1. **State the problem:** An element with an initial mass of 640 grams decays by 27.5% per minute. We want to find how much of the element remains after 5 minutes, rounded to the nearest tenth of a gram. 2. **Formula used:** The decay can be modeled by the exponential decay formula: $$m = m_0 \times (1 - r)^t$$ where: - $m$ is the remaining mass after time $t$, - $m_0$ is the initial mass, - $r$ is the decay rate per unit time (as a decimal), - $t$ is the time elapsed. 3. **Identify values:** - $m_0 = 640$ grams - $r = 27.5\% = 0.275$ - $t = 5$ minutes 4. **Calculate remaining mass:** $$m = 640 \times (1 - 0.275)^5 = 640 \times (0.725)^5$$ 5. **Calculate $(0.725)^5$ step-by-step:** $$0.725^2 = 0.525625$$ $$0.725^3 = 0.525625 \times 0.725 = 0.380078125$$ $$0.725^4 = 0.380078125 \times 0.725 = 0.275056578125$$ $$0.725^5 = 0.275056578125 \times 0.725 = 0.199442263203125$$ 6. **Multiply by initial mass:** $$m = 640 \times 0.199442263203125 = 127.642865\approx 127.6$$ 7. **Final answer:** The remaining mass after 5 minutes is approximately **127.6 grams**.