Question

Consider the following table of values, where t is the number of minutes and Q ( t ) is the amount of substance remaining in grams after t minutes. t 0 1 2 3 4 5 6 7 8 9 10 11 12 Q ( t ) 300 267 238 212 189 168 150 134 119 106 94 84 75

a. What is the initial value of Q ( t ) ?
b. What is the half-life?

c. Construct an exponential decay function Q ( t ) , where t is measured in minutes.

Q ( t ) =

d. What is Q ( 1 ) ? What does it represent? Round your answer to the nearest integer.
Q ( 1 ) = grams This represents how much of the substance is left after with x minutes.

Answer 1

Solution-

Consider the table given in the question , where values, t represents the number of minutes and Q(t) represents the amount of substance remaining in grams after t minutes.

Now,

(a)

The initial value of Q(t) is the value at t = 0.

At t =0, Q(t) = 300 grams

Hence, the initial value of Q(t) = 300 grams.

(b)

The half-life is the after which the initial value remains half.

Since intial value is 300 grams.

So, 300/2 = 150 grams is it's half value and it occur at

t = 6 minutes (from the table)

Hence, half life is 6 minutes.

(c)

Let the exponential decay function Q(t) , where t is measured in minutes be

Q(t) =Q_o(1-r)^t .....(1)

Here,

Q_o = initial value = 30p grams and

r = decay factor.

Since the respective ratio of amount for any two consecutive years is = 267/300 = 0.89

So, 1 - r = 0.89

Or r = 1-0.89 = 0.11

On putting Q_o = 300 grams and r =0.11 in equation (1), we get

Q(t)= 300(1-0.11)^t

Q(t) =300(0.89)^t

Hence, the exponential decay function amount after t minutes is

Q(t) = 300(0.89)^t ....(2)

(d)

Putting t =1 , in equation (2) we get

Q(t)= Q(1) = 300(0.89)¹ = 267 ggams

So, Q(1) = 267 grams

Q(1) represents that 267 grams of the substance is left after with 1 minutes.