The wave heights h in the open sea depend on the speed v of the wind and the length of time t that the wind has been blowing at that speed. Values of the function h = f(v, t) are recorded in feet in the following table.
the values that you need on the table are.. 20hr/30 knots= 17
20hrs/40 knots= 28
20hrs/ 50 knots= 40
Use the table to find a linear approximation to the wave height function when v is near 40 knots and t is near 20 hours.
To find a linear approximation to the wave height function, we can use the given table values to estimate the rate of change of the wave height with respect to both the speed of the wind (v) and the length of time the wind has been blowing (t).
Let's assume that the wave height function, h = f(v, t), can be approximated by a linear equation of the form:
h ≈ c + m1 * (v - 40) + m2 * (t - 20)
Where c is a constant term and m1 and m2 are constants indicating the rate of change of the wave height with respect to wind speed and time, respectively.
We can use the given table values to estimate the values of m1 and m2.
First, we can calculate m1 by finding the slope of the wave height function with respect to wind speed near 40 knots:
m1 = (h1 - h2) / (v1 - v2) = (28 - 17) / (40 - 30) = 11 / 10 = 1.1
Next, we can calculate m2 by finding the slope of the wave height function with respect to time near 20 hours:
m2 = (h2 - h3) / (t2 - t3) = (17 - 28) / (20 - 20) = -11 / 0 = undefined
Notice that the slope with respect to time is undefined because the values of h2 and h3 are the same for different wind speeds when t = 20 hours.
Since the slope with respect to time is undefined, we cannot directly estimate the rate of change of the wave height with respect to time using the given table values. However, we can still express the linear approximation as:
h ≈ c + 1.1 * (v - 40)
To find the constant term c, we can use one of the table values. Let's choose the value for 20 hours and 40 knots:
28 = c + 1.1 * (40 - 40) = c
Therefore, the linear approximation to the wave height function is:
h ≈ 28 + 1.1 * (v - 40)
This approximation is valid for values of v near 40 knots and t near 20 hours.