Solveeit Logo
Login

Question

Question: A manufacturer determines that his total cost function is\(c = \dfrac{{{q^3}}}{3} + 2q + 300\). Wher...

A manufacturer determines that his total cost function isc=q33+2q+300c = \dfrac{{{q^3}}}{3} + 2q + 300. Where q is the number of units produced.
(a)(a)Find the marginal cost function
(b)(b)Find the average cost function
(c)(c)Find the level of output at which the average cost is minimum

Explanation

Solution

Hint- Use the definitions of marginal cost, average cost, and standard procedure to find level output at average cost which is minimum.
Now the total cost is given asc=q33+2q+300c = \dfrac{{{q^3}}}{3} + 2q + 300.
So firstly let’s calculate for (a) Marginal cost function
Marginal cost function = dcdq=ddq(q33+2q+300)\dfrac{{dc}}{{dq}} = \dfrac{d}{{dq}}\left( {\dfrac{{{q^3}}}{3} + 2q + 300} \right)
The derivative gives us
dcdq=q2+2\dfrac{{dc}}{{dq}} = {q^2} + 2 As derivative of xn=nxn−1{x^n} = n{x^{n - 1}}
So the marginal cost function isq2+2{q^2} + 2.
Now let’s calculate for average cost function
Average cost function = total costnumber of units produced\dfrac{{total{\text{ cost}}}}{{number{\text{ of units produced}}}}
Average cost = cq=q33+2q+300q=q23+2+300q\dfrac{c}{q} = \dfrac{{\dfrac{{{q^3}}}{3} + 2q + 300}}{q} = \dfrac{{{q^2}}}{3} + 2 + \dfrac{{300}}{q}
So average cost function is q23+2+300q\dfrac{{{q^2}}}{3} + 2 + \dfrac{{300}}{q}…………………………………… (1)
Now to find the level output at which average cost is minimum we simply need to put the derivative of the function of average cost equal to 0
That is d(Average cost)dq=0\dfrac{{d(Average{\text{ cost)}}}}{{dq}} = 0
So we have using equation 1
ddq(q23+2+300q)=0\dfrac{d}{{dq}}\left( {\dfrac{{{q^2}}}{3} + 2 + \dfrac{{300}}{q}} \right) = 0
The derivative of this quantity is 2q3−300q2=0\dfrac{{2q}}{3} - \dfrac{{300}}{{{q^2}}} = 0
On further solving we get
2q3−900=02{q^3} - 900 = 0
Or q3=450{q^3} = 450
So the value of q=3450q{ = ^3}\sqrt {450}
Now let’s verify that this q corresponds to the minimum of the average cost function.
So ∂2(average cost)∂q2<0\dfrac{{{\partial ^2}(average{\text{ cost)}}}}{{\partial {q^2}}} < 0
Let’s substitute the values we get ∂2∂q2(q23+2+300q)>0\dfrac{{{\partial ^2}}}{{\partial {q^2}}}\left( {\dfrac{{{q^2}}}{3} + 2 + \dfrac{{300}}{q}} \right) > 0
Double derivative of this quantity is 23+600q3\dfrac{2}{3} + \dfrac{{600}}{{{q^3}}}…………………………………. (2)
Now on substitution of q=3450q{ = ^3}\sqrt {450} in above equation 2, equation 2 becomes 23+600(3450)3>0\dfrac{2}{3} + \dfrac{{600}}{{{{\left( {^3\sqrt {450} } \right)}^3}}} > 0
Clearly it is positive hence it’s verified.
Note- The problem statement of this type is purely based upon the definition conceptuality of marginal function, average cost function and minimization of average cost function. The standard procedure as mentioned above leads to the answers in such types of problems.