Problem 15 - Algebra 2 and Trigonometry

Question

Mrs. Cucci makes candy that she sells for \(\$ 8.50\) a pound. When she ships the candy to out- of town customers, she adds a flat shipping charge of \(\$ 2.00\) plus an additional \(\$ 0.50\) per pound. a. Find \(c(x),\) the cost of \(x\) pounds of candy. b. Find \(s(x),\) the cost of shipping \(x\) pounds of candy. c. Find \(t(x),\) the total bill for \(x\) pounds of candy that has been shipped, if \(t(x)=c(x)+s(x)\) d. Find \(t(5),\) the total bill for five pounds of candy that is shipped.

Step-by-Step Solution

Answer

The total cost for five pounds of candy that is shipped is \$47.

Step 1: Define the Cost Function for Candy

To find the cost of \(x\) pounds of candy, determine the cost per pound and multiply by \(x\). The candy costs \$8.50 per pound, so the cost function is: \[ c(x) = 8.50x \]

Step 2: Define the Cost Function for Shipping

Shipping has a flat charge plus a per-pound rate. Mrs. Cucci charges a flat rate of \\(2.00 plus \\)0.50 per pound. Thus, the shipping cost function is: \[ s(x) = 2 + 0.50x \]

Step 3: Calculate the Total Cost Function

The total cost function of purchasing and shipping \(x\) pounds of candy is the sum of the cost of the candy and the shipping cost. Hence, we have: \[ t(x) = c(x) + s(x) = 8.50x + (2 + 0.50x) \] Simplifying gives: \[ t(x) = 9x + 2 \]

Step 4: Find the Total Cost for 5 Pounds

Substitute \(x = 5\) into the total cost function to find \(t(5)\).\[ t(5) = 9(5) + 2 = 45 + 2 = 47 \]

Key Concepts

Linear Functions

Linear functions are a key concept in algebra. They describe relationships where a change in one variable results in a proportional change in another. In the context of our problem, these functions help model costs. A linear function is generally expressed in the format:

• \( y = mx + b \)

In this formula, \( y \) represents the output or dependent variable, \( m \) is the slope that shows the rate of change, \( x \) is the input or independent variable, and \( b \) is the y-intercept, which represents the starting value.
Let's break it down with Mrs. Cucci's candy sale. The candy cost function, \( c(x) = 8.50x \), is a linear function. Here, the slope \( m \) is \( 8.50 \), meaning for each extra pound of candy, the cost increases by \( 8.50 \) dollars. Notice there's no \( b \) term because there's no additional initial cost in the candy alone.
Understanding these linear relationships allows us to predict and calculate costs effectively, whether it is calculating the straightforward cost of candy or combining it with shipping costs.

Word Problems in Algebra

Word problems often turn algebraic concepts into real-life situations, making algebra more relatable. Mrs. Cucci’s candy sales offer a scenario where we apply algebraic functions to solve practical problems.
The critical part of solving word problems is translating the problem into equations:

• Identify what is being asked – in this case, the cost of candy, shipping, and the total cost.
• Determine what information is given – costs per pound, shipping fees, etc.
• Create equations that represent the relationships – we formed \( c(x), s(x), \text{and} t(x) \).

By organizing the information clearly, and converting the words into symbols, we simplify a potentially complex problem. This approach breaks down a real-world issue into manageable mathematical expressions we can work with.
Practicing with word problems enhances problem-solving skills. It teaches us not just to solve equations, but also to navigate real-life contexts using math.

Practical Applications of Algebra

Algebra is powerful in solving practical problems. It helps us understand and model real-life scenarios, like Mrs. Cucci’s candy business, using mathematical equations.
The candy cost function \( c(x) \) and the shipping cost function \( s(x) \) are practical tools. They don't just calculate costs; they provide a formula you can use to find costs quickly, no matter the amount of candy.
Consider the overall bill, \( t(x) = 9x + 2 \). This tells us if you know the weight, you can instantly determine the total price. For instance, for 5 pounds of candy, the cost is simply \( 9(5) + 2 = 47 \) dollars.

• Predict future prices with changes in quantity.
• Understand how pricing strategies might change with different shipping rates or sale prices.

In essence, algebra isn't just about numbers. It's a way to organize and analyze information that allows individuals to make informed decisions in everyday life, like pricing strategies for a business.