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Answer:
130.9 miles per hour.
Step-by-step explanation:
sine 16 = x / 475 where x = rate of gaining altitude.
x = 475 sin 16
= 130.9 miles per hour.
Rate of gaining altitude is ratio of height gained to time spent. The rate at which the plane is gaining altitude is 130.9 miles per hour.
How to measure the rate of change of something as some other value changes?
Suppose that we have to measure the rate of change of y as x changes, then we have:
[tex]Rate = \dfrac{y_2 - y_1}{x_2 - x_1}[/tex]
where we have
[tex]\rm when \: x=x_1, y = y_1\\when\: x = x_2, y= y_2[/tex]
Remember that, we divide by the change in independent variable so that we get some idea of how much the dependent quantity changes as we change the independent quantity by 1 unit.
(5 change per 3 unit can be rewritten as 5/3 change per 1 unit)
For the given case, the rate at which the plane is gaining altitude is the ratio of the height it gains per unit time(here per hour).
Referring to the diagram attached below, as he will walk 475 miles per hour(which is constant, this can be taken as hypotenuse's length.
The length BC is the height the plane gain each hour as he covers 475 miles in slant direction.
Using the sin ratio, and measuring from the angle A, we get:
[tex]\sin(A)= \dfrac{|BC|}{|AC|}\\\\\sin(16^\circ) = \dfrac{|BC|}{475}\\\\|BC| \approx 0.2756 \times 475 \approx 130.9 \: \rm miles\\\\[/tex]
Thus, the height gained by plane per hour is 130.9 miles.
Rate of height gaining = (new height - old height)/(1 hour)
Rate of height gaining = (old height + 130.9 miles - old height)/1 hour
Rate of height gaining = 130.9 miles per hour.
This is the rate at which the plane is gaining altitude(since it is ratio of how much height it gains to how much time is spent gaining that height)
Hence, The rate at which the plane is gaining altitude is 131.1 miles per hour.
Learn more about trigonometric ratios here:
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