Problem 50 - Algebra 2

Question

SHOPPING A certain store found that the number of customers that will attend a limited time sale can be modeled by \(N=125 \sqrt[3]{100 P t,}\) where \(N\) is the number of customers expected, \(P\) is the percent of the sale discount, and \(t\) is the number of hours the sale will last. Find the number of customers the store should expect for a sale that is 50\(\%\) off and will last four hours.

Step-by-Step Solution

Answer

The store should expect approximately 3393 customers.

Step 1: Identify the given values

We have a sale discount of 50%, which means \( P = 50 \), and the sale is planned to last for 4 hours, so \( t = 4 \).

Step 2: Substitute known values into the formula

The formula we need is \( N = 125 \sqrt[3]{100Pt} \). Substitute \( P = 50 \) and \( t = 4 \) into the formula: \( N = 125 \sqrt[3]{100 \times 50 \times 4} \).

Step 3: Calculate the product inside the cube root

First, calculate the product: \( 100 \times 50 \times 4 = 20000 \).

Step 4: Compute the cube root

Calculate the cube root of 20000: \( \sqrt[3]{20000} \approx 27.144 \).

Step 5: Calculate the expected number of customers

Finally, multiply by 125 to find \( N \): \( N = 125 \times 27.144 \approx 3393 \).

Key Concepts

Cube Root

A cube root refers to a special type of root found in mathematics. When we talk about the cube root of a number, it is the value that, when multiplied by itself twice (three times in total), gives the original number. In mathematical terms, the cube root of a number \( x \) is written as \( \sqrt[3]{x} \). Determining the cube root requires finding that value which, when multiplied three times, yields \( x \).
For instance, the cube root of \( 27 \) is \( 3 \) since \( 3 \times 3 \times 3 = 27 \).

• Cube roots can be negative or positive, unlike square roots.
• If you take the cube root of a negative number, the result is also negative.
• Understanding cube roots is crucial in modeling certain real-world scenarios, such as the one in our problem involving customer prediction during a sale.

Algebraic Expression

An algebraic expression in mathematics is a combination of numbers, variables, and arithmetic operations such as addition, subtraction, multiplication, and division. In our problem, the formula \( N = 125 \sqrt[3]{100Pt} \) serves as an algebraic expression. It models the number of customers \( N \) as a function of the percentage discount \( P \) and the time of the sale \( t \).

An important aspect of algebraic expressions is their ability to model real-world situations with variables.

• Variables like \( P \) and \( t \) allow the expression to change with different inputs, making it versatile for various scenarios.
• Algebra supports solving problems using symbols and formulas, providing a pathway to understanding how different quantities relate and affect each other.

Understanding algebraic expressions helps in recognizing patterns and relationships that can be represented mathematically.

Functions

Functions in mathematics describe a special kind of relationship between different quantities. They can be seen as machines that accept some input, do a specific operation on it, and then output a result. A function is usually expressed as \( f(x) \), where \( x \) is the input variable.

In the current exercise, the formula \( N = 125 \sqrt[3]{100Pt} \) can be viewed as a function because it defines a specific output (\( N \), number of expected customers) based on the inputs (\( P \) and \( t \), percent discount and time respectively).

• Functions ensure that each input value gives exactly one output value.
• This concept is essential for predicting and understanding outputs based on varying conditions, shown here by altering the discount and time of the sale.
• The use of functions enables problem solving in systematic and reliable ways.

Understanding functions allows us to model and predict outcomes in various contexts.

Problem Solving

Problem-solving involves the process of finding solutions to complex issues or challenges, which often requires identifying key variables, understanding the relationships between them, and using mathematical concepts or operations.

In the specific exercise given, we tackled the problem of determining how many customers a store can expect given a certain discount and sale duration. The process involved:

• Identifying the given data (percent discount \( P \) and time \( t \)).
• Substituting these values into an algebraic expression.
• Computing the necessary mathematical operations like finding the cube root and final multiplication.

This systematic approach is vital in problem-solving, making it possible to break down complex problems into manageable steps, ultimately leading to a concrete solution. Furthermore, practicing problem-solving skills enhances logical thinking and can be applied to a wide array of scenarios.