Question

(1) If the total cost of a firm is c(x) = 1.5x^3 - 5x^2 + 20x + 5 . Find the marginal cost (in dollars) when 25 units are produced and sold. (2) The number q of roller blades a firm is willing to sell per week at a price of $p is given by q = 60 square root of p + 25 + 30 for 20 < p < 100 . (i) Find dq/dp . (ii) Find the amount supplied when the price is $56. (ii) Find the instantaneous rate of change of supply with respect to price when the price is $56. 

Answer 1

(1) Given cost of firm is $C(x)=1.5 x^{3}-5 x^{2}+20 x+5$

 

$\begin{aligned} \text { Marginal cost }=M . C &=\frac{d C}{d x} \\ \Rightarrow M . C . &=\frac{d}{d x}\left(1.5 x^{3}-5 x^{2}+20 x+5\right) \\ &=3 \times 1.5 x^{2}-10 x+20 \\ &=4.5 x^{2}-10 x+20 \\ \text { Now M.C. } \mid x=25 &=4.5(25)^{2}-10 \times 25+20 \\ &=4.5 \times 625-250+20 \\ &=2812.5-250+20 \\ &=\ 2582.5 \end{aligned}$

(2) Given supply - price function

 

$q=60 \sqrt{p+25}+300 \quad \text { for } \quad 20 \leqslant p \leqslant 100\\ i) \frac{d q}{d p}=60 \cdot \frac{1}{2}(p+25)^{\frac{1}{2}-1}\\ =\frac{30}{\sqrt{p+25}}\\$

 

ii) Amount of Supply when price is $ 56

 

$\begin{aligned} q &=60 \sqrt{56+25}+300 \\ &=60 \sqrt{81}+300 \\ &=60 \times 9+300 \\ &=540+300 \\ &=840 \end{aligned}$

 

iii) Instantaneous rate of change of supply with respect to price when price is $ 56

 

$=\left.\frac{d q}{d p}\right|_{p=56}=\frac{30}{\sqrt{56+25}}=\frac{30}{9}=\frac{10}{3}=3.33$