Problem 37 - Algebra 2

Question

The number of sports magazines sold can be estimated by \(n=\frac{3500 a^{2}}{a^{2}+100},\) where \(a\) is the amount of money spent on advertising in hundreds of dollars and \(n\) is the number of subscriptions sold. About how many subscriptions will be sold if \(\$ 1500\) is spent on advertising?

Step-by-Step Solution

Answer

Approximately 2423 subscriptions will be sold.

Step 1: Identify the Value of 'a'

The problem states that \(1500 is spent on advertising. Since 'a' is the amount of money spent in hundreds of dollars, we need to convert \)1500 to hundreds. Divide 1500 by 100 to get the value of 'a'. Thus, \( a = \frac{1500}{100} = 15 \).

Step 2: Substitute 'a' into the Formula

Now that we have \( a = 15 \), substitute it into the given formula for subscriptions: \( n = \frac{3500 \times 15^{2}}{15^{2} + 100} \).

Step 3: Calculate \( a^2 \)

Calculate \( 15^2 \) which is \( 225 \). So, \( a^2 = 225 \).

Step 4: Calculate the Numerator

Substitute \( a^2 = 225 \) into the numerator of the formula: \( 3500 \times 225 \). This results in \( 787500 \).

Step 5: Calculate the Denominator

Add \( 225 \) to \( 100 \) in the denominator: \( 225 + 100 = 325 \).

Step 6: Compute the Number of Subscriptions

Divide the result of the numerator by the denominator: \( n = \frac{787500}{325} \). Calculating this gives \( n = 2423.08 \). Since 'n' represents the number of subscriptions, round to the nearest whole number to get \( n \approx 2423 \).

Key Concepts

Advertising Budget

Advertising plays a crucial role in how well a product sells. In mathematical modeling, we often explore the relationship between advertising budget and sales. Here, the advertising budget is represented as \( a \), but it's important to note that \( a \) is in hundreds of dollars. So when you see \( a = 15 \), it means the budget is actually \( 1500 \) dollars. This step is all about scaling the real-life budget into manageable numbers for mathematical calculations. Understanding this scaling is essential:

• Convert dollars into hundreds by dividing by 100.
• Realize that the formula provides an illustration of how changing your budget affects sales.

By depicting the effects of an advertising budget under such a formula, businesses can adjust their spending to optimize outcomes based on predicted subscriptions or sales.

Maximization

Maximization is about getting the most out of something given constraints. In our scenario, it involves determining the highest possible number of subscriptions sold, given a certain advertising budget. This directly applies to finding the optimal \( a \) that maximizes \( n \).Some key aspects of maximization include:

• Understanding the relationship between budget \( a \) and subscriptions \( n \).
• Recognizing the quadratic nature of the formula where \( n \) grows as \( a \) increases, but at diminishing rates.

The formula used models diminishing returns, hinting that as you spend more, each additional dollar yields fewer new sales. Practically, this is a key insight for businesses in resource allocation.

Number of Subscriptions Sold

The number of subscriptions sold, denoted as \( n \), represents the primary outcome of interest when considering advertising efforts. Here, it is determined by a specific quadratic expression.To find \( n \):

• Substitute the scaled advertising budget into the formula \( n = \frac{3500a^2}{a^2+100} \).
• Calculate each component: \( a^2 \) for the numerator, and the sum for the denominator.
• Perform division to get \( n \), rounding to the nearest whole number for final count.

This approach provides realistic estimates of sales effects from different levels of advertising. It not only helps businesses in planning but also in forecasting market reach and business growth.