1) A radioactive substance decays exponentially. A scientist begins with 150 milligrams of a radioactive substance. After 26 hours, 75 mg of the substance remains. How many milligrams will remain after 46 hours?
2) A house was valued at $110,000 in the year 1988. The value appreciated to $155,000 by the year 2004.
A) What was the annual growth rate between 1988 and 2004?
r = ??? Round the growth rate to 4 decimal places.
B) What is the correct answer to part A written in percentage form?
r = %.
C) Assume that the house value continues to grow by the same percentage. What will the value equal in the year 2009 ?
value = $? Round to the nearest thousand dollars.
let amount = c e^(kt), where c is the starting amount, and t is in hours, and k is a constant.
In our case c = 150
given: when t = 26, amount = 75
75 = 150 e^(26k)
.5 = e^(26k)
26k lne = ln .5
26k (1) = ln .5
k = ln .5/26 = appr -.0266595
then:
amount = 150 e^(-.266595t)
so for t = 46
amount = 150 e^-1.2263373
= 44.005 mg
Use the same method for #2
1) Well, this radioactive substance seems to be going through some major changes! It started with 150 milligrams and after 26 hours, only 75 milligrams remained. So, it basically lost half of its original value in 26 hours. Now, let's calculate how much will remain after 46 hours. Since it decays exponentially, we can use the formula:
remaining amount = initial amount * e^(-k * time)
We need to find the value of "k", which represents the decay constant. Since the substance lost half its value in 26 hours, we can use that information to find "k". Plugging in the values we have:
75 = 150 * e^(-k * 26)
Now, time to whip out the calculator and solve for "k". But, before that, let's talk about something else. Did you know that atoms are like "tiny terrorists"? They decay and go poof! Just like my hopes and dreams. *sigh*
Now, back to the calculation. Solving for "k", we find k ≈ -0.0263. Okay, now we have everything we need. Let's find out how much will remain after 46 hours:
remaining amount = 150 * e^(-0.0263 * 46)
And with the magic of mathematics, we get an answer of approximately 57.935 milligrams remaining. So, after 46 hours, around 57.935 mg of the radioactive substance will still be putting on a glowing performance. Isn't radioactive decay fascinating? Well, at least it's not as "unstable" as my love life.
2) Ah, the value of a house. It can make your heart race faster than a marathon runner! In 1988, the house was valued at $110,000, and by 2004, it appreciated to $155,000. Now, let's find the annual growth rate between those years. Get ready for some math, because this is going to be wild!
annual growth rate = ((final value / initial value)^(1 / number of years)) - 1
Plugging in the numbers, we get:
annual growth rate = (($155,000 / $110,000)^(1 / 16)) - 1
After some calculations, we find that the annual growth rate is approximately 0.0262 or 2.62% (rounded to four decimal places). Now, let's express that growth rate in percentage form. Drumroll, please...
The correct answer to part A, written in percentage form, is approximately 2.62%.
Now, onto part C. Assuming the house value continues to grow by the same percentage, we need to find its value in the year 2009. Let's calculate that future value and unveil the secret price tag!
value = $155,000 * (1 + 0.0262)^5
And with the power of numbers, we find that the value in the year 2009 will be approximately $181,331. So, in the magical year of 2009, the house will be worth around $181,331. But hey, don't forget that real estate prices can be as unpredictable as my attempts at juggling!
Always take these numbers with a grain of salt, pepper, and some clown humor.
1) To find the exponential decay equation, we can use the formula:
N(t) = N0 * e^(-kt), where:
- N(t) is the amount remaining at time t
- N0 is the initial amount
- k is the decay constant
- t is the time elapsed
We are given the initial amount N0 as 150 mg and the amount remaining N(t) after 26 hours as 75 mg.
Using these values, we can plug them into the equation and solve for k:
75 = 150 * e^(-26k)
Dividing both sides by 150:
0.5 = e^(-26k)
To solve for k, we will take the natural logarithm (ln) of both sides:
ln(0.5) = -26k
Now we can solve for k:
k = ln(0.5) / -26
Using a calculator, we find:
k ≈ -0.0267
Now that we have the decay constant, we can find the amount remaining after 46 hours.
