Math / Calculus

QuestionHow long will it take for an invertment to triple, if intcrest is compounded continously at 4 t - ?
It will take? years before the invectment trives
Round to the reares tentn of a year

Studdy Solution

STEP 1

What is this asking? How long does it take for an investment to triple if it grows continuously at a rate of 4%? Watch out! Remember continuous compounding uses a special formula, not the usual compound interest one!
Also, don't forget to convert the percentage to a decimal.

STEP 2

1. Set up the continuous compound interest formula.
2. Solve for time.

STEP 3

The formula for continuous compound interest is $A = Pe^{rt}$ , where $A$ is the final amount, $P$ is the principal (initial investment), $r$ is the interest rate (as a decimal), and $t$ is the time in years.

STEP 4

We want the investment to triple, so the final amount ( $A$ ) will be three times the principal ( $P$ ).
We can write this as $A = 3P$ .
The interest rate is 4%, which as a decimal is $r = 0.04$ .

STEP 5

Let's plug these values into our formula: $3P = Pe^{0.04t}$ .

STEP 6

Notice that we can divide both sides of the equation $3P = Pe^{0.04t}$ by $P$ .
This gives us $3 = e^{0.04t}$ .
See! The initial amount invested doesn't actually matter, we just care about tripling it!

STEP 7

To get $t$ out of the exponent, we'll take the natural logarithm (ln) of both sides: $\ln(3) = \ln(e^{0.04t})$ .
Remember, the natural logarithm is the inverse of the exponential function, so they undo each other.

STEP 8

Using the property $\ln(a^b) = b\cdot\ln(a)$ , we can simplify the right side of the equation: $\ln(3) = 0.04t \cdot \ln(e)$ .
Since $\ln(e) = 1$ (because e to the power of 1 equals e), we get $\ln(3) = 0.04t$ .

STEP 9

Now, we can isolate $t$ by dividing both sides by $0.04$ : $t = \frac{\ln(3)}{0.04}$ .

STEP 10

Using a calculator, we find that $\ln(3) \approx 1.0986$ .
So, $t \approx \frac{1.0986}{0.04} \approx 27.465$ .

STEP 11

Rounding to the nearest tenth gives us $t \approx 27.5$ years.

STEP 12

It will take approximately 27.5 years for the investment to triple.

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QuestionHow long will it take for an invertment to triple, if intcrest is compounded continously at 4 t - ?
It will take? years before the invectment trives
Round to the reares tentn of a year

Studdy Solution

STEP 1

What is this asking? How long does it take for an investment to triple if it grows continuously at a rate of 4%? Watch out! Remember continuous compounding uses a special formula, not the usual compound interest one!
Also, don't forget to convert the percentage to a decimal.

STEP 2

1. Set up the continuous compound interest formula.
2. Solve for time.

STEP 3

The formula for continuous compound interest is $A = Pe^{rt}$ , where $A$ is the final amount, $P$ is the principal (initial investment), $r$ is the interest rate (as a decimal), and $t$ is the time in years.

STEP 4

We want the investment to triple, so the final amount ( $A$ ) will be three times the principal ( $P$ ).
We can write this as $A = 3P$ .
The interest rate is 4%, which as a decimal is $r = 0.04$ .

STEP 5

Let's plug these values into our formula: $3P = Pe^{0.04t}$ .

STEP 6

Notice that we can divide both sides of the equation $3P = Pe^{0.04t}$ by $P$ .
This gives us $3 = e^{0.04t}$ .
See! The initial amount invested doesn't actually matter, we just care about tripling it!

STEP 7

To get $t$ out of the exponent, we'll take the natural logarithm (ln) of both sides: $\ln(3) = \ln(e^{0.04t})$ .
Remember, the natural logarithm is the inverse of the exponential function, so they undo each other.

STEP 8

Using the property $\ln(a^b) = b\cdot\ln(a)$ , we can simplify the right side of the equation: $\ln(3) = 0.04t \cdot \ln(e)$ .
Since $\ln(e) = 1$ (because e to the power of 1 equals e), we get $\ln(3) = 0.04t$ .

STEP 9

Now, we can isolate $t$ by dividing both sides by $0.04$ : $t = \frac{\ln(3)}{0.04}$ .

STEP 10

Using a calculator, we find that $\ln(3) \approx 1.0986$ .
So, $t \approx \frac{1.0986}{0.04} \approx 27.465$ .

