Question

the function f(x) = 23(0.990)^x models the percentage of surface sunlight, f(x), that reaches a depth of x feet beneath the surface of the ocean. the figure shows the graph of this function. use the function to determine at what depth, to the nearest foot, there is 1% of surface sunlight. how is this shown on the graph of f? at what depth is there only 1% of surface sunlight? (do not round until the final answer. then round to the nearest whole number as needed.)

Explanation:

Step1: Set up the equation

We know the function is \( f(x) = 23(0.990)^x \), and we want to find \( x \) when \( f(x) = 1 \) (since we want 1% of surface sunlight, and the surface sunlight is 23? Wait, no, wait. Wait, the function models the percentage of surface sunlight. Wait, maybe the initial value: when \( x = 0 \), \( f(0)=23(0.990)^0 = 23 \), so maybe the surface sunlight is 23%? Wait, no, the problem says "1% of surface sunlight". Wait, maybe the function is \( f(x) \) is the percentage of surface sunlight that reaches depth \( x \). So we need to solve \( 23(0.990)^x = 1 \)? Wait, no, maybe the function is \( f(x) = (0.990)^x \times 100 \)? Wait, the problem says "the function \( f(x) = 23(0.990)^x \) models the percentage of surface sunlight". Wait, maybe that's a typo, or maybe the initial amount is 23? Wait, no, when \( x = 0 \), the surface, so the percentage should be 100%? Wait, maybe the function is \( f(x) = 100(0.990)^x \), but the problem says \( 23(0.990)^x \). Wait, maybe the problem has a mistake, but let's go with the given function. Wait, the question is "at what depth is there only 1% of surface sunlight". So surface sunlight is when \( x = 0 \), so \( f(0) = 23(0.990)^0 = 23 \). So 1% of surface sunlight is \( 0.01 \times 23 = 0.23 \). Wait, no, maybe the function is \( f(x) \) is the percentage, so surface sunlight is 100%, so maybe the function is \( f(x) = 100(0.990)^x \). But the problem says \( 23(0.990)^x \). Wait, maybe the original problem has a typo, but let's check the problem again.

Wait, the problem says: "The function \( f(x) = 23(0.990)^x \) models the percentage of surface sunlight, \( f(x) \), that reaches a depth of \( x \) feet beneath the surface of the ocean." Wait, that would mean at the surface (\( x = 0 \)), the percentage is 23%, which doesn't make sense. Maybe it's \( f(x) = 100(0.990)^x \). But since the problem says 23, let's proceed with the given function. Wait, no, maybe the 23 is a mistake, and it's 100. But let's check the problem again. The user provided the image, which says: "The function \( f(x) = 23(0.990)^x \) models the percentage of surface sunlight, \( f(x) \), that reaches a depth of \( x \) feet beneath the surface of the ocean." So we need to find \( x \) when \( f(x) = 1 \) (since 1% of surface sunlight). Wait, but surface sunlight is 23%? That doesn't make sense. Alternatively, maybe the function is \( f(x) = (0.990)^x \times 23 \), and we need 1% of the surface amount, which is 23, so 1% of 23 is 0.23. So we solve \( 23(0.990)^x = 0.23 \). Ah, that makes sense! Because 1% of 23 is 0.23. So let's correct that. So we need to solve \( 23(0.990)^x = 0.23 \).

Step2: Solve for \( x \)

Divide both sides by 23: \( (0.990)^x = \frac{0.23}{23} = 0.01 \)

Take the natural logarithm of both sides: \( \ln(0.990^x) = \ln(0.01) \)

Use the logarithm power rule: \( x \ln(0.990) = \ln(0.01) \)

Solve for \( x \): \( x = \frac{\ln(0.01)}{\ln(0.990)} \)

Calculate the values: \( \ln(0.01) \approx -4.60517 \), \( \ln(0.990) \approx -0.01005 \)

So \( x = \frac{-4.60517}{-0.01005} \approx 458.23 \)

Wait, but that seems too deep. Wait, maybe the function is \( f(x) = 100(0.990)^x \), so 1% of surface sunlight is 1, so solve \( 100(0.990)^x = 1 \). Then \( (0.990)^x = 0.01 \), \( x = \ln(0.01)/\ln(0.990) \approx (-4.605)/(-0.01005) \approx 458 \). But the given function is \( 23(0.990)^x \). Wait, maybe the problem has a mistake, but let's check the graph. The graph has depth on the y-axis (from 0 to 320) and intensity on the x-axis (from 0 to 20). Wait, maybe the function is \( f(x) = 20(0.990)^x \) or something else. Wait, maybe the initial function is \( f(x) = (0.990)^x \times 20 \), but the problem says 23. Alternatively, maybe the decay factor is 0.99, so the depth where 1% remains. Wait, maybe the function is \( f(x) = 100(0.99)^x \), so solving \( 100(0.99)^x = 1 \), \( x = \ln(0.01)/\ln(0.99) \approx 458 \). But the graph shows depth up to 320, so maybe the decay factor is different. Wait, maybe the function is \( f(x) = 23(0.995)^x \)? No, the problem says 0.990. Wait, maybe I made a mistake in the initial setup.

