Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

A zircon crystal is measured for age using uranium-235 to lead-207 decay (half-life is 704 million years). Assuming that there was no lead207 initially, and you measure the elemental composition and find 35,000,000 lead-207 atoms and 1,100,000 uranium-235 atoms, how old is the zircon crystal?

$6$ . A zircon crystal is measured for age using uranium $-235$ to lead $-207$ decay (half-life is $704$ million years). Assuming that there was no lead $207$ initially, and you measure the elemental composition and find $35$ , $000$ , $000$ lead $-207$ atoms and $1$ , $100$ , $000$ uranium $-235$ atoms, how old is the zircon crystal?

View step-by-step help

View step-by-step help

Exponential growth and decay: word problems

Full solution

Q.

6

. A zircon crystal is measured for age using uranium

-235

to lead

-207

decay (half-life is

704

million years). Assuming that there was no lead

207

initially, and you measure the elemental composition and find

35

,

000

,

000

lead

-207

atoms and

1

,

100

,

000

uranium

-235

atoms, how old is the zircon crystal?

Calculate Initial Amount: Calculate the initial amount of uranium $-235$ . Since the half-life of uranium $-235$ is $704$ million years, and we know the current amounts of lead $-207$ and uranium $-235$ , we can find the initial amount of uranium $-235$ . Initial uranium $-235$ $=$ current uranium $-235$ $+$ lead $-207$ $= 1,100,000 + 35,000,000 = 36,100,000$ .
Determine Number of Half-Lives: Determine the number of half-lives that have passed. $\newline$ Using the formula for decay, $N = N_0 \times (1/2)^{t/T}$ , where $N$ is the remaining amount of uranium $-235$ , $N_0$ is the initial amount, $t$ is the time elapsed, and $T$ is the half-life. $\newline$ Rearranging the formula to solve for $t$ , we get $t = T \times \log_2(N_0/N)$ . $\newline$ Substituting the values, $t = 704 \times \log_2(36,100,000 / 1,100,000)$ .
Calculate Logarithm and Age: Calculate the logarithm and the age. $\newline$ Calculating $\log_2(36,100,000 / 1,100,000)$ = $\log_2(32.8181818)$ ≈ $5$ . $035$ . $\newline$ Then, $t = 704 \times 5.035$ ≈ $3544$ . $64$ million years.

More problems from Exponential growth and decay: word problems

Question

Solve. Write your answer as an integer or a fraction in simplest form.

\newline

5^x=1

\newline

x=

Posted 2 months ago

Question

Steven opened a savings account and deposited

\$100.00

as principal. The account earns

9\%

interest, compounded annually. What is the balance after

5

\newline

Use the formula

A = P(1 + \frac{r}{n})^{nt}

A

is the balance (final amount),

P

is the principal (starting amount),

r

is the interest rate expressed as a decimal,

n

is the number of times per year that the interest is compounded, and

t

is the time in years.

\newline

Round your answer to the nearest cent.

\newline

\$

Posted 2 months ago

Question

Use the following function rule to find

f(1)

\newline

f(x) = 12(7)^x + 8

\newline

f(1) =

Posted 1 month ago

Question

The town of Valley View conducted a census this year, which showed that it has a population of

1,800

people. Based on the census data, it is estimated that the population of Valley View will grow by

12\%

\newline

Write an exponential equation in the form

y = a(b)^x

that can model the town population,

y

x

decades after this census was taken.

\newline

Use whole numbers, decimals, or simplified fractions for the values of

a

b

\newline

y =

Posted 2 months ago

Question

Solve. Round your answer to the nearest thousandth.

\newline

7 = 9^x

\newline

x =

Posted 2 months ago

Question

Solve. Round your answer to the nearest thousandth.

\newline

7 = e^x

\newline

x =

Posted 1 month ago

Question

Solve. Simplify your answer.

\newline

\log u = 1

\newline

u =

Posted 2 months ago

Question

Solve. Simplify your answer.

\newline

7\log x = 7

\newline

x =

Posted 1 month ago

Question

g(t) = 3^t

change over the interval from

t = 3

t = 4

\newline

\newline

g(t)

3

\newline

g(t)

3

\newline

g(t)

3\%

\newline

g(t)

200\%

Posted 2 months ago

Question

f(x)

6

over every unit interval in

x

f(0) = 0

\newline

Which could be a function rule for

f(x)

\newline

\newline

f(x) = 6^x

\newline

f(x) = 6x

\newline

f(x) = \frac{1}{6^x}

\newline

f(x) = \frac{x}{6}

Posted 2 months ago