Understanding Scale and Distance Measurement on Plans

This problem involves correcting a distance measured on a plan when a wrong scale was used. It highlights the importance of using the correct scale factor to relate distances on a map or plan to actual distances on the ground.

A scale on a plan is a ratio that represents the relationship between a distance on the plan and the corresponding distance on the ground. It can be expressed in various ways, such as a statement (e.g., 1 cm = 50 m) or a representative fraction (e.g., 1:5000).

Analyzing the Problem with Wrong and Correct Scales

We are given two scales:

• Correct Scale: 1 cm on the plan represents 50 m on the ground.
• Wrong Scale: 1 cm on the plan represents 30 m on the ground.

The engineer measured a distance and obtained 600 m. However, this 600 m is the distance *calculated* using the wrong scale (1 cm = 30 m), not the actual distance on the ground or the length on the plan itself. To find the true distance, we first need to determine the actual length of the line drawn on the plan.

Step-by-Step Calculation of True Distance

Step 1: Determine the Length on the Plan using the Wrong Scale

The measured distance (600 m) was obtained by applying the wrong scale. If 1 cm on the plan represents 30 m on the ground according to the wrong scale, we can find the length on the plan that corresponds to the measured 600 m.

Let the length on the plan be \(L_p\).

Using the wrong scale:

\(1 \text{ cm} \text{ represents } 30 \text{ m}\)

\(L_p \text{ cm} \text{ represents } 600 \text{ m}\)

So, \(L_p \times 30 \text{ m/cm} = 600 \text{ m}\)

Solving for \(L_p\):

\(L_p = \frac{600 \text{ m}}{30 \text{ m/cm}}\)

\(L_p = 20 \text{ cm}\)

This means the line drawn on the plan is 20 cm long.

Step 2: Calculate the True Distance using the Correct Scale

Now that we know the actual length of the line on the plan is 20 cm, we can use the correct scale to find the true distance on the ground between the two locations.

Using the correct scale:

\(1 \text{ cm} \text{ represents } 50 \text{ m}\)

The plan length is \(L_p = 20 \text{ cm}\).

True Distance \(D_{true}\) = Length on Plan \(\times\) Correct Scale Factor

\(D_{true} = 20 \text{ cm} \times 50 \text{ m/cm}\)

\(D_{true} = 1000 \text{ m}\)

Therefore, the true distance between the two locations is 1000 m.

Summary of Calculation

The problem can be summarized as converting the measured distance (obtained using a wrong scale) back to the plan length, and then converting that plan length to the true ground distance using the correct scale.

\(\text{Measured Distance (using wrong scale)} \xrightarrow{\text{Convert using Wrong Scale}} \text{Length on Plan} \xrightarrow{\text{Convert using Correct Scale}} \text{True Distance}\)

\(600 \text{ m } (\text{using wrong scale } 1 \text{ cm} = 30 \text{ m}) \implies \text{Plan Length} = \frac{600}{30} = 20 \text{ cm}\)

\(\text{Plan Length } 20 \text{ cm } (\text{using correct scale } 1 \text{ cm} = 50 \text{ m}) \implies \text{True Distance} = 20 \times 50 = 1000 \text{ m}\)

Revision Table: Scales and Measurements

Additional Information on Scale Correction in Surveying

Errors due to using a wrong scale are common and require correction to find the true ground distance. The fundamental principle is always to work with the actual length measured on the plan.

• Representative Fraction (RF): Scales can also be expressed as RF, e.g., 1:5000 for 1 cm = 50 m (since 50 m = 5000 cm). If the scale is 1 cm = X meters, the RF is 1 : (X * 100).
• True Distance Formula: A common formula for scale correction is:
  \(\text{True Distance} = \text{Measured Distance} \times \left(\frac{\text{Wrong Scale}}{\text{Correct Scale}}\right)\)
  However, this formula applies when 'Wrong Scale' and 'Correct Scale' refer to Representative Fractions or their reciprocals (scale factors). In this case, the scales are given as ground distances per unit plan length. Let's see how our method relates:
  Plan Length \(= \frac{\text{Measured Distance}}{\text{Wrong Scale Factor}} = \frac{600}{30}\)
  True Distance \(= \text{Plan Length} \times \text{Correct Scale Factor} = \left(\frac{600}{30}\right) \times 50 = 20 \times 50 = 1000 \text{ m}\).
  Alternatively, we can think of the scale as a ratio of ground distance to plan distance.
  Wrong Scale Ratio = \(30 \text{ m/cm}\)
  Correct Scale Ratio = \(50 \text{ m/cm}\)
  The measured distance (600m) is obtained by \(L_p \times 30\).
  The true distance is \(L_p \times 50\).
  So, True Distance = \(L_p \times 50 = \left(\frac{600}{30}\right) \times 50 = 600 \times \frac{50}{30} = 600 \times \frac{5}{3} = 200 \times 5 = 1000 \text{ m}\).
  This confirms our step-by-step approach is correct and equivalent.
• Effect of Wrong Scale: If you use a smaller scale (like 1 cm = 30m is smaller than 1 cm = 50m in terms of ground coverage per cm) when the true scale is larger, distances calculated will be less than the true distance for the same plan length. Conversely, using a larger wrong scale makes calculated distances greater than the true distance. In our case, using 1 cm = 30m (smaller scale factor, larger ground represented per cm) incorrectly on a plan meant for 1 cm = 50m (larger scale factor, smaller ground represented per cm) led to a calculated distance (600m) that was smaller than the true distance (1000m). Wait, let me rephrase. 1 cm = 30 m represents ground *less* than 1 cm = 50 m. So 1 cm = 30 m is a *larger scale* than 1 cm = 50 m (larger scale shows more detail, covers less area). If you use a larger scale (1:3000) when the map is of a smaller scale (1:5000), the measured distance will be smaller than the true distance. Yes, 600m < 1000m. That aligns.

Understanding the relationship between plan distance, ground distance, and the scale factor is crucial for accurate measurements in surveying and mapping.