Question

Why does a spanner have a long handle?

Answer 1

Hint We need to investigate the force required to operate a (hypothetical) short handled spanner with respect to a long handled spanner. Torque is proportional to the distance between the force and the point (or centre) of rotation. So from here we can find the reason for a spanner to have a long handle.
Formula used: In this solution we will be using the following formulae;
 $ T = Fr $ where $ T $ is the torque applied about a point, $ F $ is the force generating the torque and $ r $ is the perpendicular length between the force and the point of rotation.

Complete step by step answer
Whenever we try to open an over-tightened faucet or attempt to spin an old rigid fan, we apply forces at the very end of their handles (the blades for the fan) to increase our chances of success. Similarly to make a fan spin the fastest, we apply forces at the end of the blades. In everyday life, this is common sense derived from experience. However, in actuality, what we hope to increase when applying the forces closer to the end (instead of closer to the centre) is the quantity called torque.
Torque is the quantity which tends to set a body in rotational motion. It is that quantity required to make a body rotate about an axis, similar to how forces are required to make a body accelerate. This torque is proportional to the distance between the force and the point (or centre) of rotation.
Hence for a spanner too, whose purpose is to lose a bolt by rotating it about the threaded screw until it comes off, we need to apply torque at the end of the handle. Thus, a long handled spanner, for a particular force, will be able to generate a larger torque as compared to a short spanner
Hence, spanners have long handles to generate larger torques.

Note
Alternatively, we may reason mathematically as follows: The relationship between forces and torque is given by $ T = Fr $ where $ T $ is the torque applied about a point, $ F $ is the force generating the torque and $ r $ is the perpendicular length between the point of application of the force and the axis of rotation of the object.
Assuming for a torque $ T $ required to rotate a bolt, we must apply a force $ F = \dfrac{T}{l} $ applied at its end (where $ l $ is the length of the spanner). Hence, for a short length $ l $ the force required is larger. But for a long length $ l $ the force required is little. Thus, a longer handle will be preferred, so it requires the minimum amount of force.