Question

7 a special type of timber can be bought in 3 lengths. the lengths that can be bought and the price of each plank are shown in the table below.

length of plank in metres	2.5	3.5	4.5

sophie needs to buy enough of this type of timber so that by sawing the planks she will have 10 pieces of length 1.50m, 10 pieces of length 2.00m and 10 pieces of length 3.00m.
(a) what is the least that sophie can pay for the timber that she needs? 2
sophie’s project was successful and she decided to repeat it. unfortunately, when she went back to the timber merchant the prices had changed. the new prices are in the following table.

length of plank in metres	2.5	3.5	4.5

(b) what is the least she could now pay to get enough timber for her needs? 3

Explanation:

Step1: Calculate total required length

Total length = $(10\times1.5)+(10\times2.0)+(10\times3.0) = 15+20+30 = 65$ m

Step2: Analyze (a) plank yield & cost efficiency

First, calculate cost per meter for each plank:
- 2.5m: $\frac{2.50}{2.5} = \$1.00$/m
- 3.5m: $\frac{3.50}{3.5} = \$1.00$/m
- 4.5m: $\frac{4.50}{4.5} = \$1.00$/m
All planks have same cost per meter. Now find optimal cuts:
- 4.5m plank: can be cut into one 3.0m + one 1.5m (total 4.5m, no waste)
- 3.5m plank: can be cut into one 2.0m + one 1.5m (total 3.5m, no waste)
- 2.5m plank: can be cut into one 2.0m + one 0.5m (waste) OR one 1.5m + one 1.0m (waste), so better to use other planks first.

We need 10 of 3.0m: use 10 pieces of 4.5m planks, which also gives 10 of 1.5m (fulfilling the 10 1.5m pieces).
Remaining need: 10 of 2.0m. Use 10 pieces of 3.5m planks, which also gives 10 extra 1.5m (we don't need these, but it's the minimal cost since cost per meter is same).
Total cost (a): $(10\times4.50)+(10\times3.50) = 45 + 35 = 80$
Alternative: We can also mix to reduce total planks? Wait, 4.5m can also be 2x2.0m + 0.5m waste, but that's worse. 3.5m can't make 3.0m. So the above is minimal.

Step3: Analyze (b) plank cost efficiency

Calculate cost per meter:
- 2.5m: $\frac{2.50}{2.5} = \$1.00$/m
- 3.5m: $\frac{4.50}{3.5} \approx \$1.29$/m
- 4.5m: $\frac{6.00}{4.5} \approx \$1.33$/m
The 2.5m plank is the most cost-effective. Now find optimal cuts:
- 2.5m plank: can be cut into one 2.0m + one 0.5m (waste) OR one 1.5m + one 1.0m (waste). But also, 4.5m plank can make one 3.0m + one 1.5m (no waste), which is better than using 2.5m planks for 3.0m (would need two 2.5m planks for one 3.0m, cost $\$5.00$ vs $\$6.00$? Wait no: two 2.5m planks cost $2\times2.50=\$5.00$, which is cheaper than one 4.5m plank ($\$6.00$) for a 3.0m piece (with 2m waste from two 2.5m). Wait, no: 2x2.5m = 5m, cut one 3.0m, leaving 2.0m, which can be used for the 2.0m pieces. That's better.

Wait, let's re-plan (b):
We need 10x3.0m, 10x2.0m, 10x1.5m.
Option for 3.0m: Use a 2.5m plank + cut 3.0m? No, 2.5m is shorter. Oh right! 3.0m is longer than 2.5m, so we can't get a 3.0m piece from a 2.5m plank. So 3.0m pieces must come from 3.5m or 4.5m planks.
- 3.5m plank for 3.0m: leaves 0.5m waste, cost $\$4.50$ per 3.0m piece.
- 4.5m plank for 3.0m: leaves 1.5m, which is exactly the 1.5m piece we need. Cost $\$6.00$ per 3.0m + 1.5m set.

