The radioactive isotope carbon-14 is present in small quantities in all life forms, and it is constantly replenished until the organism dies, after which it decays to stable carbon-12 at a rate proportional to the amount of carbon-14 present, with a half-life of 5557 years. Suppose C(t) is the amount of carbon-14 present at time t.
(a) Find the value of the constant k in the differential equation C′=−kC.
k=
(b) In 1988 three teams of scientists found that the Shroud of Turin, which was reputed to be the burial cloth of Jesus, contained about 91 percent of the amount of carbon-14 contained in freshly made cloth of the same material[1]. How was old the Shroud of Turin in 1988, according to these data?
Age =
Answers
Therefore, according to the data from 1988, the age of the Shroud of Turin is approximately 20,206,118 years.
(a) To find the value of the constant k in the differential equation C' = (-kC), we can use the fact that carbon-14 has a half-life of 5557 years. The half-life is the time it takes for half of the initial amount of carbon-14 to decay.
Using the formula for exponential decay, we have:
C_(t) = C₀ × e^{-kt},
where C₀ is the initial amount of carbon-14 at time t = 0.
Since the half-life is 5557 years, we know that after 5557 years, the amount of carbon-14 is reduced to half. Therefore, we have:
C_(5557) = C₀ × (1/2) = C₀ × e^{(-k) × 5557}.
Dividing the equation by C₀, we get:
1/2 = e^{(-k) × 5557}.
To solve for k, we take the natural logarithm of both sides:
ln(1/2) = (-k) × 5557.
ln(1/2) is equal to (-ln(2)), so we have:
(-ln(2)) = (-k) × 5557.
Simplifying, we find:
k = ln(2) / 5557.
Therefore, the value of the constant k in the differential equation C' = (-kC) is approximately k ≈ 0.00012427.
(b) In 1988, the Shroud of Turin was found to contain about 91 percent of the amount of carbon-14 contained in freshly made cloth of the same material. We can use this information to determine the age of the Shroud of Turin in 1988.
Let's denote the amount of carbon-14 in the freshly made cloth as C₀ (initial amount), and the amount of carbon-14 in the Shroud of Turin in 1988 as C_(1988).
We know that C_(1988) is 91% of C₀. So we have:
C_(1988) = 0.91 × C₀.
Using the exponential decay formula, we have:
C_(t) = C₀ × e^{-kt}.
Substituting t = 1988 and C_(t) = C_(1988), we get:
C_(1988) = C₀ × e{(-k) × 1988).
Substituting C_(1988) = 0.91 × C₀, we have:
0.91 × C₀ = C₀ × e^{(-k) × 1988}.
Canceling out C₀ on both sides, we get:
0.91 = e^{(-k) × 1988}.
Taking the natural logarithm of both sides, we have:
ln(0.91) = (-k )× 1988.
Solving for k, we find:
k =( -ln(0.91)) / 1988.
Using the previously found value of k ≈ 0.00012427, we can calculate the age of the Shroud of Turin in 1988:
Age = 1988 / k.
Substituting the value of k, we have:
Age ≈ 1988 / (ln(0.91) / 1988).
Age ≈ 1988 × (1988 / ln(0.91)).
Calculating the approximate value, we find:
Age ≈ 1988 × (1988 / (-0.093169)) ≈ (-20,206,118) years.
Therefore, according to the data from 1988, the age of the Shroud of Turin is approximately 20,206,118 years.
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Related Questions
You plan to retire in 30 years. After that, you need $75,000 per year for 20 years (first withdraw at t=31 ). At the end of these 20 years, you will enter a retirement home where you will stay for the rest of your life. As soon as you enter the retirement home, you will need to make a single payment of 2 million. You want to start saving in an account that pays you 8% interest p.a. Therefore, beginning from the end of the first year (t=1), you will make equal yearly deposits into this account for 30 years. You expect to receive $350,000 inheritance at t=30 from your late uncle and you will deposit this money to your retirement account. What should be the yearly deposits?
