Item 5
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=hx. Which of the following best describes the relationship between dθdt, the rate of change of θ with respect to time, and dxdt, the rate of change of x with respect to time?
fix your typo, and then use the chain rule
tanθ = h/x
sec^2θ dθ/dt = -h/x^2 dx/dt
if h were also changing, then the quotient rule would also come into play:
sec^2θ dθ/dt = (x dh/dt - h dx/dt)/x^2
Note that since h is constant in your problem, dh/dt = 0
I can not see what you pasted of course
tanθ=h/x, I suspect not h x
h is constant
d/dt tan θ = sec^2 θ dθ/dt = -h/x^2 dx/dt
so
dθ/dt = -(1/sec^2θ) h/x^2 dx /dt
but sec = 1/cos
so
dθ/dt = -cos^2 θ * (h/x^2) dx/dt
and cos θ = x / (h^2+x^2)
etc
To find the relationship between dθ/dt (the rate of change of θ with respect to time) and dx/dt (the rate of change of x with respect to time), we can use implicit differentiation.
Given that tanθ = hx, we can differentiate both sides of the equation with respect to time t:
d(tanθ)/dt = d(hx)/dt
Using the chain rule, we can rewrite the left side as:
sec^2θ * dθ/dt
And using the product rule to differentiate the right side:
h * dx/dt + x * dh/dt
Now let's substitute these results back into the equation:
sec^2θ * dθ/dt = h * dx/dt + x * dh/dt
Simplifying this equation, we have:
dθ/dt = (h * dx/dt + x * dh/dt) / sec^2θ
Since sec^2θ is equal to 1/cos^2θ, we can rewrite the equation as:
dθ/dt = (h * dx/dt + x * dh/dt) * cos^2θ
So, the best description of the relationship between dθ/dt and dx/dt is:
dθ/dt = (h * dx/dt + x * dh/dt) * cos^2θ
To determine the relationship between dθ/dt and dx/dt, we need to differentiate the equation tanθ = hx with respect to time t.
Differentiating both sides of the equation with respect to t, we get:
sec^2(θ) * dθ/dt = h * dx/dt
Now, we need to express tan(θ) in terms of x. Since tan(θ) = h * x, we can substitute it into the equation:
sec^2(θ) * dθ/dt = h * dx/dt
Next, we need to simplify the equation using trigonometric identities. Recall that sec^2(θ) = 1 + tan^2(θ). Substituting this in the equation, we get:
(1 + tan^2(θ)) * dθ/dt = h * dx/dt
Now, substitute tan^2(θ) with (hx)^2:
(1 + (hx)^2) * dθ/dt = h * dx/dt
Expand the equation further:
(1 + h^2 * x^2) * dθ/dt = h * dx/dt
Finally, divide both sides of the equation by (1 + h^2 * x^2) to isolate dθ/dt:
dθ/dt = (h * dx/dt) / (1 + h^2 * x^2)
Therefore, the relationship between dθ/dt (the rate of change of θ with respect to time) and dx/dt (the rate of change of x with respect to time) is given by:
dθ/dt = (h * dx/dt) / (1 + h^2 * x^2)
This equation represents the relationship between the two rates of change in the given scenario.