Math

Question Complete the exponential decay table: Original Amount: 19,00019,000, Decay Rate: 13%13\% per year, Years: 1818, Final Amount after 1818 years of decay.

Studdy Solution

STEP 1

Assumptions
1. The original amount is 19,000.<br/>2.Thedecayrateperyearis1319,000.<br />2. The decay rate per year is 13%.<br />3. The number of years, x,is18.<br/>4.Thedecayisexponential,followingtheformula, is 18.<br />4. The decay is exponential, following the formula A = P \cdot e^{(-rt)},where, where Aisthefinalamount, is the final amount, Pistheoriginalamount, is the original amount, risthedecayrate,and is the decay rate, and t$ is the time in years.

STEP 2

First, we need to convert the decay rate from a percentage to a decimal to use in the formula.
13%=0.1313\% = 0.13

STEP 3

Now, we will plug in the values into the exponential decay formula.
A=Pe(rt)A = P \cdot e^{(-rt)}

STEP 4

Substitute the given values into the formula.
A=19,000e(0.1318)A = 19,000 \cdot e^{(-0.13 \cdot 18)}

STEP 5

Calculate the exponent part of the formula.
0.1318=2.34-0.13 \cdot 18 = -2.34

STEP 6

Now, calculate the value of e(2.34)e^{(-2.34)} using a calculator or a software tool that can handle exponential functions.

STEP 7

After calculating the exponential part, we get the following value (rounded to five decimal places for precision):
e(2.34)0.09666e^{(-2.34)} \approx 0.09666

STEP 8

Multiply the original amount by the calculated exponential value to find the final amount after 18 years of decay.
A=19,0000.09666A = 19,000 \cdot 0.09666

STEP 9

Calculate the final amount.
A19,0000.096661836.54A \approx 19,000 \cdot 0.09666 \approx 1836.54

STEP 10

Round the final amount to the nearest whole number if necessary, as currency is usually expressed in whole units.
A1837A \approx 1837
The final amount after 18 years of decay is approximately $1,837.

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