calculus

The rate at which an amount of a radioactive substance decays is modeled by the differential equation dA/dt = kA, where A is the mass in grams, t is the time in years, and k is a constant. Answer the following.

a) If a 100-gram sample of the radioactive substance decays to 95 grams after 1 year, find an equation that can model the mass of the of the sample after t years.
b) Find the mass of the sample after 50 years.
c) The half-life of a substance is the amount of time it takes for a sample to decay to half its original size. Find the half-life of the radioactive substance.

  1. 👍
  2. 👎
  3. 👁
  1. let the equation be
    A = 100 e^(kt)

    a) if amount = 95
    95 = 100 e^(1k)
    .95 = e^k
    k = ln .95

    so A = 100 e^(ln.95 t)
    when t = 50
    A = 100 e^(50ln.95) = 7.69 g are left

    for half-life time, only 50 g of the original 100g would remain
    50 = 100 e^(ln.95 t)
    .5 = e^(ln.95 t)
    ln.95t = ln.5
    t = ln.5/ln.95 = appr13.5 years

    1. 👍
    2. 👎

Respond to this Question

First Name

Your Response

Similar Questions

  1. calculus

    A sample of a radioactive substance decayed to 94.5% of its original amount after a year. (Round your answers to two decimal places.) (a) What is the half-life of the substance? (b) How long would it take the sample to decay to

  2. precalculus

    Twenty percent of a radioactive substance decays in ten years. By what percent does the substance decay each year?

  3. Calc

    A sample of a radioactive substance decayed to 93.5% of its original amount after a year. a) What is the half-life of the substance? ? years (b) How long would it take the sample to decay to 10% of its original amount? ? years

  4. Calculus-Modeling Equations

    An unknown radioactive element decays into non-radioactive substances. In 420 days the radioactivity of a sample decreases by 39 percent. 1.What is the half-life of the element? 2.How long will it take for a sample of 100 mg to

  1. Exponential Modeling

    The half-life of a radioactive substance is one day, meaning that every day half of the substance has decayed. Suppose you have 100 grams of this substance. How many grams of the substance would be left after a week?

  2. math

    Newton’s law of cooling states that for a cooling substance with initial temperature T0, the temperature T(t) after t minutes can be modeled by the equation T(t)=Ts+(T0−Ts)e−kt, where Ts is the surrounding temperature and k

  3. Calculus

    The radioactive element polonium decays according to the law given below where Q0 is the initial amount and the time t is measured in days. Q(t) = Q0 · 2-(t/140) If the amount of polonium left after 700 days is 45 mg, what was

  4. Math

    A radioactive substance decays exponentially. A scientist begins with 130 milligrams of a radioactive substance. After 20 hours, 65 mg of the substance remains. How many milligrams will remain after 24 hours?

  1. algebra 2

    In the formula A(t) = A0ekt, A is the amount of radioactive material remaining from an initial amount A0 at a given time t, and k is a negative constant determined by the nature of the material. An artifact is discovered at a

  2. precal

    If the half-life of a certain radioactive substance is 2000 years, estimate how many years must elapse before only 35% of the radioactive substance remains?

  3. pre calculus

    if 17 percent of a radioactive substance decays in 12 hours, what is the half-life of the substance? Round your answer to one decimal place.

  4. ap calculus

    suppose that the amount in grams of a radioactive substance present at time t (in years) is given by A (t) = 160e-.70t. Find the rate of decay of the quantity present at the time when t = 4

You can view more similar questions or ask a new question.