To locate a rising gradient line of 1 in 20 from a point ‘a’ on 100 m contour on the map at a scale of 1 ∶ 1000 with contour interval of 5 m, the radius of the arc be drawn is:

Question

To locate a rising gradient line of 1 in 20 from a point &lsquo;a&rsquo; on 100 m contour on the map at a scale of 1 &ratio; 1000 with contour interval of 5 m, the radius of the arc be drawn is:

Prepp Team · Accepted Answer

Gradient Line Calculation on Map

This problem involves using map scale and contour information to locate a specific gradient line. We are given a starting point on a 100 m contour, a required rising gradient of 1 in 20, a map scale of 1:1000, and a contour interval of 5 m.

The objective is to find the radius of an arc to be drawn from the starting point to locate the point on the next contour line that achieves the specified gradient.

Understanding the Gradient and Contour Interval

A gradient of 1 in 20 means that for every 20 units moved horizontally, there is a vertical rise of 1 unit.
The starting point is on the 100 m contour. The next higher contour line will be at 100 m + 5 m = 105 m, as the contour interval is 5 m.
The vertical difference between the starting point and the next contour is the contour interval, which is 5 m. This vertical difference corresponds to the 'rise' in our gradient calculation.

Calculating Horizontal Distance on Ground

We know the gradient is the ratio of vertical distance (rise) to horizontal distance. Let H be the horizontal distance on the ground corresponding to a vertical rise of 5 m with a gradient of 1 in 20.

The gradient formula is:

\(\text{Gradient} = \frac{\text{Vertical Distance}}{\text{Horizontal Distance}}\)

Given Gradient = 1/20 and Vertical Distance = 5 m:

\(\frac{1}{20} = \frac{5 \text{ m}}{H}\)

Now, we can solve for H:

\(H = 5 \text{ m} \times 20\)

\(H = 100 \text{ m}\)

So, the horizontal distance on the ground required to rise 5 m with a 1 in 20 gradient is 100 m.

Converting Ground Distance to Map Distance

The map scale is given as 1:1000. This means that 1 unit of distance on the map represents 1000 units of the same distance on the ground.

We need to find the distance on the map that represents 100 m on the ground. Let R be the required radius on the map.

\(\text{Map Scale} = \frac{\text{Distance on Map}}{\text{Distance on Ground}}\)

\(\frac{1}{1000} = \frac{R}{100 \text{ m}}\)

Solving for R:

\(R = \frac{1}{1000} \times 100 \text{ m}\)

\(R = \frac{100}{1000} \text{ m}\)

\(R = 0.1 \text{ m}\)

The options are given in cm. Let's convert the map distance from meters to centimeters:

\(0.1 \text{ m} = 0.1 \times 100 \text{ cm}\)

\(R = 10 \text{ cm}\)

The radius of the arc to be drawn on the map is 10 cm. This arc will be drawn from point 'a' on the 100 m contour, and its intersection with the 105 m contour will indicate the location of a point with the desired 1 in 20 rising gradient.

Summary of Calculation Steps

Identify the vertical rise needed, which is the contour interval (5 m).
Use the given gradient (1 in 20) to calculate the corresponding horizontal distance on the ground for this vertical rise (100 m).
Use the map scale (1:1000) to convert the horizontal ground distance (100 m) into the distance on the map (0.1 m).
Convert the map distance to the unit used in the options (cm), resulting in 10 cm.

Comparing this result with the given options, we find that 10 cm matches one of the options.