Solve the problem.
-Sunlight is absorbed in water, and as a result the light intensity in oceans, lakes, and ponds decreases exponentially with depth. The percentage of visible light $P$ (in decimal form) at a depth of $x$ meters is given by $P = e ^ { - k x }$, where $k$ is a constant related to the clarity and other physical properties of the water. The model for a particular lake is $P = e ^ { - 0.1284 x }$. Determine the depth at which the light intensity is $80 \%$ of the value from the surface for this lake. Round to the nearest tenth of a meter.
A) $0.8 \mathrm {~m}$
B) $34.1 \mathrm {~m}$
C) $1.7 \mathrm {~m}$
D) $16.2 \mathrm {~m}$

Question

Quizplus · Accepted Answer

To find the depth at which the light intensity is 80% of the surface value, set $P = 0.8$ and solve for $x$ in the equation $0.8 = e^{-0.1284x}$. Taking the natural logarithm of both sides gives $\ln(0.8) = -0.1284x$, leading to $x = \frac{\ln(0.8)}{-0.1284}$, which approximately equals $1.7$ meters when rounded to the nearest tenth.

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