Question

the function f(x) = 20(0.986)^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. (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 \( f(x) = 20(0.986)^x \), and we want to find \( x \) when \( f(x)=1 \). So we set up the equation:

$$ 1 = 20(0.986)^x $$

Step2: Solve for \( (0.986)^x \)

Divide both sides of the equation by 20:

$$ \frac{1}{20}=0.986^x $$

$$ 0.05 = 0.986^x $$

Step3: Take the natural logarithm of both sides

Using the property of logarithms \( \ln(a^b)=b\ln(a) \), we take \( \ln \) of both sides:

$$ \ln(0.05)=\ln(0.986^x) $$

$$ \ln(0.05)=x\ln(0.986) $$

Step4: Solve for \( x \)

Divide both sides by \( \ln(0.986) \):

$$ x = \frac{\ln(0.05)}{\ln(0.986)} $$

Calculate the values:
\( \ln(0.05)\approx - 2.9957 \)
\( \ln(0.986)\approx - 0.0141 \)
Then \( x=\frac{-2.9957}{-0.0141}\approx212.46 \)

Step5: Round to the nearest whole number

Rounding \( 212.46 \) to the nearest whole number gives \( 212 \).

Answer:

\( 212 \)