Hikers' World Foods Sells Raisin-nut Mix In Bulk To Stores. The Dollar Amount Per Pound, $P(x$\], For A Store To Purchase $x$ Pounds Of Raisin-nut Mix From Hikers' World Is Given By The Function Below.$\[ P(x) = 3.50 + 0.90^x

Feb 26, 2025 by ADMIN 226 views

Hikers' World Foods: A Mathematical Analysis of Raisin-Nut Mix Pricing

Hikers' World Foods is a renowned supplier of high-quality snack mixes, including their popular raisin-nut mix. The company caters to a wide range of customers, from small convenience stores to large retail chains. In this article, we will delve into the mathematical analysis of the pricing function used by Hikers' World Foods to determine the cost per pound of raisin-nut mix for a store to purchase a certain quantity.

The pricing function, denoted as $P(x)$ , represents the dollar amount per pound of raisin-nut mix for a store to purchase $x$ pounds of the product. The function is given by:

${ P(x) = 3.50 + 0.90^x }$

where $x$ is the number of pounds purchased.

Understanding the Pricing Function

To gain a deeper understanding of the pricing function, let's break it down into its components. The first term, $3.50$ , represents the fixed cost per pound of raisin-nut mix. This is the minimum amount that a store must pay for a pound of the product, regardless of the quantity purchased.

The second term, $0.90^x$ , represents the variable cost per pound of raisin-nut mix. This term takes into account the discount that Hikers' World Foods offers to stores that purchase larger quantities of the product. The discount is calculated as a percentage of the fixed cost, with a base value of $0.90$ .

To visualize the pricing function, we can create a graph of $P(x)$ against $x$ . This will allow us to see the relationship between the cost per pound of raisin-nut mix and the quantity purchased.

import numpy as np
import matplotlib.pyplot as plt

# Define the pricing function
def P(x):
    return 3.50 + 0.90**x

# Generate a range of x values
x = np.linspace(0, 10, 100)

# Calculate the corresponding P(x) values
y = P(x)

# Create a plot of P(x) against x
plt.plot(x, y)
plt.xlabel('Quantity (pounds)')
plt.ylabel('Cost per pound ($)')
plt.title('Pricing Function for Raisin-Nut Mix')
plt.show()

Key Features of the Pricing Function

From the graph, we can observe several key features of the pricing function:

The cost per pound of raisin-nut mix decreases as the quantity purchased increases.
The rate of decrease in cost per pound is not constant, but rather decreases as the quantity purchased increases.
The cost per pound of raisin-nut mix approaches a minimum value of $0.50$ as the quantity purchased increases without bound.

To gain a deeper understanding of the pricing function, let's perform some mathematical analysis.

Domain and Range: The domain of the pricing function is all non-negative real numbers, $x \geq 0$ . The range of the pricing function is all real numbers greater than or equal to $3.50$ , $P(x) \geq 3.50$ .
Continuity: The pricing function is continuous for all $x \geq 0$ .
Differentiability: The pricing function is differentiable for all $x > 0$ .

To determine the optimal quantity of raisin-nut mix to purchase, we can use the concept of optimization. Specifically, we want to find the value of $x$ that minimizes the cost per pound of raisin-nut mix.

Let's define the cost function as:

${ C(x) = P(x) \cdot x }$

where $C(x)$ is the total cost of purchasing $x$ pounds of raisin-nut mix.

To find the minimum value of $C(x)$ , we can take the derivative of $C(x)$ with respect to $x$ and set it equal to zero:

${ \frac{dC(x)}{dx} = P'(x) \cdot x + P(x) = 0 }$

Solving for $x$ , we get:

${ x = -\frac{P(x)}{P'(x)} }$

Substituting the expression for $P(x)$ , we get:

${ x = -\frac{3.50 + 0.90^x}{0.90^x \ln(0.90)} }$

Solving for $x$ , we get:

${ x \approx 5.32 }$

This means that the optimal quantity of raisin-nut mix to purchase is approximately $5.32$ pounds.

In conclusion, the pricing function used by Hikers' World Foods to determine the cost per pound of raisin-nut mix for a store to purchase a certain quantity is a complex function that takes into account both fixed and variable costs. By analyzing the pricing function graphically and mathematically, we can gain a deeper understanding of the relationship between the cost per pound of raisin-nut mix and the quantity purchased. Additionally, we can use optimization techniques to determine the optimal quantity of raisin-nut mix to purchase.
Hikers' World Foods: A Q&A on Raisin-Nut Mix Pricing

In our previous article, we delved into the mathematical analysis of the pricing function used by Hikers' World Foods to determine the cost per pound of raisin-nut mix for a store to purchase a certain quantity. In this article, we will answer some frequently asked questions (FAQs) related to the pricing function and its implications for stores.

Q: What is the fixed cost per pound of raisin-nut mix?

A: The fixed cost per pound of raisin-nut mix is $3.50.

Q: What is the variable cost per pound of raisin-nut mix?

A: The variable cost per pound of raisin-nut mix is $0.90^x$ , where $x$ is the number of pounds purchased.

Q: How does the pricing function change as the quantity purchased increases?

A: As the quantity purchased increases, the cost per pound of raisin-nut mix decreases. The rate of decrease in cost per pound is not constant, but rather decreases as the quantity purchased increases.

Q: What is the minimum cost per pound of raisin-nut mix?

A: The minimum cost per pound of raisin-nut mix is $0.50, which is approached as the quantity purchased increases without bound.

Q: How can stores optimize their purchases to minimize costs?

A: Stores can use optimization techniques to determine the optimal quantity of raisin-nut mix to purchase. By analyzing the pricing function and its implications, stores can make informed decisions to minimize costs and maximize profits.

Q: What are the implications of the pricing function for stores with varying purchasing power?

A: The pricing function has different implications for stores with varying purchasing power. Stores with limited purchasing power may benefit from purchasing smaller quantities of raisin-nut mix, while stores with larger purchasing power may benefit from purchasing larger quantities.

Q: Can the pricing function be modified to accommodate different pricing strategies?

A: Yes, the pricing function can be modified to accommodate different pricing strategies. For example, stores may want to offer discounts for bulk purchases or charge more for smaller quantities.

Q: How can Hikers' World Foods use the pricing function to inform their pricing strategy?

A: Hikers' World Foods can use the pricing function to inform their pricing strategy by analyzing the cost per pound of raisin-nut mix for different quantities purchased. By understanding the relationship between quantity and cost, Hikers' World Foods can make informed decisions about pricing and inventory management.