A plane flying with a constant speed of 24 km/min passes over a ground radar station at an altitude of 5km and climbs at an angle of 35 degrees. At what rate, in km/min, is the distance from the plane to the radar station increasing 4 minutes later?
I know side A=5 but how do i get side B and C and with angle 35 What equation i need to do my implicit differentiation? plz help
X = 24 t cos 35
dX/dt = 24 cos 35
Y = 5 + 24 sin 35
dY/dt = 25 sin 35
R^2 = X^2 + Y^2
You want dR/dt when t = 4
2R dR/dt = 2X dX/dt + 2Y dY/dt
Compute X, Y and R when t=4 and then solve for dR/dt
To solve for side B and side C, we can use the trigonometric functions sine and cosine. In this case, side B represents the horizontal distance traveled by the plane, and side C represents the vertical distance traveled by the plane.
Let's start by considering the equation X = 24t cos 35. The value of t represents the time in minutes since the plane passed over the radar station. By substituting t = 4 into this equation, we can find the value of X.
X = 24 * 4 * cos 35 ≈ 96 * cos 35
Next, let's consider the equation Y = 5 + 24 sin 35. By substituting t = 4, we can find the value of Y.
Y = 5 + 24 * sin 35 ≈ 5 + 24 * sin 35
Now that we know X and Y, we can calculate R, the distance from the plane to the radar station, using the Pythagorean theorem. The equation is R^2 = X^2 + Y^2.
R^2 = (96 * cos 35)^2 + (5 + 24 * sin 35)^2
Finally, to find the rate at which the distance from the plane to the radar station is increasing, we need to find dR/dt. We can use implicit differentiation to do this.
Differentiating the equation R^2 = (96 * cos 35)^2 + (5 + 24 * sin 35)^2 with respect to t, we get:
2R * dR/dt = 2X * dX/dt + 2Y * dY/dt
Now, let's substitute the values of X, Y, and R that we found earlier when t = 4 into this equation:
2R * dR/dt = 2X * (24 * cos 35) + 2Y * (24 * sin 35)
Now, we can solve for dR/dt by rearranging the equation:
dR/dt = (2X * (24 * cos 35) + 2Y * (24 * sin 35)) / (2R)
Substitute the values of X, Y, and R that we calculated earlier into this equation to find the rate at which the distance from the plane to the radar station is increasing when t = 4.