Math / Algebra

QuestionThe amount of carbon-14 present in animal bones after $t$ years is given by $A(t) = A_0 e^{-0.00012t}$ . A sample of fossil had 28% of the carbon 14 of a contemporary living sample. Estimate the age of the sample.
The age of the sample is ___ years. (Round to the nearest year as needed.)

Studdy Solution

STEP 1

What is this asking? We're figuring out how old a fossil is based on how much carbon-14 is left, knowing that carbon-14 decays over time. Watch out! Don't mix up the initial amount with the current amount.
Also, make sure to round to the nearest year at the very end!

STEP 2

1. Set up the equation
2. Isolate the exponential
3. Solve for $t$

STEP 3

The problem says the fossil has \textbf{28%} of the carbon-14 of a living sample.
This means $A(t)$ is $0.28$ times the initial amount, $A_0$ .
So we can write $A(t) = 0.28 \cdot A_0$ .

STEP 4

We know the formula for carbon-14 decay is $A(t) = A_0 e^{-0.00012t}$ .
Let's substitute the expression for $A(t)$ we just found: $0.28 \cdot A_0 = A_0 e^{-0.00012t}$

STEP 5

To get the exponential term by itself, we can divide both sides of the equation by $A_0$ .
This is valid since $A_0$ represents the initial amount of carbon-14, which is a positive value. $\frac{0.28 \cdot A_0}{A_0} = \frac{A_0 e^{-0.00012t}}{A_0}$ $0.28 = e^{-0.00012t}$

STEP 6

To get $t$ out of the exponent, we can take the natural logarithm (ln) of both sides: $\ln(0.28) = \ln(e^{-0.00012t})$

STEP 7

Remember that $\ln(e^x) = x$ , so we have: $\ln(0.28) = -0.00012t$

STEP 8

Now, we can divide both sides by $-0.00012$ to solve for $t$ : $t = \frac{\ln(0.28)}{-0.00012}$

STEP 9

Using a calculator, we find: $t \approx \frac{-1.272965675}{-0.00012}$ $t \approx 10608.04729$

STEP 10

The problem asks us to round to the nearest year, so our final answer is approximately $\textbf{10608}$ years.

STEP 11

The age of the sample is approximately 10608 years.

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QuestionThe amount of carbon-14 present in animal bones after $t$ years is given by $A(t) = A_0 e^{-0.00012t}$ . A sample of fossil had 28% of the carbon 14 of a contemporary living sample. Estimate the age of the sample.
The age of the sample is ___ years. (Round to the nearest year as needed.)

Studdy Solution

STEP 1

What is this asking? We're figuring out how old a fossil is based on how much carbon-14 is left, knowing that carbon-14 decays over time. Watch out! Don't mix up the initial amount with the current amount.
Also, make sure to round to the nearest year at the very end!

STEP 2

1. Set up the equation
2. Isolate the exponential
3. Solve for $t$

STEP 3

The problem says the fossil has \textbf{28%} of the carbon-14 of a living sample.
This means $A(t)$ is $0.28$ times the initial amount, $A_0$ .
So we can write $A(t) = 0.28 \cdot A_0$ .

STEP 4

We know the formula for carbon-14 decay is $A(t) = A_0 e^{-0.00012t}$ .
Let's substitute the expression for $A(t)$ we just found: $0.28 \cdot A_0 = A_0 e^{-0.00012t}$

STEP 5

To get the exponential term by itself, we can divide both sides of the equation by $A_0$ .
This is valid since $A_0$ represents the initial amount of carbon-14, which is a positive value. $\frac{0.28 \cdot A_0}{A_0} = \frac{A_0 e^{-0.00012t}}{A_0}$ $0.28 = e^{-0.00012t}$

STEP 6

To get $t$ out of the exponent, we can take the natural logarithm (ln) of both sides: $\ln(0.28) = \ln(e^{-0.00012t})$

STEP 7

Remember that $\ln(e^x) = x$ , so we have: $\ln(0.28) = -0.00012t$

STEP 8

Now, we can divide both sides by $-0.00012$ to solve for $t$ : $t = \frac{\ln(0.28)}{-0.00012}$

STEP 9

Using a calculator, we find: $t \approx \frac{-1.272965675}{-0.00012}$ $t \approx 10608.04729$

STEP 10

The problem asks us to round to the nearest year, so our final answer is approximately $\textbf{10608}$ years.

