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The corollas of the spotted poverty bush (Eremophila abietina) have acute lobes (sense 5)
An angle of 45 degrees is an acute angle (sense 6)
All the internal angles of an acute triangle (sense 7) measure less than 90 degrees

From Late Middle English acūte (of a disease or fever: starting suddenly and lasting for a short time; of a humour: irritating, sharp), from Latin acūta, from acūtus (sharp, sharpened), perfect passive participle of acuō (to make pointed, sharpen, whet), from acus (needle, pin), from Proto-Indo-European *h₂eḱ- (sharp). The word is cognate to ague (acute, intermittent fever).

As regards the noun, which is derived from the verb, compare Middle English acūte (severe but short-lived fever; of blood: corrosiveness, sharpness; musical note of high pitch).