Question

the function ( f(x) = 200 cdot 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. how is this shown on the graph of ( f )? if there is 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 \( f(x) = 200(0.986)^x \) and we want to find \( x \) when \( f(x)=12 \). So we set up the equation:

$$ 12 = 200(0.986)^x $$

Step2: Divide both sides by 200

Divide each side of the equation by 200 to isolate the exponential term:

$$ \frac{12}{200}=0.986^x $$

Simplify \( \frac{12}{200} \) to get \( 0.06 = 0.986^x \)

Step3: Take the natural logarithm of both sides

Take the natural log of both sides to use the logarithm property \( \ln(a^b)=b\ln(a) \):

$$ \ln(0.06)=\ln(0.986^x) $$

Which simplifies to:

$$ \ln(0.06)=x\ln(0.986) $$

Step4: Solve for \( x \)

Divide both sides by \( \ln(0.986) \) to solve for \( x \):

$$ x = \frac{\ln(0.06)}{\ln(0.986)} $$

Calculate the values: \( \ln(0.06)\approx - 2.8134 \) and \( \ln(0.986)\approx - 0.0141 \)
Then \( x=\frac{-2.8134}{-0.0141}\approx199.53 \)

Step5: Round to the nearest whole number

Rounding \( 199.53 \) to the nearest whole number gives \( 200 \)

Answer:

\( 200 \)