Math  /  Algebra

QuestionThe half-life of a certain type of soft drink is 7 hours. If a person drinks 51 millifters of this drink, the formula y=51(0.5)17y=51(0.5)^{\frac{1}{7}} tells the amount of the drink left in the person's system after thours. How m the soft drink is in the person's system after 8 hours?
The person's system contains \square \square of the soft drink. (Type an integer or decimal rounded to the nearest tenth as needed.)

Studdy Solution

STEP 1

What is this asking? How much of a special soft drink is left in someone's body after 8 hours, if it starts at 51 milliliters and halves every 7 hours? Watch out! Don't mix up the starting amount, the time passed, and the half-life!

STEP 2

1. Plug in the Time
2. Calculate the Exponent
3. Calculate the Decay Factor
4. Calculate the Final Amount

STEP 3

Alright, so we've got this formula y=51(0.5)t7y = 51(0.5)^{\frac{t}{7}}, where yy is the amount of soft drink left and tt is the time in hours.
We want to know how much is left after **8 hours**, so let's **plug in** t=8t = 8.

STEP 4

This gives us y=51(0.5)87y = 51(0.5)^{\frac{8}{7}}.
See how we just **replaced** the tt with the **8**?
Awesome!

STEP 5

Now, let's **calculate** that exponent: 87\frac{8}{7}.
This tells us how many **half-lives** have passed.

STEP 6

871.14\frac{8}{7} \approx 1.14.
So, a little more than one half-life has gone by.

STEP 7

Next, we **calculate** the **decay factor**: (0.5)1.14(0.5)^{1.14}.
This tells us what fraction of the soft drink is left after 1.14 half-lives.

STEP 8

(0.5)1.140.46(0.5)^{1.14} \approx 0.46.
This means about **46%** of the soft drink is still in the person's system.
Makes sense since it's been a bit more than one half-life!

STEP 9

Finally, we **multiply** the **starting amount** by the **decay factor** to find out how much soft drink is left: 510.4651 \cdot 0.46.

STEP 10

510.4623.551 \cdot 0.46 \approx 23.5.
So, after 8 hours, there's about **23.5 milliliters** of the soft drink left in the person's system.

STEP 11

The person's system contains **23.5** milliliters of the soft drink.

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