The intensity, I, of light passing through water can be modelled by the equation I=10^1-0.13, where x is the depth of water in metres. Most aquatic plants require a light intensity of 4.2 units for strong growth. Determine the depth of water at which most aquatic plants receive the required light.
I need help. This assignment consisting of a bunch of questions is due tomorrow.
shouldn't there be an x somewhere in the formula?
To determine the depth of water at which aquatic plants receive the required light intensity, we need to solve the equation:
I = 10^(1 - 0.13x) = 4.2
Let's solve this step by step:
Step 1: Rearrange the equation to isolate x:
10^(1 - 0.13x) = 4.2
Step 2: Take the logarithm of both sides, in base 10:
log(10^(1 - 0.13x)) = log(4.2)
Step 3: Use the logarithmic property log(a^b) = b * log(a):
(1 - 0.13x) * log(10) = log(4.2)
Since log(10) = 1, we can simplify the equation to:
(1 - 0.13x) = log(4.2)
Step 4: Solve for x by isolating it:
1 - 0.13x = log(4.2)
-0.13x = log(4.2) - 1
Step 5: Divide both sides of the equation by -0.13 to solve for x:
x = (log(4.2) - 1) / -0.13
Now, use a calculator to evaluate the right side of the equation to find the value of x.
Please note that the value of x that you obtain might need to be rounded to an appropriate number of decimal places, depending on the level of precision required for your assignment.
No problem! Let's start by understanding the equation and what it represents. The equation I = 10^(1-0.13x) represents the intensity of light, I, passing through water as a function of the depth of water, x, in meters.
To determine the depth of water at which most aquatic plants receive the required light intensity of 4.2 units, we can set up the equation as follows:
4.2 = 10^(1-0.13x)
To solve this equation, we need to isolate the variable x. We can do this by taking logarithms on both sides of the equation. Since the logarithm of a number is the exponent to which another fixed number, the base, must be raised to produce that number, we can use the logarithm with base 10 (also known as the common logarithm or log).
Taking the logarithm on both sides of the equation, we have:
log(4.2) = log(10^(1-0.13x))
Using the logarithmic property log(a^b) = b * log(a), the equation becomes:
log(4.2) = (1-0.13x) * log(10)
Since log(10) is equal to 1, our equation simplifies to:
log(4.2) = 1 - 0.13x
Next, we isolate the variable x by subtracting log(4.2) from both sides:
log(4.2) - 1 = -0.13x
Dividing both sides of the equation by -0.13, we get:
x = (log(4.2) - 1) / -0.13
Using a calculator, we can evaluate the right-hand side of the equation to find the value of x:
x ≈ 6.72 meters (rounded to two decimal places)
Therefore, most aquatic plants receive the required light intensity of 4.2 units at a depth of approximately 6.72 meters.