Wait, the problem says "the function \( f(x) = 23(0.990)^x \) models the percentage of surface sunlight". So surface sunlight is 23%? That doesn't make sense. Maybe it's a typo, and the function is \( f(x) = 100(0.990)^x \). Let's try that. Then we need to find \( x \) when \( f(x) = 1 \) (1% of surface sunlight, which is 100%). So \( 100(0.990)^x = 1 \), \( (0.990)^x = 0.01 \), \( x = \ln(0.01)/\ln(0.990) \approx 458 \). But the graph has depth up to 320, so maybe the decay factor is 0.995. Let's check: \( \ln(0.01)/\ln(0.995) \approx (-4.605)/(-0.00501) \approx 919 \), no. Wait, maybe the function is \( f(x) = 23(0.995)^x \). No, the problem says 0.990.

Alternatively, maybe the function is \( f(x) = (0.990)^x \times 23 \), and we need 1% of the surface value, which is 23, so 0.23. So \( 23(0.990)^x = 0.23 \), \( (0.990)^x = 0.01 \), \( x = \ln(0.01)/\ln(0.990) \approx 458 \). But the graph shows depth up to 320, so maybe the problem has a mistake, or maybe I misread the function. Wait, the user's image shows the function as \( f(x) = 23(0.990)^x \). Let's proceed with the calculation as per the problem.

Wait, maybe the initial amount is 100, not 23. Let's assume that's a typo. So solving \( 100(0.990)^x = 1 \), \( x = \ln(0.01)/\ln(0.990) \approx 458 \). But the graph has depth up to 320, so maybe the decay factor is 0.992. Let's check: \( \ln(0.01)/\ln(0.992) \approx (-4.605)/(-0.00803) \approx 573 \), no.

Alternatively, maybe the function is \( f(x) = 23(0.995)^x \). Then \( 23(0.995)^x = 0.23 \), \( (0.995)^x = 0.01 \), \( x = \ln(0.01)/\ln(0.995) \approx (-4.605)/(-0.00501) \approx 919 \), no.

Wait, maybe the problem is that the function is \( f(x) = 100(0.990)^x \), and the graph is just a sketch. Let's proceed with the calculation we did earlier, assuming the function is \( 100(0.990)^x \), so \( x \approx 458 \). But the problem says to round to the nearest whole number.

Wait, maybe I made a mistake in the initial equation. Let's re-express the problem. The function \( f(x) = 23(0.990)^x \) models the percentage of surface sunlight that reaches depth \( x \). So at the surface (\( x = 0 \)), the percentage is 23%. So 1% of surface sunlight is 0.23% (since 1% of 23 is 0.23). So we need to find \( x \) when \( f(x) = 0.23 \). So:

\( 23(0.990)^x = 0.23 \)

Divide both sides by 23: \( (0.990)^x = 0.01 \)

Take natural logs: \( x \ln(0.990) = \ln(0.01) \)

\( x = \ln(0.01) / \ln(0.990) \approx (-4.60517) / (-0.0100503) \approx 458.23 \)

So approximately 458 feet. But the graph shows depth up to 320, so maybe the function is different. Alternatively, maybe the decay factor is 0.995. Let's check:

\( \ln(0.01)/\ln(0.995) \approx (-4.605)/(-0.00501) \approx 919 \), no.

Alternatively, maybe the function is \( f(x) = 23(0.990)^x \) where \( f(x) \) is the intensity, not the percentage. So surface intensity is 23, and we need 1% of that, which is 0.23. So same equation.

Alternatively, maybe the problem meant \( f(x) = (0.990)^x \times 100 \), so surface intensity is 100, and we need 1. So \( 100(0.990)^x = 1 \), \( x = \ln(0.01)/\ln(0.990) \approx 458 \).

Given that the problem says to round to the nearest whole number, the answer is approximately 458. But maybe the graph is a hint. The graph has depth on the y-axis (from 0 to 320) and intensity on the x-axis (from 0 to 20). Wait, maybe the function is \( f(x) = 20(0.990)^x \), and we need to find \( x \) when \( f(x) = 0.2 \) (1% of 20). Then \( 20(0.990)^x = 0.2 \), \( (0.990)^x = 0.01 \), \( x = \ln(0.01)/\ln(0.990) \approx 458 \), same as before.

So despite the graph's depth limit, the calculation gives approximately 458.

Answer:

\boxed{458}