Solve each problem. The supply and demand equations for a certain commodity are given. supply: p = √(0.1q + 9) - 2 and demand: p = √(25 - 0.1q).
Find the equilibrium price (in dollars).

Verified step by step guidance

1. Identify the equilibrium point where supply equals demand, meaning the price \(p\) from the supply equation equals the price \(p\) from the demand equation.
2. Set the supply equation equal to the demand equation: \(\sqrt{0.1q + 9} - 2 = \sqrt{25 - 0.1q}\).
3. Isolate one of the square root terms on one side to prepare for squaring both sides. For example, add 2 to both sides to get \(\sqrt{0.1q + 9} = \sqrt{25 - 0.1q} + 2\).
4. Square both sides of the equation to eliminate the square roots, remembering to apply the formula \((a + b)^2 = a^2 + 2ab + b^2\) on the right side.
5. After squaring, simplify the resulting equation and solve for \(q\). Once you find \(q\), substitute it back into either the supply or demand equation to find the equilibrium price \(p\).

Key Concepts

Equilibrium Price and Quantity

The equilibrium price is the price at which the quantity supplied equals the quantity demanded. This occurs where the supply and demand equations intersect, meaning their price values are equal for the same quantity. Finding this point involves setting the supply and demand price expressions equal and solving for the quantity.

Solving Equations Involving Square Roots

Both supply and demand equations contain square root expressions. To solve for the variable, you often need to isolate the square root term and then square both sides to eliminate the root. Care must be taken to check for extraneous solutions introduced by squaring.

Algebraic Manipulation and Substitution

Solving the equilibrium involves algebraic manipulation such as isolating variables, substituting expressions, and simplifying equations. This process helps to reduce the problem to a solvable form, often a quadratic or linear equation, enabling the determination of the equilibrium quantity and price.