91Ó°ÊÓ

A child has 64 blocks that are 1 -inch cubes. She wants to arrange the blocks into a solid rectangle \(h\) blocks long and \(w\) blocks wide. There is a relationship between \(h\) and \(w\) that is determined by the restriction that all 64 blocks must go into the rectangle. A rectangle \(h\) blocks long and \(w\) blocks wide uses a total of \(h \times w\) blocks. Thus \(h w=64\). Applying some elementary algebra, we get the relationship we need: $$ w=\frac{64}{h} . $$ a. Use a formula to express the perimeter \(P\) in terms of \(h\) and \(w\). b. Using Equation (2.3), find a formula that expresses the perimeter \(P\) in terms of the height only. c. How should the child arrange the blocks if she wants the perimeter to be the smallest possible? d. Do parts \(b\) and \(c\) again, this time assuming that the child has 60 blocks rather than 64 blocks. In this situation the relationship between \(h\) and \(w\) is \(w=60 / h\). (Note: Be careful when you do part c. The child will not cut the blocks into pieces!)

Step by step solution

01

Express Perimeter in Terms of h and w

The perimeter of a rectangle is given by the formula:\[ P = 2h + 2w \] where \( h \) is the height and \( w \) is the width.

02

Substitute for w in Terms of h

We know from the problem that \( w = \frac{64}{h} \). Substitute this expression into the perimeter formula:\[ P = 2h + 2\left(\frac{64}{h}\right) \] to express \( P \) in terms of only \( h \).

03

Simplify Perimeter Formula in Terms of h

Simplify the equation from Step 2:\[ P = 2h + \frac{128}{h} \] This equation represents the perimeter in terms of \( h \) alone.

04

Minimize Perimeter for 64 Blocks

To find the minimum perimeter, evaluate \( h \) as factors of 64. The factors of 64 are 1, 2, 4, 8, 16, 32, 64. Let’s consider pairs (\( h, w \)) that satisfy \( hw = 64 \): - \( h = 1, w = 64 \) gives \( P = 130 \) - \( h = 2, w = 32 \) gives \( P = 68 \) - \( h = 4, w = 16 \) gives \( P = 40 \) - \( h = 8, w = 8 \) gives \( P = 32 \) The smallest perimeter is \( 32 \) when \( h = w = 8 \).

05

Express Perimeter in Terms of h for 60 Blocks

Using the relationship \( w = \frac{60}{h} \), substitute into the perimeter formula:\[ P = 2h + 2\left(\frac{60}{h}\right) \] Simplify to: \[ P = 2h + \frac{120}{h} \] which gives the perimeter in terms of \( h \) for 60 blocks.

06

Minimize Perimeter for 60 Blocks

With 60 blocks, we examine the integer factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60. Evaluate perimeter \( P \) for these factors:- \( h = 4, w = 15 \) gives \( P = 38 \)- \( h = 5, w = 12 \) gives \( P = 34 \)- \( h = 6, w = 10 \) gives \( P = 32 \)For the smallest perimeter of 32, choose \( h = 6 \) and \( w = 10 \).

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Perimeter Formulas

Perimeter is a measure of the distance around a shape, which is vital in mathematics for various calculations. For rectangles, the perimeter is the sum of all its sides. Specifically, the formula we use is:

• \( P = 2h + 2w \)

where \(h\) represents the height, and \(w\) indicates the width. This equation arises from adding the length of all sides of the rectangle, doubling the height and width because there are two of each. In our exercise, understanding this formula lets us compute how much total border surrounds a rectangular shape formed out of 64 blocks.
By substituting the expression for \(w\) (such as \(w = \frac{64}{h}\)), students learn how to reflect changes in one dimension on another, keeping the total perimeter constant.

Factorization

Factorization in mathematics involves breaking down a number into its essential components or factors. This is key for solving problems related to dimensions, such as the rectangle in our exercise. Consider a number like 64; its factors are those numbers which, when multiplied together, return 64, such as:

• 1 and 64
• 2 and 32
• 4 and 16
• 8 and 8

To find the possible configurations of our block rectangle, you explore these factor pairs to get corresponding dimensions \((h, w)\), ensuring each completely utilizes the fixed amount of 64 blocks. This process highlights how dimensional constraints work and why factor pairs are crucial in defining variable relationships.

Optimization Problems

Optimization aims to make the best or most efficient use of resources. Often, in mathematics, it involves making calculations to minimize or maximize a particular quantity. In the context of our exercise, it's about minimizing the perimeter for specific \(h\) and \(w\). After substituting \(w = \frac{64}{h}\) back into the perimeter formula, we explore factor pairs of 64, calculating the perimeter for each to identify the smallest value:

• \((h, w) = (1, 64)\), perimeter is 130
• \((h, w) = (2, 32)\), perimeter is 68
• \((h, w) = (4, 16)\), perimeter is 40
• \((h, w) = (8, 8)\), perimeter is 32

This approach provides insights into choosing the most efficient rectangular arrangement without altering the overall block count.

Rectangular Arrangements

Rectangular arrangements are about organizing elements into rectangles, leveraging their geometric properties fully. When dealing with tangible items like blocks, these arrangements depend on maintaining certain relations, such as ensuring every block is used effectively. In this exercise, the child's arrangement decision focuses on factor-pair configurations:- Blocks are fixed; rearranging is only about changing which dimension is longer or equal.- Here, equal dimensions, as in \(h = w = 8\), yield the smallest perimeter.
Thinking in terms of rectangular arrangements helps students understand spatial constraints, evolve problem-solving strategies for optimal configurations, and appreciate how algebraic principles come to life in real-world scenarios. This grounding in tangible math exercises gives clarity and structural insight into seemingly abstract algebraic challenges.