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Maths-

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Question

The half-life of a certain substance in an aquatic environment is about 150 years. Which of the following exponential functions best models the amount $A left parenthesis t right parenthesis$ , in grams, of the substance $t$ years after 200 grams of the substance is applied to the aquatic environment? (The half-life is the length of time needed for an amount of the substance to decrease to one-half of that amount.)

$A (t) = 150 {(\frac{1}{2})}^{\frac{t}{200}}$
$A (t) = 150 {(\frac{1}{2})}^{200 t}$
$A (t) = 200 {(\frac{1}{2})}^{150 t}$
$A (t) = 200 {(\frac{1}{2})}^{\frac{t}{150}}$

The correct answer is: $A left parenthesis t right parenthesis equals 200 open parentheses fraction numerator 1 over denominator 2 end fraction close parentheses to the power of fraction numerator t over denominator 150 end fraction end exponent$

The given question states that, for a certain substance, half of its total life (total time to get completely used up/ over) is 150 years. i.e. after, 150 years, the amount at which it had started will be used upto 50%.
We have to find the function for such a substance, when the starting point amount = 200 grams.
Step-by-step solution:-
As per the given information, at t = 0, A(t) = 200 -----------------. (Condition i)
∴ substituting t = 0 in the given equations, we should get 200-
A. $blank A left parenthesis 0 right parenthesis blank equals 150 open parentheses fraction numerator 1 over denominator 2 end fraction close parentheses to the power of fraction numerator 0 over denominator 200 end fraction end exponent equals 150 open parentheses fraction numerator 1 over denominator 2 end fraction close parentheses to the power of 0 end exponent equals 150 left parenthesis 1 right parenthesis$ ----------------- (Anything raised to 0 = 1)
B. $A left parenthesis 0 right parenthesis blank equals 150 open parentheses fraction numerator 1 over denominator 2 end fraction close parentheses to the power of 200 blank cross times blank 0 end exponent equals 150 open parentheses fraction numerator 1 over denominator 2 end fraction close parentheses to the power of 0 end exponent equals 150 left parenthesis 1 right parenthesis$ ---------------- (Anything raised to 0 = 1)
C. $blank A open parentheses 0 close parentheses equals 200 open parentheses fraction numerator 1 over denominator 2 end fraction close parentheses to the power of 150 blank cross times blank 0 end exponent equals 200 open parentheses fraction numerator 1 over denominator 2 end fraction close parentheses to the power of 0 end exponent equals 200 left parenthesis 1 right parenthesis$ ----------------- (Anything raised to 0 = 1)
D. $A left parenthesis 0 right parenthesis blank equals 200 open parentheses fraction numerator 1 over denominator 2 end fraction close parentheses to the power of fraction numerator 0 over denominator 150 end fraction end exponent equals 200 open parentheses fraction numerator 1 over denominator 2 end fraction close parentheses to the power of 0 end exponent equals 200 open parentheses 1 close parentheses$ ----------------- (Anything raised to 0 = 1)
∴ only Options C & D satisfy the first condition.
Hence, Options A & B are eliminated
Also, after 150 years, Amount left = 50% of 200 = 100 ----------------- (Condition ii)
∴ substituting t =150 in the equations C & D, we should get 100-
C. $A left parenthesis 150 right parenthesis equals 200 open parentheses fraction numerator 1 over denominator 2 end fraction close parentheses to the power of 150 blank cross times blank 150 end exponent equals 200 open parentheses fraction numerator 1 over denominator 2 end fraction close parentheses to the power of 300 end exponent$ ----------------- (very huge number, cannot be calculated)
D. $A left parenthesis 0 right parenthesis equals 200 open parentheses fraction numerator 1 over denominator 2 end fraction close parentheses to the power of fraction numerator 150 over denominator 150 end fraction end exponent equals 200 open parentheses fraction numerator 1 over denominator 2 end fraction close parentheses to the power of 1 end exponent equals 200 open parentheses fraction numerator 1 over denominator 2 end fraction close parentheses equals 100$ ----------------- (Anything raised to 0 = 1)
Since Option D satisfies the 2nd condition, It is the correct answer.
Final Answer:-
Option D is the correct Option.

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