Question

A sloth moves 3m to the right and then climbs 4m up a tree. How many degrees above the ground is the sloth compared to its initial position? Assume rightward is the positive x-direction and upward is the positive y-direction.

Answer 1

The sloth is approximately 53.13° above the ground relative to its initial position, calculated by using the arctangent function on the lengths of the right-angled triangle formed by its horizontal and vertical movements.

Step-by-step explanation:

The question involves determining the angle above the ground at which a sloth has climbed, relative to its initial position. The sloth first moves 3 meters to the right and then 4 meters up a tree. To find the angle, we can use the concept of a right-angled triangle where the horizontal and vertical displacements (3m and 4m respectively) can be considered as the base and the height.

To find the angle θ above the ground the sloth has climbed with respect to its initial position, we can use the arctangent function (tan-1), which is the inverse of the tangent function and gives us an angle when we know the opposite and adjacent sides of a right-angled triangle.

The formula to find the angle in degrees is θ = tan-1(opposite/adjacent). Here, the opposite side is 4m (up a tree), and the adjacent side is 3m (to the right). By plugging the values into the formula, we get θ = tan-1(4/3).

Using a calculator, we find that θ ≈ 53.13°. Therefore, the sloth is approximately 53.13° above the ground compared to its initial position.