SOLUTION: Rocco and Biff are two koala bears frolicking in a meadow. Suddenly, a tasty clump of eucalyptus falls to the ground, catching their attention. Biff glances at Rocco, who appears t

Question 1204688: Rocco and Biff are two koala bears frolicking in a meadow. Suddenly, a tasty clump of eucalyptus falls to the ground, catching their attention. Biff glances at Rocco, who appears to be 15 m away, then over to the eucalyptus, which appears to be 18 m away. From Biff’s point of view, Rocco and the eucalyptus are separated by an angle of 45°. Rocco’s top running speed is 1.0 m/s, but Biff can run one and a half times as fast. Can Biff beat Rocco to the eucalyptus? State any assumptions you make.
Answer by math_tutor2020(3817) (Show Source): You can put this solution on YOUR website!

B = Biff's starting location
R = Rocco's starting location
E = eucalyptus location

Distances or side lengths:
EB = 18 meters
BR = 15 meters

Given angle
angle EBR = 45 degrees

Let's add a new point A.
This will be placed on segment BR such that segment EA is perpendicular to segment BR.

Point A splits triangle REB into two right triangles EAB and EAR.

Angle ABE = 45 degrees, which leads to angle AEB = 45 degrees as well.

Use the 45-45-90 triangle template, or the trig ratio cosine, to determine that segment AB = EB/sqrt(2) = 18/sqrt(2) = 12.7279 approximately.
The segment EA will have the same length because triangle EAB is isosceles.

Then,
AB + AR = BR
12.7279 + AR = 15
AR = 15 - 12.7279
AR = 2.2721 approximately

Focus your attention on triangle EAR.
It is a right triangle with these approximate leg lengths
EA = 12.7279
AR = 2.2721

Use the pythagorean theorem to find the hypotenuse ER.
(EA)^2 + (AR)^2 = (ER)^2
ER = sqrt( (EA)^2 + (AR)^2 )
ER = sqrt( (12.7279)^2 + (2.2721)^2 )
ER = 12.9291
Like the other decimal values, this is approximate.

Another way to find segment ER is to use the Law of Cosines on triangle REB.

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We have these key distances or segment lengths
EB = 18
ER = 12.9291 (approx)

Biff needs to travel 18 meters, and his speed is 1.5*1 = 1.5 m/s since it is 1.5 times faster compared to Rocco's speed (of 1 m/s).

Biff's travel time is:
time = distance/rate
time = EB/1.5
time = 18/1.5
time = 12 seconds

Rocco needs to travel along segment ER = 12.9291, and his speed is 1 m/s, so,
time = distance/rate
time = ER/1
time = 12.9291/1
time = 12.9291

-----------------------------------------------------------

We have these time values
Biff = 12 seconds
Rocco = 12.9291 seconds approximately

Because these time values are so close together, it's practically a tie.

If we could only select one bear, then the winner is Biff

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The key assumption made is that each koala bear travels at their top speed right from the start.
This is of course not realistic.
The bears need time to build up to their top speed.
There's no guarantee they maintain the top speed even after reaching it.
The koalas also need to slow down when arriving at the eucalyptus, or else they would overshoot the goal.

For a more realistic way to solve this, we'd need to use a kinematic equation. However, such a topic is usually dealt with in physics classrooms.

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