Mathematics, 14.02.2020 19:33 JarretteWhitaker

The concentration, C, in ng/ml, of a drug in the blood as a function of the time, t, in hours since the drug was administered is given by C = 15te−0.2t. The area under the concentration curve is a measure of the overall effect of the drug on the body, called the bioavailability. Find the bioavailability of the drug between t =0 and t =3

Answers: 3

Show answers

Answers

Answer from: kahnerp6448

$\displaystyle \int\limits^3_0 {15te^{-0.2t}} \, dt = 45.713 \ \text{ng/mL}$

General Formulas and Concepts:

Calculus

Differentiation

DerivativesDerivative Notation

Derivative Property [Multiplied Constant]: $\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)$

Basic Power Rule:

f(x) = cxⁿf’(x) = c·nxⁿ⁻¹

Integration

Integrals

Integration Rule [Fundamental Theorem of Calculus 1]: $\displaystyle \int\limits^b_a {f(x)} \, dx = F(b) - F(a)$

Integration Property [Multiplied Constant]: $\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx$

U-Substitution

Integration by Parts: $\displaystyle \int {u} \, dv = uv - \int {v} \, du$

[IBP] LIPET: Logs, inverses, Polynomials, Exponentials, Trig

Area of a Region Formula: $\displaystyle A = \int\limits^b_a {[f(x) - g(x)]} \, dx$

Step-by-step explanation:

*Note:

It is given that the area under the concentration curve is equal to the bioavailability.

Step 1: Define

Identify

$\displaystyle C = 15te^{-0.2t} \\\left[ 0 ,\ 3 \right]$

Step 2: Integrate Pt. 1

Substitute in variables [Area of a Region Formula]: $\displaystyle \int\limits^3_0 {15te^{-0.2t}} \, dt$ [Integral] Rewrite [Integration Property - Multiplied Constant]: $\displaystyle \int\limits^3_0 {15te^{-0.2t}} \, dt = 15 \int\limits^3_0 {te^{-0.2t}} \, dt$

Step 3: Integrate Pt. 2

Identify variables for integration by parts using LIPET.

Set u: $\displaystyle u = t$ [u] Differentiate [Basic Power Rule]: $\displaystyle du = dt$ Set dv: $\displaystyle dv = e^{-0.2t} \ dt$ [dv] Exponential Integration [U-Substitution]: $\displaystyle v = \frac{-e^{-0.2t}}{0.2}$

Step 4: Integrate Pt. 3

[Integral] Integration by Parts: $\displaystyle \int\limits^3_0 {15te^{-0.2t}} \, dt = 15 \Bigg[ \frac{-te^{-0.2t}}{0.2} \bigg| \limits^3_0 - \int\limits^3_0 {\frac{-e^{-0.2t}}{0.2}} \, dt \Bigg]$ [Integral] Rewrite [Integration Property - Multiplied Constant]: $\displaystyle \int\limits^3_0 {15te^{-0.2t}} \, dt = 15 \Bigg[ \frac{-te^{-0.2t}}{0.2} \bigg| \limits^3_0 + \frac{1}{0.2} \int\limits^3_0 {e^{-0.2t}} \, dt \Bigg]$

Step 5: Integrate Pt. 4

Identify variables for u-substitution.

Set u: $\displaystyle u = -0.2t$ [u] Differentiation [Basic Power Rule, Multiplied Constant]: $\displaystyle du = -0.2 \ dt$ [Limits] Switch: $\displaystyle \left \{ {{t = 3 ,\ u = -0.2(3) = -0.6} \atop {t = 0 ,\ u = -0.2(0) = 0}} \right.$

Step 6: Integrate Pt. 5

[Integral] Rewrite [Integration Property - Multiplied Constant]: $\displaystyle \int\limits^3_0 {15te^{-0.2t}} \, dt = 15 \Bigg[ \frac{-te^{-0.2t}}{0.2} \bigg| \limits^3_0 - \frac{1}{0.04} \int\limits^3_0 {-0.2e^{-0.2t}} \, dt \Bigg]$ [Integral] U-Substitution: $\displaystyle \int\limits^3_0 {15te^{-0.2t}} \, dt = 15 \Bigg[ \frac{-te^{-0.2t}}{0.2} \bigg| \limits^3_0 - \frac{1}{0.04} \int\limits^{-0.6}_0 {e^{u}} \, du \Bigg]$ [Integral] Exponential Integration: $\displaystyle \int\limits^3_0 {15te^{-0.2t}} \, dt = 15 \Bigg[ \frac{-te^{-0.2t}}{0.2} \bigg| \limits^3_0 - \frac{1}{0.04}(e^u) \bigg| \limits^{-0.6}_0 \Bigg]$ Evaluate [Integration Rule - Fundamental Theorem of Calculus 1]: $\displaystyle \int\limits^3_0 {15te^{-0.2t}} \, dt = 15 \Bigg[ -8.23217 - \frac{1}{0.04}(-0.451188) \Bigg]$ Simplify: $\displaystyle \int\limits^3_0 {15te^{-0.2t}} \, dt = 45.713$

Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Integration

Answer from: Quest

step-by-step explanation: a reaction is first order. if the initial reactant concentration is 0.0200 m, and 25.0 days later the concentration is 6.25 x 10-4 m, then its half-life is:

Answer from: Quest

Ithink if ur looking for a ratio it would be 12: 3

Another question on Mathematics

Mathematics, 21.06.2019 15:30

Consider the system: y = 3x + 5 y = ax + b what values for a and b make the system inconsistent? what values for a and b make the system consistent and dependent? explain

Answers: 1

Answer

Mathematics, 21.06.2019 18:00

Solve this and show you’re work step by step ! -5 3/4+3h< 9 1/4 -

Answers: 1

Answer

Mathematics, 21.06.2019 18:30

The measure of one angle of an octagon is two times smaller that of the other seven angles. what is the measure of each angle?

Answers: 3

Answer

Mathematics, 21.06.2019 23:30

Afactory buys 10% of its components from suppliers b and the rest from supplier c. it is known that 6% of the components it buys are faulty. of the components brought from suppliers a,9% are faulty and of the components bought from suppliers b, 3% are faulty. find the percentage of components bought from supplier c that are faulty.

Answers: 1

Answer

You know the right answer?

The concentration, C, in ng/ml, of a drug in the blood as a function of the time, t, in hours since...

Questions

Mathematics, 18.08.2021 01:50

Una fracción multiplicada por 2 resulta 4 /15. ¿Por cuánto hay que dividir dicha fracción para obtener 1/3 ? * 5 puntos 2/15 -2/5 -2/15 2/5...

Mathematics, 18.08.2021 01:50

A bakery van has an original value of $40,000. The van is depreciating at a rate of 9% each year. About how much will this van be worth 5 years later...

Mathematics, 18.08.2021 01:50

marvin says that there are more white triangles than black squares do you agree or disagree explain your reasoning...