The depth d in feet of the water in a bay at is given by d(t) = 3/2 sin (5πt/31) + 23 where t is time. Graph the depth of the water as a function of time. What is the maximum depth of the water to the nearest tenth of a foot? (Enter only the number.)
The depth d in feet of the water in a bay at is given by d(t) = 3/2 sin (5πt/31) + 23 where t is time. Graph the depth of the water as a function of time. What is the maximum depth of the water to the nearest tenth of a foot? (Enter only the number.)
since the maximum of sin(x) is 1, d(t) achieves a maximum of 3/2 + 23.
To graph the depth of the water as a function of time, we can plot several points and then connect them to form a curve.
First, let's find some key points on the graph by evaluating the depth of the water at specific time intervals. We'll use a time interval of 1 unit for simplicity.
When t = 0, we have:
d(0) = (3/2)sin(5π(0)/31) + 23
d(0) = (3/2)sin(0) + 23
d(0) = (3/2)(0) + 23
d(0) = 23
So at t = 0, the depth of the water is 23 feet.
Similarly, let's find the depth at t = 1, t = 2, t = 3, and t = 4.
When t = 1:
d(1) = (3/2)sin(5π(1)/31) + 23
d(1) = (3/2)sin(5π/31) + 23
When t = 2:
d(2) = (3/2)sin(5π(2)/31) + 23
d(2) = (3/2)sin(10π/31) + 23
When t = 3:
d(3) = (3/2)sin(5π(3)/31) + 23
d(3) = (3/2)sin(15π/31) + 23
When t = 4:
d(4) = (3/2)sin(5π(4)/31) + 23
d(4) = (3/2)sin(20π/31) + 23
Now, plot these points on a graph, where the x-axis represents time and the y-axis represents depth. Connect the points to create a smooth curve.
Once you have the graph, you can visually determine the maximum depth of the water. Look for the highest point on the curve, which corresponds to the maximum depth. Read the y-coordinate of that point, and round it to the nearest tenth of a foot.
If you prefer a more accurate approach, you can also calculate the maximum depth algebraically. The function for depth is d(t) = (3/2)sin(5πt/31) + 23. Since the sine function oscillates between -1 and 1, the maximum value of the depth occurs when sin(5πt/31) is equal to 1. In this case, the maximum depth would be (3/2)(1) + 23.
Evaluating this expression gives:
Max depth = (3/2)(1) + 23
Max depth = 3/2 + 23
Max depth ≈ 23.5 feet
Therefore, the maximum depth of the water is approximately 23.5 feet, rounding to the nearest tenth.