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Chapter 1: Problem 16

A set of blocks contains some that are 1 inch high and some that are 2 inches high. How many ways are there to make a stack of blocks 15 inches high?

Short Answer

Expert verified
There are 8 ways to arrange the blocks to build a stack of 15 inches.

Step by step solution

01

Identify the Problem

We want to find the total number of distinct arrangements of blocks that result in a stack height of 15 inches, using blocks that are 1 inch high and blocks that are 2 inches high.
02

Define Variables

Let the number of 1-inch blocks be denoted as \( x \) and the number of 2-inch blocks as \( y \). The goal is to find solutions for \( x \) and \( y \) such that the stack height is exactly 15 inches.
03

Set Up the Equation

The equation representing the total height built by the blocks is:\[x + 2y = 15\]Here, \( x \) represents the number of 1-inch blocks and \( y \) the number of 2-inch blocks.
04

Determine Possible Values for \( y \)

Since 2-inch blocks cannot exceed half the total height (or 7.5), the maximum possible value for \( y \) is 7. Thus, \( y \) can be any integer from 0 to 7.
05

Solve for \( x \)

For each integer \( y \) from 0 to 7, solve for \( x = 15 - 2y \). Ensure that \( x \) remains a non-negative integer to represent a feasible number of 1-inch blocks.
06

Count the Solutions

Calculate the number of different ways to assign values to \( x \) and \( y \) by iterating over all valid \( y \) values:- If \( y = 0 \), then \( x = 15 \).- If \( y = 1 \), then \( x = 13 \).- If \( y = 2 \), then \( x = 11 \).- Continue this until \( y = 7 \), where \( x = 1 \).Each pair \((x, y)\) is a distinct solution.
07

Total Count of Arrangements

After determining all possible values, count the pair solutions. For each integer \( y \) from 0 to 7, there is a valid arrangement, resulting in 8 possible solutions.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Problem-solving
In the context of combinatorics, problem-solving often involves figuring out how to arrange or choose items in a way that meets specific criteria. To tackle this, we break down the question at hand into manageable parts. In our exercise, the question is about stacking blocks to reach a specified height of 15 inches. By systematically approaching this question, one can uncover the various arrangements possible. Breaking down the problem step-by-step helps to simplify a complex issue, guiding towards a solution in an organized fashion. Problem-solving in mathematics involves creativity, logic, and patience, ensuring that every possibility is considered.
Algebraic equations
Algebraic equations are powerful tools in problem-solving, enabling us to represent real-world scenarios mathematically. In the block stacking problem, we created an equation to model the situation: \[x + 2y = 15\]. Here, \(x\) is the number of 1-inch blocks, and \(y\) is the number of 2-inch blocks. This equation represents the total height of the stack, which must be 15 inches.
  • The equation helps to outline the constraints of our problem.
  • The variables \(x\) and \(y\) can be adjusted while still achieving the required stack height.
By solving this equation, we determine the possible combinations of block heights that fulfill the conditions set by the problem. Understanding how to form and solve equations sets the foundation for finding correct solutions to similar problems.
Mathematical modeling
Mathematical modeling involves converting real-world problems into a mathematical language to understand and solve them better. In this task, we modeled the block stacking problem using the equation: \[x + 2y = 15\]. This model reflects the height constraint of the stack and includes the variables for different block types. Mathematical models help:
  • Simplify complex situations by creating representations we can manipulate mathematically.
  • Analyze relationships, predict outcomes, and test different scenarios.
By using this model, we take a structured approach to examine all possible combinations of 1-inch and 2-inch blocks. By adjusting these variables within the model, solutions become more apparent, clearing the path to solving the problem effectively.
Integer Solutions
Integer solutions mean finding whole number values for variables that satisfy the given equation. In our exercise, the equation \[x + 2y = 15\] only considers integer values due to the nature of the blocks being physical objects that cannot be fractional. The number of 1-inch blocks, \(x\), and the number of 2-inch blocks, \(y\), must each be whole numbers.
  • Potential values for \(y\) range from 0 to 7, as 2-inch blocks cannot exceed half of 15 inches.
  • For each \(y\), calculate \(x\) as \(15 - 2y\) and ensure \(x\) remains non-negative.
By iterating through valid values of \(y\), we find corresponding values for \(x\) that make the equation true. This exploration results in 8 different pairs, showing all the integer solutions that meet the conditions of the task. This process of finding integer solutions is crucial for combinatorial problems, where items cannot be divided.

