Because of the angle strain present in cyclopropanes, they tend to open up in the presence of nearby radicals. Show a mechanism for the following reaction that demonstrates this principle.

Mullins 1st Edition
Ch. 8 - Alkenes I: Properties and Electrophilic Additions
Problem 68bWhen using sulfuric acid, but in the absence of other nucleophiles like water or bromide ion, less stable alkenes can be isomerized to their more stable isomer. Provide a mechanism for these acid-catalyzed isomerization reactions. [This is one illustration of the principle of microscopic reversibility.]
(b) 
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Key Concepts
Acid-Catalyzed Isomerization
Microscopic Reversibility
Carbocation Stability
Predict the product of each of the following hydroboration–oxidation or oxymercuration–reduction reactions used in the modern synthetic organic chemistry literature (modified to use reagents we are used to seeing).
(b) A two-step hydroboration–oxidation was used to prepare a silanediol peptidomimetic as a serine protease inhibitor (Org. Lett. 2012, 14, 4422–4425).
When using sulfuric acid, but in the absence of other nucleophiles like water or bromide ion, less stable alkenes can be isomerized to their more stable isomer. Provide a mechanism for these acid-catalyzed isomerization reactions. [This is one illustration of the principle of microscopic reversibility.]
(c)
How many stereoisomers are possible for the following alkenes?
(d)
When using sulfuric acid, but in the absence of other nucleophiles like water or bromide ion, less stable alkenes can be isomerized to their more stable isomer. Provide a mechanism for these acid-catalyzed isomerization reactions. [This is one illustration of the principle of microscopic reversibility.]
(a)
If you react carvone with a single equivalent of HBr, only one product is produced, even though carvone has two carbon–carbon double bonds. Explain this observation.