I was working on some questions for homework and for some reason cannot understand how to approach this question
A camera, located 5 km from the launch pad, is tracking the rocket that is fired straight up. When the height of the rocket is 12 km, the camera is rotating at the rate of 1/48 radians per second. What is the speed of the rocket at that instant? Give your answer in km/sec. ?
Any help would be much appreciated thanks!
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To solve this problem, you can use the concept of related rates. Here's how you can approach it:
1. Identify the given information:
- The distance from the camera to the launch pad is 5 km.
- The height of the rocket is 12 km.
- The camera is rotating at a rate of 1/48 radians per second.
2. Determine what values you need to find:
- The speed of the rocket at the given height.
3. Establish a relationship between the given variables:
- The distance from the camera to the rocket forms a right triangle with the height of the rocket as the opposite side and the distance from the camera to the launch pad as the hypotenuse.
4. Differentiate the equation with respect to time:
- To relate the rates of change, differentiate the equation with respect to time. You will need to use implicit differentiation.
5. Solve for the unknown rate:
- Once you've differentiated the equation, plug in the given values and solve for the unknown rate. In this case, the unknown rate is the speed of the rocket.
6. Convert the units:
- The final step is to convert the speed from the given unit of radians per second to kilometers per second. This can be done by using the formula: 1 radian = radius of the circle. Since the camera is 5 km away from the rocket, you can multiply the speed in radians per second by the radius (5 km) to get the speed in kilometers per second.
By following these steps, you should be able to solve the problem and find the speed of the rocket at the given height of 12 km.