SAT Math: M(t) = 480(0.68)^(t/18) Interpreting 0.68 Base in Exponential Decay

Exact Question Text

The mass $M(t)$, in grams, of a radioactive isotope remaining after $t$ days is modeled by:

> $$ M(t) = 480 \cdot (0.68)^{\frac{t}{18}} $$

Which of the following statements is the best interpretation of the number $0.68$ in this context?

- A) The mass decreases by $32\%$ every $18$ days.

- B) The mass decreases by $68\%$ every $18$ days.

- C) The mass decreases by $32\%$ every $1$ day.

- D) The mass decreases by $68\%$ every $1$ day.

Step-by-Step Algebraic Solution

1. Standard Exponential Decay Form:

$$ M(t) = M_0 (1 - r)^{\frac{t}{T}} $$

• Initial mass $M_0 = 480$
• Base $= 1 - r = 0.68 \implies r = 1 - 0.68 = 0.32 \implies 32\%\text{ decrease}$
• Denominator in exponent $= 18 \implies \text{applies every } 18\text{ days}$

1. Correct Answer: A (The mass decreases by 32% every 18 days)