Question

A quantity with an initial value of 980 decays exponentially at a rate of 1% every 2 days. What is the value of the quantity after 5 weeks, to the nearest hundredth?​

Answer 1

To find the time for 90% decay of iodine-131 to Xe-131 with a decay constant of 0.138 d⁻¹, use the formula N = N_0e^{-kt}. Set N to 10% of N_0, solve for t after taking the natural logarithm of both sides. The time t in days is the result of ln(0.1) / -0.138.

Step-by-step explanation:

To determine how long it will take for 90% of the iodine-131 in a 0.500 M solution to decay to Xenon-131 (Xe-131), we need to use the first-order decay formula which is represented as N = N_0e^{-kt}, where N is the final quantity, N_0 is the initial quantity, k is the decay constant, and t is the time. With a decay constant of 0.138 d⁻¹, we want to find the time when the quantity is reduced to 10% of the initial concentration (because 90% has decayed).

We start by expressing the final amount (10% of the initial) as N = 0.1N_0. Substituting into the formula, we get 0.1N_0 = N_0e^{-0.138t}. Dividing both sides by N_0 gives us 0.1 = e^{-0.138t}. Taking the natural logarithm of both sides leads to ln(0.1) = -0.138t.

Solving for t gives us t = ln(0.1) / -0.138. Evaluating this gives us the time required for 90% decay. Make sure to convert the natural logarithm result to days if necessary using a calculator.

Answer 2

An exponential decay function has the form y = a*b^(-kt) where a is the initial value, b is the base of the exponential function (in this case, 1 - the decay rate as a decimal), k is the decay rate and t is the time.

Given that the initial value is 980, the decay rate is 1% every 2 days and the time is 5 weeks. The first step is to convert the time to the same unit of time as the decay rate, in this case, days: 5 weeks * 7 days/week = 35 days.

The decay rate per day is 1/100 because 1% as a decimal is 0.01, so the decay rate per day is 0.01/2 = 0.005

Now we can plug in the values in the function:

y = 980 * (1 - 0.005)^(-0.005*35)

y = 980 * 0.995^(-17.5)

y ≈ 574.64

So, the value of the quantity after 5 weeks to the nearest hundredth is 574.64

It's worth noting that in this case, the base (1 - decay rate) is less than 1, which implies that the value of the quantity decreases over time.