A product costs $429 Today. Write an expression for the cost is t days if the price is reduced at the end of each day by: a. $6 a day b. 4.5% a day

Question

Natural Logarithmic and Exponential Functions.

Answer 1

Ken:

 

(a) Heuristically, the price at the start of the second day ( t = 2) is 429 - 6 =  429 - 6(1), at the start of  the third day (t = 3), it is 429 - 12 = 429 -6(2).  Generally, therefore at the end of the t-th day, the cost is 429 - 6(t - 1).

 

The expression for the cost in t days C(t), say, is C(t) = 429-6(t-1), t=1(today), t=2(next day),....

 

(a) Heuristically, the price at the start of the second day ( t = 2) is 429 - 429 * 0.045 = 429(1- 0.045), at the start of the third day (t = 3), it is 429(1-0.045) - 429(1-0.045)(1-0.045) = 429(1- 0.045)²  and so on. Generally, therefore at the end of the t-th day, the cost is 429(1-0.045)^{t - 1} -.

 

The expression for the cost in t days D(t), say, this time, is D(t) = 429(1-0.045)^{t - 1} .  t=1(today), t=2(next day),....

 

Remember, costs C(t) and D(t) above are start-of-of-the-day costs.

 

Dr. G.

Answer 2

Let p be the price and t be the time in days.

 

a) since we take off $6 each day, p = 429 - 6t. Check to make sure it works. after 1 day, p=429-6 =423. After 2 days, you would deduct 2 6s so p=429-6*2. so this works .

 

b)when you deducr a %, you need to realize that this deduction is the same as (100-x)% of the original price. For example, if the price is reduced by 10% it is the same as 90% of the original price. A 15% reduction is the same as 85% of the original price ......

So, a 4.5% reduction is the same as 95.5% of the original price or .955*p.

If that price is reduced again you'd get .955 * (.955 p) or p*(.955)^2.

If that price is reduced again you'd get .955 * (p*.955^2) = p*(.955)^3

 

So if the price is reduced for n days, you get p = 429*(.955)^t . Let's check that this gives you the desired result. after 1 day, the price is calculated by p = 429 *.955 just like above. after 3 days, the price is calculated by p = 429 *.955^3.

 

There is a problem with these sometimes. Often the variable for time is defined differently and your formula must be changed. If t represent the number of days the item is on a shelf, then t-1 represents the number of reductions. For example, on day 2, there was only one reduction, on day 3, there were 2 reductions, day 4 would be 3 reductions.

Therefore your formula would be p = 429*(.955)^(t-1).

 

Hope this helped.