Question

Need help with the 3 questions below. Thank you in advance!

1. A can of soda is taken from a room temperature where the temp. is 72F and placed into a refrigerator where the temp. is 42F. Assume that the soda cools according to Newton's Law of Cooling.

a. Assume that after 5mins. the temp. of the soda is 60F. Find the temp. of the soda after t mins. (find k to four decimal places).

b. Find the amount of time for the soda to cool to 50F.

2. Suppose an experimental population of fruit flies increases according to the law of exponential growth. The initial population was 100 and there were 300 flies after the fourth day.

a. Find the function that produces the number of flies at a time t

b. Find the time required for there to be 10,000 flies.

3. Assume that a payment of $20,000 is due in five years based on 4% interest compounded continously. Find the present value of this payment.

Answer 1

according to newtons law of cooling , the rate of variation of temperature (T) with respect to time dT/dt = K(72-T)

hence dT/((T-42)= Kdt

at t=0 T= 72F

ln(T-42)= Kt+C at t= 0 T=72

Ln30 = C

C= 3.40

ln( (T-42) = Kt +3.40

at t= 5min, T=60F

ln(60-42)= K*5+3.40, K=-0.10193 T/sec

hence ln (T-42)= -0.10193t+3.40

for T=50

ln(50-42)= -0.10193t+3.40,

t=12 vmin

2. Let the variation of flies with time be represented as

Y= Ce^(kt)

K and C are constants

at t=0 Y=100 Y= C* exp(0), C= 100

Hecne the original equation becomes Y= 100*e^(kt)

t=4 Y=300, 300 = 100*e^(4k)

e^(4K)= 3

taking ln

4K = ln 3

K= ln3/4 =0.275

hence Y = 100*e(^0.275t)

for Y=10000

10000 = 100*e^(0.275t)

t= 16.75 days

3. for the case of interest compounded continuously

S ( Future worth)= P(Present worth)* e^(rn)

S= $52000

r= rate of interest =4/100, n = number of years =5

52000= Pe^(0.04*5),

P= $42574