Problem 30 - Algebra 2

Question

PROFIT Kara is planning to set up a booth at a local festival to sell her paintings. She determines that the amount of profit she will make is determined by the function \(P(x)=11 x-100\) , where \(x\) is the number of paintings she sells. How much profit will Kara make if she sells 35 of her paintings?

Step-by-Step Solution

Answer

Kara will make a profit of $285 if she sells 35 paintings.

Step 1: Identify the Function

The profit function is given as \(P(x) = 11x - 100\), where \(x\) is the number of paintings sold.

Step 2: Substitute the Number of Paintings

Kara plans to sell 35 paintings, so substitute \(x = 35\) into the profit function: \(P(35) = 11(35) - 100\).

Step 3: Calculate the Product

First, compute the product of 11 and 35: \(11 \times 35 = 385\).

Step 4: Apply the Expression

Substitute the product back into the profit function: \(P(35) = 385 - 100\).

Step 5: Final Calculation

Subtract 100 from 385 to find the profit: \(385 - 100 = 285\).

Key Concepts

Profit Calculation

Understanding profit calculation is essential for anyone involved in business or sales. Profit is the monetary gain obtained when revenues exceed the costs associated with producing or purchasing the product. In Kara's scenario, she's selling paintings and wants to determine how much money she will earn after covering all expenses related to her stall. The profit function given, \(P(x) = 11x - 100\), is a linear function that helps calculate profit based on the number of paintings sold, \(x\).

• The expression \(11x\) represents the money earned from selling \(x\) paintings, where each painting is sold for $11.
• The term \(-100\) is the fixed cost associated with her set-up at the festival.

Effectively, Kara’s profit depends linearly on the number of paintings sold minus the cost incurred. By evaluating this function for different values of \(x\), she can estimate her profits at varying sales levels.

Substitution in Functions

Substitution in functions is a straightforward yet powerful tool in algebra to find specific values of the function. It involves replacing the variable in a function with a given number to calculate the function's value. In Kara's case, she wants to determine her profit when selling a particular number of paintings. To predict the exact profit:

• Start with the function \(P(x) = 11x - 100\).
• Substitute \(x\) with \(35\) because Kara plans to sell 35 paintings.
• This gives the equation \(P(35) = 11(35) - 100\).

With substitution, you quickly transform a general equation into a specific calculation, making it easier to find the desired result or answer.

Step-by-Step Problem Solving

Approaching problems with a step-by-step method simplifies complex tasks into manageable chunks. This approach is particularly helpful in math to avoid mistakes and ensure clarity. Let's breakdown the problem-solving process demonstrated in Kara's exercise:

• **Step 1:** Identify the function that needs evaluation. In this case, it is \(P(x) = 11x - 100\).
• **Step 2:** Substitute the known value into the function, which is \(x = 35\).
• **Step 3:** Compute any arithmetic within the function, like multiplication or addition.
• **Step 4:** Apply results back into the original function and solve for the final answer.

By following these structured steps, Kara successfully calculates her profit without skipping critical mathematical processes.

Basic Arithmetic Operations

Basic arithmetic operations such as addition, subtraction, multiplication, and division form the foundation of mathematical problem solving. These operations are widely used in everyday scenarios like budgeting, shopping, and even in Kara’s profit calculation. Here’s how they apply:

• **Multiplication:** Kara multiplies 11 by 35, as represented by \(11 \times 35 = 385\), to calculate the total earnings from sales.
• **Subtraction:** After finding the product, Kara subtracts the fixed cost of \(-100\) from the total earnings to determine the actual profit: \(385 - 100 = 285\).

These operations are the building blocks that helped Kara calculate exact results effortlessly. Learning and practicing these operations enhances speed and efficiency in tackling everyday math problems.