The relationship between the elapsed time \( t \), in minutes, since the medicine is ingested, and the number of harmful bacteria remaining in the body, \( H_{\text {minute }}(t) \), is modeled by the following function: \( H_{\text {minute }}(t)=500,000,000 \cdot(0.2)^{t} \) Complete the following sentence about the rate of change in the number of harmful bacteria remaining in the body in seconds. Round your answer to two decimal places. Every second, the number of harmful bacteria remaining in the body decays by a factor of

To find the rate of change in the number of harmful bacteria remaining in the body in seconds, we need to first express the function \( H_{\text{minute}}(t) \) in terms of seconds. Since \( t \) is in minutes, we can convert it to seconds by noting that 1 minute = 60 seconds. Therefore, if \( t \) is in minutes, then \( t \) in seconds is \( t_{\text{seconds}} = 60t \). The function in terms of seconds becomes: \[ H_{\text{seconds}}(s) = 500,000,000 \cdot (0.2)^{\frac{s}{60}} \] where \( s \) is the time in seconds. Next, we need to find the rate of change of \( H_{\text{seconds}}(s) \) with respect to \( s \). This requires us to differentiate the function with respect to \( s \). Let's calculate the derivative \( \frac{dH_{\text{seconds}}}{ds} \) and then evaluate the decay factor per second. The decay factor can be found by evaluating the ratio of the number of bacteria at \( s \) and \( s + 1 \): \[ \text{Decay Factor} = \frac{H_{\text{seconds}}(s+1)}{H_{\text{seconds}}(s)} \] Now, let's perform the calculations. 1. Differentiate \( H_{\text{seconds}}(s) \). 2. Calculate the decay factor. I'll proceed with these calculations. Simplify the expression by following steps: - step0: Solution: \(500000000\times 0.2^{\frac{s}{60}}\) - step1: Convert the expressions: \(500000000\left(\frac{1}{5}\right)^{\frac{s}{60}}\) - step2: Multiply the terms: \(500000000\times 5^{-\frac{s}{60}}\) Calculate or simplify the expression \( (500000000 * (0.2)^((s+1)/60)) / (500000000 * (0.2)^(s/60)) \). Simplify the expression by following steps: - step0: Solution: \(\frac{\left(500000000\times 0.2^{\frac{\left(s+1\right)}{60}}\right)}{\left(500000000\times 0.2^{\frac{s}{60}}\right)}\) - step1: Remove the parentheses: \(\frac{500000000\times 0.2^{\frac{s+1}{60}}}{500000000\times 0.2^{\frac{s}{60}}}\) - step2: Multiply by \(a^{-n}:\) \(\frac{500000000\times 0.2^{\frac{s+1}{60}}\times 0.2^{-\frac{s}{60}}}{500000000}\) - step3: Reduce the fraction: \(0.2^{\frac{s+1}{60}}\times 0.2^{-\frac{s}{60}}\) - step4: Multiply: \(0.2^{\frac{s+1}{60}-\frac{s}{60}}\) - step5: Calculate: \(0.2^{\frac{1}{60}}\) - step6: Evaluate the power: \(0.973533\) 1. The derivative of the function \( H_{\text{seconds}}(s) \) is: \[ \frac{dH_{\text{seconds}}}{ds} = 500,000,000 \times 5^{-\frac{s}{60}} \] This indicates how the number of harmful bacteria changes with respect to time in seconds. 2. The decay factor, which represents the ratio of the number of bacteria remaining after one second, is approximately: \[ \text{Decay Factor} \approx 0.973533 \] Thus, we can complete the sentence as follows: Every second, the number of harmful bacteria remaining in the body decays by a factor of approximately **0.97** (rounded to two decimal places).