Question

Removing a Stump A farmer wishes to remove a stump from a field by pulling it out with his tractor. Having removed many stumps before, he estimates that he will need 6 tons (12,000 pounds) of force to remove the stump. However, his tractor is only capable of pulling with a force of 7000 pounds, so he asks his neighbor to help. His neighbor's tractor can pull with a force of 5500 pounds. They attach the two tractors to the stump with a $40^{\circ}$ angle between the forces, as shown in the figure. (a) Assuming the farmer's estimate of a needed 6-ton force is correct, will the farmer be successful in removing the stump? (b) Had the farmer arranged the tractors with a $25^{\circ}$ angle between the forces, would he have been successful in removing the stump? (IMAGE CANT COPY)

Name: Problem 96, Chapter 8: Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry
Uploaded: 2020-05-02T02:05:19-08:00
Duration: 2 min 1 s
Channel: Julie Silva

    Removing a Stump A farmer wishes to remove a stump from a field by pulling it out with his tractor. Having removed many stumps before, he estimates that he will need 6 tons (12,000 pounds) of force to remove the stump. However, his tractor is only capable of pulling with a force of 7000 pounds, so he asks his neighbor to help. His neighbor's tractor can pull with a force of 5500 pounds. They attach the two tractors to the stump with a $40^{\circ}$ angle between the forces, as shown in the figure.
(a) Assuming the farmer's estimate of a needed 6-ton force is correct, will the farmer be successful in removing the stump?
(b) Had the farmer arranged the tractors with a $25^{\circ}$ angle between the forces, would he have been successful in removing the stump?
(IMAGE CANT COPY)

Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry

Michael Sullivan 4th Edition

Chapter 8, Problem 96

Instant Answer

Solved by Expert Julie Silva

05/02/2020

Step 1

The farmer's tractor can exert a force of 7000 pounds, and the neighbor's tractor can exert a force of 5500 pounds. The angle between the two forces is given as $40^\circ$ for part (a) and $25^\circ$ for part (b). Show more…

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Removing a Stump A farmer wishes to remove a stump from a field by pulling it out with his tractor. Having removed many stumps before, he estimates that he will need 6 tons (12,000 pounds) of force to remove the stump. However, his tractor is only capable of pulling with a force of 7000 pounds, so he asks his neighbor to help. His neighbor's tractor can pull with a force of 5500 pounds. They attach the two tractors to the stump with a $40^{\circ}$ angle between the forces, as shown in the figure. (a) Assuming the farmer's estimate of a needed 6-ton force is correct, will the farmer be successful in removing the stump? (b) Had the farmer arranged the tractors with a $25^{\circ}$ angle between the forces, would he have been successful in removing the stump? (IMAGE CANT COPY)

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Key Concepts

Law of Cosines

The law of cosines is used to determine the magnitude of the resultant force when two forces act at an angle. It generalizes the Pythagorean theorem to non-right triangles, providing a formula that incorporates the magnitudes of the two forces and the cosine of the angle between them.

Resultant Force Calculation

Calculating the resultant force is crucial in scenarios where multiple forces are applied simultaneously. It involves using vector addition and trigonometry to assess whether the combined force exceeds a certain threshold, such as overcoming resistance or achieving motion in mechanical contexts.

Vector Addition of Forces

This concept involves summing individual force vectors to obtain a single net or resultant force. In practical applications, forces acting at angles must be added vectorially rather than by simple arithmetic, taking into account both magnitude and direction to determine the overall effect.

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