The rate at which a quantity of a radioactive substance decays is proportional to the quantity of the substance. The constant of proportionality λ is called the decay constant or decay rate (λ, like most physical constants, is conventionally represented by a positive number. That's why we call it a decay rate. If a quantity were increasing, we would use the words "growth rate" and also represent that by a positive number). Write the initial value problem that describes the amount N(t) of the radioactive substance remaining after time t, if the initial quantity N(0) is 6.

λN           N(0) = 6

N(t) = N₀e^(λt)

Applying the inital value condition

N(t) = 6e^(λt)

Step-by-step explanation:

Summarizing  the information briefly and stating the  variables in the problem.

t = time elapsed during the decay

N(t) = the amount of the radioactive substance remaining after time t

λ= The constant of proportionality  is called the decay constant or decay rate

Given the initial conditions

N(0) = N₀ = 6

The rate at which a quantity of a radioactive substance decays () is proportional to the quantity of the substance (N)  and λ is the constant of proportionality  is called the decay constant or decay rate :

λN          

N(t) = N₀e^(λt) ......equ 1

substituting the value of  N₀ = 6 into equation 1

N(t) = 6e^(λt)

so firstly, we need to isolate the y variables to be able to solve these inequalities. firstly, add 0.5x on both sides of the first inequality and subtract 1.5x on both sides of the second inequality:

now since the slope is positive for the first inequality, this means that the line going upwards belongs to the first inequality, and the line going downwards belongs to the second inequality.

next, since y ≥ in the first inequality, this means that the shaded region will be above the first inequality's line, thus shading regions a and b.

next, since y ≤ in the second inequality, this means that the shaded region is going to be below this line, thus shading regions a and c.

since both lines shade region a, this means that region a is the solution.

step-by-step explanation:

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