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3 years ago

A nurse infuses a dose in 50 minutes. She needs to infuse the dose at a 20% faster rate. How much time does she need ?

Mathematics

1 answer:

Katen [24]3 years ago

8 0

Answer: The nurse needs 40 minutes to infuse the dose at a 20% faster rate.

Step-by-step explanation:

In order to calculate how much time the nurse needs to infuse the dose, you have to calculate how much is the 20%

The 20% of 50 minutes is obtained multiplying 50 by the factor that represents the 20% which is $\frac{20}{100}$

$(\frac{20}{100})(50)$ = 10 minutes

Now, the time she needs to infuse the dose at a faster rate is obtained as follows:

50 minutes - 10 minutes = 40 minutes.

Notice that you have to subtract the 20% of 50 minutes to the original time to obtain a 20% faster rate.

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

⊕ = XOR

∨ = OR

∧ = AND

~ = NOT

↔ = IFF (meaning "if and only if")

→ = IMP (short for "implies")

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)

= p AND NOT p

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))

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

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

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

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