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?
3/2 + 23
since max of sine is 1.
24.5
To graph the depth of the water as a function of time, you can start by plotting some points. Choose a few values of t and calculate the corresponding depth d(t). Then, plot these points on a graph with time on the x-axis and depth on the y-axis. By connecting these points, you will be able to see the overall shape of the graph.
Let's choose several values of t to calculate the corresponding depths:
When t = 0:
d(0) = 3/2 * sin(5π(0)/31) + 23
= 3/2 * sin(0) + 23
= 0 + 23
= 23
When t = 5:
d(5) = 3/2 * sin(5π(5)/31) + 23
= 3/2 * sin(25π/31) + 23
≈ 3/2 * 0.809 + 23
≈ 1.213 + 23
≈ 24.213
When t = 10:
d(10) = 3/2 * sin(5π(10)/31) + 23
= 3/2 * sin(50π/31) + 23
≈ 3/2 * 0.309 + 23
≈ 0.463 + 23
≈ 23.463
By plotting these values on a graph, you would get a sinusoidal curve that varies between approximately 21.3 ft and 24.6 ft. The vertical axis represents the depth of the water, and the horizontal axis represents time.
To find the maximum depth, you need to locate the highest point on the graph. In this case, the maximum value of the function occurs at the peak of the sinusoidal curve. The maximum depth is the highest value of d(t).
Since the graph is a vertical shift of the sine function, the maximum value of d(t) occurs when sin(5πt/31) reaches its maximum value of 1. Therefore, the maximum depth is obtained when d(t) = 3/2 * sin(5πt/31) + 23 = 3/2 * 1 + 23 = 3/2 + 23 = 24.5 ft.
So, the maximum depth of the water is approximately 24.5 feet to the nearest tenth.