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?
1. What is the purpose of the , after y0 in the equation?
I assume it should not be there.
Secondly, from the .9^t-1 I can tell that the amount would be decreasing. If you start with 20 mm how can you end up with 50 mg for a decreasing process?
2. Assuming he pays it off after 5 years with one payment ...
amount of loan = 80000(1.1)^5 = 128840.80
so the interest would be 48,840.80
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?
1) To find the time it takes for the initial dose to reach 50 mg, we need to solve the equation y = y0 * (0.90)^t - 1 for t, where y0 is the initial dose (20 mg) and y is the desired dose (50 mg).
Rearranging the equation, we have:
50 = 20 * (0.90)^t - 1
To solve for t, we need to isolate the exponential term:
50 + 1 = 20 * (0.90)^t
51 = 20 * (0.90)^t
Next, divide both sides of the equation by 20:
(0.90)^t = 51/20
Now, take the logarithm of both sides to get rid of the exponent:
log[(0.90)^t] = log(51/20)
Using the property of logarithms, we can move the exponent down:
t * log(0.90) = log(51/20)
Finally, solve for t by dividing both sides of the equation by log(0.90):
t = log(51/20) / log(0.90)
Using a calculator, you can find the value of t to determine how long it will take for the initial dose to reach 50 mg.
2) To calculate the interest amount, we need to use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
A = Total amount to be paid back (loan amount + interest)
P = Principal amount (loan amount)
r = Annual interest rate (10% or 0.10)
n = Number of times interest is compounded per year (quarterly, so 4 times)
t = Number of years
In this case, the principal amount (P) is $80,000 and the loan will be paid off in 5 years (t = 5). The annual interest rate is 10% (r = 0.10) and interest is compounded quarterly (n = 4).
Let's plug in these values into the compound interest formula:
A = $80,000(1 + 0.10/4)^(4*5)
Simplifying the formula, we get:
A = $80,000(1 + 0.025)^20
Now, calculate the value inside the parentheses:
(1 + 0.025)^20 ≈ 1.640627...
Finally, multiply the principal amount by the value inside the parentheses:
A ≈ $80,000 * 1.640627 ≈ $131,250.16
Therefore, the interest amount will be approximately $131,250.16 - $80,000 = $51,250.16.