Math  /  Calculus

QuestionThe intensity of sunlight below the ocean's surface decreased exponentially with depth below the surface. When the intensity at the surface is 100 units, then intensity at a depth of 3 m is 6 units. A particular plant cannot grow if the intensity of the sunlight is less than 0.001 unit, What is the maximum depth, to the nearest centimetre, at which this plant can grow?

Studdy Solution

STEP 1

What is this asking? How deep can a plant grow underwater given that sunlight fades exponentially and the plant needs a certain amount of light? Watch out! Remember that the intensity decreases *exponentially*, not linearly, and we need the answer to the nearest *centimeter*!

STEP 2

1. Define the exponential function
2. Find the constant k
3. Calculate the maximum depth

STEP 3

We know the intensity of sunlight decreases exponentially with depth.
So, we can use the exponential decay formula: I(d)=I0ekdI(d) = I_0 \cdot e^{-kd}, where I(d)I(d) is the intensity at depth dd, I0I_0 is the **initial intensity** at the surface, kk is the **decay constant**, and dd is the depth.

STEP 4

We're given that the **initial intensity** at the surface is 100\textbf{100} units, so I0=100I_0 = \textbf{100}.
Our formula becomes I(d)=100ekdI(d) = \textbf{100} \cdot e^{-kd}.

STEP 5

We're told that at a depth of 3\textbf{3} meters, the intensity is 6\textbf{6} units.
So, we have I(3)=6I(\textbf{3}) = \textbf{6}.
Let's plug this into our equation: 6=100ek3\textbf{6} = 100 \cdot e^{-k \cdot \textbf{3}}.

STEP 6

Now, we need to solve for kk.
First, divide both sides by 100\textbf{100} to isolate the exponential term: 6100=e3k\frac{\textbf{6}}{\textbf{100}} = e^{-3k}, which simplifies to 0.06=e3k0.06 = e^{-3k}.

STEP 7

To get rid of the exponential, we can take the natural logarithm (ln) of both sides: ln(0.06)=ln(e3k)\ln(0.06) = \ln(e^{-3k}).
This simplifies to ln(0.06)=3k\ln(0.06) = -3k.

STEP 8

Finally, divide both sides by 3-3 to find kk: k=ln(0.06)3k = \frac{\ln(0.06)}{-3}.
Calculating this gives us k0.957k \approx \textbf{0.957}.

STEP 9

We know the plant can't grow if the intensity is less than 0.001\textbf{0.001} units.
So, we want to find the depth dd when I(d)=0.001I(d) = \textbf{0.001}.
Our equation becomes 0.001=100e0.957d\textbf{0.001} = 100 \cdot e^{-0.957d}.

STEP 10

Divide both sides by 100\textbf{100}: 0.001100=e0.957d\frac{\textbf{0.001}}{\textbf{100}} = e^{-0.957d}, which simplifies to 0.00001=e0.957d0.00001 = e^{-0.957d}.

STEP 11

Take the natural logarithm of both sides: ln(0.00001)=ln(e0.957d)\ln(0.00001) = \ln(e^{-0.957d}).
This simplifies to ln(0.00001)=0.957d\ln(0.00001) = -0.957d.

STEP 12

Divide both sides by 0.957-0.957 to find dd: d=ln(0.00001)0.957d = \frac{\ln(0.00001)}{-0.957}.
Calculating this gives us d11.95d \approx \textbf{11.95} meters.

STEP 13

Since we need the answer to the nearest centimeter, we multiply by 100\textbf{100}: 11.95100=119511.95 \cdot \textbf{100} = \textbf{1195} centimeters.

STEP 14

The maximum depth at which the plant can grow is approximately **1195** centimeters.

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord