A manufacturer produces 1500 boxes of breakfast cereal every day.The weights of these boxes are normally distributed with a mean of 502 grams and a standard deviation of 2 grams.All boxes of cereal with a weight between 497.5 grams and 505 grams are sold. The manufacturer’s income from the sale of each box of cereal is $2.00.The manufacturer recycles any box of cereal with a weight not between 497.5 grams and 505 grams. The manufacturer’s recycling cost is $0.16 per box.A different manufacturer produces boxes of cereal with weights that are normally distributed with a mean of 350 grams and a standard deviation of 1.8 grams.This manufacturer sells all boxes of cereal that are above a minimum weight, w.They sell 97% of the cereal boxes produced.

$1500\times 0.16\times 0.079031\dots$ (M1)

Notes: Award (A1) for $1500\times 0.16\times \text{their}(1-0.920968\dots )$.

OR

$(1500-1381.45)\times 0.16$ (M1)

Notes: Award (M1) for $(1500-\text{their}1381.45)\times 0.16$.

$=(\mathrm{\$})19.0\text{(}18.9676\dots )$ (A1)(ft)(G2)

[2 marks]

(i) $0.921(0.920968\dots ,92.0968\dots \mathrm{\%})$ (G2)

Note: Award (M1) for a diagram showing the correct shaded region.

(ii) $1500\times 2\times 0.920968\dots$ (M1)

$=(\mathrm{\$})2760(2762.90\dots )$ (A1)(ft)(G2)

Note: Follow through from their answer to part (b)(i).

[4 marks]

$347(\text{grams})(346.614\dots )$ (G3)

Notes: Award (G2) for an answer that rounds to 346.

Award (G1) for $353.385\dots$ seen without working (for finding the top 3%).

[3 marks]