2. An isotope of sodium has a half-life of 15 h. A sample of this isotope has a mass of 2 g.

a. [2] Find the mass equation that remains after t hours.
b. [1] Find the mass that remains after 5 hours, to the nearest hundredth.
c. [2] How long, to the nearest hour, will the sample take to decay to a mass of 0.4 grams?
d. [2] Find the average rate of change between 15h and 45h.

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a. The mass equation that remains after t hours can be determined using the formula:

mass remaining = initial mass × (1/2)^(t/half-life)

In this case, the initial mass is 2 g and the half-life is 15 h. Substituting these values into the formula, we get:

mass remaining = 2 g × (1/2)^(t/15)

b. To find the mass that remains after 5 hours, we can substitute t = 5 into the mass equation:

mass remaining = 2 g × (1/2)^(5/15)

Simplifying the exponent, we have:

mass remaining = 2 g × (1/2)^(1/3)

Evaluating this expression, the mass remaining after 5 hours is approximately 1.319 g (rounded to the nearest hundredth).

c. To find how long it takes for the sample to decay to a mass of 0.4 grams, we can set the mass remaining equal to 0.4 g in the mass equation:

0.4 g = 2 g × (1/2)^(t/15)

Simplifying this equation, we have:

(1/2)^(t/15) = 0.2

To solve for t, we can take the logarithm of both sides of the equation:

log[(1/2)^(t/15)] = log(0.2)

Using the logarithmic property, we can bring down the exponent:

(t/15) × log(1/2) = log(0.2)

Now, we can isolate t by dividing both sides of the equation by log(1/2):

t = [15 × log(0.2)] / log(1/2)

Evaluating this expression, the sample will take approximately 8.3 hours (rounded to the nearest hour) to decay to a mass of 0.4 grams.

d. The average rate of change between 15 hours and 45 hours can be found by calculating the change in mass divided by the change in time:

average rate of change = (final mass - initial mass) / (final time - initial time)

In this case, the initial time is 15 hours, the final time is 45 hours, and the initial mass is 2 grams. To find the final mass, we can substitute t = 45 into the mass equation:

final mass = 2 g × (1/2)^(45/15)

Simplifying the exponent, we have:

final mass = 2 g × (1/2)^3

Evaluating this expression, the final mass is 0.5 grams.

Now, we can substitute these values into the average rate of change formula:

average rate of change = (0.5 g - 2 g) / (45 h - 15 h)

Simplifying this expression, the average rate of change between 15 hours and 45 hours is approximately -0.09 g/h (rounded to two decimal places).