Question

choose the formula you would use to solve the following related rates problem: a construction worker pulls a five meter plank up the side of a building under construction by means of a rope tied to one end of the plank. assume the opposite end of the plank follows a path perpendicular to the wall of the building and the worker pulls the rope at a rate of 15 meters per second. how fast is the end of the plank sliding along the ground when it is 2.5 meters from the wall of the building?
formulas (right side, from top to bottom):
h = r·sinθ
1 = sin²θ + cos²θ
a² + b² = c²
d = √(x² + y²)

Explanation:

This is a related rates problem in calculus (a subfield of Mathematics) involving a right triangle formed by the plank, the wall, and the ground. The Pythagorean theorem \(a^{2}+b^{2}=c^{2}\) (where \(c\) is the hypotenuse, and \(a,b\) are the legs) is used to model the relationship between the length of the plank (hypotenuse, constant at 5 meters), the distance from the building (one leg), and the height on the wall (the other leg). We need this formula to set up the equation for differentiation in related rates. The other formulas don't fit: \(1 = \sin^{2}\theta+\cos^{2}\theta\) is a trigonometric identity, \(h = r\cdot\sin\theta\) is for polar - height relations, and \(d=\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}\) is for distance between two points.

Answer:

\(a^{2}+b^{2}=c^{2}\)