calc problem on interest/exponential growth/decay?

Question

1) A freshly brewed cup of coffee has temperature 95degrees in a 20degree room. When its temperature is 70degrees, it is cooling at a rate of 1degree per minute. When does this occur?

2) If $500 is borrowed at 14% interest, find the amounts due at the end of 2 years if the interest is compounded (i) annually, (ii) quarterly, (iii) monthly, (iv) daily, (v) hourly, and (vi) continuously.

show work plz..thanks! =)

Answer 1

U won't solve these -- but will tell you how to do them.

1. I haven't done this kind of problem in years -- so you need to verify this.

I think the formula for this is Temp = 95*exp(-L*T). If you take the first derivative, then you can solve for when it is equal to 1. That will give you the value for L.

2. This one I know for sure -- because I am a Finance Professor. The formula you want to use is:

Future Value = A*(1+R/N)^(N*t)

where

A = amount = 500
R = rate = 14%
t = number of years = 2

N = number of compounding periods per year.
(i) N = 1
(ii) N = 4
(iii) N = 12
(iv) N = 365 or 366
(v) N = 8760 or 8784

For continuous compounding, use the equation:

Future Value = A * e^(R*t)

As an aside, you might find it interesting that bank deposits compound daily. However, they divide the rate by 260 instead of 365 -- then compound for the actual number of days. So, investing a dollar for one year ar 14% would give you:

(1+14%/360)^365

This value is actually greater than what you would get from continuous compounding.

Answer 2

For #1 see the answer to your other questions. For #2 the basic equation is P = P(0) [ 1 + r/n]^(nt) where P(0) is initial amount, r is the rate as a decimal, n is the number of compounding that occur in a year, and t is the number of years. Except for the final one..P = P(0) e^(rt) for continuous compounding. This is all in your text.

Answer 3

those all use Newton's regulation of Cooling, dT/dt = ok( T - ambient temperature). Separate the variables and combine...for a million a) use the actuality that T= one hundred fifty while t = a million/2 to discover ok and T(0) = 185. comparable concepts for the rest. Your answer to 1b) is approximately 116 minutes. word T = T(0) e^(kt) would not artwork.