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 58.5
2 52.65
Find 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.
N(t)=
To find the expression for N(t), we can use the given values of t and N(t) from the table.
We know that N(t) = a(b)^t.
From the table, when t = 1, N(t) = 58.5.
Therefore, we have 58.5 = a(b)^1.
Simplifying this equation, we get 58.5 = ab.
Similarly, when t = 2, N(t) = 52.65.
Plugging in the values, we get 52.65 = a(b)^2.
Since we already know that ab = 58.5, we can substitute this value into the equation: 52.65 = (58.5)(b)^2.
Dividing both sides of the equation by 58.5, we get (b)^2 = 52.65/58.5.
Taking the square root of both sides, we have b = sqrt(52.65/58.5).
This gives us the value of b.
Since ab = 58.5, we can substitute the known value of b into this equation to find a: a(sqrt(52.65/58.5)) = 58.5.
Dividing both sides by sqrt(52.65/58.5), we get a = 58.5 / sqrt(52.65/58.5).
Putting it all together, the expression for N(t) is N(t) = (58.5 / sqrt(52.65/58.5)) * (sqrt(52.65/58.5))^t.