QuestionFind the loss profit points for the monopolist with demand $p=152.5-3q$ and cost $TVC=0.5q^3-15q^2+175q$ , $TFC=0$ .

Studdy Solution

STEP 1

Assumptions1. The demand function is given by $p =152.5 -3q$ . The total variable cost (VC) is given by $VC =0.5q^{3} -15q^{} +175q$
3. The total fixed cost (FC) is04. The monopolist's total cost (TC) is the sum of TVC andFC5. The monopolist's profit ( $\pi$ ) is given by the formula $\pi = pq - TC$ , where $p$ is the price, $q$ is the quantity, and $TC$ is the total cost

STEP 2

First, we need to find the total cost (TC). Since the total fixed cost (FC) is0, the total cost is just the total variable cost (VC).
$TC = TVC$

STEP 3

Substitute the given TVC into the TC equation.
$TC =0.5q^{3} -15q^{2} +175q$

STEP 4

Now, we need to find the profit ( $\pi$ ). Substitute the given demand function and the total cost into the profit equation.
$\pi = (152. -3q)q - (0.q^{3} -15q^{2} +175q)$

STEP 5

implify the profit equation.
$\pi =152.5q -3q^{2} -0.5q^{3} +15q^{2} -175q$

STEP 6

Combine like terms in the profit equation.
$\pi = -0.5q^{3} +12q^{2} -22.5q$

STEP 7

The loss profit points occur when the profit is0. So, set the profit equation to0 and solve for $q$ .
$-0.5q^{3} +12q^{2} -22.5q =0$

STEP 8

This is a cubic equation and can be factored by grouping. Factor out a $q$ from each term.
$q(-0.5q^{2} +12q -22.5) =0$

STEP 9

Now, we have a quadratic equation $-.5q^{2} +12q -22.5 =$ . We can solve this using the quadratic formula $q = \frac{-b \pm \sqrt{b^{2} -4ac}}{2a}$ .

STEP 10

Substitute the coefficients $a = -0.5$ , $b =12$ , and $c = -22.5$ into the quadratic formula.
$q = \frac{-12 \pm \sqrt{(12)^{2} -4(-0.5)(-22.5)}}{2(-0.5)}$

STEP 11

implify the equation to find the values of $q$ .
The solutions to this equation will give the quantity values at which the monopolist makes no profit (i.e., the loss profit points).

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