Scientists are studying a sample of radioactive material. The amount left, in grams, after t days can be modeled by the function N(t)=a(b)t, where a and b are constants. This table shows two values of the function.t N(t)1 58.52 52.65Find an expression for N(t). Write your answer in the form N(t)=a(b)t, where a and b are integers or decimals. Do not round.
Question
Scientists are studying a sample of radioactive material. The amount left, in grams, after t days can be modeled by the function N(t)=a(b)t, where a and b are constants. This table shows two values of the function.t N(t)1 58.52 52.65Find an expression for N(t). Write your answer in the form N(t)=a(b)t, where a and b are integers or decimals. Do not round.
Solution
The given function is N(t) = a(b)^t. We are given two points on this function: (1, 58.5) and (2, 52.65).
We can use these points to form a system of equations and solve for a and b.
From the first point, we know that when t = 1, N(t) = 58.5. Substituting these values into the function gives us:
58.5 = a * b^1 58.5 = a * b
This tells us that a * b = 58.5.
From the second point, we know that when t = 2, N(t) = 52.65. Substituting these values into the function gives us:
52.65 = a * b^2
We can substitute a * b from the first equation into the second equation to get:
52.65 = 58.5 * b b = 52.65 / 58.5 b = 0.9
Substituting b = 0.9 into the first equation gives us:
58.5 = a * 0.9 a = 58.5 / 0.9 a = 65
So, the function N(t) = 65 * (0.9)^t.
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