STEP 11

Rounding to the nearest tenth gives us $t \approx 27.5$ years.

STEP 12

It will take approximately 27.5 years for the investment to triple.

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QuestionHow long will it take for an invertment to triple, if intcrest is compounded continously at 4 t - ?It will take? years before the invectment trivesRound to the reares tentn of a year

Studdy Solution

STEP 1

What is this asking? How long does it take for an investment to triple if it grows continuously at a rate of 4%? Watch out! Remember continuous compounding uses a special formula, not the usual compound interest one!Also, don't forget to convert the percentage to a decimal.

STEP 2

1. Set up the continuous compound interest formula.2. Solve for time.

STEP 3

The formula for continuous compound interest is A=Pert A = Pe^{rt} A=Pert, where AAA is the **final amount**, PPP is the **principal** (initial investment), rrr is the **interest rate** (as a decimal), and ttt is the **time** in years.

STEP 4

We want the investment to **triple**, so the final amount (AAA) will be **three times** the principal (PPP).We can write this as A=3P A = 3P A=3P.The interest rate is **4%**, which as a decimal is r=0.04 r = 0.04 r=0.04.

STEP 5

Let's plug these values into our formula: 3P=Pe0.04t 3P = Pe^{0.04t} 3P=Pe0.04t.

STEP 6

Notice that we can divide both sides of the equation 3P=Pe0.04t 3P = Pe^{0.04t} 3P=Pe0.04t by PPP.This gives us 3=e0.04t 3 = e^{0.04t} 3=e0.04t.See! The initial amount invested doesn't actually matter, we just care about tripling it!

STEP 7

To get ttt out of the exponent, we'll take the natural logarithm (ln) of both sides: ln⁡(3)=ln⁡(e0.04t) \ln(3) = \ln(e^{0.04t}) ln(3)=ln(e0.04t).Remember, the natural logarithm is the inverse of the exponential function, so they undo each other.

STEP 8

STEP 9

Now, we can isolate ttt by dividing both sides by 0.040.040.04: t=ln⁡(3)0.04 t = \frac{\ln(3)}{0.04} t=0.04ln(3)​.

STEP 10

Using a calculator, we find that ln⁡(3)≈1.0986 \ln(3) \approx 1.0986 ln(3)≈1.0986.So, t≈1.09860.04≈27.465 t \approx \frac{1.0986}{0.04} \approx 27.465 t≈0.041.0986​≈27.465.

STEP 11

Rounding to the nearest tenth gives us t≈27.5 t \approx 27.5 t≈27.5 years.

STEP 12

It will take approximately **27.5 years** for the investment to triple.

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.

QuestionHow long will it take for an invertment to triple, if intcrest is compounded continously at 4 t - ?
It will take? years before the invectment trives
Round to the reares tentn of a year

What is this asking? How long does it take for an investment to triple if it grows continuously at a rate of 4%? Watch out! Remember continuous compounding uses a special formula, not the usual compound interest one!
Also, don't forget to convert the percentage to a decimal.

1. Set up the continuous compound interest formula.
2. Solve for time.

The formula for continuous compound interest is $A = Pe^{rt}$ , where $A$ is the final amount, $P$ is the principal (initial investment), $r$ is the interest rate (as a decimal), and $t$ is the time in years.

We want the investment to triple, so the final amount ( $A$ ) will be three times the principal ( $P$ ).
We can write this as $A = 3P$ .
The interest rate is 4%, which as a decimal is $r = 0.04$ .

Let's plug these values into our formula: $3P = Pe^{0.04t}$ .

Notice that we can divide both sides of the equation $3P = Pe^{0.04t}$ by $P$ .
This gives us $3 = e^{0.04t}$ .
See! The initial amount invested doesn't actually matter, we just care about tripling it!

To get $t$ out of the exponent, we'll take the natural logarithm (ln) of both sides: $\ln(3) = \ln(e^{0.04t})$ .
Remember, the natural logarithm is the inverse of the exponential function, so they undo each other.

Now, we can isolate $t$ by dividing both sides by $0.04$ : $t = \frac{\ln(3)}{0.04}$ .

Using a calculator, we find that $\ln(3) \approx 1.0986$ .
So, $t \approx \frac{1.0986}{0.04} \approx 27.465$ .

Rounding to the nearest tenth gives us $t \approx 27.5$ years.

It will take approximately 27.5 years for the investment to triple.