Radioactive decay is the process by which an unstable element transforms into different element, typically releasing energy as it does so. For 500g of a radioactive substance with a half-life of 5.2 hours, the amount remaining is given by the formula M(t)=500(0.5) ^ t/5.2

Question

Radioactive decay is the process by which an unstable element transforms into different element, typically releasing energy as it does so. For 500g of a radioactive substance with a half-life of 5.2 hours, the amount remaining is given by the formula M(t)=500(0.5) ^ t/5.2

a) calculate the average rate of change between 8 days and 15 days.
b)calculate the approximate instantaneous rate of change when t=2hours

Answer 1

When you are in caclulus, you will have a much easier and elegant way of doing this.

For now

avgratechange=(rate8 - rate 15)/(8-15)

rate 8=500(1/2)^8 and likewise for the rate15

approximate rate at t=2
(500*(.5)^(2+e)-500(.5)^2)/e
500*(.5)^2 (.5^e -1)/e
now the limit of this as e approaches zero (I leave it to you to prove) will be.
500*.5^2 ln .5

Answer 2

(rate 5+ rate 8) / (15-8) ?

t is measured in hours

Answer 3

So use 8 days * 24 hr/day = 192 hours and 15*24 for 15 days and proceed as advised.

Answer 4

To calculate the average rate of change between 8 days and 15 days, we need to find the difference in the amount remaining over that time period and divide it by the difference in time. Let's plug the values into the formula M(t) = 500(0.5)^(t/5.2) and calculate the average rate of change.

a) Start by calculating the amount remaining at 8 days:
M(8 days) = 500(0.5)^(8/5.2)
M(8 days) ≈ 500(0.5)^1.538 ≈ 294.05

Next, calculate the amount remaining at 15 days:
M(15 days) = 500(0.5)^(15/5.2)
M(15 days) ≈ 500(0.5)^2.885 ≈ 137.82

Now, find the difference in the amount remaining:
Difference = M(15 days) - M(8 days)
Difference ≈ 137.82 - 294.05 ≈ -156.23

Next, calculate the difference in time:
Time difference = 15 days - 8 days
Time difference = 7 days

Now, divide the difference in amount remaining by the difference in time to find the average rate of change:
Average rate of change = Difference / Time difference
Average rate of change ≈ -156.23 / 7 ≈ -22.32

Therefore, the average rate of change between 8 days and 15 days is approximately -22.32 grams per day.

b) To calculate the approximate instantaneous rate of change when t = 2 hours, we need to take the derivative of the formula M(t) = 500(0.5)^(t/5.2) with respect to t.

First, rewrite the formula as:
M(t) = 500 * (0.5)^(1/5.2t)

Now, take the derivative of M(t) with respect to t:
dM/dt = 500 * (1/5.2) * (0.5)^(1/5.2t) * ln(0.5) * (1/5.2)

Since the question asks for the approximate instantaneous rate of change when t = 2 hours, substitute t = 2 into the derivative equation and calculate the value.

Instantaneous rate of change when t = 2 hours: dM/dt(t = 2) ≈ 500 * (1/5.2) * (0.5)^(1/5.2*2) * ln(0.5) * (1/5.2)

Now, plug in the values into the equation and calculate the approximate instantaneous rate of change:
dM/dt(t = 2) ≈ 500 * (1/5.2) * (0.5)^(1/10.4) * ln(0.5) * (1/5.2)

The value might be a bit complicated to calculate directly, so you can use a calculator or software to obtain the approximate value.

Radioactive decay is the process by which an unstable element transforms into different element, typically releasing energy as it does so. For 500g of a radioactive substance with a half-life of 5.2 hours, the amount remaining is given by the formula M(t)=500(0.5) ^ t/5.2

When you are in caclulus, you will have a much easier and elegant way of doing this.

(rate 5+ rate 8) / (15-8) ?

So use 8 days * 24 hr/day = 192 hours and 15*24 for 15 days and proceed as advised.

72y2-98

To calculate the average rate of change between 8 days and 15 days, we need to find the difference in the amount remaining over that time period and divide it by the difference in time. Let's plug the values into the formula M(t) = 500(0.5)^(t/5.2) and calculate the average rate of change.