Problem 66 - Algebra A Combined Function

Question

Two tractors are pulling a tree stump from a field. If two forces \(A\) and \(B\) pull at right angles \(\left(90^{\circ}\right)\) to each other, the resulting force \(R\) is given by the formula \(R=\sqrt{A^{2}+B^{2}}\). If tractor \(A\) is exerting 600 pounds of force and the resulting force is 850 pounds, find how much force tractor \(B\) is exerting.

Step-by-Step Solution

Answer

Tractor B is exerting 602.08 pounds of force.

Step 1: Identify Known Values

We need to find the force exerted by tractor B. We know the force exerted by tractor A is 600 pounds and the resulting force is 850 pounds. So, \( A = 600 \) and \( R = 850 \).

Step 2: Formula Setup

The formula for the resulting force when two forces act at right angles is \( R = \sqrt{A^2 + B^2} \). Our goal is to solve for \( B \), the force exerted by tractor B.

Step 3: Rearrange the Formula

To solve for \( B \), we first square both sides of the equation: \( R^2 = A^2 + B^2 \). Thus, \( 850^2 = 600^2 + B^2 \).

Step 4: Calculate \( R^2 \) and \( A^2 \)

Calculate \( R^2 \) and \( A^2 \) using their respective values. \( 850^2 = 722500 \) and \( 600^2 = 360000 \).

Step 5: Find \( B^2 \)

Subtract \( A^2 \) from \( R^2 \): \( B^2 = 722500 - 360000 \).

Step 6: Calculate \( B^2 \)

Perform the subtraction: \( B^2 = 362500 \).

Step 7: Solve for \( B \)

Take the square root of both sides to solve for \( B \): \( B = \sqrt{362500} \).

Step 8: Final Calculation

Calculate the square root: \( B = 602.08 \). Therefore, tractor B is exerting approximately 602.08 pounds of force.

Key Concepts

Resultant Force

When two forces act at right angles to each other, they combine to produce a single force known as the resultant force. This force can be thought of as the overall effect of the individual forces working together.
The Pythagorean Theorem is behind the formula used to calculate the resultant force, especially in cases where the forces are perpendicular. According to the theorem, the sum of the squares of the two force vectors equals the square of the resultant force. More plainly, when you have forces pulling at right angles, you can imagine a right triangle where each force is a leg and the hypotenuse is the resultant force.
To find the resultant force, use the formula:

• \( R = \sqrt{A^2 + B^2} \)

Where \( R \) is the resultant, \( A \) and \( B \) are the two forces acting at right angles. In the context of the problem, the tractors generate a force in such a way that they create an overall straight pulling force on the tree stump.

Vector Addition

Vector addition is the process of combining two or more vectors to determine a single vector's magnitude and direction. Vectors represent both magnitude (how much) and direction (which way).
In practical situations like our tractor example, vector addition helps us determine how the combination of different directional forces leads to a single force acting on an object.
For forces acting at a right angle, we don’t just add their magnitudes because each force is pulling in a different direction. This is where the Pythagorean Theorem comes into play. By treating each force as a component of a right-angled triangle, we can determine the vector sum (or resultant force) that creates the combined effect.

• The principle of vector addition provides us with a geometric visualization of how individual forces combine to create a net effect.

Force Calculation

Force calculation involves determining the impact of different forces acting on an object. In physics, especially when dealing with problems in mechanics, it is crucial to know not just the individual forces but how they interact to affect the object as a whole.
Using our exercise as an example, calculating the force exerted by tractor \( B \) requires us to manipulate and solve the equation derived from the Pythagorean Theorem:

• Step 1: Rearrange the formula to \( R^2 = A^2 + B^2 \).
• Step 2: Substitute known values \( 850^2 = 600^2 + B^2 \).
• Step 3: Calculate \( R^2 \) and \( A^2 \): \( 722500 = 360000 + B^2 \).
• Step 4: Solve for \( B^2 \) by subtracting: \( B^2 = 362500 \).
• Step 5: Find \( B \) by taking the square root: \( B = \sqrt{362500} \).

Thus, the force exerted by tractor \( B \) is approximately 602.08 pounds. This process highlights how systematically applying mathematical principles allows us to solve real-world physics problems efficiently.