Question

in a census, green haven had a population of 160.\
assuming the population of green haven decays at a rate of 4% per year,\
find the decay factor and write exponential model for this situation in terms\
of ( t ), the number of years passed.\
round to two decimal places, when necessary.\
show your work here\
hint: to add an exponent (( x^y )), type \exponent\ or press \^\\
\\( \bigcirc \\ a(t) = 160\\ (15.6816)^t \\) \\( \bigcirc \\ a(t) = 160\\ (0.96)^t \\)\
\\( \bigcirc \\ a(t) = 161\\ (0.96)^t \\) \\( \bigcirc \\ a(t) = 160\\ (3.96)^t \\)\
\
a business owner is studying the decay of her products inventory over time.\
the business starts off with 60 units of its product in their warehouse. if this\
value decays at a rate of 10% per week, how many units will be in the\
warehouse after 6 weeks? (round to the nearest integer if necessary.)\
show your work here\
hint: to add an exponent (( x^y )), type \exponent\ or press \^\\
\\( \bigcirc \\) 96 units \\( \bigcirc \\) 35 units\
\\( \bigcirc \\) 42 units \\( \bigcirc \\) 32 units

Explanation:

First Question (Exponential Decay Model for Green Haven's Population)

Step 1: Recall the exponential decay formula

The general formula for exponential decay is \( A(t) = A_0(1 - r)^t \), where \( A_0 \) is the initial amount, \( r \) is the decay rate (in decimal), and \( t \) is the time.

Step 2: Identify the values for Green Haven's population

Here, \( A_0 = 160 \) (initial population), and the decay rate \( r = 4\% = 0.04 \).

Step 3: Substitute the values into the formula

Substitute \( A_0 = 160 \) and \( r = 0.04 \) into \( A(t) = A_0(1 - r)^t \). So, \( 1 - r = 1 - 0.04 = 0.96 \). Thus, the model is \( A(t)=160(0.96)^t \).

Step 1: Recall the exponential decay formula

The formula for exponential decay is \( A(t) = A_0(1 - r)^t \), where \( A_0 \) is the initial amount, \( r \) is the decay rate (in decimal), and \( t \) is the time.

Step 2: Identify the values for the inventory

Here, \( A_0 = 60 \) (initial units), \( r = 10\% = 0.10 \) (decay rate per week), and \( t = 6 \) (number of weeks).

Step 3: Substitute the values into the formula

Substitute \( A_0 = 60 \), \( r = 0.10 \), and \( t = 6 \) into \( A(t) = A_0(1 - r)^t \). First, calculate \( 1 - r = 1 - 0.10 = 0.90 \). Then, \( A(6)=60(0.90)^6 \).

Step 4: Calculate \( (0.90)^6 \) and then multiply by 60

\( (0.90)^6 \approx 0.531441 \). Then, \( A(6)=60\times0.531441\approx 31.88646 \approx 32 \)? Wait, no, wait: Wait, 60(0.9)^6: Let's recalculate \( 0.9^6 \): \( 0.9\times0.9 = 0.81 \), \( 0.81\times0.9 = 0.729 \), \( 0.729\times0.9 = 0.6561 \), \( 0.6561\times0.9 = 0.59049 \), \( 0.59049\times0.9 = 0.531441 \). Then \( 60\times0.531441 = 31.88646 \approx 32 \)? Wait, but let's check again. Wait, maybe I made a mistake. Wait, 60(0.9)^6: Let's compute (0.9)^6: 0.9^2 = 0.81, 0.9^3 = 0.810.9 = 0.729, 0.9^4 = 0.7290.9 = 0.6561, 0.9^5 = 0.65610.9 = 0.59049, 0.9^6 = 0.590490.9 = 0.531441. Then 60*0.531441 = 31.88646, which rounds to 32. Wait, but let's check the options. The options are 96, 35, 42, 32. So 32 is an option. Wait, but let's recheck the formula. Wait, decay rate is 10% per week, so the formula is \( A(t)=60(1 - 0.10)^t = 60(0.9)^t \). For \( t = 6 \), \( A(6)=60(0.9)^6 \approx 60\times0.531441 \approx 31.886 \approx 32 \). So the answer is 32 units.

Step 1: Recall the exponential decay formula

The formula for exponential decay is \( A(t)=A_0(1 - r)^t \), where \( A_0 \) is the initial amount, \( r \) is the decay rate (as a decimal), and \( t \) is time.

Step 2: Identify the values for the inventory

Here, \( A_0 = 60 \) (initial units), \( r = 10\% = 0.10 \) (decay rate per week), and \( t = 6 \) (number of weeks).

Step 3: Substitute the values into the formula

Substitute \( A_0 = 60 \), \( r = 0.10 \), and \( t = 6 \) into \( A(t)=A_0(1 - r)^t \). So, \( 1 - r = 1 - 0.10 = 0.90 \), and \( A(6)=60(0.90)^6 \).

Step 4: Calculate \( (0.90)^6 \) and multiply by 60

Calculate \( (0.90)^6 \approx 0.531441 \). Then, \( A(6)=60\times0.531441\approx 31.89 \), which rounds to 32 (nearest integer).

Answer:

\( A(t) = 160(0.96)^t \) (Option C)

Second Question (Inventory Decay After 6 Weeks)