STEP 11

The age of the sample is approximately 10608 years.

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Studdy Solution

STEP 1

What is this asking? We're figuring out how old a fossil is based on how much carbon-14 is left, knowing that carbon-14 decays over time. Watch out! Don't mix up the initial amount with the current amount.Also, make sure to round to the nearest year at the very end!

STEP 2

1. Set up the equation2. Isolate the exponential3. Solve for ttt

STEP 3

The problem says the fossil has \textbf{28%} of the carbon-14 of a living sample.This means A(t)A(t)A(t) is 0.280.280.28 times the initial amount, A0A_0A0​.So we can write A(t)=0.28⋅A0A(t) = 0.28 \cdot A_0A(t)=0.28⋅A0​.

STEP 4

We know the formula for carbon-14 decay is A(t)=A0e−0.00012tA(t) = A_0 e^{-0.00012t}A(t)=A0​e−0.00012t.Let's substitute the expression for A(t)A(t)A(t) we just found: 0.28⋅A0=A0e−0.00012t0.28 \cdot A_0 = A_0 e^{-0.00012t}0.28⋅A0​=A0​e−0.00012t

STEP 5

STEP 6

To get ttt out of the exponent, we can take the natural logarithm (ln) of both sides: ln⁡(0.28)=ln⁡(e−0.00012t)\ln(0.28) = \ln(e^{-0.00012t})ln(0.28)=ln(e−0.00012t)

STEP 7

Remember that ln⁡(ex)=x\ln(e^x) = xln(ex)=x, so we have: ln⁡(0.28)=−0.00012t\ln(0.28) = -0.00012tln(0.28)=−0.00012t

STEP 8

Now, we can divide both sides by −0.00012-0.00012−0.00012 to solve for ttt: t=ln⁡(0.28)−0.00012t = \frac{\ln(0.28)}{-0.00012}t=−0.00012ln(0.28)​

STEP 9

Using a calculator, we find: t≈−1.272965675−0.00012t \approx \frac{-1.272965675}{-0.00012}t≈−0.00012−1.272965675​ t≈10608.04729t \approx 10608.04729t≈10608.04729

STEP 10

The problem asks us to round to the nearest year, so our final answer is approximately 10608\textbf{10608}10608 years.

STEP 11

The age of the sample is approximately **10608** years.

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

What is this asking? We're figuring out how old a fossil is based on how much carbon-14 is left, knowing that carbon-14 decays over time. Watch out! Don't mix up the initial amount with the current amount.
Also, make sure to round to the nearest year at the very end!

1. Set up the equation
2. Isolate the exponential
3. Solve for $t$

The problem says the fossil has \textbf{28%} of the carbon-14 of a living sample.
This means $A(t)$ is $0.28$ times the initial amount, $A_0$ .
So we can write $A(t) = 0.28 \cdot A_0$ .

We know the formula for carbon-14 decay is $A(t) = A_0 e^{-0.00012t}$ .
Let's substitute the expression for $A(t)$ we just found: $0.28 \cdot A_0 = A_0 e^{-0.00012t}$

To get $t$ out of the exponent, we can take the natural logarithm (ln) of both sides: $\ln(0.28) = \ln(e^{-0.00012t})$

Remember that $\ln(e^x) = x$ , so we have: $\ln(0.28) = -0.00012t$

Now, we can divide both sides by $-0.00012$ to solve for $t$ : $t = \frac{\ln(0.28)}{-0.00012}$

Using a calculator, we find: $t \approx \frac{-1.272965675}{-0.00012}$ $t \approx 10608.04729$

The problem asks us to round to the nearest year, so our final answer is approximately $\textbf{10608}$ years.

The age of the sample is approximately 10608 years.