Exponential and Logarithmic Functions
McGraw-Hill Education Campus

Evaluating an exponential function that models a real-world situation

The radioactive substance uranium-240 has a half-life of 14 hours. The amount $A(t)$ of a sample of uranium-240 remaining (in grams) after $t$ hours is given by the following exponential function.
\[
A(t)=2800\left(\frac{1}{2}\right)^{\frac{t}{14}}
\]

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: [1] grams
Amount after 30 hours:
grams

Step 1 ：The radioactive substance uranium-240 has a half-life of 14 hours. The amount $A(t)$ of a sample of uranium-240 remaining (in grams) after $t$ hours is given by the following exponential function: \[A(t)=2800\left(\frac{1}{2}\right)^{\frac{t}{14}}\]

Step 2 ：We are asked to find the initial amount in the sample and the amount remaining after 30 hours. We can find these by evaluating the function at t=0 and t=30 respectively.

Step 3 ：First, let's find the initial amount by substituting t=0 into the function: \[A(0)=2800\left(\frac{1}{2}\right)^{\frac{0}{14}}\]

Step 4 ：Solving the above equation gives us the initial amount of the sample: \[A(0)=2800\]

Step 5 ：Next, let's find the amount remaining after 30 hours by substituting t=30 into the function: \[A(30)=2800\left(\frac{1}{2}\right)^{\frac{30}{14}}\]

Step 6 ：Solving the above equation gives us the amount remaining after 30 hours: \[A(30)\approx634\]

Step 7 ：Final Answer: The initial amount is \(\boxed{2800}\) grams and the amount remaining after 30 hours is approximately \(\boxed{634}\) grams.