If θ increases at a constant rate of 3 rads per min, at what rate is x increasing in units per min when x = 3 units?

Solution:

Given,

The right angled triangle representing the problem statement is given below.

dθ/dt = 3 radians/minute.--- (1)

From the diagram above,

sinθ = x/5.

x = 5 sinθ.

dx/dt = 5cosθdθ/dt.

From (1)

dx/dt = 5cosθ(3) = 15cosθ --- (2).

Since it is a right angled triangle above,

5² = x² + base²

X = 3 units.

Base² = 5² - 3²

Base² = 25 - 9 = 16.

Base = 4 units.

cosθ = base/ hypotenuse = 4/5.

Substitute the above value in equation(2).

dx/dt = 15(4/5)

= 12 units/ minute.

Therefore,dx/dt = 12 units/ minute.

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If θ increases at a constant rate of 3 rads per min, at what rate is x increasing in units per min when x = 3 units?

Summary:

If θ increases at a constant rate of 3 rads per min, the rate at which x increases in units per min when x = 3 units is 12 units/ minute.