Question Video: Converting between Different Forms of Exponential Expressions Mathematics • Higher Education

Video Transcript

The results of a medical study showed that, in healthy adults, the half-life of caffeine is 5.7 hours. So if an adult consumes 250 milligrams of caffeine in their breakfast coffee at 6 AM, they will have approximately 125 milligrams of caffeine in their system at 11:40 AM. If a person drinks a can of cola containing 30 milligrams of caffeine, the amount of caffeine 𝐶 in their system 𝑡 hours later can be found using the equation 𝐶 is equal to 30 times one-half raised to the power of 𝑡 divided by 5.7. Write the equation in the form 𝐶 is equal to 𝐴 times 𝑏 raised to the power of 𝑡, giving values to three decimal places if necessary.

The question gives us a real word problem in terms of the half-life of caffeine in adults. We’re told in healthy adults the half-life of caffeine will be approximately 5.7 hours. We’re then given an example of this in action. We’re told if an adult consumes 250 milligrams of caffeine in their breakfast coffee at 6 AM, then after approximately 5.7 hours, they will have 125 milligrams of caffeine left in their system. This will approximately be at 11:40 AM.

And the problem we’re concerned about is a can of cola which contains 30 milligrams of caffeine. We’re then given an equation which tells us the amount of caffeine left in their system 𝐶 in milligrams 𝑡 hours after they drunk the can of cola. We’re told 𝐶 is equal to 30 times one-half raised to the power of 𝑡 divided by 5.7.

In fact, we could’ve formulated this equation ourself. However, in this case, it’s given to us. We just need to rewrite this equation in the form 𝐶 is equal to 𝐴 times 𝑏 to the power of 𝑡. And we need to give any values necessary to three decimal places.

To start, we’ll rewrite one-half as 0.5 in our equation, giving us 𝐶 is equal to 30 times 0.5 raised to the power of 𝑡 divided by 5.7. We need to rewrite this equation in the form 𝐶 is equal to 𝐴 times 𝑏 raised to the power of 𝑡. And in fact, we can see our equation is almost in this form. However, our exponent is not just 𝑡. We have 𝑡 divided by 5.7.

To rewrite this equation in this form, we’re going to need to recall one of our laws of exponents. 𝑥 raised to the power of 𝑦 times 𝑧 is equal to 𝑥 raised to the power of 𝑦 all raised to the power of 𝐶. We’re going to use this to rewrite our equation in the given form. We just need to notice our exponent of 𝑡 divided by 5.7 can be rewritten as one over 5.7 all multiplied by 𝑡. This means we could rewrite our equation as 30 times 0.5 raised to the power of one over 5.7 times 𝑡.

Now, all we need to do is apply our laws of exponents. We want our value of 𝑥 equal to 0.5, our value of 𝑦 equal to one over 5.7, and our value of 𝑧 equal to 𝑡. So by using this law of exponents, we’ve rewritten our equation as 30 times 0.5 raised to the power of one over 5.7 all raised to the power of 𝑡.

Now, all we need to do is evaluate zero raised to the power of one over 5.7. Remember, we can give our answer to three decimal places if it’s necessary. And in fact, it is necessary. If we evaluate 0.5 raised to the power of one over 5.7 to three decimal places, we get 0.885. So our equation simplifies to give us 𝐶 is approximately equal to 30 times 0.885 raised to the power of 𝑡. And this is our final answer.

Therefore, given a real word problem about the half-life of caffeine in adults, we were able to rearrange an equation given to us, 𝐶 is equal to 30 times one-half raised to the power of 𝑡 over 5.7, into the form 𝐶 is equal to 𝐴 times 𝑏 raised to the power of 𝑡. We found that 𝐶 is approximately equal to 30 times 0.885 raised to the power of 𝑡.