Final answer:
To determine after how many hours the battery power is at 75%, given that it decreases by 20% per hour, one applies the exponential decay formula and uses logarithms to solve for time, yielding approximately 1.71 hours.
Step-by-step explanation:
The question at hand involves an exponential decay model where a battery's power decreases by 20% per hour. To find after how many hours the battery power is at 75%, we need to solve the problem using the formula for exponential decay, where the final amount (P) is equal to the initial amount (P0) times the decay rate raised to the number of time periods (t).
The decay rate per hour is 80% (as it decreases by 20%), which we can express as 0.8. We set up the equation as follows: 0.75 = 0.8t, where 0.75 represents the 75% remaining power, and t is the number of hours.
To solve for t, we'll use logarithms. Taking the natural logarithm (ln) of both sides gives us ln(0.75) = t · ln(0.8). We can thus calculate t by dividing ln(0.75) by ln(0.8).
Calculating the values gives us t ≈ 1.7095. Therefore, it takes approximately 1.71 hours (after rounding off) for the battery to reach 75% power.