A plane flying with a constant speed of 23 km/min passes over a ground radar station at an altitude of 5 km 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 7 minutes later?
i got it nvm
To find the rate at which the distance from the plane to the radar station is increasing, we can use the concept of related rates. We know the speed of the plane, the distance from the radar station to the plane, and the angle at which the plane is climbing.
Let's break down the problem step-by-step:
1. Draw a diagram: Draw a right triangle to represent the situation. Label one leg of the triangle as the altitude of the plane (5 km) and the other leg as the distance from the radar station to the plane (let's call it x km). The hypotenuse represents the path of the plane as it climbs.
2. Determine what is changing and what is constant: In this problem, the altitude of the plane (5 km) and the angle of climb (35 degrees) are constant. We need to find the rate of change of x, the distance from the plane to the radar station.
3. Relate the variables: You can use trigonometry to relate the variables. Since we know the angle of climb (35 degrees) and the altitude (5 km), we can set up the equation:
tan(35 degrees) = 5 km / x km
4. Solve for x: Rearrange the equation to solve for x:
x km = 5 km / tan(35 degrees)
Use a calculator to find the exact value of x.
5. Find dx/dt: We are given that the plane is flying with a constant speed of 23 km/min. This means that the rate at which x is changing (dx/dt) is also 23 km/min.
6. Determine the rate of change of x after 7 minutes: Since dx/dt is constant, the rate of change after 7 minutes will still be 23 km/min.
So, the rate at which the distance from the plane to the radar station is increasing 7 minutes later is 23 km/min.