We need 10x3.0m and 10x1.5m: using 10 of 4.5m planks gives us all 10 3.0m and 10 1.5m pieces, cost $10\times6.00 = \$60.00$.

Now we need 10x2.0m pieces. The most cost-effective is 2.5m planks: each 2.5m plank can be cut into one 2.0m piece (with 0.5m waste). So we need 10 of 2.5m planks, cost $10\times2.50 = \$25.00$.

Total cost (b): $60 + 25 = 85$

Wait, alternative: Can we use 3.5m planks for 2.0m? 3.5m can be cut into 2.0m +1.5m, but we already have all 1.5m pieces, so that would be waste. Cost per 2.0m would be $\$4.50$, which is more expensive than $\$2.50$ from 2.5m plank. So no.

Another alternative: For the 3.0m pieces, using 3.5m planks: 10 of them would cost $10\times4.50=\$45$, but we still need 10x1.5m pieces, which would need 10x2.5m planks (cost $\$25$), total $45+25=70$? Wait no! Wait 3.5m plank cut into 3.0m leaves 0.5m, which can't be used for 1.5m. So we need to get 10x1.5m pieces from somewhere. 1.5m pieces can come from 2.5m planks: each 2.5m plank can make one 1.5m piece (with 1m waste), so 10 of them cost $\$25$. Total cost would be $45+25=70$? But wait, that's cheaper, but we have 10x0.5m waste from 3.5m planks, and 10x1m waste from 2.5m planks. But wait, is there a better way? Wait 2.5m plank can be cut into one 1.5m + one 1.0m, but we don't need 1.0m. Alternatively, can we combine? Wait 4.5m plank can be cut into two 2.0m pieces (4.0m, 0.5m waste), cost $\$6.00$ for two 2.0m pieces, which is $\$3.00$ per 2.0m, more expensive than $\$2.50$ per 2.0m from 2.5m planks.

Wait wait, no mistake here: if we use 10 3.5m planks for 3.0m pieces, cost $\$45$, then 10 2.5m planks for 1.5m pieces, cost $\$25$, then we still need 10 2.0m pieces, which would need another 10 2.5m planks, cost $\$25$. Total cost $45+25+25=95$, which is more than 85. Oh right! I forgot we need 2.0m pieces too. My mistake earlier.

So back to the original plan for (b): 10 4.5m planks (gives 10x3.0m and 10x1.5m) cost $\$60$, plus 10 2.5m planks (gives 10x2.0m) cost $\$25$, total $\$85$.

Is there a better mix? Let's see: What if we use some 3.5m planks for 2.0m +1.5m? Suppose we use x 3.5m planks, which gives x 2.0m and x 1.5m. Then we need (10-x) 1.5m from 4.5m planks, which would require (10-x) 4.5m planks (each gives 1x3.0m +1x1.5m). We need 10 3.0m, so (10-x) 4.5m planks give (10-x) 3.0m, so we need x more 3.0m, which would have to come from x 3.5m planks (each gives 1x3.0m). Wait, that would be x 3.5m planks for 3.0m, and x 3.5m planks for 2.0m+1.5m, total 2x 3.5m planks, plus (10-x) 4.5m planks. Then we need (10-x) 2.0m pieces, which would come from (10-x) 2.5m planks.

Total cost: $2x\times4.50 + (10-x)\times6.00 + (10-x)\times2.50$
$=9x + 60 -6x +25 -2.5x$
$=0.5x +85$
Since x≥0, the minimal cost is when x=0, which is $\$85$, which matches our original plan.

Another option: Use 4.5m planks for 2x2.0m (4.0m, 0.5m waste). To get 10 2.0m pieces, we need 5 4.5m planks, cost $5\times6.00=\$30$, which is more than $\$25$ for 10 2.5m planks. So no.

Answer:

(a) $\$80.00$
(b) $\$85.00$