Given demand function
P = 35 - 2x - x²
for x = 3
P = 35 - 2(3) - 3²
= 35 - 6 - 9
P = 20
Therefore,
CS = (Area of the curve below the demand curve from 0 to 3) - Area of the rectangle (20 x 3 = 60)
\(CS=\int_0^3(35-2x-x^2)dx=(20\times 3)\)
\(=\left[35x-2{x^2\over 2}-{x^3\over 3}\right]_0^3-60\)
\(=35(3)-2\left ( \frac{3^{2}}{2} \right )-\frac{3^{3}}{3}-60\)
= 105 - 9 - 9 - 60
= 27 Units

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For equilibrium P_d = P_s
\(1600-x^{2} =2 x^{2}+400  \)
\(1600-400 =2 x^{2}+x^{2}  \)
\(3 x^{2} =1200  \)
\(x^{2} =400  \)
\(x =\pm 20  \)
\(P_{\mathrm{d}} =1600-(20)^{2} \)
=1600-400 
=1200 
\(\mathrm{P}_{\mathrm{s}} =2(20)^{2}+400 \)
= 2(400) + 400 
= 800 + 400 
= 1200
Consumer's surplus:
\(\mathrm{C}_{\mathrm{s}}=\int_{0}^{20}\left(1600-x^{2}\right) \mathrm{d} x-(20 \times 1200) \)
\(=\left[1600 x-\frac{x^{3}}{3}\right]_{0}^{20}-(24000)=\left[1600(20)-\frac{20^{3}}{3}\right]-24000 \)
\(=32000-\frac{8000}{3}-24000=5,333.34\)
Producer's surplus:
\(\mathrm{P}_{\mathrm{s}} =(20 \times 1200)-\int_{0}^{20}\left(2 x^{2} +400\right) \mathrm{d} x  \)
\(=24000-\left[\frac{2 x^{3}}{3}+400 x\right]_{0}^{20}   \)
\(=24000-\left[\frac{2(20)^{3}}{3}+400(20)\right]=24000-\left[\frac{2 \times 8000}{3}+8000\right]\)
= 24000-5333.33 - 8000 = 10,666.67

i) zero
\(\mathrm{P} =60-0.2 \mathrm{Q}  \)
\(0 =60-0.2 \mathrm{Q}  \)
\(0.2 \mathrm{Q} =60  \)
\(\mathrm{Q} =\frac{60}{0.2}=\frac{600}{2}=300\)
ii) Rs. 20
\(\mathrm{P} =60-0.2 \mathrm{Q}  \)
\(20 =60-0.2 \mathrm{Q}  \)
\(0.2 \mathrm{Q} =60-20=40 ; \quad \mathrm{Q}=\frac{40}{0.2}=\frac{400}{2}=200\)
iii) Rs. 40
\(\mathrm{P} =60-0.2 \mathrm{Q}  \)
\(40 =60-0.2 \mathrm{Q}  \)
\(0.2 \mathrm{Q} =60-40  \)
\(\mathrm{Q} =\frac{20}{0.2}=\frac{200}{2}=100\)

a) Q_t = f(Pt, Y, A, N, Pc) it means the quantity of milk demand depends on the price of fresh cow milk, the average income, advertising expenditure on processed packet milk, the price of that packet milk and the populations. These are the factors which influence the demand for fresh cow milk.
b) The independent variables are price of fresh cow milk, average household income, advertising expenditure on processed packet milk, population, price of processed packet milk.
c) Q_d = a + bP + cY + dA + eN + f Pc
a shows the value of Qd when the variables P, Y, A, N, Pc are all simultaneously =0. b, c, d, c, f are called slope parameter. They show the effect on quantity demanded of changing one of the variables P, Y, A, N or Pc while holding the rest of these variables constant.

Variable	Relation to quantity demanded
P	inverse when price of fresh milk increases, demand decreases.
Y	direct when income increases the demand for fresh milk increases.
A	direct when advertising expenditure on processed milk increases the demand for fresh milk increases.
N	direct Price of packet milk increases the demand for fresh milk increases
Pc	direct Population increases the demand for fresh milk increases.