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3.6 · Q2

Q.If the demand function for a commodity is p=25−x2p = 25 - x^2, find the consumers' surplus for p0=9p_0 = 9.

Puducherry CbseNCERTSubjective· 3mImportance★★★★★
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✓ Free question

With p0=9p_0=9 the demand curve p=25−x2p=25-x^2 gives x0=4x_0=4; the consumers' surplus is ∫04p dx−p0x0=1283≈₹42.67\int_0^{4}p\,dx-p_0x_0=\dfrac{128}{3}\approx\text{₹}42.67.

Consumers' surplus: CS=∫0x0p dx−p0x0\displaystyle CS=\int_0^{x_0}p\,dx-p_0x_0.

  1. Given p=25−x2p=25-x^2, p0=9p_0=9.
  2. Equilibrium quantity: 9=25−x2⇒x2=16⇒x0=49=25-x^2\Rightarrow x^2=16\Rightarrow x_0=4.
  3. ∫04(25−x2) dx=[25x−x33]04=100−643=2363\displaystyle\int_0^4(25-x^2)\,dx=\Big[25x-\tfrac{x^3}{3}\Big]_0^4=100-\tfrac{64}{3}=\tfrac{236}{3}.
  4. CS=2363−p0x0=2363−(9)(4)=2363−36=236−1083=1283CS=\tfrac{236}{3}-p_0x_0=\tfrac{236}{3}-(9)(4)=\tfrac{236}{3}-36=\tfrac{236-108}{3}=\tfrac{128}{3}.
✓Final answer

Consumers' surplus =1283≈₹42.67=\dfrac{128}{3}\approx\text{₹}42.67.

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