The weekly demand and supply functions for Sportsman 5 × 7 tents are given by
p = −0.1x2 − x + 50
p = 0.1x2 + 2x + 30
respectively, where p is measured in dollars and x is measured in units of a hundred. Find the equilibrium quantity.

To find the equilibrium quantity, we need to determine when the demand and supply are equal. The equilibrium occurs when the demand function equals the supply function. In other words, we set the two equations equal to each other:

\[
-0.1x^2 - x + 50 = 0.1x^2 + 2x + 30
\]

Now let's solve this equation step by step.

### Step 1: Combine like terms
Move all terms to one side of the equation:

\[
-0.1x^2 - x + 50 - 0.1x^2 - 2x - 30 = 0
\]

Simplify:

\[
-0.2x^2 - 3x + 20 = 0
\]

### Step 2: Solve the quadratic equation
We now have the quadratic equation:

\[
-0.2x^2 - 3x + 20 = 0
\]

Multiply the entire equation by \(-5\) to eliminate the decimals:

\[
x^2 + 15x - 100 = 0
\]

### Step 3: Use the quadratic formula
The quadratic formula is given by:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

For the equation \(x^2 + 15x - 100 = 0\), the coefficients are \(a = 1\), \(b = 15\), and \(c = -100\). Substituting these values into the quadratic formula:

\[
x = \frac{-15 \pm \sqrt{15^2 - 4(1)(-100)}}{2(1)}
\]
\[
x = \frac{-15 \pm \sqrt{225 + 400}}{2}
\]
\[
x = \frac{-15 \pm \sqrt{625}}{2}
\]
\[
x = \frac{-15 \pm 25}{2}
\]

### Step 4: Calculate the two solutions
Now, solve for the two possible values of \(x\):

1. \(x = \frac{-15 + 25}{2} = \frac{10}{2} = 5\)
2. \(x = \frac{-15 - 25}{2} = \frac{-40}{2} = -20\)

### Step 5: Interpret the results
Since \(x\) represents the number of units in hundreds, a negative value doesn't make sense in this context. Therefore, the equilibrium quantity is:

\[
x = 5
\]

Thus, the equilibrium quantity is \(5 \times 100 = 500\) units.

Would you like further details or have any questions?

Here are 5 related questions:
1. How do we find the equilibrium price after finding the equilibrium quantity?
2. What if the demand and supply functions were linear? How would the solution process change?
3. How does the quadratic nature of the functions affect the demand and supply curves?
4. What happens if the two functions never intersect? How would that affect the equilibrium?
5. Can the quadratic formula always be used to solve supply and demand problems?

**Tip**: Always check the units of measurement in word problems to ensure the final answer is interpreted correctly!

The weekly demand and supply functions for Sportsman 5 × 7 tents are given by p = −0.1x^2 − x + 50 and p = 0.1x^2 + 2x + 30 respectively, where p is measured in dollars and x is measured in units of a hundred. Find the equilibrium quantity.

Discover how to find the equilibrium quantity for Sportsman 5x7 tents by solving quadratic demand and supply functions. This step-by-step solution demonstrates the use of algebraic techniques, including solving quadratic equations, to determine equilibrium quantity. Perfect for high school students in Grades 9-12.

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