Problem 63 Solve each problem The total p... [FREE SOLUTION]

Chapter 10: Problem 63

Solve each problem The total profit in dollars on the sale of \(x\) Electronic Tummy Trimmers is given by the polynomial function \(P(x)=x^{3}-40 x^{2}+400 x .\) Find the profit when 10 are sold. How many must be sold to get a profit of 0 dollars?

Short Answer

Expert verified

When 10 units are sold, the profit is 1000 dollars. To get a profit of 0 dollars, 0 or 20 units must be sold.

Step by step solution

- Substitute 10 into the Profit Function

To find the profit when 10 Tummy Trimmers are sold, substitute 10 for x in the polynomial function. This means we calculate \[P(10) = 10^3 - 40 \times 10^2 + 400 \times 10.\]

- Simplify the Expression

Simplify the expression from Step 1: \[P(10) = 1000 - 4000 + 4000.\]

- Calculate the Profit for 10 Units

Combine the terms to find the profit when 10 units are sold: \[P(10) = 1000.\]

- Set Profit Equation to Zero

To find how many units must be sold to get a profit of 0 dollars, set the profit function equal to zero and solve for x: \[P(x) = x^3 - 40x^2 + 400x = 0.\]

- Factor the Polynomial

Factor the polynomial to simplify the equation: \[x(x^2 - 40x + 400) = 0.\]

- Solve for x

This gives us two factors to solve: \[x = 0\] and \[x^2 - 40x + 400 = 0.\]

- Solve the Quadratic Equation

To solve \[x^2 - 40x + 400 = 0,\] we can use the quadratic formula \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.\] Here, \[a = 1, b = -40,\] and \[c = 400.\]

- Apply the Quadratic Formula

Substitute the values into the quadratic formula: \[x = \frac{40 \pm \sqrt{(-40)^2 - 4 \cdot 1 \cdot 400}}{2 \cdot 1} = \frac{40 \pm \sqrt{1600 - 1600}}{2} = \frac{40 \pm 0}{2} = 20.\]

- List All Solutions

The solutions are \[x = 0\] and \[x = 20.\] So, the Tummy Trimmers must sell either 0 or 20 units to achieve a profit of 0 dollars.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Profit calculation

To calculate the profit from selling products, we often use a polynomial function like the one given in the exercise. A polynomial function is an algebraic expression that includes variables and coefficients. For our exercise, the profit function is given by \[ P(x) = x^3 - 40x^2 + 400x. \]Here,

\( x \) is the number of Electronic Tummy Trimmers sold, and
\( P(x) \) represents the total profit from selling \( x \) Tummy Trimmers.

To find the profit from selling a specific number of units, we simply substitute that number into the polynomial function.

Substituting values

Substituting values into a polynomial function is straightforward. For our problem, we need to find the profit when 10 Tummy Trimmers are sold. Here's what we do:

Take the given polynomial function: \[ P(x) = x^3 - 40x^2 + 400x \]
Substitute \( x = 10 \) into the function: \[ P(10) = 10^3 - 40 \times 10^2 + 400 \times 10 \]
Simplify the expression: \[ P(10) = 1000 - 4000 + 4000 \]
Combine the terms: \[ P(10) = 1000. \]

So, the profit from selling 10 Tummy Trimmers is 1000 dollars.

Factoring polynomials

Factoring helps break down complex polynomial expressions into simpler parts. Let's take our profit function: \[ P(x) = x^3 - 40x^2 + 400x \]To find when the profit is 0 dollars, set this equation to zero: \[ x^3 - 40x^2 + 400x = 0 \]Since each term contains \( x \), we can factor out \( x \): \[ x(x^2 - 40x + 400) = 0. \]This gives us two factors:

\( x = 0 \)
\( x^2 - 40x + 400 = 0. \)

To solve for \( x \, \) we'll address the quadratic \( x^2 - 40x + 400. \)

Quadratic formula

To solve the quadratic equation, we use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. \]First, identify the coefficients from our equation \( x^2 - 40x + 400 = 0 \):

\( a = 1, \)
\( b = -40, \)
\( c = 400. \)

Now substitute these values into the quadratic formula: \[ x = \frac{40 \pm \sqrt{(-40)^2 - 4 \cdot 1 \cdot 400}}{2 \cdot 1} \]This simplifies to: \[ x = \frac{40 \pm \sqrt{1600 - 1600}}{2} = \frac{40 \pm 0}{2} = 20 \]So, the solutions are \( x = 0 \) and \( x = 20. \) Therefore, either 0 or 20 Tummy Trimmers need to be sold for a profit of 0 dollars.

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