### 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 ๐ถ in their system ๐ก hours later can be found using the equation ๐ถ is equal to 30 times one-half raised to the power of ๐ก divided by 5.7. Write the equation in the form ๐ถ is equal to ๐ด times ๐ raised to the power of ๐ก, 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. Weโre told in healthy adults the half-life of caffeine will be approximately 5.7 hours. Weโre then given an example of this in action. Weโre 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 weโre concerned about is a can of cola which contains 30 milligrams of caffeine. Weโre then given an equation which tells us the amount of caffeine left in their system ๐ถ in milligrams ๐ก hours after they drunk the can of cola. Weโre told ๐ถ is equal to 30 times one-half raised to the power of ๐ก divided by 5.7.

In fact, we couldโve formulated this equation ourself. However, in this case, itโs given to us. We just need to rewrite this equation in the form ๐ถ is equal to ๐ด times ๐ to the power of ๐ก. And we need to give any values necessary to three decimal places.

To start, weโll rewrite one-half as 0.5 in our equation, giving us ๐ถ is equal to 30 times 0.5 raised to the power of ๐ก divided by 5.7. We need to rewrite this equation in the form ๐ถ is equal to ๐ด times ๐ raised to the power of ๐ก. And in fact, we can see our equation is almost in this form. However, our exponent is not just ๐ก. We have ๐ก divided by 5.7.

To rewrite this equation in this form, weโre going to need to recall one of our laws of exponents. ๐ฅ raised to the power of ๐ฆ times ๐ง is equal to ๐ฅ raised to the power of ๐ฆ all raised to the power of ๐ถ. Weโre going to use this to rewrite our equation in the given form. We just need to notice our exponent of ๐ก divided by 5.7 can be rewritten as one over 5.7 all multiplied by ๐ก. This means we could rewrite our equation as 30 times 0.5 raised to the power of one over 5.7 times ๐ก.

Now, all we need to do is apply our laws of exponents. We want our value of ๐ฅ equal to 0.5, our value of ๐ฆ equal to one over 5.7, and our value of ๐ง equal to ๐ก. So by using this law of exponents, weโve rewritten our equation as 30 times 0.5 raised to the power of one over 5.7 all raised to the power of ๐ก.

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 itโs 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 ๐ถ is approximately equal to 30 times 0.885 raised to the power of ๐ก. 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, ๐ถ is equal to 30 times one-half raised to the power of ๐ก over 5.7, into the form ๐ถ is equal to ๐ด times ๐ raised to the power of ๐ก. We found that ๐ถ is approximately equal to 30 times 0.885 raised to the power of ๐ก.