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Tomtit [17]
2 years ago
11

What is the area of a sector in a circle of radius 8cm,if the angle between the radii is 30°

Mathematics
1 answer:
IceJOKER [234]2 years ago
3 0

Answer:

16.75 square cm

Step-by-step explanation:

Area  \: of  \: sector  =  \frac{ \theta}{360 \degree}  \times \pi {r}^{2}  \\  \\  =  \frac{30 \degree}{360 \degree}  \times 3.14 \times  {8}^{2}  \\  \\  =  \frac{1}{12}  \times 3.14 \times 64 \\  \\  =  \frac{1}{3}  \times 3.14 \times 16 \\  \\  =  \frac{1}{3}  \times 50.24 \\  \\  = 16.7466667 \\  \\Area  \: of  \: sector  = 16.75 \:  {cm}^{2}

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I'll use text forms of the logical symbols because copying/pasting on mobile is annoying.

⊕ = XOR

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∧ = AND

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and for logical equivalence I'll use a regular equal sign.

Then

p XOR q = (p OR q) AND NOT (p AND q)

(a) By definition of XOR, we have

p XOR p = (p OR p) AND NOT (p AND p)

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and this is tautologically FALSE.

(b) To rewrite XOR, first distribute the NOT over the AND using DeMorgan's law:

NOT (p AND q) = NOT p OR NOT q

Then we can interchange the AND and ORs using their respective distributive properties. That is,

p XOR q = (p OR q) AND (NOT p OR NOT q)

= (p AND (NOT p OR NOT q)) OR (q AND (NOT p OR NOT q))

= ((p AND NOT p) OR (p AND NOT q)) OR ((q AND NOT p) OR (q AND NOT q))

Both (p AND NOT p) and (q AND NOT q) are tautologically FALSE, so the truth value of either OR depends only on the other statements, so we have

p XOR q = (p AND NOT q) OR (q AND NOT p)

Recall the definition of IFF:

p IFF q = (p IMP q) AND (q IMP p)

Also recall that IMP is logically equivalent to

p IMP q = p OR NOT q

Then we have

p IFF q = (p OR NOT q) AND (q OR NOT p)

so that its negation is

NOT (p IFF q) = NOT ((p OR NOT q) AND (q OR NOT p))

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The statements in bold match, so p XOR q is indeed equivalent to NOT (p IFF q).

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