Question

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a patient is given a 50 mg dose of medicine. the medicines effectiveness decreases every hour at a constant rate of 40%. what is the exponential decay function that models this scenario? how much medicine will be left in the patients system after 2 hours? f(x)=50(40)^x; 80,000 mg f(x)=50(60)^x; 180,000 mg f(x)=50(0.40)^x; 8 mg f(x)=50(0.60)^x; 18 mg

Explanation:

Step1: Recall Exponential Decay Formula

The general form of an exponential decay function is \( f(x) = a(1 - r)^x \), where \( a \) is the initial amount, \( r \) is the rate of decay (as a decimal), and \( x \) is time. Here, \( a = 50 \) mg, \( r = 0.40 \) (since 40% decay means 60% remains, so \( 1 - 0.40 = 0.60 \)? Wait, no—wait, decay rate: if it decreases by 40% each hour, the remaining amount is \( 100\% - 40\% = 60\% = 0.60 \) of the previous hour. Wait, let's clarify:

Wait, exponential decay: \( f(x) = a(b)^x \), where \( b = 1 - r \) (for decay) or \( b = 1 + r \) (for growth). If the effectiveness decreases by 40% per hour, the remaining fraction is \( 1 - 0.40 = 0.60 \) per hour. Wait, but let's check the options.

Wait, the initial dose is 50 mg. Let's compute the function first.

Step2: Determine the Decay Factor

The decay rate is 40% per hour, so the remaining amount each hour is \( 100\% - 40\% = 60\% = 0.60 \) of the previous amount. So the function should be \( f(x) = 50(0.60)^x \), where \( x \) is the number of hours.

Now, let's check the amount after 2 hours. Plug \( x = 2 \) into \( f(x) = 50(0.60)^2 \).

Calculate \( (0.60)^2 = 0.36 \), then \( 50 \times 0.36 = 18 \) mg.

Now, check the options:

• First option: \( f(x) = 50(40)^x \): 40 is greater than 1, so that's growth, not decay. Eliminate.
• Second option: \( f(x) = 50(60)^x \): 60 is growth, eliminate.
• Third option: \( f(x) = 50(0.40)^x \): 0.40 is the decay rate, but that would mean remaining 40% each hour, but 40% decay means remaining 60%, so this is incorrect. Wait, no—wait, maybe I mixed up. Wait, if the decay rate is 40%, then the multiplier is \( 1 - 0.40 = 0.60 \), but let's recalculate.

Wait, let's re-express:

If the medicine decreases by 40% each hour, the amount left after 1 hour is \( 50 - 0.40 \times 50 = 50(1 - 0.40) = 50(0.60) \). After 2 hours, it's \( 50(0.60)(0.60) = 50(0.60)^2 \). So the function is \( f(x) = 50(0.60)^x \), and after 2 hours, \( 50(0.60)^2 = 50 \times 0.36 = 18 \) mg.

Looking at the options, the fourth option is \( f(x) = 50(0.60)^x \); 18 mg. That matches.

Wait, but let's check the third option: \( f(x) = 50(0.40)^x \). If we plug \( x = 2 \), \( 50(0.40)^2 = 50 \times 0.16 = 8 \) mg. But that would be if the remaining is 40% each hour, which would mean a decay rate of 60% (since 100% - 40% remaining = 60% decay). So that's incorrect. The correct decay factor is 0.60 (remaining 60% each hour), so the function is \( 50(0.60)^x \), and after 2 hours, 18 mg.

So the correct option is the fourth one: \( f(x) = 50(0.60)^x \); 18 mg.

Answer:

The correct option is the fourth one: \( f(x) = 50(0.60)^x \); 18 mg (the bottom-left option, or the fourth option in the list, depending on the layout. Assuming the options are:

1. \( f(x) = 50(40)^x \); 80,000 mg
2. \( f(x) = 50(60)^x \); 180,000 mg
3. \( f(x) = 50(0.40)^x \); 8 mg
4. \( f(x) = 50(0.60)^x \); 18 mg

Then the answer is the fourth option: \( f(x) = 50(0.60)^x \); 18 mg.