A stick n inches long must be cut into n 1-inch pieces. We could cut the stick (n-1) times, but we would like to minimize the number of times the saw is used. Several stick pieces can be stacked together and cut at the same time. Draw the sticks and illustrate their arrangement so the saw is used as few times as possible. Clearly identify the stick lengths at each cut, and clearly identify the number of cuts needed. a. Solve the problem for a 30-inch stick. (2 points) b. Solve the problem for a 33-inch stick. (2 points) c. State a general formula for the minimum number of cuts for an n-inch stick. (2 points)

a. The minimum number of 1-inch cuts for a 30-inch stick is 5.

b. The minimum number of 1-inch cuts for a 33-inch stick is 6.

c. 2ˣ ⁻¹ < n < 2ˣ

Where x shows the minimum number of cuts needed.

Where 2ˣ ⁻¹ shows the starting point of cutting for the minimum number of cuts.

Step-by-step explanation:

a. Solve the problem for a 30-inch stick.

We need to cut a 30-inch wooden stick into 1-inch pieces.

2⁴ < 30 < 2⁵

16 < 30 < 32

Which means that we need to make at least 5 cuts and the starting point of cutting should be 16 in.

1st Cut:

Cutting the wood at 16 in. gives at most 2 pieces

No. of pieces  |  size (in) |  No. of pieces  |  size (in)  

          1             |      16      |            1             |     14

2nd Cut:

Cutting the wood at 8 in. gives at most 4 pieces

No. of pieces  |  size (in) |  No. of pieces  |  size (in)  

          3             |      8      |            1             |     6

3rd Cut:

Cutting the wood at 4 in. gives at most 8 pieces

No. of pieces  |  size (in) |  No. of pieces  |  size (in)  

          7             |      4      |            1             |     2

4th Cut:

Cutting the wood at 2 in. gives at most 16 pieces (actual 15)

No. of pieces  |  size (in) |  No. of pieces  |  size (in)  

          14           |      2       |            1             |     2

5th Cut:

Cutting the wood at 1 in. gives at most 32 pieces (actual 30)

No. of pieces  |  size (in) |  No. of pieces  |  size (in)  

          28          |      1        |            2            |     1

Hence total pieces are 28 + 2 = 30 and size is 1 inch.

Therefore, the minimum number of 1-inch cuts for a 30-inch stick is 5.

b. Solve the problem for a 33-inch stick.

We need to cut a 33-inch wooden stick into 1-inch pieces.

2⁵ < 30 < 2⁶

32 < 33 < 64

Which means that we need to make at least 6 cuts and the starting point of cutting should be 32 in.

1st Cut:

Cutting the wood at 32 in. gives at most 2 pieces

No. of pieces  |  size (in) |  No. of pieces  |  size (in)  

          1             |      32      |            1             |     1

2nd Cut:

Cutting the wood at 16 in. gives at most 4 pieces (actual 3)

No. of pieces  |  size (in) |  No. of pieces  |  size (in)  

          2             |      16      |            1             |     1

3rd Cut:

Cutting the wood at 8 in. gives at most 8 pieces (actual 5)

No. of pieces  |  size (in) |  No. of pieces  |  size (in)  

          4             |      8      |            1             |     1

4th Cut:

Cutting the wood at 4 in. gives at most 16 pieces (actual 9)

No. of pieces  |  size (in) |  No. of pieces  |  size (in)  

          8           |      4       |            1             |     1

5th Cut:

Cutting the wood at 2 in. gives at most 32 pieces (actual 17)

No. of pieces  |  size (in) |  No. of pieces  |  size (in)  

          16          |      2       |            1            |     1

6th Cut:

Cutting the wood at 1 in. gives at most 64 pieces (actual 33)

No. of pieces  |  size (in) |  No. of pieces  |  size (in)  

          32          |      1       |            1            |     1

Hence total pieces are 32 + 1 = 30 and size is 1 inch.

Therefore, the minimum number of 1-inch cuts for a 33-inch stick is 6.

c. State a general formula for the minimum number of cuts for an n-inch stick.

2ˣ ⁻¹ < n < 2ˣ

Where x shows the minimum number of cuts needed.

Where 2ˣ ⁻¹ shows the starting point of cutting for minimum number of cuts.

3< x< 5

step-by-step explanation:

|x - 4| < 1

there are two solutions, one positive and one negative.   remember to flip the inequality when taking the negative solution

x-4 < 1       x-4 > -1

add 4 to each side

x-4+4 < 1+4       x-4+4 > -1+4

x< 5                 x> 3

3< x< 5

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step-by-step explanation: