Answer:
Step-by-step explanation:
To establish independence between two events, we must understand the concept of independence:
Events are deemed independent if 
, which means that the occurrence of one does not influence the probability of the other event.
In this situation, the only selection that aligns with the independence criterion is 
. Other options do not comply with the independent events definition.
The Huey family is buying a winter cover for their above-ground pool, which is rectangular and measures 18 ft x 9 ft. They wish for the cover to extend 15 inches over the pool on each side.
y2 = C1xe^(4x) Step-by-step explanation: Knowing that y1 = e^(4x) satisfies the differential equation y'' - 8y' + 16y = 0, we need to derive the second solution y2 using the reduction of order technique. Let y2 = uy1. Since y2 is a solution to the differential equation, it holds that y2'' - 8y2' + 16y2 = 0. By substituting for y2, its derivatives become y2 = ue^(4x), y2' = u'e^(4x) + 4ue^(4x), and y2'' = u''e^(4x) + 8u'e^(4x) + 16ue^(4x). Plugging these into the differential equation gives us u''e^(4x) = 0. Let w = u', so w' = u''. This results in w' e^(4x) = 0, leading to w' = 0. Integrating gives w = C1. Since w = u', this implies u' = C1, and integrating once more results in u = C1x. Therefore, y2 = ue^(4x) becomes y2 = C1xe^(4x), which is the second solution.
Answer:
P(t) = 1000e^(0.01155)t
Step-by-step explanation:
The population of the barangay can be modeled using the exponential function;
P(t) = P0e^kt
P(t) reflects the population after t years
P0 denotes the initial population
t indicates the time
With an initial population of 1000, we set P0 = 1000
Given that the population doubles every 60 years, at t = 60, it holds that P(t) = 2P0
Inserting that into the equation yields
2P0 = P0e^k(60)
2 = e^60k
Taking the natural logarithm of both sides
ln2 = lne^60k
ln2 = 60k
k = ln2/60
k = 0.01155
Inserting the determined k value and P0 into the function gives
P(t) = 1000e^(0.01155)t
Thus, the exponential model for the population of the barangay is
P(t) = 1000e^(0.01155)t