6587.25
7198.40
8066.36
8744.81
Answers
The yearly deposit needed to achieve the retirement goal is approximately $17,650.23. None of the given options match this amount, so the correct answer is not provided in the given options.
To calculate the yearly deposits needed, we can use the concept of future value of an annuity. The future value formula for an annuity is given by:
FV = P * [(1 + r)^n - 1] / r
Where:
FV = Future value of the annuity
P = Yearly deposit amount
r = Interest rate per period
n = Number of periods
In this case, the future value needed is $2 million, the interest rate is 8% (0.08), and the number of periods is 30 years. We need to solve for the yearly deposit amount (P).
Using the given formula:
2,000,000 = P * [(1 + 0.08)^30 - 1] / 0.08
Simplifying the equation:
2,000,000 = P * [1\(.08^3^0 -\) 1] / 0.08
2,000,000 = P * [10.063899 - 1] / 0.08
2,000,000 = P * 9.063899 / 0.08
Dividing both sides by 9.063899 / 0.08:
P = 2,000,000 / (9.063899 / 0.08)
P ≈ 2,000,000 / 113.298737
P ≈ 17,650.23
Therefore, the yearly deposit needed to achieve the retirement goal is approximately $17,650.23. None of the given options match this amount, so the correct answer is not provided in the given options.
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based on the experimental probability about how many times will yosef draw a green marble if he draws a total of 75 marbles
Answers
The four small circles are identical. The total area of the small circles is 36π square inches. What is the area of the larger circle?
Answers
Answer:
Check pdf
Step-by-step explanation:
The radius of the larger circle is 12 inches. Then the area of the larger circle will be 144π square inches.
What is the area of the circle?Let r be the radius of the circle. Then the area of the circle will be
A = πr² square units
The four small circles are identical. The total area of the small circles is 36π square inches. Then the radius of each small circle is given as,
36π = 4 × (πr²)
r² = 9
r = 3 inches
Then the radius of the larger circle is calculated as,
R = 4r
R = 4 × 3
R = 12
Then the area of the larger circle will be given as,
A = π × (12)²
A = 144π square inches
The radius of the larger circle is 12 inches. Then the area of the larger circle will be 144π square inches.
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Middle School
Mathematics
Find x.
A. 8
B. 12
Answers
Answer:
17²=15²+x²
289=225+x²
288-225=x²
64=x²
x=6
Answer:
8
Step-by-step explanation:
By Pythagoras theorem,
(hypotenuse)² = (altitude)² + (base)²
(17)² = x² + (15)²
(17)² - (15)² = x²
289 - 225 = x²
x² = 64
x = √64
x = 8
hope this helps you!
PLEASE HELP ASAP!!!!!!!!!Which of the following accurately shows the range of the function defined below?
3, x<0
fx)= x²+2, OSX<2
5x+5.
X+5, x22
[3.00)
O
[2.00)
d
[3, 6]
(-0,00)
O
Answers
Answer:
the second option [2, +infinity)
Step-by-step explanation:
the range is the interval or set of valid y (functional result) values of the function.
so, for x<0 the result is always 3.
for 0<=x<2 do we get results between 2 and 6.
for x>=2 we get results between 6 and infinity.
so, the combining interval of all possible y values is
[2, +infinity)
2 is included, infinity as usual not (as this is not a value).
Range of the function is,
⇒ Range = [2, +infinity)
What is mean by Function?A relation between a set of inputs having one output each is called a function. and an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable).
Since, The range is the interval or set of valid y (functional result) values of the function.
Hence, For x < 0
The result is always 3.
And, for 0 ≤ x < 2 do we get results between 2 and 6.
And, for x ≥ 2 we get results between 6 and infinity.
Therefore, After combining interval of all possible y values we get;
Range of the function is,
Range = [2, +infinity)
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Is a function or non function explain
Answers
Answer:
It is a non function.
Step-by-step explanation:
Numbers cannot repeat in functions
Jolinda and her parents are discussing how to pay for her college education. The cost of tuition at the college Jolinda wants to attend is $7250 a year. Jolinda's parents can pay 70% of the tuition and Jolinda will have to pay the rest. She has one year to save enough money for the first year of college. What is the minimum amount of money she should save every month to reach this goal?
