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To solve this problem, let’s break it down step by step:

### Given:
- Two forces are acting at right angles.
  - Force \( F_1 = 31 \) N
  - Force \( F_2 = 48 \) N

### Part 1: Finding the Magnitude of the Resultant Force
Since the forces are perpendicular, we can use the Pythagorean theorem to find the magnitude \( R \) of the resultant force:

\[
R = \sqrt{F_1^2 + F_2^2}
\]

Substitute the values:

\[
R = \sqrt{31^2 + 48^2}
\]
\[
R = \sqrt{961 + 2304}
\]
\[
R = \sqrt{3265}
\]
\[
R \approx 57.14
\]

Rounding to the nearest integer as instructed, the magnitude of the resultant force is:

\[
R \approx 57 \text{ N}
\]

### Part 2: Finding the Angle with the Smaller Force
Let’s find the angle \( \theta \) that the resultant force makes with the smaller force \( F_1 \). We can use the tangent function since we have opposite and adjacent forces:

\[
\tan \theta = \frac{F_2}{F_1}
\]

So,

\[
\theta = \arctan \left(\frac{48}{31}\right)
\]

Calculate \( \theta \):

\[
\theta \approx \arctan(1.5484)
\]
\[
\theta \approx 57.99^\circ
\]

Rounding to the nearest integer, the angle is:

\[
\theta \approx 58^\circ
\]

### Summary:
- The magnitude of the resultant force is **57 N**.
- The angle with the smaller force is **58 degrees**.

Would you like further details on any part of this solution? 

Here are 5 related questions for further practice:
1. What would the resultant force be if the forces acted at 120 degrees to each other instead?
2. How would you calculate the angle if the forces were not perpendicular?
3. If one of the forces increased to 60 N, how would the resultant change?
4. What trigonometric functions are helpful for calculating angles between forces?
5. Can you find the components of the resultant force along the axes of the original forces?

**Tip:** When working with perpendicular forces, the Pythagorean theorem is often your go-to method for finding the resultant force.

Two forces of 31 N (newtons) and 48 N act on an object at right angles. Find the magnitude of the resultant and the angle that it makes with the smaller force.

Learn how to calculate the magnitude of the resultant force and the angle it makes with the smaller force when two forces (31 N and 48 N) act at right angles. This problem involves trigonometry and the Pythagorean theorem. Suitable for high school students in Grades 10-12.

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