The radioactive substance uranium-240 has a half-life of 14 hours. The amount of a sample of uranium- 240
remaining (in grams) after hours is given by the following exponential function.

Find the initial amount in the sample and the amount remaining after 30 hours.
Round your answers to the nearest gram as necessary.
Initial amount:
Amount after 30 hours: grams

Ask by Sanders Harper. in the United States

Mar 15,2025

Question

The radioactive substance uranium-240 has a half-life of 14 hours. The amount of a sample of uranium- 240
remaining (in grams) after hours is given by the following exponential function.

Find the initial amount in the sample and the amount remaining after 30 hours.
Round your answers to the nearest gram as necessary.
Initial amount:
Amount after 30 hours: grams

Ask by Sanders Harper. in the United States

Mar 15,2025

UpStudy AI · Accepted Answer

To find the initial amount in the sample and the amount remaining after 30 hours, we need to use the given exponential function $$ A(t) = 5600\left(\frac{1}{2}ight)^{\frac{t}{14}} $$.

1. To find the initial amount in the sample, we need to find $$ A(0) $$ by substituting $$ t = 0 $$ into the function.

2. To find the amount remaining after 30 hours, we need to find $$ A(30) $$ by substituting $$ t = 30 $$ into the function.

Let's calculate these values step by step.
Evaluate the expression by following steps:
- step0: Evaluate:
  $$5600\left(\frac{1}{2}ight)^{\frac{t}{14}}$$
- step1: Substitute:
  $$5600\left(\frac{1}{2}ight)^{\frac{0}{14}}$$
- step2: Divide the terms:
  $$5600\left(\frac{1}{2}ight)^{0}$$
- step3: Evaluate the power:
  $$5600	imes 1$$
- step4: Multiply:
  $$5600$$
Substitute $$ t = 0 $$ into the function $$ A(t) = 5600\left(\frac{1}{2}ight)^{\frac{t}{14}} $$ to find the initial amount in the sample.

Now, let's calculate the amount remaining after 30 hours by substituting $$ t = 30 $$ into the function.
Evaluate the expression by following steps:
- step0: Evaluate:
  $$5600\left(\frac{1}{2}ight)^{\frac{t}{14}}$$
- step1: Substitute:
  $$5600\left(\frac{1}{2}ight)^{\frac{30}{14}}$$
- step2: Reduce the fraction:
  $$5600\left(\frac{1}{2}ight)^{\frac{15}{7}}$$
- step3: Evaluate the power:
  $$5600	imes \frac{\sqrt[7]{64}}{8}$$
- step4: Multiply:
  $$700\sqrt[7]{64}$$
Calculate or simplify the expression $$ 700\sqrt[7]{64} $$.
Calculate the value by following steps:
- step0: Calculate:
  $$700\sqrt[7]{64}$$
Calculate or simplify the expression $$ 1268.01313 $$.
Numbers by following steps:
- step0: Solution:
  $$1268.01313$$
The initial amount in the sample is 5600 grams.

The amount remaining after 30 hours is approximately 1268 grams.
Initial amount: 5600 grams Amount after 30 hours: 1268 grams

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