Problem 32 - Algebra Form and Function

Question

Give the domain and range of the functions described. Let \(T=w(r)\) give the total number of minutes of radio advertising bought in a month by a small company as a function of the rate \(r\) in \$/minute. Rates vary depending on the time of day the ad runs, ranging from $$ 40\( to $$ 100\) per minute. The company's monthly radio advertising budget is $$ 2500$.

Step-by-Step Solution

Answer

Answer: The domain of the function is $$40 \le r \le 100$$, and the range is $$25 \le T(w(r)) \le 62.5$$.

Step 1: Determine the domain of the function

The domain of the function includes all possible rates per minute for radio advertising, which ranges from $$ 40\( to $$ 100\). Therefore, the domain of \(T(w(r))\) is $$ 40 \le r \le 100$.

Step 2: Determine the range of the function based on the company's budget constraint

The total cost for the advertising is given by the product of the rate and the number of minutes purchased. If the company has a budget of $$ 2500$, we can find the range of total advertising time by considering two scenarios: 1. Scenario 1: At the lowest possible rate (i.e. $$ 40/min$), the company will purchase the most minutes. In this case, the number of minutes purchased is: $$ T(w(r)) = \frac{2500}{40} \approx 62.5$$ 2. Scenario 2: At the highest possible rate (i.e. $$ 100/min$), the company will purchase the least minutes. In this case, the number of minutes purchased is: $$ T(w(r)) = \frac{2500}{100} = 25$$ The total minutes of radio advertising bought lies between 25 and 62.5 minutes. Therefore, the range of \(T(w(r))\) is \(25 \le T(w(r)) \le 62.5\).

Key Concepts

Radio Advertising Budget

The radio advertising budget is the total amount of money a company allocates for purchasing radio advertisement minutes within a month. In this exercise, the budget is a constraint that limits how many minutes can be bought based on the rate per minute.
With an advertising budget of $2500, the company must determine how to best allocate this spending to cover various rates, which range from $40 to $100 per minute depending on what time of day the advertisement runs.
The budget operates as a financial boundary that dictates the maximum potential cost of advertising. This establishes the essential constraints required for calculating the domain and range of the function in this problem.

Rate Per Minute

The rate per minute is the price charged for a single minute of radio advertising. This rate fluctuates based on the time of day the ad broadcast occurs, which impacts how many ad minutes the company can afford given their budget.

The rate per minute acts as a critical component in determining both the domain and range of the function. Here's why:

• The domain of the function is defined by this rate, specifically ranging from $40 to $100 per minute as stated in the problem.
• The range of advertising minutes purchased will depend on how low or high this rate is. A lower rate allows for more minutes within the same budget, while a higher rate limits the number of minutes.

Effectively understanding these rates is key to knowing how much advertising time the company can buy for its investment in different scenarios.

Function Constraints

Function constraints refer to the limitations or conditions that must be maintained within a function. For this exercise, the constraints are defined by the rate per minute and the company's radio advertising budget.

• The rates must not drop below $40 or exceed $100 per minute.
• The total spend must not surpass the $2500 budget.

These constraints are essential in solving for the domain and range of the function. The constraints illustrate real-world limits, ensuring the calculations maintain practical relevance, like how much ad time is financially feasible and at what rates. Adhering to these constraints in mathematical modeling helps ensure that the results are realistic and applicable to the company’s advertising goals.

Piecewise Functions

Piecewise functions are mathematical expressions defined by different sub-functions, each applied to a certain interval of the independent variable (in this case, the rate per minute). This problem doesn't explicitly introduce a piecewise function but lays the groundwork for understanding how different rates apply over specific intervals.

In advertising budget scenarios, piecewise functions can be useful to model situations where rates change during different times of the day or vary based on other conditions.

• For instance, different times might have different rates, thereby splitting into multiple function intervals
• This type of function is versatile for real-world applications, such as pricing strategies that vary by time slot.

Understanding piecewise functions is valuable for comprehending the complexity of varying rates in scenarios like radio advertising, where the budget needs to be effectively allocated based on differing time periods or conditions.