If the radius were increased to 40 x  its length, what would be the new force? ___________________ x original force.

To solve the problem, we need to apply some knowledge from physics related to force and the properties of materials, particularly focusing on the relationship between force, radius, and stress.

1. Break Down the Problem

Let's denote:

• $F$ = original force
• $r$ = original radius
• $40r$ = new radius (40 times the original radius)

The relationship in question likely pertains to some physical law, such as those governing structural forces or pressure, depending on the context implied (though it isn't explicitly stated). If the force scales with the area (as it often does in contexts of pressure), we need to consider how the area changes with the radius.

2. Relevant Concepts

The area $A$ of a circle can be calculated using the formula: $A = \pi r^2$

When the radius is increased to $40r$, the new area $A'$ becomes: $A' = \pi (40r)^2 = \pi \times 1600r^2 = 1600 \pi r^2$

3. Analysis and Detail

The original area $A$ is: $A = \pi r^2$ Therefore, the ratio of the new area to the original area is: $\frac{A'}{A} = \frac{1600 \pi r^2}{\pi r^2} = 1600$

Assuming that the force is evenly distributed across this area and increases proportionally with area (as in the context of hydraulic systems or similar), the new force $F'$ can be represented as: $F' = 1600F$

4. Verify and Summarize

Thus, the new force when the radius is increased to $40$ times its original length is $1600$ times the original force.

Final Answer

The new force is $1600$ x original force.