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Chapter 1: Problem 135

Profit \(\quad\) A small-appliance manufacturer finds that the profit \(P\) (in dollars) generated by producing \(x\) microwave ovens per week is given by the formula \(P=\frac{1}{10} x(300-x)\) provided that \(0 \leq x \leq 200 .\) How many ovens must be manufactured in a given week to generate a profit of \(\$ 1250 ?\)

Short Answer

Expert verified
Manufacture 50 ovens per week to generate a $1250 profit.

Step by step solution

01

Set up the Equation

We start with the profit formula given by the manufacturer: \( P=\frac{1}{10} x(300-x) \). We need to find the value of \( x \) when the profit \( P \) is \$1250. Substitute \( P = 1250 \) into the equation: \( 1250 = \frac{1}{10} x(300-x) \).
02

Eliminate the Fraction

To simplify the equation, multiply both sides by 10 to eliminate the fraction: \( 12500 = x(300-x) \).
03

Expand and Rearrange

Expand the equation: \( 12500 = 300x - x^2 \). To make it easier to solve, rearrange the equation into a standard quadratic form: \( x^2 - 300x + 12500 = 0 \).
04

Use the Quadratic Formula

To solve for \( x \), use the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Identify coefficients as \( a = 1 \), \( b = -300 \), and \( c = 12500 \). Substitute these into the formula.
05

Calculate the Discriminant

Calculate the discriminant: \( b^2 - 4ac = (-300)^2 - 4 \cdot 1 \cdot 12500 \). This results in \( 90000 - 50000 = 40000 \).
06

Solve for x

Substitute the discriminant back into the quadratic formula: \( x = \frac{300 \pm \sqrt{40000}}{2} \). Calculate \( \sqrt{40000} \), which is 200. Then, solve for the two values of \( x \): \( x = \frac{300 + 200}{2} \) and \( x = \frac{300 - 200}{2} \).
07

Determine Valid Solutions

Solve for \( x \): \( x = 250 \) and \( x = 50 \). However, we need to ensure \( x \) is within the constraint \( 0 \leq x \leq 200 \). Thus, only \( x = 50 \) is valid.

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Key Concepts

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

Profit Maximization
Profit maximization is a core goal for businesses, aiming to generate the highest profit possible given certain constraints. In this exercise, the small-appliance manufacturer wants to know how many microwave ovens should be made to reach a specific profit of $1250. Profit maximization involves balancing production, costs, and sales to achieve optimal results.
  • The formula given helps determine how different levels of production will affect profits.
  • To maximize profit, it is crucial to understand how changes in the number of units produced impact overall earnings.
  • By analyzing the profit equation, you can identify the production level that will yield the desired profit.
In practice, managers use such models to forecast and plan production strategies that align with their financial goals.
Quadratic Formula
The quadratic formula is a critical tool for solving quadratic equations, which take the form: \[ ax^2 + bx + c = 0 \], where \(a\), \(b\), and \(c\) are constants. This formula provides solutions for the variable \(x\) and is given by:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
  • This formula is particularly useful when factoring is challenging or coefficients are complex.
  • It helps in finding out not just one but potentially two possible values for \(x\), as quadratic equations can have up to two solutions.
  • In our exercise, we used the quadratic formula to determine how many microwave ovens need to be produced to meet the profit goal.
Using the quadratic formula ensures that you are accurately solving for \(x\) when dealing with quadratic relationships in profit or any other mathematical model.
Discriminant
The discriminant is an important component of the quadratic formula and plays a critical role in determining the nature of the solutions of a quadratic equation. It is derived from the part of the quadratic formula under the square root: \[ b^2 - 4ac \]
  • The value of the discriminant tells us whether the equation will have one, two, or no real solutions.
  • If the discriminant is positive, as in our given problem where it calculates to 40000, there are two distinct real solutions.
  • A zero discriminant means one real solution exists, while a negative discriminant implies no real solutions in the set of real numbers.
Understanding the discriminant is crucial as it provides insight into the nature of the roots of the quadratic equation and aids in interpreting the potential outcomes before solving for \(x\).
Mathematical Modelling
Mathematical modeling is a powerful technique that involves using mathematics to represent real-world situations. In the context of this exercise, mathematical modelling allows us to formulate an equation to predict profits based on production levels.
  • This approach provides a systematic way to analyze and solve problems like the given profit equation.
  • By representing the problem mathematically, you can make informed decisions and predictions, which are crucial in business environments.
  • It involves identifying relationships between different elements, such as production and profit, and using equations to find optimal solutions.
Through mathematical modelling, complex real-life problems are translated into manageable mathematical equations, which can be solved to gain valuable insights into the scenario.

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Most popular questions from this chapter

Rationalize Put each fractional expression into standard form by rationalizing the denominator. (a) \(\frac{12}{\sqrt{3}}\) (b) \(\sqrt{\frac{12}{5}}\) (c) \(\frac{8}{\sqrt[3]{5^{2}}}\)

Plot the points \(M(6,8)\) and \(A(2,3)\) on a coordinate plane. If \(M\) is the midpoint of the line segment \(A B,\) find the coordinates of \(B\) Write a brief description of the steps you took to find \(B\) and your reasons for taking them.

Area of a Region Find the area of the region that lies outside the circle \(x^{2}+y^{2}=4\) but inside the circle $$ x^{2}+y^{2}-4 y-12=0 $$

Show that the equation represents a circle, and find the center and radius of the circle. $$2 x^{2}+2 y^{2}-3 x=0$$

Radicals Simplify the expression, and eliminate any negative exponents(s). Assume that all letters denote positive numbers. (a) \(\sqrt[5]{x^{3} y^{2}} \sqrt[19]{x^{4} y^{16}}\) (b) \(\frac{\sqrt[3]{8 x^{2}}}{\sqrt{x}}\)
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