Determine the quantity 'x' that the supplier is willing to make available in the market: \[ p=-0.1 x^{2}-x+50 \] The quantity $x$ (in units of a hundred) that the supplier is willing to make available in the market is related to the unit price by the relation \[ p=0.1 x^{2}+6 x+10 \] consumer's surplus $\$$ producer's surplus \$

Step 1 ：The problem is asking for the quantity 'x' that the supplier is willing to make available in the market. This is a quadratic equation problem. The quantity 'x' can be found by setting the equation to zero and solving for 'x'.

Step 2 ：The quadratic formula can be used to solve for 'x'. The quadratic formula is given by: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] where 'a', 'b', and 'c' are the coefficients of the quadratic equation.

Step 3 ：In this case, 'a' is -0.1, 'b' is -1, and 'c' is 50. So, we substitute these values into the quadratic formula.

Step 4 ：Calculate the discriminant (D) using the formula \[D = b^2 - 4ac\]. Substituting the values, we get D = 21.0

Step 5 ：Substitute 'a', 'b', and 'D' into the quadratic formula to find the values of 'x'. The solutions to the equation are x1 = 17.9128784747792 and x2 = -27.9128784747792.

Step 6 ：However, since 'x' represents the quantity that the supplier is willing to make available in the market, it cannot be negative. Therefore, the only valid solution is x1 = 17.9128784747792.

Step 7 ：Final Answer: The quantity 'x' that the supplier is willing to make available in the market is \(\boxed{17.9128784747792}\).