Question

Find the break - even points for company X, which sells all it produces, if the variable cost per unit is $3, fixed costs are $2 and YT R = 5√q, where q is the number of thousands of units of output produced.

Answer 1

Given:

The variable cost per unit is $3, fixed costs are $2.

The revenue function is:

$Y_{TR}=5\sqrt{q}$

where q is the number of thousands of units of output produced.

To find:

The break - even points for company X.

Solution:

The variable cost per unit is $3, fixed costs are $2.

So, the cost function is:

Total cost = Fixed cost + Variable cost × Quantity

$Y_{TC}=2+3q$

The revenue function is:

$Y_{TR}=5\sqrt{q}$

At break - even points the profit is zero. It means the cost and revenue are equal.

$Y_{TC}=Y_{TR}$

$2+3q=5\sqrt{q}$

Squaring both sides, we get

$(2+3q)^2=(5\sqrt{q})^2$

$2^2+2(2)(3q)+(3q)^2=25q$

$4+12q+9q^2-25q=0$

$4-13q+9q^2=0$

Splitting the middle term, we get

$4-4q-9q+9q^2=0$

$4(1-q)-9q(1+q)=0$

$(4-9q)(1-q)=0$

Using zero product property, we get

$4-9q=0$ and $1-q=0$

$q=\dfrac{4}{9}$ and $q=1$

$q\approx 0.444$ and $q=1$

Therefore, the break even points are 0.444 and 1.