Problem 75 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

The quantity demanded \(x\) (measured in units of a thousand) of the Sentinel smoke alarm/week is related to its unit price \(p\) (in dollars) by the equation $$ p=\frac{30}{0.02 x^{2}+1} \quad(0 \leq x \leq 10) $$ If the unit price is set at \(\$ 10\), what is the quantity demanded?

Step-by-Step Solution

Answer

When the unit price is set at $10, the quantity demanded is 10 thousand units of Sentinel smoke alarm every week.

Step 1: Write down the original equation

The equation relating the unit price \(p\) and quantity demanded \(x\) is given as: $$ p = \frac{30}{0.02x^2+1} $$

Step 2: Substitute the unit price value

We are given that the unit price \(p\) is set at $10. Substitute this value into the equation: $$ 10 = \frac{30}{0.02x^2+1} $$

Step 3: Solve for x

Now we need to solve the equation for the quantity demanded \(x\). First, multiply both sides by \((0.02x^2 + 1)\) to get rid of the fraction: $$ 10(0.02x^2 + 1) = 30 $$ Now, distribute the \(10\) on the left side of the equation: $$ 0.2x^2 + 10 = 30 $$ Next, subtract \(30\) from both sides to set the equation equal to zero: $$ 0.2x^2 - 20 = 0 $$ Now, divide both sides by \(0.2\) to make the coefficient of \(x^2\) equal to \(1\): $$ x^2 - 100 = 0 $$ Finally, solve for \(x\) by taking the square root of both sides of the equation: $$ x = \pm \sqrt{100} $$ Since \(0 \leq x \leq 10\), only the positive square root is valid: $$ x = \sqrt{100} = 10 $$

Step 4: Conclusion

When the unit price is set at $10, the quantity demanded is 10 thousand units of Sentinel smoke alarm every week.

Key Concepts

Quantity Demanded

In economics, the term 'quantity demanded' refers to the total amount of goods or services that consumers are willing to purchase at a given price during a specific time period. It's a fundamental concept that illustrates the consumer's desire to buy certain quantities of a commodity based on price levels.

For instance, if the price of a product increases, the quantity demanded typically decreases because consumers may feel that the product is less affordable or not worth the higher price. Conversely, if the price drops, consumers might purchase more, assuming the product offers greater value for their money. In our exercise, we're dealing with a mathematical demand function that correlates unit price and quantity demanded in a directly inversible manner. This relationship is crafted into a formula where a specific unit price would yield a tangible quantity demanded.

Unit Price Equation

A 'unit price equation' in mathematics defines the relationship between the price of a single unit of a product and other variables, commonly the quantity of the product demanded. It expresses how the product price will fluctuate with changes in consumer demand or other market conditions.

The equation provided in the exercise,
\[ p=\frac{30}{0.02x^2+1} \],
exemplifies this type of formula. In this specific case, the unit price 'p' changes inversely with a quadratic function of the quantity demanded 'x'. Such equations are pivotal in understanding how pricing can be optimized for revenue generation while ensuring market demands are met.

Solving Quadratic Equations

Solving quadratic equations is a fundamental skill in mathematics that involves finding the values of an unknown variable that make the equation true. These equations are recognizable by the highest power of the variable being squared (i.e., raised to the power of 2).

The standard form of a quadratic equation is \( ax^2+bx+c=0 \). The solution process can involve factoring, completing the square, or using the quadratic formula. In our example, we simplified the quadratic equation through a series of algebraic steps to isolate the variable, and then extracted the square root to find the value of 'x'.

In real-world contexts, understanding how to manipulate and solve these equations allows us to analyze and make predictions in diverse fields such as physics, engineering, economics, and biology.