A missile rises vertically from a point on the ground 75,000 feet from a radar station. If the missile is rising at the rate of 16,500 feet per minute at the instant when it is 38,000 feet high, what is the rate of change(in radians per minute) of the missile's angle of elevation from the radar station at this instant?
TIA
Tan Theta= h/75K
d Tan Theta/dt * dTheta/dt = dh/dt * 1/75K
You know dh/dt (given).The derivative of Tan theta is sec^2 Theta. Which you can solve for given the triangle.
So you know everything, solve for dTheta/dt
Well, if we break down the problem, we can start with highlighting the given values. We have:
Distance from the radar station to the missile, D = 75,000 feet
Rate of change of distance, dh/dt = 16,500 feet per minute
Height of the missile, h = 38,000 feet
To find the rate of change of the missile's angle of elevation, we need to find dTheta/dt.
Using the formula Tan Theta = h/D, we can calculate that:
Tan Theta = 38,000/75,000
Now, taking the derivative of both sides with respect to time (t), we get:
Sec^2 Theta * dTheta/dt = (dh/dt * D - h * dD/dt) / D^2
We know that dh/dt = 16,500 feet per minute and D = 75,000 feet. To find dD/dt, we need to consider the geometry of the situation.
Since the missile is rising vertically, the distance D will always be the hypotenuse of a right triangle formed by the ground, the missile, and the radar station. This means that the angle Theta will be the angle of elevation from the radar station to the missile.
Now, using the Pythagorean theorem, we can find the other side of the triangle:
D^2 = h^2 + 75,000^2
Differentiating both sides with respect to time (t), we get:
2D * dD/dt = 2h * dh/dt
Plugging in the values, we have:
2(75,000) * dD/dt = 2(38,000) * 16,500
Simplifying and solving for dD/dt, we get:
dD/dt = (38,000 * 16,500) / 75,000
Now, we have all the values to substitute back into our previous equation:
Sec^2 Theta * dTheta/dt = (16,500 * 75,000 - 38,000 * (38,000 * 16,500) / 75,000) / (75,000^2)
Finally, we can solve for dTheta/dt by dividing both sides by Sec^2 Theta:
dTheta/dt = (16,500 * 75,000 - 38,000 * (38,000 * 16,500) / 75,000) / (75,000^2 * Sec^2 Theta)
Just plug in the value of Theta (which you can find using the given height and distance), and you'll have the rate of change of the missile's angle of elevation from the radar station.
To find dTheta/dt, let's solve for Theta first:
Tan Theta = h/75,000
Differentiating both sides of the equation with respect to t:
Sec^2 Theta * dTheta/dt = dh/dt * (1/75,000)
Since we are given dh/dt and h (38,000 feet), we can substitute these values into the equation to solve for dTheta/dt.
Sec^2 Theta * dTheta/dt = (16,500 ft/min) * (1/75,000)
To find Sec^2 Theta, we need to use the Pythagorean Identity:
Tan^2 Theta + 1 = Sec^2 Theta
Rearranging the equation, we can write:
Sec^2 Theta = 1 + Tan^2 Theta
Substituting this into our equation:
(1 + Tan^2 Theta) * dTheta/dt = (16,500 ft/min) * (1/75,000)
Now, we can substitute the given height (38,000 feet) into the original equation:
Tan Theta = (38,000 ft)/75,000
Solving for Tan Theta:
Theta = Tan^(-1)((38,000 ft)/75,000)
Once we have the value of Theta, we can substitute it back into the equation:
(1 + (Tan(Tan^(-1)((38,000 ft)/75,000)))^2) * dTheta/dt = (16,500 ft/min) * (1/75,000)
Simplifying the equation:
(1 + ((38,000 ft)/75,000)^2) * dTheta/dt = (16,500 ft/min) * (1/75,000)
Now we can solve for dTheta/dt by isolating it:
dTheta/dt = (16,500 ft/min) * (1/75,000) / (1 + ((38,000 ft)/75,000)^2)
Calculate the value to find the rate of change of the missile's angle of elevation from the radar station at this instant.
To find the rate of change of the missile's angle of elevation, you can use the given information and apply the chain rule of differentiation.
Let's start by expressing the equation in terms of the variables:
We have: tan(θ) = h/75,000, where h represents the height of the missile.
Differentiating both sides with respect to time t, we get:
sec^2(θ) * dθ/dt = (dh/dt) / 75,000
Now, we need to find the values of sec^2(θ) and (dh/dt).
To find the value of sec^2(θ), we can use the fact that tan(θ) = h/75,000.
Rearranging, we have:
θ = arctan(h/75,000)
Taking the derivative of both sides with respect to h, we obtain:
dθ/dh = 1 / (1 + (h/75,000)^2)
Now, substitute the height value given in the problem: h = 38,000.
dθ/dh = 1 / (1 + (38,000/75,000)^2)
Now we have the value of dθ/dh.
Next, substitute the rate of change of height (dh/dt) given in the problem: dh/dt = 16,500 feet per minute.
Finally, substitute these values into the equation:
sec^2(θ) * dθ/dt = (16,500 / 75,000) / (1 + (38,000/75,000)^2)
Simplify the equation and solve for dθ/dt:
sec^2(θ) * dθ/dt = (16,500 / 75,000) / (1 + (38,000/75,000)^2)
Now, you can calculate dθ/dt, the rate of change of the missile's angle of elevation from the radar station at this instant.