Chapter 4, Problem 1EA

Suppose you plan to build a raised flower bed using landscape timbers, which come in 8-ft lengths. You want the bed’s outer dimensions to be 6 ft by 4 ft. and you will use three layers of timbers. The timbers are 6 in, by 6 in. in cross section, so if you make the bottom and top layers with 6-tt lengths on the sides and lengths on the ends, and the middle layer with 5-ft lengths on the sides and 4-ft lengths on the ends, you could build the bed out of the following lengths.

Plan A
Length	Number Needed
3 ft	4
4 ft	2
5 ft	2
6 ft	4

(a) What is the smallest number of timbers you can buy to build your bed? How will you lay out the cuts? How much wood will you waste?

(b) If you overlap the corners in a different way, you can build the bed with this plan:

Plan B
Length	Needed
3 ft	2
4 ft	4
5 ft	4
6 ft	2

Does plan B allow you to build the bed with fewer 8-ft timbers?

(c) What is the smallest length for the uncut timbers that would allow you to build the bed with no waste?

To determine

The minimum number of timber used to build a bed, the layouts the cuts, and the wasted wood.

Answer to Problem 1EA

Solution: The minimum timber used is 8 timbers and the wastage of timber is 10 ft.

Explanation of Solution

Given: The provided data:

$\begin{array}{l}\text{Plan A}\\ \begin{array}{cc}\text{Length}& \text{Number Needed}\\ 3\text{ft}& 4\\ \text{4ft}& 2\\ \text{5ft}& 2\\ \text{6ft}& 4\end{array}\end{array}$

Explanation:

The length of the timber is 8 ft.

In the provided plan A:

4 blocks of length 3 ft, 2 blocks of length 4 ft, 2 blocks of length 5 ft, and 4 blocks of length 6 ft are needed.

Now, if a timber of length 8 ft is cut in two pieces of length 5 and 3 ft, then there is a requirement of 2 blocks of length 5 ft, so 2 timbers are required and cut into length of 5 and 3 ft, which also fulfilled the requirement of 2 blocks of length 3 ft.

As 4 blocks are required of length 3 ft but 2 blocks are provided from the above cutting, 1 timber is cut to get 2 more blocks to length 3 ft, and there will be a wastage of 2 ft timber.

As 2 blocks are required of length 4 ft, 1 timber is cut into two pieces of length 4 ft each.

Now, 4 blocks are required of length 6 ft each; therefore, 4 timbers are required to get 4 blocks of 6 ft each, and there is a wastage of 2 ft block from each timber.

Thus, the total number of timber used is equal to $2+1+1+4=8$, and the wasted timber is equal to $2\text{ft+4}\times \text{2}\text{ft=10 ft}$.

Conclusion: The total timber used is 8 timbers, and the wastage of timber is 10 ft.

To determine

The minimum number of timber used to build a bed, the layouts the cuts, the wasted wood, and whether it is less than that used in part (a).

Answer to Problem 1EA

Solution: Still, the minimum timber used is 8 timbers and the wastage of timber is 10 ft.

Explanation of Solution

Given: The provided data:

$\begin{array}{l}\text{Plan B}\\ \begin{array}{cc}\text{Length}& \text{Number Needed}\\ 3\text{ft}& 2\\ \text{4ft}& 4\\ \text{5ft}& 4\\ \text{6ft}& 2\end{array}\end{array}$

Explanation:

The length of the timber is 8 ft.

In the provided plan A:

2 blocks of length 3 ft, 4 blocks of length 4 ft, 4 blocks of length 5 ft, and 2 blocks of length 6 ft are needed.

Now, if a timber of length 8 ft is cut in two pieces of length 5 and 3 ft, there is a requirement of 4 blocks of length 5 ft, so 4 timbers are required and cut into length of 5 and 3 ft, which also fulfilled the requirement of 2 blocks of length 3 ft, and there is a wastage of 6 ft timber.

As 4 blocks are required of length 4 ft. a timber is to be cut into two pieces of length 4 ft each; thus, 2 timbers are required to get 4 blocks of length 4 ft each.

As 2 blocks are required of length 6 ft, 2 timbersare cut into length of 6 and 2ft, and there is a wastage of 2 blocks of length of 2 ft each.

Thus, the total number of timber used is equal to $4+2+2=8$, and the wasted timber is equal to $\text{6}\text{ft+2}\times \text{2}\text{ft=10 ft}$.

Conclusion: Still, the total timber used is 8 timbers, and the wastage of timber is 10 ft.

To determine

The smallest length of the uncut timber to build the bed without wastage of timber.

Answer to Problem 1EA

Solution: If the uncut timber is of 9 ft, there will be no wastage of timber.

Explanation of Solution

Given: Information provided in part (a) and part (b)

Explanation:

The required lengths of the timber blocks are 3, 4, 5, and 6ft.

So, when the length of the uncut timber is 8 ft, there is wastage in cutting a block of the required length; then, to minimize the waste, the smallest length of timber that is required is 9 ft because when the timber is of 9 ft, then if it is cut into two pieces of length 5 and 4ft, and timber can be cut into two pieces of length 3 and 6ft, which implies that there is no wastage of timber.

Conclusion: The smallest length of uncut timber that minimizes the wastage is 9 ft.