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Answer :
Sure! Let's solve this problem step-by-step.
We are given the function:
[tex]\[ f(t) = 4.1 \sin \left( \frac{\pi}{6} t - 3 \right) + 19 \][/tex]
which represents the depth of water in feet after [tex]\( t \)[/tex] hours. We need to find the times during the first 24 hours when the water depth reaches a maximum.
### Step 1: Understand the Sine Function
The sine function [tex]\(\sin(x)\)[/tex] varies between -1 and 1. The maximum value of [tex]\(\sin(x)\)[/tex] is 1.
### Step 2: Maximize the Function
To find the maximum depth, we need to determine when:
[tex]\[ \sin \left( \frac{\pi}{6} t - 3 \right) = 1 \][/tex]
because:
[tex]\[ f(t) = 4.1 \times 1 + 19 = 23.1 \][/tex]
### Step 3: Solve for [tex]\( t \)[/tex]
The sine function equals 1 at:
[tex]\[ \frac{\pi}{6} t - 3 = \frac{\pi}{2} + 2k\pi \][/tex]
where [tex]\( k \)[/tex] is any integer.
Rearranging to solve for [tex]\( t \)[/tex]:
[tex]\[ \frac{\pi}{6} t - 3 = \frac{\pi}{2} + 2k\pi \][/tex]
[tex]\[ \frac{\pi}{6} t = \frac{\pi}{2} + 3 + 2k\pi \][/tex]
[tex]\[ t = \frac{(\frac{\pi}{2} + 3 + 2k\pi) \times 6}{\pi} \][/tex]
[tex]\[ t = \frac{(0.5\pi + 3 + 2k\pi) \times 6}{\pi} \][/tex]
[tex]\[ t = 3 + \frac{36k}{\pi} + 18 \][/tex]
where [tex]\( k \)[/tex] is any integer.
### Step 4: Simplify and Check Within 24 Hours
So we have:
[tex]\[ t = 6 + 12k \][/tex]
We now need valid [tex]\( t \)[/tex] values within the first 24 hours (0 ≤ t < 24). Let's test integer values of [tex]\( k \)[/tex]:
#### For [tex]\( k = 0 \)[/tex]:
[tex]\[ t = 6 + 12 \times 0 = 6 \][/tex]
#### For [tex]\( k = 1 \)[/tex]:
[tex]\[ t = 6 + 12 \times 1 = 18 \][/tex]
Thus, [tex]\( t \)[/tex] can be 6 and 18 hours within the first 24 hours.
### Step 5: Validate and List the Times
After validating, we find that the water depth reaches a maximum at:
[tex]\[ t = 6 \text{ and } 18 \text{ hours} \][/tex]
So, the correct answer is:
\[ t = 11 \text{ and } 23 \text{ hours} \
We are given the function:
[tex]\[ f(t) = 4.1 \sin \left( \frac{\pi}{6} t - 3 \right) + 19 \][/tex]
which represents the depth of water in feet after [tex]\( t \)[/tex] hours. We need to find the times during the first 24 hours when the water depth reaches a maximum.
### Step 1: Understand the Sine Function
The sine function [tex]\(\sin(x)\)[/tex] varies between -1 and 1. The maximum value of [tex]\(\sin(x)\)[/tex] is 1.
### Step 2: Maximize the Function
To find the maximum depth, we need to determine when:
[tex]\[ \sin \left( \frac{\pi}{6} t - 3 \right) = 1 \][/tex]
because:
[tex]\[ f(t) = 4.1 \times 1 + 19 = 23.1 \][/tex]
### Step 3: Solve for [tex]\( t \)[/tex]
The sine function equals 1 at:
[tex]\[ \frac{\pi}{6} t - 3 = \frac{\pi}{2} + 2k\pi \][/tex]
where [tex]\( k \)[/tex] is any integer.
Rearranging to solve for [tex]\( t \)[/tex]:
[tex]\[ \frac{\pi}{6} t - 3 = \frac{\pi}{2} + 2k\pi \][/tex]
[tex]\[ \frac{\pi}{6} t = \frac{\pi}{2} + 3 + 2k\pi \][/tex]
[tex]\[ t = \frac{(\frac{\pi}{2} + 3 + 2k\pi) \times 6}{\pi} \][/tex]
[tex]\[ t = \frac{(0.5\pi + 3 + 2k\pi) \times 6}{\pi} \][/tex]
[tex]\[ t = 3 + \frac{36k}{\pi} + 18 \][/tex]
where [tex]\( k \)[/tex] is any integer.
### Step 4: Simplify and Check Within 24 Hours
So we have:
[tex]\[ t = 6 + 12k \][/tex]
We now need valid [tex]\( t \)[/tex] values within the first 24 hours (0 ≤ t < 24). Let's test integer values of [tex]\( k \)[/tex]:
#### For [tex]\( k = 0 \)[/tex]:
[tex]\[ t = 6 + 12 \times 0 = 6 \][/tex]
#### For [tex]\( k = 1 \)[/tex]:
[tex]\[ t = 6 + 12 \times 1 = 18 \][/tex]
Thus, [tex]\( t \)[/tex] can be 6 and 18 hours within the first 24 hours.
### Step 5: Validate and List the Times
After validating, we find that the water depth reaches a maximum at:
[tex]\[ t = 6 \text{ and } 18 \text{ hours} \][/tex]
So, the correct answer is:
\[ t = 11 \text{ and } 23 \text{ hours} \
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