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Most popular questions from this chapter

Suppose \(\bar{p}\) must approximate \(p\) with reative error at most \(10^{-3}\). Find the largest interval in which \(\bar{p}\) must. lie if \(p=900\).

Let the sequence \(\left\langle a_{n}\right\rangle\) be defined by $$ \begin{aligned} a_{n+1} &=\frac{1}{4}\left|5 a_{n}^{2}-30 a_{n}+25\right|, \quad n \geq 1 \\ a_{0} &=\frac{17+2 \sqrt{7}}{5} \end{aligned} $$ (a) Calculate \(a_{1}, a_{2}\) and \(a_{3}\) exactly. (b) Find \(a_{20}\) and \(a_{51}\) exactly. (c) Write a recursive function that calculates the \(n^{t h}\) term of the sequence. Your function should have one argument, \(n\). Write a program that calls this function to calculate \(a_{1}, a_{2}, a_{3}, a_{20},\) and \(a_{51}\). (d) Write a function that uses a for loop to calculate the \(n^{\text {th }}\) term of the sequence. Your function should have one argument, \(n .\) Write a program that calls this function to calculate \(a_{1}, a_{2}, a_{3}, a_{20}\) and \(a_{51}\) (e) Which code is simpler (recursive or nonrecursive)? (f) Which function is faster? (g) Which code is more accurate, and why? (h) Which function is better, and why? (i) Do you trust either function to calculate \(a_{\text {Goo }} \mathrm{ac}-\) curately? If not, why not?

Some linearly convergent sequences and their limits are given. Find the (fastest) rate of convergence of the form \(O\left(\frac{1}{n^{p}}\right)\) or \(O\left(\frac{1}{a^{n}}\right)\) for each. If this is not possible, suggest a reasonable rate of convergence. (a) \(6, \frac{6}{7}, \frac{6}{49}, \frac{6}{343}, \frac{6}{2401}, \ldots \rightarrow 0\) (b) \(\left\langle\frac{11 n-2}{n+3}\right\rangle \rightarrow 11\) (c) \(\left\langle\frac{\sin n}{\sqrt{n}}\right\rangle \rightarrow 0\), (d) \(\left\langle\frac{4}{10^{n}+35 n+9}\right\rangle \rightarrow 0^{[\text {[s] }}\) (e) \(\left\langle\frac{1}{n}+\frac{1}{n^{2}}\right\rangle \rightarrow 0\) (f) \(\left\langle\frac{4}{10^{n}-35 n-9}\right\rangle \rightarrow 0^{[s]}\) (g) \(\left\langle\frac{2 n}{\sqrt{n^{2}+3 n}}\right\rangle \rightarrow 2^{[A]}\) (h) \(\left\langle\frac{5^{n}-2}{5^{n}+3}\right\rangle \rightarrow 1\) (i) \(\langle\sqrt{n+47}-\sqrt{n}\rangle \rightarrow 0^{[A]}\) (j) \(\left\langle\frac{n^{2}}{3 n^{2}+1}\right\rangle \rightarrow \frac{1}{3}\) (k) \(\left\langle\frac{\pi}{e^{n}-\pi^{n}}\right\rangle \rightarrow 0\) (1) \(\left\langle\frac{n^{2}}{2^{n}}\right\rangle \rightarrow 0^{[\mathbb{S}]}\) (m) \(\left\langle\frac{7+\cos (5 n)}{n^{3}+1}\right\rangle \rightarrow 0\) (n) \(\left\langle\frac{8 n^{2}}{3 n^{2}+12}+\frac{n}{3 n+10}\right\rangle \rightarrow 3\) (o) \(\left\langle\frac{2 n^{2}+3 n}{1-n^{2}}\right\rangle \rightarrow-2^{[A]}\) (p) \(\left\langle\frac{3 n^{5}-5 n}{1-n^{5}}\right\rangle \rightarrow-3\)

Compute the absolute error and relative error in approximations of \(p\) by \(\bar{p}\). (a) \(p=\sqrt{2}, \bar{p}=1.414\) (b) \(p=10^{\pi}, \tilde{p}=1400\) (c) \(p=9 !, \tilde{p}=\sqrt{18 \pi}(9 / e)^{9}\)

Compute the absolute and relative errors in the approximation of \(\pi\) by \(3 .\)
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