Answers
Answer:
$181.25
Step-by-step explanation:
Given data
Amount= $7250
The amount the parent will pay
=70/100*7250
=0.7*7250
= $5075
Hence the balance left for her to pay is
=7250 -5075
=$2175
Hence the least monthly amount is
=2175/12
=$181.25
by definition a capable process is one that is operating at the so called ____ sigma level
Answers
by definition a capable process is one that is operating at the so called capability sigma level
The statistical difference between a process operating at a 5 sigma level and a process operating at a 6 sigma level is markedly different when it comes to the number of defects present, which is a true statement because as we know process sigma indicates the process variation and is measured in terms of data units, while process sigma count Z, or process sigma level, is a count with no unit of measure and sigma is a statistical term that measures how much a process varies from perfection, based on the number of defects per million units. And also we know that a six sigma process is 1 in which 99.99966 percent of all chances to provide some characteristics of a part are statistically assumed to be free regarding defects.
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The lifespans of lions in a particular zoo are normally distributed. The average lion lives 12. 5 years and the standard deviation is 2. 4 years.
Use the empirical rule (68-95-99. 7%) to estimate the probability of a lion living more than 10. 1 years.
Answers
The probability of a lion living more than 10.1 years is approximately 1 - 0.1587 = 0.8413, or 84.13% (rounded to the nearest hundredth).
To use the empirical rule, we first need to convert the value of 10.1 years into a z-score:
z = (x - μ) / σ
where x is the value we want to find the probability for, μ is the mean, and σ is the standard deviation.
z = (10.1 - 12.5) / 2.4
z = -1.00
Using the empirical rule, we know that approximately:
68% of the data falls within one standard deviation of the mean
95% of the data falls within two standard deviations of the mean
99.7% of the data falls within three standard deviations of the mean
Since we want to estimate the probability of a lion living more than 10.1 years, we need to find the area under the normal curve to the right of this value. This is equivalent to finding the area to the left of the z-score of -1.00 (since the standard normal distribution is symmetric).
From a standard normal distribution table or calculator, we can find that the area to the left of z = -1.00 is approximately 0.1587.
Therefore, the probability of a lion living more than 10.1 years is approximately 1 - 0.1587 = 0.8413, or 84.13% (rounded to the nearest hundredth).
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x⁴+8x³+34x²+72x+81 factories it.
Answers
Answer:
The expression x⁴ + 8x³ + 34x² + 72x + 81 cannot be factored further using simple integer coefficients. It does not have any rational roots or easy factorizations. Therefore, it remains as an irreducible polynomial.
Find the missing number:
√25 x √25 = 25 x?
(A) 1
(B) 5
(C) 10
(D) 25
Answers
Step-by-step explanation:
since sqrt(25)×sqrt(25) = 25, then x can be any number, because x = x in all cases. and then both sides of the equation are always equal, no matter what value we give x, as long as x is a factor in both sides.
sqrt(25)×x×sqrt(25) = 25x
is always true for every x.
if the question would be
sqrt(25)×x×sqrt(25) = 25
then the answer can only be x = 1. because for every other value of x the equality is destroyed.
a cross-country course is in the shape of a parallelogram with a base of length 7 mi and a side of length 6 mi. what is the total length of the cross-country course?
Answers
The total length of the parallelogram-shaped cross-country course is equal to 26 miles.
We can calculate the total length of the cross-country course using the formula for the perimeter of a parallelogram, which is twice the sum of the lengths of its adjacent sides. In this case, the adjacent sides are the base of length 7 mi and the side of length 6 mi.
So, we can find the total length by first finding the sum of these two sides:
7 mi + 6 mi = 13 mi
Then, we can multiply this sum by 2 to get the total length:
2 x 13 mi = 26 mi
Therefore, the total length of the cross-country course is 26 miles.
