1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
olasank [31]
3 years ago
14

A breeder reactor converts uranium-238 into an isotope of plutonium-239 at a rate proportional to the amount of uranium-238 pres

ent at any time. After 10 years, 0.03% of the radioactivity has dissipated (that is, 0.9997 of the initial amount remains). Suppose that initially there is 180 pounds of this substance. Find the half-life. (Round your answer to the nearest whole number.)
Mathematics
1 answer:
topjm [15]3 years ago
7 0

Let U(t) denote the amount of uranium-238 in the reactor at time t. As conversion to plutonium-239 occurs, the amount of uranium will decrease, so the conversion rate is negative. Because the rate is proportional to the current amount of uranium, we have

\dfrac{\mathrm dU}{\mathrm dt}=-kU

where k>0 is constant. Separating variables and integrating both sides gives

\dfrac{\mathrm dU}U=-k\,\mathrm dt\implies\ln|U|=-kt+C\implies U=Ce^{-kt}

Suppose we start some amount u. This means that at time t=0 we have U(0)=u, so that

u=Ce^{-0k}\implies C=u\implies U=ue^{-kt}

We're given that after 10 years, 99.97% of the original amount of uranium remains. This means (if t is taken to be in years) for some starting amount u,

0.9997u=ue^{-10k}\implies k=-\dfrac{\ln(0.9997)}{10}

The half-life is the time t_{1/2} it takes for the starting amount u to decay to half, 0.5u:

0.5u=ue^{-kt_{1/2}}\implies t_{1/2}=-\dfrac{\ln(0.5)}k=\dfrac{10\ln2}{\ln(0.9997)}

or about 23,101 years. Notice that it doesn't matter what the actual starting amount is, the half-life is independent of that.

You might be interested in
Find the area of the triangle below. <br> Be sure to include the correct unit in your answer.
qaws [65]
Is there a picture???????
8 0
3 years ago
What is the perimeter?
Alex_Xolod [135]
9x+3 y I’m in math hope that helps man
8 0
3 years ago
Read 2 more answers
What are the missing operations?
Ronch [10]

Answer:

Solution in photo

Step-by-step explanation:

6 0
3 years ago
in a proportional relationship the ratio y/x is constant. Show that this ratio is not constant for the equation y=a-14
Colt1911 [192]

To show that the ratio \frac{y}{x} is not constant for the function y=a-14 we only need to evaluate this function at 2 points and then show that the ratio is not the same for those two points. Note that here a represents x.  

To find the counter examples here, I will use the values a=20  and a=28. For a = 20 , the equation tells us that the output is

y=20-14=6. For this value of a the point is point is (20,6). The ratio \frac{y}{x} =\frac{6}{20}. For the a=28, y=28-14=14. For this value of a the  point is (28,14) . The ratio \frac{y}{x} =\frac{14}{28} =\frac{1}{2}. Clearly this ratio is not constant for all ordered pairs.

8 0
3 years ago
Triangle JKL is dilated from the origin at a scale factor of 10 to create triangle J′K′L′.Select the option that completes the s
Aneli [31]

Given: Triangle JKL is dilated from the origin at a scale factor of 10 to create triangle J′K′L′.

Required: Correct option that complete the statement.

Explanation:

Now, triangle is dilated from the origin at a scale factor of 10.

So, the sides will become larger, but angles will remain the same.

Hence, the triangles will be similar as sides will be in proption (SSS rule).

So, The triangles are similar because their corresponding sides are proptional and their corresponding angles are congurent.

Final Answer: Second group is correct answer.

5 0
1 year ago
Other questions:
  • Who good at the math
    8·1 answer
  • A random sample of records of electricity usage of homes in the month of July gives the amount of electricity used and size (in
    11·2 answers
  • Find the volume of the sphere to the nearest whole value. The volume is ________ in3
    5·1 answer
  • If the three angles of a triangle have equal measures, find that measure. The measure of each angle is degrees.
    12·1 answer
  • Many places of business will not take a check, but will take a credit card.<br><br> True<br> False
    9·1 answer
  • Stephen is making curtains for several windows in his aunt’s house. He determines that he needs 8.5 yards of fabric for each win
    14·1 answer
  • 4600 X 1.04 to the power of 7?????
    9·2 answers
  • Which shape has a larger area: a rectangle that is 7 inches by 3/4 inch,or a square with side length of 2 and 1/2 inches? Show y
    6·1 answer
  • Colin drank 75% of a 12.8 ounce container of milk. how many ounces of milk did he drink?
    9·1 answer
  • Solve (−7) ⋅ (−4). Please hurry :D<br><br> -28<br> -11<br> 28<br> 11
    7·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!