a quantity with an initial value of 590 decays exponentially at a rate such that the quantity cuts in half…

Explanation:

Step1: Convert time to seconds

0.1 minutes is equal to (0.1\times60 = 6) seconds.

Step2: Recall the exponential decay formula for half - life

The formula for exponential decay with half - life (t_{1/2}) is (N = N_0\times\left(\frac{1}{2}\right)^{\frac{t}{t_{1/2}}}), where (N_0) is the initial quantity, (t) is the time elapsed, and (t_{1/2}) is the half - life. Here, (N_0 = 590), (t = 6) seconds, and (t_{1/2}=4) seconds.

Step3: Substitute the values into the formula

Substitute (N_0 = 590), (t = 6), and (t_{1/2}=4) into the formula (N = N_0\times\left(\frac{1}{2}\right)^{\frac{t}{t_{1/2}}}). We get (N=590\times\left(\frac{1}{2}\right)^{\frac{6}{4}}). First, simplify the exponent: (\frac{6}{4}=\frac{3}{2} = 1.5). Then, (\left(\frac{1}{2}\right)^{1.5}=\frac{1}{2^{1.5}}=\frac{1}{\sqrt{2^3}}=\frac{1}{\sqrt{8}}\approx\frac{1}{2.8284}\approx0.3536).

Step4: Calculate the final value

Multiply (590) by (0.3536): (N = 590\times0.3536=208.624\approx208.62)

Answer:

(208.62)