitute for b in any of the equations

3564=a+(14*90)

3564=a+1260

a=3564-1260

a=2304

Hence the total cost equation is

TC=2304+14Q

At equilibrium TC =TR

50P=2304+14Q

50P-14Q=2304

36Q=2304

Q=64

468

step-by-step explanation:

take the logarithm of both sides (the base of the log doesn't matter):

apply the exponent property - this says that :

distribute the log terms on both sides:

group like terms together:

divide through both sides by the coefficient on :

you can do some additional rewriting here using the properties of the logarithm to simplify things if you like: