1). Using the function: y=y0,(.90)^t-1. In this equation y0 is the amount of initial dose and y is the amount of medication still available t hours after drug is administered. Suppose 20mg of the drug is administered. How long will it take for this initial dose to reach 50mg?

2).a contractor needs $80000.00 to finish one of the biuldings. He is able to borrow at 10% per year compounded quarterly. how much will interest amount to if he pays off the loan in 5 years?

the y0 should actually be y with a little 0 at the bottom right corner and as i said from the equation it is the amount of initial dose. Plus the initial dose is 200mg not 20 sorry. so how long would it take to get from 200mg to 50mg?

As I mentioned the last time you posted this question, what is the purpose of the comma after the y0?

You did not answer that question, so I will just ignore the comma.

y = y0 (.9)^^(t-1)
50 = 200 (.9)^(t-1)
.25 = (.9)^(t-1)
ln .25 = (t-1) ln.9
t-1 = ln.25/ln.9 = 13.16
t = 14.16 hrs

I had answered the second part of your question before.

http://www.jiskha.com/display.cgi?id=1353294829

To solve the first problem, which is finding the time it takes for the initial dose of 200mg to reach 50mg using the function y=y₀(.90)^t-1, you can set up the equation as follows:

50 = 200(.90)^t-1

To solve for t, you need to isolate the variable t. Here's how:

1. Divide both sides of the equation by 200 to cancel out the coefficient:
50/200 = (.90)^t-1

2. Simplify the left side of the equation:
0.25 = (.90)^t-1

3. Take the logarithm of both sides of the equation. You can use any base for the logarithm, but the most common base is 10 (log base 10, denoted as "log"):
log(0.25) = log[(.90)^t-1]

4. Apply the logarithmic property to bring down the exponent:
log(0.25) = (t-1)log(0.90)

5. Divide both sides of the equation by log(0.90) to isolate t:
(t-1) = log(0.25) / log(0.90)

6. Add 1 to both sides of the equation:
t = log(0.25) / log(0.90) + 1

Now, you can use a calculator or software with logarithmic functions to find the value of t.

For the second problem, calculating the interest amount when borrowing $80,000 at 10% per year compounded quarterly, you can use the compound interest formula:

A = P(1 + r/n)^(nt)

Where:
A = the final amount (loan + interest)
P = the principal amount (initial loan)
r = the annual interest rate (expressed as a decimal)
n = the number of times that interest is compounded per year
t = the number of years

Let's calculate the interest amount if the loan is paid off in 5 years:

P = $80,000
r = 10% or 0.10 (10% expressed as a decimal)
n = 4 (quarterly compounding)
t = 5 years

A = $80,000(1 + 0.10/4)^(4*5)
A = $80,000(1 + 0.025)^(20)
A = $80,000(1.025)^20
A ≈ $128,425.54

To calculate the interest amount, subtract the principal amount from the final amount:
Interest Amount = A - P
Interest Amount = $128,425.54 - $80,000
Interest Amount ≈ $48,425.54

Therefore, the interest amount will be approximately $48,425.54 if the loan is paid off in 5 years.