Step-by-step answer:
The base of the exponential function is set at 1.29 for a period of 7 days, which is expressed as
f(x) = 86*(1.29)^x
To determine the daily rate, divide the variable x by 7 (keeping x as the number of weeks), resulting in
f(x) = 86*1.29^(x/7)
Applying the exponent rule, b^(x/a) = b^(x*(1/a)) = (b^(1/a))^x
we can simplify by setting b=1.29, a=7 to arrive at
f(x) = 86*(1.29^(1/7))^x
f(x) = 86*(1.037)^x since evaluating 1.29^(1/7) yields approximately 1.037
Rounding 1.037 to 1.04 gives a (VERY) rough estimate function
f(x) = 86 * (1.04^x)
1.04 is only an approximation because 1.04^7 is expected to return 1.29, it actually results in 1.316; meanwhile, 1.037^7 returns 1.2896, which is much closer to 1.29.
Customers served in 0.5 hours : 3.6 versus x customers served in 7.5 hours
-------------------- = --------------------
0.5 hours 7.5 hours
Applying cross multiplication:
3.6 multiplied by 7.5 equals x times 0.5
Dividing both sides by 0.5:
x = (3.6 * 7.5) / 0.5
x = 54
So, you assisted 54 customers in 7.5 hours.
Answer:
Expiration Date: 1/17/2017
Expiration Time: 4:00am
Preparation Date: 12/3/2016
Preparation Time: 4:00am
Initial Usage Date: 12/7/2016
Detailed Breakdown:
An illustrative depiction of the question has been provided in an image format for clarity.
From the information given, it is noted that her store order arrived on 12/3/2016 at 4am, confirming that both the prep date and time are 12/3/2016 and 4am respectively. The product has a printed expiration date of 1/17/2017, logically indicating that its expiration time is also 4am, in line with the prep time; adding 24 hours leads us back to the same time on the expiration date. Furthermore, we were informed that she utilized the product on 12/7/2016, which marks the initial use date. Based on this information, we can summarize as follows:
Expiration Date: 1/17/2017
Expiration Time: 4:00am
Preparation Date: 12/3/2016
Preparation Time: 4:00am
Initial Usage Date: 12/7/2016
The maximum area that can be enclosed is 64 ft². To achieve the largest area while minimizing the perimeter, the dimensions should be as equal as possible. Allocating 32 feet of fencing for four sides gives us 8 feet per side, resulting in a square with a side length of 8; thus, the area equals 8*8 = 64.
The function can be expressed as:
f(x) = log(-20x + 12√x)
To ascertain the maximum value, differentiate the equation with respect to x and set the derivative to zero. The procedure unfolds as follows.
The differentiation formula is:
d(log u)/dx = du/u ln(10)
Thus,
d/dx = (-20 + 6/√x)/(-20x + 12√x)(ln 10) = 0
-20 + 6/√x = 0
6/√x = 20
From which we derive x = (6/20)² = 9/100
Therefore,
f(x) = log(-20(9/100)+ 12√(9/100)) = 0.2553
The function's maximum value is 0.2553.
