Math / Calculus

QuestionThe half-life of a drug in the bloodstream is 9 hours. What fraction of the original drug dose remains in 12 hours? in 24 hours?
What fraction of the original drug dose remains in 12 hours? $\square$ (Do not round until the final answer. Then round to the nearest hundredth as needed.)

Studdy Solution

STEP 1

1. The half-life of the drug is 9 hours.
2. We need to calculate the fraction of the original drug dose remaining after 12 hours.
3. The decay of the drug follows an exponential decay model.

STEP 2

1. Understand the concept of half-life and its implications for exponential decay.
2. Develop the exponential decay formula based on the half-life.
3. Calculate the fraction of the drug remaining after 12 hours.
4. Round the final answer to the nearest hundredth.

STEP 3

The half-life of a substance is the time required for half of the substance to decay or be eliminated from the system. For this problem, every 9 hours, the amount of the drug in the bloodstream is reduced by half.

STEP 4

The exponential decay formula can be written as: $N(t) = N_0 \left(\frac{1}{2}\right)^{\frac{t}{T}}$ where $N(t)$ is the amount remaining at time $t$ , $N_0$ is the initial amount, and $T$ is the half-life.

STEP 5

Substitute the given values into the formula: - $t = 12$ hours - $T = 9$ hours
$N(12) = N_0 \left(\frac{1}{2}\right)^{\frac{12}{9}}$

STEP 6

Simplify the exponent: $\frac{12}{9} = \frac{4}{3}$
$N(12) = N_0 \left(\frac{1}{2}\right)^{\frac{4}{3}}$

STEP 7

Calculate the fraction: $\left(\frac{1}{2}\right)^{\frac{4}{3}} \approx 0.39685$

STEP 8

Round the final answer to the nearest hundredth: $0.39685 \approx 0.40$
The fraction of the original drug dose remaining in 12 hours is:
$\boxed{0.40}$

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QuestionThe half-life of a drug in the bloodstream is 9 hours. What fraction of the original drug dose remains in 12 hours? in 24 hours?
What fraction of the original drug dose remains in 12 hours? $\square$ (Do not round until the final answer. Then round to the nearest hundredth as needed.)

Studdy Solution

STEP 1

1. The half-life of the drug is 9 hours.
2. We need to calculate the fraction of the original drug dose remaining after 12 hours.
3. The decay of the drug follows an exponential decay model.

STEP 2

1. Understand the concept of half-life and its implications for exponential decay.
2. Develop the exponential decay formula based on the half-life.
3. Calculate the fraction of the drug remaining after 12 hours.
4. Round the final answer to the nearest hundredth.

STEP 3

The half-life of a substance is the time required for half of the substance to decay or be eliminated from the system. For this problem, every 9 hours, the amount of the drug in the bloodstream is reduced by half.

STEP 4

The exponential decay formula can be written as: $N(t) = N_0 \left(\frac{1}{2}\right)^{\frac{t}{T}}$ where $N(t)$ is the amount remaining at time $t$ , $N_0$ is the initial amount, and $T$ is the half-life.

STEP 5

Substitute the given values into the formula: - $t = 12$ hours - $T = 9$ hours
$N(12) = N_0 \left(\frac{1}{2}\right)^{\frac{12}{9}}$

STEP 6

Simplify the exponent: $\frac{12}{9} = \frac{4}{3}$
$N(12) = N_0 \left(\frac{1}{2}\right)^{\frac{4}{3}}$

STEP 7

Calculate the fraction: $\left(\frac{1}{2}\right)^{\frac{4}{3}} \approx 0.39685$

STEP 8

Round the final answer to the nearest hundredth: $0.39685 \approx 0.40$
The fraction of the original drug dose remaining in 12 hours is:
$\boxed{0.40}$

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QuestionThe half-life of a drug in the bloodstream is 9 hours. What fraction of the original drug dose remains in 12 hours? in 24 hours?What fraction of the original drug dose remains in 12 hours? □\square□ (Do not round until the final answer. Then round to the nearest hundredth as needed.)

Studdy Solution

STEP 1

1. The half-life of the drug is 9 hours.2. We need to calculate the fraction of the original drug dose remaining after 12 hours.3. The decay of the drug follows an exponential decay model.

STEP 2

1. Understand the concept of half-life and its implications for exponential decay.2. Develop the exponential decay formula based on the half-life.3. Calculate the fraction of the drug remaining after 12 hours.4. Round the final answer to the nearest hundredth.

STEP 3

The half-life of a substance is the time required for half of the substance to decay or be eliminated from the system. For this problem, every 9 hours, the amount of the drug in the bloodstream is reduced by half.

STEP 4

The exponential decay formula can be written as: N(t)=N0(12)tT N(t) = N_0 \left(\frac{1}{2}\right)^{\frac{t}{T}} N(t)=N0​(21​)Tt​ where N(t) N(t) N(t) is the amount remaining at time t t t, N0 N_0 N0​ is the initial amount, and T T T is the half-life.

STEP 5

Substitute the given values into the formula: - t=12 t = 12 t=12 hours - T=9 T = 9 T=9 hoursN(12)=N0(12)129 N(12) = N_0 \left(\frac{1}{2}\right)^{\frac{12}{9}} N(12)=N0​(21​)912​

STEP 6

Simplify the exponent: 129=43 \frac{12}{9} = \frac{4}{3} 912​=34​N(12)=N0(12)43 N(12) = N_0 \left(\frac{1}{2}\right)^{\frac{4}{3}} N(12)=N0​(21​)34​

STEP 7

Calculate the fraction: (12)43≈0.39685 \left(\frac{1}{2}\right)^{\frac{4}{3}} \approx 0.39685 (21​)34​≈0.39685

STEP 8

Round the final answer to the nearest hundredth: 0.39685≈0.40 0.39685 \approx 0.40 0.39685≈0.40 The fraction of the original drug dose remaining in 12 hours is:0.40 \boxed{0.40} 0.40​

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QuestionThe half-life of a drug in the bloodstream is 9 hours. What fraction of the original drug dose remains in 12 hours? in 24 hours?
What fraction of the original drug dose remains in 12 hours? $\square$ (Do not round until the final answer. Then round to the nearest hundredth as needed.)

1. The half-life of the drug is 9 hours.
2. We need to calculate the fraction of the original drug dose remaining after 12 hours.
3. The decay of the drug follows an exponential decay model.

1. Understand the concept of half-life and its implications for exponential decay.
2. Develop the exponential decay formula based on the half-life.
3. Calculate the fraction of the drug remaining after 12 hours.
4. Round the final answer to the nearest hundredth.

The exponential decay formula can be written as: $N(t) = N_0 \left(\frac{1}{2}\right)^{\frac{t}{T}}$ where $N(t)$ is the amount remaining at time $t$ , $N_0$ is the initial amount, and $T$ is the half-life.

Substitute the given values into the formula: - $t = 12$ hours - $T = 9$ hours
$N(12) = N_0 \left(\frac{1}{2}\right)^{\frac{12}{9}}$

Simplify the exponent: $\frac{12}{9} = \frac{4}{3}$
$N(12) = N_0 \left(\frac{1}{2}\right)^{\frac{4}{3}}$

Calculate the fraction: $\left(\frac{1}{2}\right)^{\frac{4}{3}} \approx 0.39685$

Round the final answer to the nearest hundredth: $0.39685 \approx 0.40$
The fraction of the original drug dose remaining in 12 hours is:
$\boxed{0.40}$