Let θ (in radians) be an acute angle in a right triangle and let x and y, respectively, be the lengths of the sides adjacent to and opposite θ. suppose also that x and y vary with time. at a certain instant x=9 units and is increasing at 9 unit/s, while y=5 and is decreasing at 19 units/s. how fast is θ changing at that instant?

negative sign indicates decreasing.

Explanation:

Given:

Using trigonometric relation of the sides and angles of a right angled triangle:

...........................(1)

Differentiate this eq. w.r.to time 't':

............(2)

Now , putting the respective values in eq. (2)

......................(3)

Now, determine cos θ from the triangle:

Putting this value in eq. (3)

negative sign indicates decreasing.

Explanation:

According to question

tan θ = y / x

Differentiate with respect to t on both the sides

  .... (1)

According to question,

tan θ = 5 / 9

So, Sec θ = 10.3 / 9 = 1.14

dx/dt = 9 units/s

dy/dt = 19 units/s

Substitute the values in equation (1), we get

dθ/dt = 1.2 units/s

the total distance traveled by the car is

30km + 49km = 79 km .

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