Question

A zoologist in the Galapagos Islands is located on a boat anchored 400 meters off shore tracking a rarely seen purple iguana walking along the shore. He is using a powerful mounted telescope to track the animal. Assuming the shoreline is straight and the iguana is moving along the shore in a straight line at the rate of 2 km/hr, how fast must the zoologist rotate the telescope to track the iguana when the iguana is 1 km from the point on the shore nearest to the boat? Convert your answer to degrees/minute.

Answer 1

To track the iguana, the zoologist needs to rotate the telescope at a certain speed to keep it in view. We can use the concept of angular velocity to determine how fast the telescope needs to rotate. The angular velocity is the rate at which an object rotates around an axis. In this case, the axis is the zoologist's position on the boat and the object is the iguana.

Step-by-step explanation:

To track the iguana, the zoologist needs to rotate the telescope at a certain speed to keep it in view. We can use the concept of angular velocity to determine how fast the telescope needs to rotate. The angular velocity is the rate at which an object rotates around an axis. In this case, the axis is the zoologist's position on the boat and the object is the iguana.

First, let's convert the speed of the iguana from kilometers per hour to meters per minute. Since there are 1000 meters in a kilometer and 60 minutes in an hour, the speed of the iguana is 2 km/hr * (1000 m/km) / (60 min/hr) = 33.33 m/min.

Next, we can use the concept of similar triangles to find the angular velocity. The distance from the boat to the point on the shore nearest to the boat is 400 meters, and the distance from the iguana to that point is 1000 meters. Since the iguana is moving in a straight line, we can calculate the ratio of the distances as 1000/400 = 2.5.

Finally, we can use the formula for angular velocity, which is the ratio of the linear velocity (33.33 m/min) to the radius of rotation (400 meters). Therefore, the angular velocity is 33.33 m/min / 400 m = 0.0833 rad/min. To convert this to degrees/minute, we can multiply by the conversion factor 180°/π radians, so the angular velocity is approximately 4.77°/min.

Answer 2

The zoologist must rotate the telescope at a speed of 10.909 degrees per minute to track the iguana.

Step-by-step explanation:

To find the speed at which the zoologist must rotate the telescope to track the iguana, we can use the concept of angular velocity. The angular velocity is defined as the rate at which an object rotates about a specific axis. It is usually measured in radians per second (rad/s) or degrees per second (°/s). In this case, we need to find the angular velocity in degrees per minute (°/min).

1. First, we need to determine the distance between the boat and the point on the shore nearest to the boat. This distance can be found using the Pythagorean theorem. Let's call this distance 'd'. Using the given values, we have: d = sqrt(400^2 + 1000^2) = 1100 meters.
2. Next, we need to find the time it takes for the iguana to cover this distance. Since the iguana is moving at a rate of 2 km/hr, or 2000 meters per hour, we can divide the distance by the speed to get the time. Let's call this time 't'. We have: t = d / (2000/60) = 33 minutes.
3. Finally, we can calculate the angular velocity using the formula: angular velocity = 360° / t. Plugging in the value of 't', we get: angular velocity = 360° / 33 = 10.909 °/min.

Therefore, the zoologist must rotate the telescope at a speed of 10.909 degrees per minute to track the iguana.