Amount after time t A (t) = yobt
Where:
yo = starting amount
b = growth constant (percent remaining)
t = number of time units
It is well known that superman can withstand as much as 3 grams of Green Kryptonite indefinitely. However, exposure to more than 3 grams for more than 13 hours is fatal. Assume that Kryptonite decays at an hourly rate of 5.35%
If superman is trapped in a room with 6 grams of Kryptonite, will he survive? (Exposure to the deadly rays makes him too weak to escape)
To determine if Superman will survive being trapped in a room with 6 grams of Kryptonite, we can use the given information and the formula:
Amount after time t A (t) = yo * b^t
Here, yo represents the starting amount (6 grams), b indicates the growth constant (5.35% decay rate), and t represents the number of time units (in hours).
Let's calculate the amount of Kryptonite remaining after 13 hours:
A (13) = 6 * (1 - 0.0535)^13
Using a calculator or by simplifying the equation, we find:
A (13) = 6 * (0.9465)^13
≈ 6 * 0.480615557
≈ 2.883693342
After 13 hours, there will be approximately 2.88 grams of Kryptonite remaining. Since this is within the safe limit of 3 grams, Superman will survive being trapped in the room with 6 grams of Kryptonite for 13 hours.
To determine if Superman will survive being trapped in a room with 6 grams of Kryptonite, we can use the given information and the formula A(t) = yob^t.
In this case, the starting amount (yo) is 6 grams, and the growth constant (b) is -5.35% (since Kryptonite decays at a rate of 5.35% per hour). We need to find out if the amount of Kryptonite remaining after 13 hours (t) is less than or equal to 3 grams, which is the maximum amount Superman can withstand indefinitely.
Substituting the values into the formula, we have A(t) = 6 * (1 - 0.0535)^13. Let's calculate this:
A(13) = 6 * (1 - 0.0535)^13
A(13) ≈ 6 * (0.9465)^13
A(13) ≈ 6 * 0.516
A(13) ≈ 3.096
The calculation shows that after 13 hours of Kryptonite decay, there would be approximately 3.096 grams remaining. Since this is slightly more than 3 grams, it means Superman will not survive being trapped in a room with 6 grams of Kryptonite for over 13 hours.