The picture above shows a cylindrical hay bale. Often, they will be snugly covered by black or white plastic, but as a boy in England, growing up on a farm, I saw haystacks in fields shaped like cylindrical houses with conical roofs. Elsewhere, they just consisted of bales of hay stacked in rectangular formations. No matter what the shape, if they are too large, they can self-ignite (and you thought spontaneous combustion was just an urban legend!). Note the following:

An inexperienced farmer decided to save time and effort by storing his hay in one very large hemispherical haystack of radius R m instead of several smaller ones. Given the following information: (i) hay produces heat at a rate of b (J/h)/m³; (ii) the heat escapes at a rate of a (J/h)/m²; and (iii) there is no heat loss through the base,

Compare the answers to Question 1 with the...

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