Question

David and Emily are climbing El Capitan, a big cliff wall in Yosemite National Park. David is on the ground holding the rope attached to a carabiner (a rope “pulley” that is on the wall) above Emily as she climbs. When Emily stops to rest, David wonders how high she has climbed. The rope is attached to his waist, about 3 feet off the ground, and he has let out 48 feet of rope which goes up to the carabiner and then back down the wall to Emily’s harness. The rope at David’s waist makes a 55° angle with the ground and he is standing 20 feet away from the base of the wall.
Assuming that the rope is taut (pulled tight), approximately how long is the rope between David and the carabiner above Emily?

How high up the wall has Emily climbed? Describe your method.

Answer 1

Solution:

Part (1) Assuming that the rope is taut (pulled tight), approximately how long is the rope between David and the carabiner above Emily?

As per the above figure

$\cos 55\degree$ $=$ $\frac{20}{x}$

$x*$ $\cos 55\degree$ $=$ 20

$x$ $=$ $\frac{20}{cos\thinspace 55\degree}$

$x=$ $\frac{20}{.5735}$

$x=34.87$ feet

Part (2) How high up the wall has Emily climbed? Describe your method.

$\tan 55\degree =$ $\frac{y}{20}$

$y=$ $20 * \tan 55\degree$

$=20*1.428$

$=28.56$ feet

So Height of the wall $= 28.56 + 3$

$=31.56$ feet

As per the question length of the rope = 48 feet.

Rope between Emily and carabiner $=$ Total length of the rope $-$ x

$=$ 48 $-$ 34.87

$=$ 13.13 feet

So Emily Climbed $=$ $31.56 − 13.13$

$=$ $18.43$ feet