3.5 Real World Problems of Quadratic Equations and Inequalities

3.5 Real World Problems of Quadratic Equations and Inequalities

Problems that lead to quadratic equations in one variable are solved by:

1. Letting the unknown(s) be $x$ (or $y$, etc.).
2. Translating the problem into symbols and forming the equation or inequality from the given conditions.

Example 9

The length of a room is $3$ metres greater than its breadth. If the area is $180$ square metres, find the length and breadth.

Solution

Let breadth $= x$ metres and length $= (x + 3)$ metres. Then

Breadth cannot be negative, so $x = 12$. Length $= 15$ metres.

Hence breadth $= 12$ m and length $= 15$ m.

Example 10

A company manufactures laptops with weekly profit function (in thousands of dollars) $P(x) = -x^2 + 2x + 3$, where $x$ is the number of laptops produced (in hundreds). Find the range of production levels where the company makes at least \4000$ profit.

Solution

We need $P(x) \ge 4$:

This holds only when $(x - 1)^2 = 0$, i.e. $x = 1$.

The company makes exactly \4000$profit when$100$laptops are produced ($x = 1$). There is no production level where profit exceeds$$4000$.