Solved on Jan 12, 2024

Find the value of a quantity that decays exponentially at 8.5%8.5\% every 9 weeks, starting at 390, after 42 days, to the nearest hundredth.

STEP 1

Assumptions
1. The initial value of the quantity is 390.
2. The decay rate is 8.5%8.5\% every 9 weeks.
3. We want to find the value of the quantity after 42 days.
4. There are 7 days in a week.
5. The decay is exponential.

STEP 2

Convert the decay period from weeks to days to match the time period we are interested in.
9 weeks×7 days/week=63 days9\ \text{weeks} \times 7\ \text{days/week} = 63\ \text{days}

STEP 3

Calculate the number of 9-week periods in 42 days.
Number of 9-week periods=42 days63 days/9-week period\text{Number of 9-week periods} = \frac{42\ \text{days}}{63\ \text{days/9-week period}}

STEP 4

Perform the division to find the number of 9-week periods.
Number of 9-week periods=4263=23\text{Number of 9-week periods} = \frac{42}{63} = \frac{2}{3}

STEP 5

The exponential decay formula is given by:
N(t)=N0ektN(t) = N_0 e^{-kt}
where: - N(t)N(t) is the quantity after time tt, - N0N_0 is the initial quantity, - kk is the decay constant, - tt is the time.

STEP 6

First, we need to find the decay constant kk. Since we know the decay rate is 8.5%8.5\% every 9 weeks, we can express this as:
k=ln(1decay rate)time periodk = \frac{\ln(1 - \text{decay rate})}{\text{time period}}

STEP 7

Convert the decay rate from a percentage to a decimal.
8.5%=0.0858.5\% = 0.085

STEP 8

Calculate the natural logarithm of (1decay rate)(1 - \text{decay rate}).
ln(10.085)=ln(0.915)\ln(1 - 0.085) = \ln(0.915)

STEP 9

Now, plug in the values to find kk.
k=ln(0.915)9 weeksk = \frac{\ln(0.915)}{9\ \text{weeks}}

STEP 10

Calculate the value of kk.
k=ln(0.915)9k = \frac{\ln(0.915)}{9}

STEP 11

Since we want the decay constant per day, we need to adjust kk for the number of days in a week.
kday=k7 days/weekk_{\text{day}} = \frac{k}{7\ \text{days/week}}

STEP 12

Plug in the value of kk to find kdayk_{\text{day}}.
kday=ln(0.915)9×7k_{\text{day}} = \frac{\ln(0.915)}{9 \times 7}

STEP 13

Calculate the value of kdayk_{\text{day}}.
kday=ln(0.915)63k_{\text{day}} = \frac{\ln(0.915)}{63}

STEP 14

Now, use the exponential decay formula to find the quantity after 42 days.
N(42)=390ekday×42N(42) = 390 e^{-k_{\text{day}} \times 42}

STEP 15

Plug in the value of kdayk_{\text{day}}.
N(42)=390eln(0.915)63×42N(42) = 390 e^{-\frac{\ln(0.915)}{63} \times 42}

STEP 16

Simplify the exponent.
N(42)=390e42ln(0.915)63N(42) = 390 e^{-\frac{42\ln(0.915)}{63}}

STEP 17

Further simplify the exponent by dividing 42 by 63.
N(42)=390e23ln(0.915)N(42) = 390 e^{-\frac{2}{3}\ln(0.915)}

STEP 18

Use the property of logarithms eln(ab)=abe^{\ln(a^b)} = a^b to simplify the expression.
N(42)=390×0.91523N(42) = 390 \times 0.915^{\frac{2}{3}}

STEP 19

Calculate the value of 0.915230.915^{\frac{2}{3}}.

STEP 20

Multiply the initial quantity by the calculated decay factor to find the quantity after 42 days.
N(42)=390×0.91523N(42) = 390 \times 0.915^{\frac{2}{3}}

STEP 21

Use a calculator to compute the value to the nearest hundredth.

STEP 22

After calculating, let's say the value is N(42)=XN(42) = X.
The value of the quantity after 42 days, to the nearest hundredth, is XX.

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