Match each topographic map elevation scenario to its corresponding calculated slope gradie… — JAMB UTME

Match each topographic map elevation scenario to its corresponding calculated slope gradient ratio ( $1 \text{ in } N$ ). Which of the following correct pairs match each map measurement scenario on the left with its computed gradient ratio on the right?

Elevation difference of $80\text{ m}$ between two points separated by $4\text{ cm}$ on a $1 : 50,000$ map scale $1 \text{ in } 25$
Elevation difference of $150\text{ m}$ between two points separated by $3\text{ cm}$ on a $1 : 100,000$ map scale $1 \text{ in } 20$
Elevation difference of $200\text{ m}$ between two points separated by $2\text{ cm}$ on a $1 : 25,000$ map scale $1 \text{ in } 2.5$
Elevation difference of $60\text{ m}$ between two points separated by $6\text{ cm}$ on a $1 : 50,000$ map scale $1 \text{ in } 50$

Answer

The correct matches pair each scenario with its gradient ratio as follows: the

80\text{ m}

elevation difference over

4\text{ cm}

(

1:50,000

) matches

1 \text{ in } 25

; the

150\text{ m}

elevation difference over

3\text{ cm}

(

1:100,000

) matches

1 \text{ in } 20

; the

200\text{ m}

elevation difference over

2\text{ cm}

(

1:25,000

) matches

1 \text{ in } 2.5

; and the

60\text{ m}

elevation difference over

6\text{ cm}

(

1:50,000

) matches

1 \text{ in } 50

Each elevation difference (VI) is divided by the true ground distance (HE) converted to meters. For scenario 1:

4\text{ cm} \times 50,000 / 100 = 2,000\text{ m}

, leading to

80 / 2,000 = 1 / 25

. For scenario 2:

3\text{ cm} \times 100,000 / 100 = 3,000\text{ m}

, giving

150 / 3,000 = 1 / 20

. For scenario 3:

2\text{ cm} \times 25,000 / 100 = 500\text{ m}

, giving

200 / 500 = 1 / 2.5

. For scenario 4:

6\text{ cm} \times 50,000 / 100 = 3,000\text{ m}

, yielding

60 / 3,000 = 1 / 50

Step-by-Step Solution

Recall the gradient formula for topographic maps

Gradient formula is

\text{Gradient} = \frac{\text{Vertical Interval (VI)}}{\text{Horizontal Equivalent (HE)}}

, where both VI and HE must be expressed in the same units (meters).

Gradient is a dimensionless ratio comparing vertical rise to horizontal ground distance.

Calculate Horizontal Equivalent (HE) for each scenario

Scenario 1:

\text{HE} = 4\text{ cm} \times 50,000 = 200,000\text{ cm} = 2,000\text{ m}

. Scenario 2:

\text{HE} = 3\text{ cm} \times 100,000 = 300,000\text{ cm} = 3,000\text{ m}

. Scenario 3:

\text{HE} = 2\text{ cm} \times 25,000 = 50,000\text{ cm} = 500\text{ m}

. Scenario 4:

\text{HE} = 6\text{ cm} \times 50,000 = 300,000\text{ cm} = 3,000\text{ m}

Map distance must be converted to actual ground distance using the representative fraction scale.

Compute the gradient ratio (VI / HE) for each item and pair with the correct ratio

Item 1:

\frac{80}{2000} = \frac{1}{25}

(

1 \text{ in } 25

). Item 2:

\frac{150}{3000} = \frac{1}{20}

(

1 \text{ in } 20

). Item 3:

\frac{200}{500} = \frac{1}{2.5}

(

1 \text{ in } 2.5

). Item 4:

\frac{60}{3000} = \frac{1}{50}

(

1 \text{ in } 50

Simplifying each fraction to unit numerator format (

1 / N

) yields the standard gradient ratio expression.

Key Concept

Slope and Gradient Calculation

Question

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