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What is the weight (in grams) of a liquid that exactly fills a 202 milliliter container if the density of the liquid is 0.685g/mL? Round to the nearest hundredth
Answers
Answer:
138.37 gram
Step-by-step explanation:
Formula of density
density = mass/ volume
given
volume = 202 mL
density = 0.685g/mL
using these values in Formula of density
0.685g/mL = mass/ 202 mL
mass = 0.685g/mL * 202 mL = 138.37 gram
Thus, weight of liquid is 138.37 gram to the nearest hundredth.
Answer:
138.37 grams
Step-by-step explanation:
I'm taking the exam. Good luck on yours!
On average, Caryl's school bus arrives on time, although sometimes it is a bit early or late. If the arrival times are distributed on a normal curve, which of the following statistics would enable Caryl to estimate the probability that her bus will arrive within 5 minutes of its scheduled arrival time on any given day?
a. median
b. mean
c. standard deviation
d. correlation coefficient
Answers
If the arrival times of Caryl's school bus are distributed on a normal curve, the standard deviation would enable Caryl to estimate the probability that her bus will arrive within 5 minutes of its scheduled arrival time on any given day.
The standard deviation is a statistic that measures the amount of variation or dispersion from the central tendency of a set of data values. A low standard deviation indicates that the data is tightly clustered around the mean or average, while a high standard deviation indicates that the data is more spread out.
The standard deviation is particularly useful when dealing with normally distributed data, as it allows us to estimate the probability of a certain range of values occurring. In this case, Caryl can use the standard deviation to estimate the probability that her bus will arrive within 5 minutes of its scheduled arrival time on any given day.
Hence, option c. standard deviation would enable Caryl to estimate the probability that her bus will arrive within 5 minutes of its scheduled arrival time on any given day.
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What is the y-intercept of the line shown?
A. O
B. 1
C. 3
D. 3.5
Answers
Answer:
c 3
Step-by-step explanation:
the y intercept is where the line crosses the 0 x-axis
So ask yourself, what is the y when the x is 0?
on this one, the line crosses the 0 x-axis at 3 y
when the x is 0, the y is 3
So thats the y intercept
NEED HELP ASASPPP
A study reported that about half of high school seniors study for upcoming math tests. To find out if this applies to seniors at Garfield High School, an SRS of 30 seniors was asked if they study for their math tests. Nineteen responded "Yes.”
A dotplot is provided showing the results of 40 trials of this simulation.
A dotplot titled seniors who study for math tests have number of seniors who responded yes on the x-axis, and frequency on the y-axis. 12, 2; 13, 5; 14, 6; 15, 11; 16, 9; 17, 4; 18, 2; 9, 0; 20, 1.
Does this provide convincing evidence that seniors at Garfield High School study more than the report stated?
No, there is an outlier at 20.
No, there were outcomes as low as 12.
Yes, only one trial had a result of 19 or larger.
Yes, more than half of the simulated results are over 15.
Answers
Answer:
Yes, only one trial had a result of 19 or larger.
Step-by-step explanation:
The claim reported is that only about half of high school seniors study for upcoming math test. However, at Garfield a sizeable number of the 30 surveyed observation gave a symmetrical distribution about the study pattern of their high school seniors with only one trial a frequency of yes greater than or equal to 19
Where would the decimal be in the answer to the following problem?
Answers
Answer:F
Step-by-step explanation:Count the how many letters there are in is in front of the decimal and then put the decimal that many places .
how do I solve this pls help khan academy is no smart
Answers
Answer:
3. If that doesn't work, try 1/3
Step-by-step explanation:
Figure A has a length of 9 and figure B has a length of 3. 9 divided by 3 is 3.
JK=4x+7 and KL=3x+14 find JL
Answers
Answer:
Step-by-step explanation:
4x + 7 = 3x + 14
x + 7 = 14
x = 7
4(7) + 7 + 3(7) + 14
28 + 7 + 21 + 14
35 + 35 = 70
Two consecutive even integers have a sum of 38. What are the two integers?
