A volcanic eruption deposits ash that thins with distance according to the formula $T(d)=5e^{-0.05d}$ cm, where $d$ is the distance from the volcano in kilometres. Crops can tolerate up to $2$ cm of ash; for each centimetre of ash above this threshold, yield is reduced by $20$% per centimetre. The affected agricultural region is a circular area of radius $40$ km centred on the volcano, and the average crop value is $£200{,}000$ per km² per year.

(a) Determine the radius (in km) within which ash thickness exceeds the 2 cm tolerance.
(b) Calculate the total economic loss (in £) for the year caused by the ash fall.
(c) A mitigation programme costing $£20$ million would reduce ash thickness everywhere by 30 %. Should the region implement the programme? Show the net benefit (or loss) in £.

Question

A volcanic eruption deposits ash that thins with distance according to the formula $T(d)=5e^{-0.05d}$ cm, where $d$ is the distance from the volcano in kilometres. Crops can tolerate up to $2$ cm of ash; for each centimetre of ash above this threshold, yield is reduced by $20$% per centimetre. The affected agricultural region is a circular area of radius $40$ km centred on the volcano, and the average crop value is $£200{,}000$ per km² per year.

(a) Determine the radius (in km) within which ash thickness exceeds the 2 cm tolerance.
(b) Calculate the total economic loss (in £) for the year caused by the ash fall.
(c) A mitigation programme costing $£20$ million would reduce ash thickness everywhere by 30 %. Should the region implement the programme? Show the net benefit (or loss) in £.

Accepted Answer

Yes; net benefit ≈ £13 million

Gce Model Questions Vol1 2023 Geography Question 28

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