Question

A right triangle has base meters and height meters where his constant and changes with respect to time measured in seconds. The angle measured in radians, is defined by tan�� = h/pi. Which of the following best describes the relationship between the rate of change of with respect to time, and d/dx (x) the rate of change of with respect to time?

Answer 1

The relationship between the rate of change of θ with respect to time (dθ/dt) and d/dx (x) the rate of change of h with respect to time is given by dθ/dt = (1/π) * d/dx (h).

Step-by-step explanation:

The tangent of an angle θ in a right triangle is defined as the ratio of the height (h) to half of the base (π/2). Mathematically, tan(θ) = h/(π/2). To find the relationship between the rate of change of θ with respect to time (dθ/dt) and d/dx (x) the rate of change of h with respect to time, we differentiate both sides of the equation with respect to time t.

Starting with the given equation:

`\[ \tan(θ) = (h)/((\pi)/(2)) \]`

Differentiating both sides with respect to time t:

`\[ \sec^2(θ) \cdot (dθ)/(dt) = (1)/((\pi)/(2)) \cdot (dh)/(dt) \]`

Solving for dθ/dt, we get:

`\[ (dθ)/(dt) = (2)/(\pi) \cdot (dh)/(dt) \]`

Therefore, the relationship is:

`\[ (dθ)/(dt) = (2)/(\pi) \cdot (dh)/(dt) \]`

This shows that the rate of change of the angle θ with respect to time is related to the rate of change of the height h with respect to time by a factor of (2/π). The derivative d/dx (x) does not directly appear in this relationship; instead, it is the rate of change of the height h with respect to time that is linked to the rate of change of the angle θ with respect to time.