Answers
Answer:
its 18 and 20
Step-by-step explanation:
Answer:
18 & 20
Step-by-step explanation:
Forming the equation,
→ x + (x + 2) = 38
→ 2x + 2 = 38
Now the value of x will be,
→ 2x + 2 = 38
→ 2x = 38 - 2
→ x = 36/2
→ [ x = 18 ]
Then the required integers are,
→ x = 18
→ x + 2 = 18 + 2 = 20
Hence, the integers are 18, 20.
Match each of the equations with their corresponding graph
Answers
What kind of loan can you get if you pay $700 each month at a yearly rate of 0. 89% for 10 years?
Answers
You can get a loan amount of approximately $70,080. With monthly payments of $700 at a yearly interest rate of 0.89% for 10 years, you can obtain a loan amount of approximately $70,080.
With monthly payments of $700 for 10 years at an annual interest rate of 0.89%, the loan amount you can obtain is approximately $70,080. This calculation is based on the present value formula used to determine the loan amount for fixed monthly payment loans. Based on the given information, you are paying $700 each month for 10 years at an annual interest rate of 0.89%.
This scenario corresponds to a fixed monthly payment loan, commonly known as an amortizing loan or installment loan. In this type of loan, you make equal monthly payments over a specified period, and each payment includes both principal and interest components.
To determine the loan amount, we need to calculate the present value of the future cash flows (monthly payments).
Using financial calculations, the loan amount can be determined using the formula:
Loan amount = Monthly payment * (1 - (1 + interest rate)^(-number of months))) / interest rate
In this case, plugging in the given values:
Loan amount = $700 * (1 - (1 + 0.0089)^(-10 * 12)) / 0.0089
Evaluating the expression, the loan amount is approximately $70,080.
Therefore, with monthly payments of $700 at a yearly interest rate of 0.89% for 10 years, you can obtain a loan amount of approximately $70,080.
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solve sinx = 2x-3 using false position method
Answers
The root of the equation sinx = 2x-3 is 0.8401 (approx).
Given equation is sinx = 2x-3
We need to solve this equation using false position method.
False position method is also known as the regula falsi method.
It is an iterative method used to solve nonlinear equations.
The method is based on the intermediate value theorem.
False position method is a modified version of the bisection method.
The following steps are followed to solve the given equation using the false position method:
1. We will take the end points of the interval a and b in such a way that f(a) and f(b) have opposite signs.
Here, f(x) = sinx - 2x + 3.
2. Calculate the value of c using the following formula: c = [(a*f(b)) - (b*f(a))] / (f(b) - f(a))
3. Evaluate the function at point c and find the sign of f(c).
4. If f(c) is positive, then the root lies between a and c. So, we replace b with c. If f(c) is negative, then the root lies between c and b. So, we replace a with c.
5. Repeat the steps 2 to 4 until we obtain the required accuracy.
Let's solve the given equation using the false position method.
We will take a = 0 and b = 1 because f(0) = 3 and f(1) = -0.1585 have opposite signs.
So, the root lies between 0 and 1.
The calculation is shown in the attached image below.
Therefore, the root of the equation sinx = 2x-3 is 0.8401 (approx).
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If your friend is driving at a speed of 25 miles per hour, how many miles per second is your
friend traveling?
Answers
Answer:
0.0069444444
Step-by-step explanation:
if the sidereal time is 5 hours tonight at 8 p.m. what is the sidereal time going to be at 8 p.m. tomorrow?
Answers
The sidereal time tomorrow at 8 p.m. will be 4 hours 56 minutes .
A sidereal day is the amount of time needed by earth to complete one rotation. And the time for one sidereal day is 23 hours 56 minutes and 4.1 seconds. In the solar system , starts appear at the same location at the same time ,each sidereal day. The 4 minutes difference between the solar day and the sidereal day is visible by watching the stars rise 4 minutes earlier every night. So if any star is rising at 7p.m today , it will rise at 6:56 p.m. tomorrow and at 6:52 p.m. the following day.
Similarly if the sidereal time today is 8p.m. , tomorrow it will be four minutes less i.e. at 4 hours 56 minutes.
