Half-life question?

Question

If Grandma’s cookies have a half life of 2 hours and she bakes 3 dozen cookies, how many hours will pass until only two cookies remain uneaten? Round your answer to the nearest minute.

Is it 8 hours and 22 minutes?

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Answer 1 · 2018-01-18T20:25:06

$8$ $8$ h $20$ $20$ min

$y (t) = a \cdot {(\frac{1}{2})}^{0.5 t}$ $y(t) = a*(1/2)^(0.5t)$

(where $t$ $t$ is in hours)

e.g. $y (2) = {(\frac{1}{2})}^{1} \cdot 36 = \frac{1}{2} \cdot 36$ $y(2) = (1/2)^1 * 36 = 1/2 * 36$

(this is the number after the first half-life.)

$3$ $3$ dozen $= 36$ $= 36$

$a = 36$ $a = 36$

if $y (t) = 2$ $y(t) = 2$ :

$2 = a \cdot {(\frac{1}{2})}^{0.5 t}$ $2 = a*(1/2)^(0.5t)$

$2 = 36 \cdot {(\frac{1}{2})}^{0.5 t}$ $2 = 36*(1/2)^(0.5t)$

${(\frac{1}{2})}^{0.5 t} = \frac{1}{18}$ $(1/2)^(0.5t) = 1/18$

$0.5 t = {log}_{\frac{1}{2}} (\frac{1}{18})$ $0.5t = log_(1/2)(1/18)$

$0.5 t = 4.17$ $0.5t = 4.17$

$t = 8.339 . .$ $t = 8.339..$

$t = 8.339$ $t = 8.339$ h $= 8$ $= 8$ h $20$ $20$ min

(this answer is very close to $8$ $8$ h $22$ $22$ min, so you may have rounded differently, but used the right method.)