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The given problem asks us to find the angle \(\theta\) that minimizes the force \(F\) needed to pull a weight along a horizontal surface. The force is defined as:

\[
F = \frac{fm}{f \sin \theta + \cos \theta}
\]

where:
- \( f \) is the coefficient of friction (given as 0.75),
- \( m \) is the mass of the object,
- \( \theta \) is the angle of elevation at which the force is applied.

To minimize \( F \), we will differentiate \( F \) with respect to \(\theta\) and set the derivative equal to zero.

### Step-by-Step Solution:
1. **Substitute the given value of \( f \) into the formula:**

   \[
   F = \frac{0.75m}{0.75 \sin \theta + \cos \theta}
   \]

2. **Simplify the equation:**

   \[
   F = \frac{m}{\sin \theta + \frac{\cos \theta}{0.75}}
   \]
   
   This can be rewritten as:
   
   \[
   F = \frac{m}{\sin \theta + \frac{4}{3} \cos \theta}
   \]

3. **Let’s find the derivative of \( F \) with respect to \( \theta \):**

   To minimize \( F \), we use the quotient rule. If \( F(\theta) = \frac{u}{v} \), then:
   
   \[
   F'(\theta) = \frac{u'v - uv'}{v^2}
   \]
   
   Here, \( u = m \) (which is a constant, so \( u' = 0 \)) and \( v = \sin \theta + \frac{4}{3} \cos \theta \).

   Therefore, we get:

   \[
   F'(\theta) = \frac{0 \cdot v - m \cdot \left( \cos \theta - \frac{4}{3} \sin \theta \right)}{v^2} = \frac{-m (\cos \theta - \frac{4}{3} \sin \theta)}{(\sin \theta + \frac{4}{3} \cos \theta)^2}
   \]

4. **Set \( F'(\theta) = 0 \):**

   \[
   -m (\cos \theta - \frac{4}{3} \sin \theta) = 0
   \]
   
   Simplifying:

   \[
   \cos \theta - \frac{4}{3} \sin \theta = 0
   \]
   
   \[
   \cos \theta = \frac{4}{3} \sin \theta
   \]
   
   \[
   \tan \theta = \frac{3}{4}
   \]
   
   So, the angle \( \theta \) that minimizes the force is given by:

   \[
   \theta = \tan^{-1} \left( \frac{3}{4} \right)
   \]

5. **Calculate the value of \( \theta \):**

   \[
   \theta = \tan^{-1} \left( \frac{3}{4} \right) \approx 36.87^\circ
   \]

Hence, the angle that minimizes the force is approximately \( 36.87^\circ \).

Would you like a detailed explanation of any specific step? Here are some follow-up questions:

1. What is the importance of minimizing force in practical applications?
2. How does the coefficient of friction affect the force required?
3. How would the answer change if the coefficient of friction was different?
4. What is the physical interpretation of the angle \(\theta\) in this context?
5. How can this concept be applied in other areas such as engineering or physics?

**Tip:** Understanding the effect of the angle on force reduction is crucial for optimizing mechanical systems like pulley setups and incline planes.

A force pulls a weight along a horizontal surface according to F = (fm) / (f sinθ + cosθ), where f is the coefficient of friction, m is the mass of the object, and θ is the angle of elevation at which the force is applied. Find the angle that minimizes the force when the coefficient of friction is 0.75.

Learn how to calculate the angle that minimizes the force applied to pull a weight along a horizontal surface with friction. This problem involves key trigonometric and calculus concepts, including the use of the quotient rule for differentiation. Coefficient of friction is given as 0.75, and the angle is found to be approximately 36.87 degrees.

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