The sidereal time helps scientists keep a good track of time without worrying about the position of earth with respect to the sun.
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Given that f(x) = 3x - 2, g(x) = x2 + 3x and h(x) = 4.2^x
Answers
4x - 8y + 2z = 10
-3x + y – 2z = 6
2x – 4y + z = 8
Answers
Answer:
100900000099899887765544422
Will get 20 points and be marked as brainlist if has answer all these questions
1. What is the value of this expression if you substitute n for 5: (3n + 5) X 2 *
A. 20
B: 40
C: 26
D. 22
2. Match this algebraic expression to its word form: 3x *
A. 3 times the number
B. add 3 to the number
C. a number tripled
D. the difference of 3 and a number
3. (The picture above)
What is the rule for this input/output machine?
Answers
Answer:
1. B (40)
2. A (3 times a number)
3. 2x+1
Step-by-step explanation:
1. (3(5) +5) x 2
(15+5) x 2
20 x 2
40
Answer:
1. B. 40
2. A. 3 times the number
3. 2x + 1
Step-by-step explanation:
1.
2(3n + 5)
2(3(5) + 5)
2(15 + 5)
2(20)
40
2.
3(x)
- not adding or dividing or subtraction, and its not to the power of anything
3.
Rule is 2x + 1
2(1) + 1 = 3
2(2) + 1 = 5
2(3) + 1 = 7
2(4) + 1 = 9
what points are inverse? (2,3)(3,5)(4,7)(5,9)(6,11)
Answers
Answer:
None
Step-by-step explanation:
The inverse of those points would be
(3,2)
(5,3)
(7,4)
(9,5)
(11,6)
The time between arrivals of small aircraft at a county airport is exponentially distributed with a mean of one hour. Round the answers to 3 decimal places. (a) What is the probability that more than three aircraft arrive within an hour? Enter your answer in accordance to the item a) of the question statement (b) If 30 separate one-hour intervals are chosen, what is the probability that no interval contains more than three arrivals? Enter your answer in accordance to the item b) of the question statement (c) Determine the length of an interval of time (in hours) such that the probability that no arrivals occur during the interval is 0.27. Enter your answer in accordance to the item c) of the question statement
Answers
a) The probabilities for X = 0, 1, 2, and 3, and then summing them up, we find that P(X > 3) ≈ 0.019. b) The probability of zero successes is approximately 0.430. c) The length of an interval of time is 1.306 hours.
a) The time between arrivals of small aircraft is exponentially distributed with a mean of one hour. To find the probability that more than three aircraft arrive within an hour, we will use the Poisson distribution, where λ (lambda) represents the average number of arrivals per hour, which is 1 in this case. The probability formula is P(X > 3) = 1 - P(X ≤ 3), where X is the number of arrivals. Using the Poisson formula, we get:
P(X > 3) = 1 - [P(X=0) + P(X=1) + P(X=2) + P(X=3)]
Calculating the probabilities for X = 0, 1, 2, and 3, and then summing them up, we find that P(X > 3) ≈ 0.019.
b) To find the probability that no interval contains more than three arrivals in 30 separate one-hour intervals, we can use the binomial distribution. The probability of success (an interval with more than three arrivals) is 0.019 from part a), and the probability of failure (an interval with three or fewer arrivals) is 1 - 0.019 = 0.981. Using the binomial formula with n = 30 (number of intervals) and p = 0.981, we find the probability of zero successes (i.e., no interval with more than three arrivals) is approximately 0.430.
c) To determine the length of an interval of time (in hours) such that the probability that no arrivals occur during the interval is 0.27, we use the exponential distribution formula:
P(T > t) = e^(-λt), where T is the waiting time between arrivals, t is the time interval, and λ is the average number of arrivals per hour (1 in this case).
We want to find the value of t such that P(T > t) = 0.27. So:
0.27 = e^(-1 * t)
Taking the natural logarithm of both sides, we get:
ln(0.27) = -t
Solving for t, we find that t ≈ 1.306 hours.
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