Question

A right triangle has base x and height h meters where h is constant

Answer 1

An equation that best describes the relationship between dθ/dt, the rate of change of θ with respect to time, and dh/dt, the rate of change of h with respect to time is: A.
`(d\theta)/(dt) =((-h)/(x^(2) +h^2) )(dx)/(dt) \;radians \;per \;second`

In Mathematics, the tangent trigonometric ratio can be modeled by the following formula;

Tan(θ) = Opp/Adj

Where:

• Adj represents the adjacent side of a right-angled triangle.
• Opp represents the opposite side of a right-angled triangle.
• θ represents the angle.

Based on the information provided about the right triangle with base x meters and height h meters, we have the following tangent trigonometric ratio:

tanθ = h/x

Note:
`sec^2\theta =1+tan^2\theta` and
`tan^2\theta=(h^2)/(x^(2) )`

By taking the first derivative of θ with respect to t, we have:

`sec^2\theta \cdot (d\theta )/(dt) =(-h)/(x^2) \cdot (dx)/(dt) \\\\(d\theta )/(dt) =(-h)/(x^2sec^2\theta) \cdot (dx)/(dt)\\\\(d\theta )/(dt) =(-h)/(x^2(1+tan^2\theta)) \cdot (dx)/(dt)\\\\(d\theta )/(dt) =(-h)/(x^2(1+(h^2)/(x^2) )) \cdot (dx)/(dt)\\\\(d\theta )/(dt) =(-h)/(x^2+h^2) } \cdot (dx)/(dt)`

Complete Question:

A right triangle has base x meters and height h meters, where h is constant and x changes with respect to time t, measured in seconds. The angle θ, measured in radians, is defined by tanθ=h/x.

Which of the following best describes the relationship between dθ/dt, the rate of change of θ with respect to time, and dx/dt, the rate of change of h with respect to time?

`a.\;(d\theta)/(dt) =((-h)/(x^(2) +h^2)) (dx)/(dt) \;radians \;per \;second\\\\b.\;(d\theta)/(dt) =((h )/(x^(2) +h^2) )(dx)/(dt) \;radians \;per \;second\\\\c.\;(d\theta)/(dt) =(\frac{-h}{x\sqrt{x^(2) +h^2} }) (dx)/(dt) \;radians \;per \;second\\\\d.\;(d\theta)/(dt) =(\frac{h}{x\sqrt{x^(2) +h^2} } )(dx)/(dt) \;radians \;per \;second`