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ASU Prep

G04 Reflection Assignment: A

Evaluating an Exponential Function that Models a Real-World Situation

The radioactive substance uranium-240 has a half-life of 14 hours. The amount [tex]$A(t)$[/tex] of a sample of uranium-240 remaining (in grams) after [tex]$t$[/tex] hours is given by the following exponential function:

[tex]\[ A(t) = 4700\left(\frac{1}{2}\right)^{\frac{t}{14}} \][/tex]

Find the initial amount in the sample and the amount remaining after 60 hours. Round your answers to the nearest gram as necessary.

Initial amount: [tex]$\square$[/tex] grams

Amount after 60 hours: [tex]$\square$[/tex] grams


Sagot :

To solve the problem of finding the initial amount and the remaining amount of uranium-240 after 60 hours, let's break it down step-by-step using the given exponential function.

The function is:

[tex]\[ A(t) = 4700 \left(\frac{1}{2}\right)^{\frac{t}{14}} \][/tex]

### Step 1: Finding the Initial Amount

The initial amount is the amount when time [tex]\( t = 0 \)[/tex].

Substitute [tex]\( t = 0 \)[/tex] into the function:

[tex]\[ A(0) = 4700 \left(\frac{1}{2}\right)^{\frac{0}{14}} \][/tex]

Simplify the exponent:

[tex]\[ A(0) = 4700 \left(\frac{1}{2}\right)^0 \][/tex]

Any number raised to the power of 0 is 1:

[tex]\[ A(0) = 4700 \cdot 1 \][/tex]

So, the initial amount is:

[tex]\[ \boxed{4700} \, \text{grams} \][/tex]

### Step 2: Finding the Amount Remaining After 60 Hours

Now, we need to find the amount remaining after [tex]\( t = 60 \)[/tex] hours.

Substitute [tex]\( t = 60 \)[/tex] into the function:

[tex]\[ A(60) = 4700 \left(\frac{1}{2}\right)^{\frac{60}{14}} \][/tex]

First, calculate the exponent:

[tex]\[ \frac{60}{14} \approx 4.2857 \][/tex]

Then, evaluate the exponential expression [tex]\(\left(\frac{1}{2}\right)^{4.2857}\)[/tex]:

[tex]\[ \left(\frac{1}{2}\right)^{4.2857} \approx 0.0513 \][/tex]

Now multiply this value by the initial amount:

[tex]\[ A(60) = 4700 \cdot 0.0513 \approx 240.97 \][/tex]

When rounded to the nearest gram, we get:

[tex]\[ A(60) \approx 241 \, \text{grams} \][/tex]

So, the amount remaining after 60 hours is:

[tex]\[ \boxed{241} \, \text{grams} \][/tex]

### Final Answers

- Initial amount: [tex]\(\boxed{4700} \, \text{grams}\)[/tex]
- Amount after 60 hours: [tex]\(\boxed{241} \, \text{grams}\)[/tex]

This detailed step-by-step explanation ensures proper understanding of how the initial and remaining amounts are computed using the given exponential decay function.
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