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√(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√(q)`

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

`Y_(TC)=Y_(TR)`

`2+3q=5√(q)`

Squaring both sides, we get

`(2+3q)^2=(5√(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=(4)/(9)` and
`q=1`

`q\approx 0.444` and
`q=1`

Therefore, the break even points are 0.444 and 1.