Find the surface area of each different face of this book. Record the surface area for each face. Calculate the total surface area. With a partner, arrange your two books so that t... Find the surface area of each different face of this book. Record the surface area for each face. Calculate the total surface area. With a partner, arrange your two books so that they are touching and have the least possible total surface area. What is this total surface area? Then arrange the books so that they are touching but have the largest possible total surface area. What is this total surface area? How does changing the arrangement of the books affect the surface area?
Understand the Problem
The question is a set of instructions on how to measure the surface area of a book. It asks how the arrangement of multiple books affects the total surface area. The key concept here is understanding how different configurations minimize or maximize the combined surface area.
Answer
Answer for screen readers
Let $l$, $w$, and $h$ be the length, width and height of a single book. The total surface area of a single book is $SA_{single} = 2(lw + lh + wh)$. The total surface area of two separate books is $2 \times SA_{single} = 4(lw + lh + wh)$. The minimum surface area when the books are touching is $SA_{min} = 4(lw + lh + wh) - 2lw = 2lw + 4lh + 4wh$. The maximum surface area when the books are touching is $SA_{max} = 4(lw + lh + wh) - 2wh = 4lw + 4lh + 2wh$.
Steps to Solve
- Measure each face of a single book and calculate the surface area
A book has 6 faces. Measure the length, width, and thickness (height) of the book using a ruler. Let's call them $l$, $w$, and $h$ respectively. The surface area of each face can then be calculated.
- Calculate the total surface area of a single book
The total surface area ($SA_{single}$) of the book is the sum of the areas of all six faces. Since opposite faces have the same area, we have: $SA_{single} = 2(lw + lh + wh)$
- Calculate the total surface area of two books
The total surface area of two separate books is simply twice the surface area of a single book: $SA_{two_separate} = 2 \times SA_{single}$
- Determine the arrangement that minimizes the total surface area when two books are touching
To minimize the total surface area, the two books should be placed such that the largest faces are touching each other. This means joining two faces with area $l \times w$. The total surface area of the two books touching at their largest faces is: $SA_{min} = 2 \times SA_{single} - 2 \times (l \times w)$
- Determine the arrangement that maximizes the total surface area when two books are touching
To maximize the total surface area, the two books should be placed such that the smallest faces are touching each other. This means joining two faces with area $w \times h$. The total surface area of the two books touching at their smallest faces is: $SA_{max} = 2 \times SA_{single} - 2 \times (w \times h)$
- Analyze how the arrangement affects the surface area
Changing the arrangement of the books affects the total surface area by changing the amount of overlapping surface. The greater the overlapping surface, the smaller the total surface area.
Let $l$, $w$, and $h$ be the length, width and height of a single book. The total surface area of a single book is $SA_{single} = 2(lw + lh + wh)$. The total surface area of two separate books is $2 \times SA_{single} = 4(lw + lh + wh)$. The minimum surface area when the books are touching is $SA_{min} = 4(lw + lh + wh) - 2lw = 2lw + 4lh + 4wh$. The maximum surface area when the books are touching is $SA_{max} = 4(lw + lh + wh) - 2wh = 4lw + 4lh + 2wh$.
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