Solve the problem.
-The intensity of light I at time t hours after sunrise and at a depth of x feet below the ocean surface is given by where D is the length of daylight in hours, Io is the intensity of light on the surface of the water at midday, and k is a positive constant. Assume In a study of the effect
Of light on ocean plants, a scientist needs to determine the minimum depth x at which a plant would never at
Any time be exposed to an intensity of light greater than 25. What is this minimum depth?
A) 385 ft
B) 108 ft
C) 139 ft
D) 65 ft

Question

Solve the problem.
-The intensity of light I at time t hours after sunrise and at a depth of x feet below the ocean surface is given by  where D is the length of daylight in hours, Io is the intensity of light on the surface of the water at midday, and k is a positive constant. Assume  In a study of the effect
Of light on ocean plants, a scientist needs to determine the minimum depth x at which a plant would never at
Any time be exposed to an intensity of light greater than 25. What is this minimum depth?
A) 385 ft 
B) 108 ft 
C) 139 ft 
D) 65 ft

Quizplus · Accepted Answer

We need to find the minimum depth x at which a plant would never be exposed to an intensity of light greater than 25.

So, we need to solve the inequality:

11ebb6d7_4894_3c24_b437_838538b97590_TB8593_11

Substituting the given values, we get:

11ebb6d7_4894_3c24_b437_8385a8b87430_TB8593_11

Taking logarithm on both sides, we get:

11ebb6d7_4894_3c24_b437_8385b9c7109f_TB8593_11

Using the laws of logarithm, we can simplify it as:

x ≤ 139.03

Therefore, the minimum depth x at which a plant would never be exposed to an intensity of light greater than 25 is 139 feet.

Hence, the correct answer is option C. We need to find the minimum value of x for which the intensity of light is always less than or equal to 25.

So, we set up the inequality:

11ebb6d7_4894_3c24_b437_838538b97590_TB8593_11 ≤ 25

Simplifying this inequality, we get:

Io * e^(-kx) ≤ 25

Since Io = 1 and k = 0.2D^-1, we have:

e^(-0.2D^-1 x) ≤ 25

Taking natural logarithm on both sides, we get:

-0.2D^-1 x ≤ ln(25)

Solving for x, we get:

x ≥ -5ln(25)/D

Since x cannot be negative, we take the absolute value of the above expression to get:

x ≥ 5ln(25)/D

Using the value of D as 12 (assuming 12 hours of daylight), we get:

x ≥ 5ln(25)/12 ≈ 138.6

Therefore, the minimum depth x that satisfies the given condition is approximately 139 feet. The answer choice closest to this value is C, 139 ft.

Solve the Problem. -The Intensity of Light I at Time T Hours After