A substance decays according to the formula , where is in seconds.
(a) What is its half-life? (Do not round off.)
The half-life is Number seconds
(b) By what percent does it decay each second? (Report to 2 decimal places.)
It decays by Number % per second

Ask by Parsons Cole. in the United States

Mar 20,2025

Question

A substance decays according to the formula , where is in seconds.
(a) What is its half-life? (Do not round off.)
The half-life is Number seconds
(b) By what percent does it decay each second? (Report to 2 decimal places.)
It decays by Number % per second

Ask by Parsons Cole. in the United States

Mar 20,2025

UpStudy AI · Accepted Answer

1. **Half-life**

The decay formula is given by  
   $$
   Q = 36\left(0.5\right)^{t/36.7}.
   $$
   The half-life is the time $$ t $$ when $$ Q $$ becomes half of its initial value. Since the initial value is $$ 36 $$, we have  
   $$
   \frac{36}{2} = 18.
   $$
   Substitute $$ t = T_{1/2} $$ in the formula:
   $$
   18 = 36\left(0.5\right)^{T_{1/2}/36.7}.
   $$
   Dividing both sides by $$ 36 $$ gives:
   $$
   0.5 = \left(0.5\right)^{T_{1/2}/36.7}.
   $$
   Since the bases are the same, the exponents must be equal:
   $$
   \frac{T_{1/2}}{36.7} = 1 \quad \Longrightarrow \quad T_{1/2} = 36.7.
   $$
   Therefore, the half-life is $$ 36.7 $$ seconds.

2. **Percent Decay per Second**

The decay per second is found by considering the decay factor when $$ t = 1 $$. For $$ t = 1 $$, the formula becomes:
   $$
   Q(1) = 36\left(0.5\right)^{1/36.7}.
   $$
   The factor of decay per second is:
   $$
   r = \left(0.5\right)^{1/36.7}.
   $$
   Expressing the exponentiation in terms of the natural exponential function:
   $$
   r = \exp\left(\frac{1}{36.7}\ln(0.5)\right).
   $$
   Since $$\ln(0.5) \approx -0.693147$$, we have:
   $$
   r \approx \exp\left(\frac{-0.693147}{36.7}\right) = \exp(-0.0188824) \approx 0.98129.
   $$
   The fraction decayed in one second is:
   $$
   1 - r \approx 1 - 0.98129 = 0.01871.
   $$
   Therefore, the percent decay per second is:
   $$
   0.01871 \times 100\% \approx 1.87\%.
   $$

- (a) The half-life is $$ 36.7 $$ seconds.
- (b) It decays by approximately $$ 1.87\% $$ per second.
- (a) The half-life is 36.7 seconds. - (b) It decays by 1.87% per second.

A substance decays according to the formula , where is in seconds.
(a) What is its half-life? (Do not round off.)
The half-life is Number seconds
(b) By what percent does it decay each second? (Report to 2 decimal places.)
It decays by Number % per second

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A substance decays according to the formula , where is in seconds. (a) What is its half-life? (Do not round off.) The half-life is Number seconds (b) By what percent does it decay each second? (Report to 2 decimal places.) It decays by Number % per second

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A substance decays according to the formula , where is in seconds.
(a) What is its half-life? (Do not round off.)
The half-life is Number seconds
(b) By what percent does it decay each second? (Report to 2 decimal places.)
It decays by Number % per second