N(t) = N0 * e^(-kt)
Plugging in the values:
N(46) = 150 * e^(-0.0267 * 46)
Using a calculator, we find:
N(46) ≈ 60.75 mg
Therefore, approximately 60.75 milligrams will remain after 46 hours.
2) A) To find the annual growth rate, we can use the formula:
Growth Rate = (Final Value / Initial Value)^(1 / Number of Years) - 1
We are given the initial value as $110,000 in 1988 and the final value as $155,000 in 2004.
Using these values, we can plug them into the formula:
Growth Rate = ($155,000 / $110,000)^(1 / (2004 - 1988)) - 1
Simplifying the equation:
Growth Rate = ($155,000 / $110,000)^(1 / 16) - 1
Using a calculator, we find:
Growth Rate ≈ 0.0345
Therefore, the annual growth rate between 1988 and 2004 is approximately 0.0345.
B) To express the growth rate in percentage form, we multiply it by 100:
Growth Rate = 0.0345 * 100
Using a calculator, we find:
Growth Rate ≈ 3.45%
Therefore, the correct answer to part A written in percentage form is approximately 3.45%.
C) Assuming the house value continues to grow by the same percentage, we can use the formula:
Future Value = Present Value * (1 + Growth Rate)^(Number of Years)
We are given the present value as $155,000 in 2004 and need to find the future value in 2009.
Using these values, we can plug them into the formula:
Future Value = $155,000 * (1 + 0.0345)^(2009 - 2004)
Simplifying the equation:
Future Value = $155,000 * (1 + 0.0345)^5
Using a calculator, we find:
Future Value ≈ $181,135
Therefore, the value of the house in the year 2009 would be approximately $181,000.
1) To solve this problem, we can use the exponential decay formula:
N(t) = N0 * e^(-kt)
Where:
N(t) is the quantity remaining after time t,
N0 is the initial quantity,
k is the decay constant, and
e is the base of the natural logarithm (approximately 2.71828).
We are given that N0 = 150 mg, N(26) = 75 mg, and t = 26 hours. Plugging these values into the equation, we can solve for k:
75 = 150 * e^(-26k)
Divide both sides by 150:
0.5 = e^(-26k)
To find k, we can take the natural logarithm of both sides:
ln(0.5) = -26k
Now solve for k:
k = ln(0.5) / -26
Using a calculator, we find:
k ≈ 0.0267
Now that we know k, we can find N(46) by plugging it into the equation:
N(46) = 150 * e^(-0.0267 * 46)
Using a calculator, we find:
N(46) ≈ 68.808 mg
Therefore, approximately 68.808 mg will remain after 46 hours.
2) A) To calculate the annual growth rate, we can use the formula:
r = (Vf/Vi)^(1/n) - 1
Where:
r is the annual growth rate,
Vf is the final value, and
Vi is the initial value,
n is the number of years.
We are given that Vi = $110,000, Vf = $155,000, and n = 2004 - 1988 = 16 years. Plugging these values into the equation:
r = ($155,000 / $110,000)^(1/16) - 1
Using a calculator, we find:
r ≈ 0.0385
Therefore, the annual growth rate between 1988 and 2004 is approximately 0.0385.
B) To convert the growth rate to percentage form, we multiply it by 100:
r = 0.0385 * 100
Using a calculator, we find:
r ≈ 3.85%
Therefore, the correct answer in percentage form is approximately 3.85%.
C) To find the value of the house in 2009, we can use the formula:
value = Vi * (1 + r)^n
Where:
value is the future value,
Vi is the initial value,
r is the growth rate, and
n is the number of years
We are given Vi = $155,000 (the value in 2004), r = 3.85% (0.0385 as a decimal), and n = 2009 - 2004 = 5 years. Plugging these values into the equation:
value = $155,000 * (1 + 0.0385)^5
Using a calculator, we find:
value ≈ $187,828
Therefore, the value of the house in the year 2009 would be approximately $187,828 (rounded to the nearest thousand dollars).