Find the producers' surplus at a price level of \( \bar{p}=\$ 64 \) for the price-supply equation below, \( p=S(x)=10+0.1 x+0.0003 x^{2} \) The producers' surplus is 3 , (Round to the nearest integer as needed.)

To find the producers' surplus at a price level of \( \bar{p} = 64 \) for the given price-supply equation \( p = S(x) = 10 + 0.1x + 0.0003x^2 \), we need to follow these steps: 1. **Find the quantity supplied \( x \) at the price level \( \bar{p} \)**: We set the supply equation equal to the price level: \[ 64 = 10 + 0.1x + 0.0003x^2 \] Rearranging gives: \[ 0.0003x^2 + 0.1x + 10 - 64 = 0 \] \[ 0.0003x^2 + 0.1x - 54 = 0 \] 2. **Solve the quadratic equation**: We can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 0.0003 \), \( b = 0.1 \), and \( c = -54 \). First, calculate the discriminant: \[ b^2 - 4ac = (0.1)^2 - 4(0.0003)(-54) = 0.01 + 0.0648 = 0.0748 \] Now, apply the quadratic formula: \[ x = \frac{-0.1 \pm \sqrt{0.0748}}{2 \cdot 0.0003} \] \[ x = \frac{-0.1 \pm 0.2735}{0.0006} \] Calculating the two possible values for \( x \): \[ x_1 = \frac{-0.1 + 0.2735}{0.0006} \approx \frac{0.1735}{0.0006} \approx 289.17 \] \[ x_2 = \frac{-0.1 - 0.2735}{0.0006} \text{ (not relevant since it gives a negative quantity)} \] Thus, we take \( x \approx 289 \). 3. **Calculate the producers' surplus**: The producers' surplus is given by the area above the supply curve and below the price level up to the quantity supplied. It can be calculated as: \[ \text{Producers' Surplus} = \bar{p} \cdot x - \int_0^x S(t) \, dt \] First, calculate \( \bar{p} \cdot x \): \[ \bar{p} \cdot x = 64 \cdot 289 \approx 18496 \] Next, we need to compute the integral \( \int_0^{289} S(t) \, dt \): \[ S(t) = 10 + 0.1t + 0.0003t^2 \] The integral is: \[ \int S(t) \, dt = \int (10 + 0.1t + 0.0003t^2) \, dt = 10t + 0.05t^2 + 0.0001t^3 + C \] Evaluating from 0 to 289: \[ \int_0^{289} S(t) \, dt = \left[ 10t + 0.05t^2 + 0.0001t^3 \right]_0^{289} \] \[ = 10(289) + 0.05(289^2) + 0.0001(289^3) \] Calculate each term: \[ 10(289) = 2890 \] \[ 0.05(289^2) = 0.05(83521) = 4176.05 \] \[ 0.0001(289^3) = 0.0001(289^3) = 0.0001(24137569) = 2413.7569 \] Now sum these: \[ \int_0^{289} S(t) \, dt \approx 2890 + 4176.05 + 2413.7569 \approx 9480.8069 \] 4. **Calculate the producers' surplus**: \[ \text{Producers' Surplus} = 18496 - 9480.8069 \approx 9015.1931 \] Rounding to the nearest integer, the producers' surplus is approximately: \[ \boxed{9015} \]