Question

Alfred Juarez owns a small publishing house specializing in Latin American poetry. His fixed cost to produce a typical poetry volume is $525, and his total cost to produce 1000 copies of the book is $2675. His books sell for $4.95 each.
(A) Find the linear cost function for Alfred's book production.
(B) How many poetry books must he produce and sell in order to break even?
(C) How many books must he produce and sell to make a profit of $1000?

Answer 1

Answer:

(A)  $C(x)=2.15x+525$

(B) 188 books

(C) 545 books

Step-by-step explanation:

We have been given that Alfred Juarez's fixed cost to produce a typical poetry volume is $525, and his total cost to produce 1000 copies of the book is $2675.

(A) The cost function will be in form $C(x)=ax+b$, where, a is cost of each copy, x is number of books and b is fixed cost.

Upon substituting our given information, we will get:

$2675=1000a+525$

Let us solve for a.

$2675-525=1000a$

$2150=1000a$

$a=\frac{2150}{1000}=2.15$

Therefore, the cost function would be $C(x)=2.15x+525$.

(B) Since each book sells for $4.95, so amount earned by selling x books would be $4.95x$

Revenue function would be $R(x)=4.95x$

We know that break-even is a point, where cost is equal to revenue or when there is a 0 profit.

$R(x)=C(x)\\\\4.95x=2.15x+525$

$4.95x-2.15x=525$

$2.8x=525$

$x=\frac{525}{2.8}$

$x=187.5\approx 188$

Therefore, Alfred must produce 188 poetry books to break even.

(C) We know that profit is equal to difference of revenue and cost.

$\text{Profit}=\text{Revenue}-\text{Cost}$

$P(x)=4.95x-(2.15x+525)$

$1000=4.95x-(2.15x+525)$

$1000=4.95x-2.15x-525$

$1000=2.8x-525$

$1000+525=2.8x$

$1525=2.8x$

$x=\frac{1525}{2.8}$

$x=544.642857\approx 545$

Therefore, Alfred must produce and sell 545 books to make a profit of $1000.