Problem 39 - Calculus Early Transcendentals: Pearson New International Edition

Question

A grocer stacks oranges in a pyramidlike pile. If the bottom layer is rectangular with 10 rows of 16 oranges and the top layer has a single row of oranges, how many oranges are in the stack?

Step-by-Step Solution

Answer

There are 880 oranges in the stack.

Step 1: Understanding the Problem

The problem involves understanding a pyramid-like structure of oranges where each consecutive layer has fewer rows and columns than the one below. The bottom layer has 10 rows of 16 oranges, forming a rectangular shape, and the topmost layer has 1 row of oranges. We need to find the total number of oranges in all layers.

Step 2: Identify the Number of Layers

In this pyramid structure, the number of layers equals the number of rows in the base layer because each layer loses one row from the bottom to the top. Since the base layer has 10 rows, there will be 10 layers total.

Step 3: Calculate Oranges in Each Layer

Each layer of the pyramid has oranges based on its position. The n-th layer from the top has 16 oranges per row and (10-n+1) rows. The general formula for oranges in the n-th layer is: \[O_n = 16 \times (10 - n + 1)\].

Step 4: Compute the Total Oranges

Sum up the oranges from all the layers. The total number of oranges can be computed by the series:\[\text{Total Oranges} = \sum_{n=1}^{10} 16 \times (11 - n)\]. This simplifies as:\[16 \times (10 + 9 + 8 + \ldots + 1)\].

Step 5: Sum of the Series

The sequence inside the series (10, 9, ..., 1) is an arithmetic series. The sum of an arithmetic series is given by:\[S = \frac{n}{2} \times (a + l)\], where \(n\) is the number of terms, \(a\) is the first term, and \(l\) is the last term. Here, \(n = 10\), \(a = 10\), \(l = 1\). Thus, \[S = \frac{10}{2} \times (10 + 1) = 55\].

Step 6: Final Solution

Multiply the sum by 16 to find the total number of oranges:\[16 \times 55 = 880\]. Therefore, the stack has a total of 880 oranges.

Key Concepts

Pyramid Structure

When stacking objects such as oranges in a pyramid-like manner, we often create layers that resemble a tiered system, decreasing in size from the bottom to the top. In the context of the problem, this pyramid structure begins with a broad base and narrows as you ascend through each layer.

This type of arrangement allows each layer to support the next, reducing the number of objects as you move upward. The layers in this structure are essentially steps made of oranges, with each layer diminishing by one row until reaching the topmost layer. Understanding this pattern is crucial as it aids in visualizing the quantity of oranges needed per layer and simplifies calculations.

Layered Stacking

Layered stacking is a method of organizing items in consecutively decreasing amounts. Here, each layer in the pyramid is directly above its lower counterpart and slightly smaller, following specific rules reducing the number of rows.

In the given problem, each consecutive layer loses one row compared to the previous layer, starting from 10 rows at the bottom and culminating in one row at the very top. This creates a stacked structure where the number of rows in each subsequent layer is essential for calculating the total number of oranges.

• The base or bottom layer has 10 rows of oranges.
• Each successive layer has one less row.
• This pattern makes it easy to determine the number of oranges in each layer.

Understanding layered stacking is fundamental to calculating how many oranges are in the entire pyramid.

Series Summation

When finding the total number of oranges in the stack, we use a mathematical technique called series summation. This involves adding up all the oranges from each layer according to a defined pattern.

The series we're dealing with here is arithmetic, meaning each term decreases by a constant amount. The series is created based on the number of rows in each layer, which decreases from 10 to 1. The sum of this series is calculated using the formula for arithmetic series:

• We start with 10 rows for the first layer.
• Each row consists of a fixed number of oranges (16 in this problem).
• The sequence forms an arithmetic series: 10, 9, 8, ..., down to 1.
• The sum of this series is straightforward to compute using the formula: \[S = \frac{n}{2} \times (a + l)\], where \(n = 10\), \(a = 10\), and \(l = 1\).

Using this formula efficiently gives us the sum for calculating total oranges.

Calculus Problem Solving

While this particular problem doesn't directly involve calculus, its structured approach and logical reasoning are relevant to problem solving in calculus. Calculus often requires breaking complex problems into manageable parts, similar to deconstructing the pyramid into layers to calculate the total number of oranges.

By systematically understanding the problem, identifying patterns – in this case, an arithmetic sequence – and employing appropriate mathematical formulas, we develop skills that are crucial in solving calculus-related problems. The essence of problem-solving in calculus and similar exercises lies in:

• Identifying the core structure or pattern in the given problem.
• Applying known formulas to compute values efficiently.
• Understanding and verifying each step in solving the problem.

Approaching the pyramid problem with this mindset helps prepare for more complex calculus scenarios by encouraging logical and precise thinking.