Question

A grapefruit farmer in central Florida is trying to decide whether to take protective action to limit damage to his crop in the event that the overnight temperature falls to a level well below freezing. He is concerned that if the temperature falls sufficiently low and he fails to make an effort to protect his grapefruit? trees, he runs the risk of losing his entire? crop, which is worth approximately? $75,000. Based on the latest forecast issued by the National Weather? Service, the farmer estimates that there is a? 60% chance that he will lose his entire crop if it is left unprotected.? Alternatively, the farmer can insulate his fruit by spraying water on all of the trees in his orchards. This? action, which would likely cost the farmer C? dollars, would prevent total devastation but might not completely protect the grapefruit trees from incurring some damage as a result of the unusually cold overnight temperatures. The file below contains the assessed distribution of possible damages? (in dollars) to the insulated fruit in light of the cold weather forecast. Of? course, this farmer seeks to minimize the expected total cost of coping with the threatening weather.

Grapefruit.xlsx

Damage to grapefruit crop if insulated	Probability
$0	0.30
$5,000	0.15
$10,000	0.10
$15,000	0.15
$20,000	0.30
Total worth	$75,000
P(lose entire crop if unprotected)	0.6

?a) Find the maximum value of C below which the farmer will choose to insulate his crop in hopes of limiting damage as result of the unusually cold weather. (please show each step; i know the answer is 35,000 but i dont understand how)

Answer 1

Solution

Expected damage if not insulated = 75000 x 0.6 = 45000 ……………………………(1)

Expected damage if insulated = 10000 as detailed below ……………………………(2)

Damage, x	0	5000	10000	15000	20000	Total
Probability p(x)	0.3	0.15	0.1	0.15	0.3	1.0
x. p(x)	0	750	1000	2250	6000	10000

(1) and (2) => gain due to insulating = reduction in expected damage = 45000- 10000 = 35000.

So, so far as the cost of insulating is less than or equal to $35,0000, the farmer would minimize the damage. Thus, the maximum value of C below which the farmer will choose to insulate his crop in hopes of limiting damage as result of the unusually cold weather is

$35,0000 ANSWER

DONE