p = 6.55

UpStudy Free Solution:

To determine the prices \(p\) that yield a revenue of $8140, given the demand equation \(p = 106 - 0.08x\) and the revenue equation \(R = xp\), we need to solve for \(p\) when \(R = 8140\).

### Step-by-Step Solution:

1. Revenue Equation:

\[R = xp\]

2. Demand Equation:

\[p = 106 - 0.08x\]

3. Substitute the demand equation into the revenue equation:

\[R = x( 106 - 0.08x) \]

Given \(R = 8140\):

\[8140 = x( 106 - 0.08x) \]

4. Rearrange to form a quadratic equation:

\[8140 = 106x - 0.08x^ 2\]

\[0.08x^ 2 - 106x + 8140 = 0\]

5. Solve the quadratic equation using the quadratic formula \(x = \frac { - b \pm \sqrt { b^ 2 - 4ac} } { 2a} \):

\[a = 0.08, \quad b = - 106, \quad c = 8140\]

\[x = \frac { - ( - 106) \pm \sqrt { ( - 106) ^ 2 - 4 \cdot 0.08 \cdot 8140} } { 2 \cdot 0.08} \]

\[x = \frac { 106 \pm \sqrt { 11236 - 2604.8} } { 0.16} \]

\[x = \frac { 106 \pm \sqrt { 8631.2} } { 0.16} \]

Calculate the two solutions for \(x\):

\[x_ 1\approx 1243.15\]

\[x_ 2\approx 81.85\]

6. Find the corresponding prices \(p\) for each \(x\) using the demand equation:

\[p_ 1 = 106 - 0.08 \times 1243.15 \approx 106 - 99.45 \approx 6.55\]

So, the prices \(p\) that yield a revenue of $8140 are approximately:

- Lowest such price: \(\approx 6.55\) dollars

Supplemental Knowledge:

When solving quadratic equations, the quadratic formula is a powerful tool. It helps find the roots (solutions) of the equation by using the coefficients of the quadratic equatio \(ax^ 2 + bx + c = 0\). In many real-world applications, such as revenue optimization, these roots can represent critical points where the objective function reaches specific values.

Need help with complex quadratic equations or other challenging homework problems? UpStudy is here for you! As the world's top multidisciplinary solving and learning platform, UpStudy offers instant, accurate, and step-by-step solutions across various subjects including mathematics, chemistry, physics, and biology.