A vessel is crossing a channel. The depth of the water(measured in metres) varies with time and is represented by the following equation:
d(t)=2.5sin(0.523t)+2.9
a. create a graph showing the depth of the water over 24 hours.
b. what is the period of this function and what does it represent about the varying depths of the water.
c. the vessel requires a depth of at least 1.3m of water to cross the channel. They want to make 2 trips/day with each trip lasting at least 5 hours. What times could these trips be scheduled to allow for safe navigation?
(a) see
http://www.wolframalpha.com/input/?i=plot+2.5sin%280.523t%29%2B2.9+for+t%3D0+to+24
(b) since sin(kt) has period 2π/k,
sin(.523t) has period 2π/.523 = 12.01
That is, the tide cycles every 12 hours.
(c) look at the graph and see where d(t) >= 1.3
You want intervals at least 5 hours long
Is there somewhere a person can see the answer?
a. To create a graph showing the depth of the water over 24 hours, you need to plot the values of the equation d(t) = 2.5sin(0.523t) + 2.9 for the time duration of 24 hours.
You can create an x-axis to represent time in hours and a y-axis to represent the depth of the water in meters. Then, choose regular intervals of time (e.g., every hour) and calculate the corresponding depth of the water using the equation. Plot these points on the graph and connect them to get a smooth curve.
b. The period of the function can be determined from the formula of the sine function, which is given as y = A*sin(Bx + C) + D. In this case, B represents the coefficient of t, which is 0.523.
The period (P) of a sine function is given by P = (2π) / |B|. Therefore, in this equation, the period would be P = (2π) / |0.523| ≈ 12.04.
The period represents the time it takes for the function to complete one full cycle. In this context, it represents the time it takes for the depth of the water to go through a complete cycle of variation.
c. To find the times when the vessel could schedule the trips for safe navigation, we need to consider two conditions: the depth of at least 1.3m and the duration of each trip lasting at least 5 hours.
1. Depth of at least 1.3m: Set the equation d(t) ≥ 1.3 and solve for t.
2.5sin(0.523t) + 2.9 ≥ 1.3
2.5sin(0.523t) ≥ -1.6
sin(0.523t) ≥ -0.64
Taking arcsin on both sides, we get:
0.523t ≥ arcsin(-0.64)
Now, divide both sides by 0.523:
t ≥ arcsin(-0.64) / 0.523
Using a calculator, find the value of arcsin(-0.64) and divide it by 0.523 to get the minimum value of t.
2. Duration of each trip lasting at least 5 hours: To find the possible times for the trips, we can consider intervals of time within the 24-hour period where the depth of the water is consistently above 1.3m for at least 5 hours. This can be determined by observing the graph obtained in part (a). Identify the consecutive segments where the curve is consistently above 1.3m and has a duration of at least 5 hours.
The intersection of these two conditions will give you the possible times the trips could be scheduled for safe navigation, allowing enough depth for 5 hours or more.
Note: Keep in mind any additional factors that might affect safe navigation (e.g., tidal changes, weather conditions).