Question

Element X decays radioactively with a half life of 15 minutes. If there are 340 grams of Element X, how long, to the nearest tenth of a minute, would it take the element to decay to 6 grams?

Answer 1

Answer: 57 min

i used a calculator.

Answer 2

t=87.36643 ≈ 87.4 minutes

Explanation:

y=a(.5)^{\frac{t}{h}}

y=a(.5)

h

t

​

y=6 \hspace{15px} a=340 \hspace{15px} h=15 \hspace{15px} t=?

y=6a=340h=15t=?

6=340(.5)^{\frac{t}{15}}

6=340(.5)

15

t

​

\frac{6}{340}=\frac{340(.5)^{\frac{t}{15}}}{340}

340

6

​

=

340

340(.5)

15

t

​

​

0.0176471=(.5)^{\frac{t}{15}}

0.0176471=(.5)

15

t

​

\log(0.0176471)=\log((.5)^{\frac{t}{15}})

log(0.0176471)=log((.5)

15

t

​

)

\log(0.0176471)=\frac{t}{15}\log(.5)

log(0.0176471)=

15

t

​

log(.5)

Power Rule.

15\log(0.0176471)=t\log(.5)

15log(0.0176471)=tlog(.5)

Multiply by 15.

\frac{15\log(0.0176471)}{\log(.5)}=\frac{t\log(.5)}{\log(.5)}

log(.5)

15log(0.0176471)

​

=

log(.5)

tlog(.5)

​

Divide by log(.5).

t=\frac{-26.299915}{-0.30103}

t=

−0.30103

−26.299915

​

t=87.36643\approx 87.4 \text{ minutes}

t=87.36643≈87.4 minutes