Question

find the producers surplus at a price level of \\( \bar{p} = \\$51 \\) for the price-supply equation below.\\( p = s(x) = 10 + 0.1x + 0.0003x^2 \\)\\( \dots \\)\\( \text{the producers surplus is } \\$\boxed{5577} \\).\\( \text{(round to the nearest integer as needed.)} \\)

Explanation:

Step1: Find the quantity \( x \) when \( p = 51 \)

Set \( S(x)=51 \), so \( 10 + 0.1x + 0.0003x^{2}=51 \).
Rearrange to \( 0.0003x^{2}+0.1x - 41 = 0 \).
Multiply by 10000 to eliminate decimals: \( 3x^{2}+1000x - 410000 = 0 \).
Use quadratic formula \( x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a} \), where \( a = 3 \), \( b = 1000 \), \( c=-410000 \).
Discriminant \( \Delta=1000^{2}-4\times3\times(-410000)=1000000 + 4920000 = 5920000 \).
\( \sqrt{\Delta}\approx2433.11 \).
\( x=\frac{-1000\pm2433.11}{6} \). Take positive root: \( x=\frac{1433.11}{6}\approx238.85 \), so \( x_0\approx239 \) (or keep more decimals for accuracy).

Step2: Calculate Producer Surplus (PS)

The formula for producer surplus is \( PS=\bar{p}x_0-\int_{0}^{x_0}S(x)dx \).
First, integrate \( S(x)=10 + 0.1x + 0.0003x^{2} \):
\( \int S(x)dx=\int(10 + 0.1x + 0.0003x^{2})dx=10x + 0.05x^{2}+0.0001x^{3}+C \).
Evaluate from 0 to \( x_0 \): \( \int_{0}^{x_0}S(x)dx=10x_0 + 0.05x_0^{2}+0.0001x_0^{3} \).
Now, \( \bar{p}x_0 = 51x_0 \).
So \( PS = 51x_0-(10x_0 + 0.05x_0^{2}+0.0001x_0^{3})=41x_0 - 0.05x_0^{2}-0.0001x_0^{3} \).
Substitute \( x_0\approx238.85 \) (more accurately, let's use \( x_0 \) from quadratic solution without rounding early):
From \( 0.0003x^{2}+0.1x - 41 = 0 \), \( x=\frac{-0.1+\sqrt{0.1^{2}+4\times0.0003\times41}}{2\times0.0003}=\frac{-0.1+\sqrt{0.01 + 0.0492}}{0.0006}=\frac{-0.1+\sqrt{0.0592}}{0.0006}=\frac{-0.1 + 0.2433}{0.0006}=\frac{0.1433}{0.0006}\approx238.83 \).
Now calculate \( \int_{0}^{238.83}S(x)dx=10(238.83)+0.05(238.83)^{2}+0.0001(238.83)^{3} \)
\( = 2388.3+0.05\times57040.77+0.0001\times13627000 \)
\( = 2388.3 + 2852.04+1362.7 \)
\( = 2388.3+4214.74 = 6603.04 \).
\( \bar{p}x_0 = 51\times238.83 = 12180.33 \).
\( PS = 12180.33 - 6603.04 = 5577.29\approx5577 \).

Answer:

\(\$5577\)