The radioactive element polonium decays according to the law given below where Q

Question

The radioactive element polonium decays according to the law given below where Q_0 is the initial amount and the time t is measured in days. Q(t) = Q_0 middot 2^-(t/140) If the amount of polonium left after 700 days is 30 mg, what was the initial amount present? _____ mg Metro Department Store found that t weeks after the end of a sales promotion the volume of sales was given by S(t) = B + Ae^-kt (0 lessthanorequalto t lessthanorequalto 4) where B = 42,000 and is equal to the average weekly volume of sales before the promotion. the sales volumes at the end of the first and third weeks were $83, 420 and $62, 040, respectively. Assume that the sales volume is decreasing exponentially. (a) Find the decay constant k. (Round your answer to five decimal places.) k = _____ (b) Find the sales volume at the end of the fourth week. (Round your answer to the nearest whole number.) $ _____ (c) How fast is the sales volume dropping at the end of the fourth week? (Round your answer to the nearest dollar.) $ _____ per week

Explanation / Answer

The radioactive element polonium decays according to the law shown below where Q0 is the initial amount and the time t is measured in days.

Q(t) = Q0 • 2-(t/140)

Substituting for Q(t) and t in the equation we get

60 = Q0 x 2-(700/140)

So 60 = Q0 x 2-5

60 = Q0/32

60 x 32 = Q0

Q0 = 1920

5. We are given the following information:

B = 42000 (a constant amount, so you can plug it in)
S(1) = 83580 and S(3) =62040 are two data points. when we put that in equation we get:

(a) S(t) = B + Ae-kt , (0 t 4)
83580 = 42000 + Ae(1k)
41580 = Aek .............(1)

62040 = 42000 + Ae(3k)
20040 = Ae(3k)   ............(2)

Dividing Equation (1) by Equation (2):
2.074 = e(-2k)
ln 2.074 = -2k
k = (ln 2.074)/-2 = -0.364
Now we need to find A, so we will use Eq1 for this:
41580 = Ae(-.0.364)
41580/e(-0.364) = A

A = 41580/ 0.694 = $59913

Thus, the function is S(T) = 42000 + 59913e(-0.364T)

(b) The sales volume at the end of the fourth week will be
S(4) = 42000 + 59913e(-0.364 x 4)
= $42000 + 59913e-1.456 = $55969.73

(c) How fast is the sales volume dropping at the end of the fourth week

S' (T) = -0.364 x 59913e(-0.364T)
S '(T) = -21808e(-0.364T)
S '(4) = -21808e(-0.364 x 4) = -$5084.90 / week

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