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(a) Consider the initial-value problem dA/dt = kA, A(0) = A0 as the model for the decay of a radioactive substance. Show that, i

murzikaleks [220]

Answer:

a) $t = -\frac{ln(2)}{k}$

b) See the proof below

$A(t) = A_o 2^{-\frac{t}{T}}$

c) $t = 3T \frac{ln(2)}{ln(2)}= 3T$

Explanation:

Part a

For this case we have the following differential equation:

$\frac{dA}{dt}= kA$

With the initial condition $A(0) = A_o$

We can rewrite the differential equation like this:

$\frac{dA}{A} =k dt$

And if we integrate both sides we got:

$ln |A|= kt + c_1$

Where $c_1$ is a constant. If we apply exponential for both sides we got:

$A = e^{kt} e^c = C e^{kt}$

Using the initial condition $A(0) = A_o$ we got:

$A_o = C$

So then our solution for the differential equation is given by:

$A(t) = A_o e^{kt}$

For the half life we know that we need to find the value of t for where we have $A(t) = \frac{1}{2} A_o$ if we use this condition we have:

$\frac{1}{2} A_o = A_o e^{kt}$

$\frac{1}{2} = e^{kt}$

Applying natural log we have this:

$ln (\frac{1}{2}) = kt$

And then the value of t would be:

$t = \frac{ln (1/2)}{k}$

And using the fact that $ln(1/2) = -ln(2)$ we have this:

$t = -\frac{ln(2)}{k}$

Part b

For this case we need to show that the solution on part a can be written as:

$A(t) = A_o 2^{-t/T}$

For this case we have the following model:

$A(t) = A_o e^{kt}$

If we replace the value of k obtained from part a we got:

$k = -\frac{ln(2)}{T}$

$A(t) = A_o e^{-\frac{ln(2)}{T} t}$

And we can rewrite this expression like this:

$A(t) = A_o e^{ln(2) (-\frac{t}{T})}$

And we can cancel the exponential with the natural log and we have this:

$A(t) = A_o 2^{-\frac{t}{T}}$

Part c

For this case we want to find the value of t when we have remaining $\frac{A_o}{8}$

So we can use the following equation:

$\frac{A_o}{8}= A_o 2^{-\frac{t}{T}}$

Simplifying we got:

$\frac{1}{8} = 2^{-\frac{t}{T}}$

We can apply natural log on both sides and we got:

$ln(\frac{1}{8}) = -\frac{t}{T} ln(2)$

And if we solve for t we got:

$t = T \frac{ln(8)}{ln(2)}$

We can rewrite this expression like this:

$t = T \frac{ln(2^3)}{ln(2)}$

Using properties of natural logs we got:

$t = 3T \frac{ln(2)}{ln(2)}= 3T$

8 0

2 years ago

A pressure cooker contains water and steam in equilibrium at a pressure greater than atmo-spheric pressure. How does this greate

Amanda [17]

Answer:

As the pressure inside the pressure cooker increases as it is heated, the temperature also increases.

Explanation:

A pressure cooker contains water and steam in equilibrium at a pressure greater than atmospheric pressure.

According to the ideal gas law, as the pressure increases, the temperature also increases.

As the pressure inside the pressure cooker increases as it is heated, the temperature also increases.

6 0

3 years ago

In question 8 from “Problems 2”, you calculated that an archery arrow shot with an initial velocity of 45 m/s at an angle of 10

Zigmanuir [339]

The distance traveled by the arrow and horizontal velocity are directly proportional, thus when the distance increases, an increase in initial velocity will allow the arrow to hit the target.

<h3>Time of motion of the projectile</h3>

If the horizontal distance the projectile would travel before hitting the target is 70.67 m, the time of motion of the projectile is calculated as;

X = Vx(t)

t = X/Vx

t = X/Vcosθ

t = (70.67) / (45 x cos10)

t = 1.6 s

When the archery target is moved further away from the archer, the archer needs increase the initial velocity of the arrow assuming the angle of projection is constant.

Since the distance and horizontal velocity are directly proportional to each other, thus when the distance increases, an increase in initial velocity will allow the arrow to hit the target.

Learn more about velocity of projectile here: brainly.com/question/12870645

6 0

1 year ago

Calculate the volume of 10g of helium ( M= 4kg/kmol) at 25C and 600 mmHg

Pachacha [2.7K]

Answer:

T=273+25=298 K

n= m/M = 10/ 4 = 2.5

R=0.08206 L.atm /mol/k

760mmHg = 1 atm therefore

600mmHg = X atm

760 X = 600mmHg

X = 600/760 = 0.789 atm

P = 0.789 atm

V= ?

PV= nRT

0.789 V = 2.5 × 0.08206 × 298

V= 2.5 × 0.08206 ×298 / 0.789

V= 77.48 L

I hope I helped you ^_^

8 0

2 years ago

True or false - The vector for a negative acceleration points in the opposite direction when

alexira [117]

Explanation:

the ratio between two circles vqjoizxcvu g çm

7 0

1 year ago