All practice questions

Question 1361Question →

A cross-section is constructed from a topographic map drawn at a horizontal scale of $1 : 50\,000$ . If the vertical scale of the cross-section is set such that $1\text{ cm}$ represents $100\text{ m}$ of elevation, what is the vertical exaggeration of the cross-section?

Show answer & explanation

Answer: 5

Answer

The vertical exaggeration of the cross-section is 5.

To find the vertical exaggeration, both scales must be in representative fraction format. The horizontal scale is

1 : 50\,000

. The vertical scale (

1\text{ cm}

to

100\text{ m}

) equals

1 : 10\,000

. Dividing the horizontal denominator (

50\,000

) by the vertical denominator (

10\,000

) gives a vertical exaggeration of 5.

Step-by-Step Solution

1

Convert the vertical scale to representative fraction form

Vertical Scale = 1 : 10,000

Both scales must be expressed as dimensionless ratios (RF) in matching units to determine vertical enlargement.

2

Divide the horizontal scale denominator by the vertical scale denominator

VE = 50,000 / 10,000 = 5

Vertical exaggeration quantifies how many times larger the vertical scale is compared to the horizontal scale.

Key Concept

Vertical Exaggeration in Topographic Profiles

Question 1362Question →

A drainage basin outlined on a topographic map with a scale of $1:50,000$ has a total stream network length of $45\text{ cm}$ and a basin area of $36\text{ cm}^2$ measured directly from the map sheet. What is the actual drainage density of the basin in $\text{km}/\text{km}^2$ ?

Show answer & explanation

Answer: 2.5

Answer

The actual drainage density of the river basin is

2.5\text{ km}/\text{km}^2

.

To calculate the true drainage density, map linear measurements and area measurements must first be converted to ground units using the scale

1:50,000

(

1\text{ cm} = 0.5\text{ km}

,

1\text{ cm}^2 = 0.25\text{ km}^2

). Total ground stream length is

45 \times 0.5 = 22.5\text{ km}

and total ground basin area is

36 \times 0.25 = 9\text{ km}^2

. Dividing stream length by basin area yields

2.5\text{ km}/\text{km}^2

.

Step-by-Step Solution

1

Convert the measured total stream network length from map units (cm) to real-world kilometers.

Ground stream length

L = 45\text{ cm} \times 0.5\text{ km/cm} = 22.5\text{ km}

.

At a scale of

1:50,000

,

1\text{ cm}

on the map represents

50,000\text{ cm} = 0.5\text{ km}

on the ground.

2

Convert the measured basin area from square centimeters to square kilometers.

Ground basin area

A = 36\text{ cm}^2 \times (0.5\text{ km})^2 = 36 \times 0.25\text{ km}^2 = 9\text{ km}^2

.

The areal scale factor is the square of the linear scale factor (

1\text{ cm}^2 = 0.25\text{ km}^2

).

3

Divide the total ground stream length by the total ground basin area.

Drainage density

D_d = \frac{22.5\text{ km}}{9\text{ km}^2} = 2.5\text{ km}/\text{km}^2

.

Drainage density measures stream channel length per unit area within a river basin.

Key Concept

Drainage Density and Scale Conversion in Basin Analysis

Estimated Time:2m 0s

Question 1363Question →

On a topographical map of a river basin drawn to a Representative Fraction (R.F.) scale of $1 : 40,000$ , the distance along a proposed drainage channel between two agricultural settlements measures $14.5\text{ cm}$ . What is the actual ground distance of the drainage channel in kilometers?

Show answer & explanation

Answer: 5.8

Answer

The actual ground distance of the drainage channel is

5.8\text{ km}

.

The correct answer of

5.8\text{ km}

is obtained by multiplying the map distance of

14.5\text{ cm}

by the R.F. denominator

40,000

(

14.5 \times 40,000 = 580,000\text{ cm}

) and then converting centimeters to kilometers by dividing by

100,000

(

580,000 \div 100,000 = 5.8\text{ km}

).

Step-by-Step Solution

1

Calculate the total ground distance in centimeters

580,000\text{ cm}

According to the R.F. scale of

1 : 40,000

,

1\text{ cm}

on the map represents

40,000\text{ cm}

on the ground.

2

Convert the ground distance from centimeters to kilometers

5.8\text{ km}

Since

1\text{ km} = 100,000\text{ cm}

, divide the total distance in centimeters by

100,000

.

Key Concept

Ground distance calculation using Representative Fraction (R.F.) scale

Estimated Time:1m 15s

Question 1364Question →

In a morphometric analysis of a river basin, a hydrologist counts $32$ first-order streams and $8$ second-order streams using Strahler's stream ordering system. What is the bifurcation ratio between the first-order and second-order streams?

Show answer & explanation

Answer: 4

Answer

The bifurcation ratio between the first-order and second-order streams is 4.

The bifurcation ratio (

R_b

) is obtained by dividing the number of streams of a given order (

N_1 = 32

) by the number of streams of the next higher order (

N_2 = 8

). Therefore,

R_b = \frac{32}{8} = 4

.

Step-by-Step Solution

1

Identify the number of streams of order

u

and order

u+1

N_1 = 32

and

N_2 = 8

The bifurcation ratio measures the ratio of the number of stream segments of a given order to the number of segments of the higher order.

2

Calculate the bifurcation ratio using

R_b = \frac{N_1}{N_2}

R_b = \frac{32}{8} = 4

Dividing the count of first-order streams by the count of second-order streams gives the ratio of stream branching.

Key Concept

Bifurcation Ratio in River Basin Morphometry

Question 1365Question →

A topographic map drawn at a scale of $1 : 25,000$ shows a hillside where point P on a ridge has an elevation of $580\text{ m}$ and point Q near a river valley has an elevation of $430\text{ m}$ . If the straight-line distance between P and Q measured on the map is $7.2\text{ cm}$ , what is the value of $x$ when the average gradient between the two points is expressed in the ratio form $1 : x$ ?

Show answer & explanation

Answer: 12

Answer

The numerical value of

x

in the gradient ratio

1 : x

is 12.

To find the gradient ratio

1 : x

, first calculate the Vertical Interval (VI) by subtracting

430\text{ m}

from

580\text{ m}

, which gives

150\text{ m}

. Next, calculate the Horizontal Equivalent (HE) by multiplying the

7.2\text{ cm}

map distance by the scale factor of

25,000

, giving

180,000\text{ cm}

or

1,800\text{ m}

. Finally, divide VI by HE:

\frac{150\text{ m}}{1,800\text{ m}} = \frac{1}{12}

. Expressed in the form

1 : x

,

x = 12

.

Step-by-Step Solution

1

Calculate the Vertical Interval (VI)

VI = 150 m

Subtract the lower elevation (430 m) from the higher elevation (580 m) to determine the vertical height difference.

2

Calculate the Horizontal Equivalent (HE) in meters

HE = 1,800 m

Multiply the map measurement of 7.2 cm by 25,000 to obtain ground distance in cm (180,000 cm), then divide by 100 to convert to meters.

3

Compute the gradient ratio and solve for x

Gradient = 1 / 12, giving x = 12

Divide the Vertical Interval by the Horizontal Equivalent: 150 m / 1,800 m = 1 / 12.

Key Concept

Slope and Gradient Calculation from Topographic Maps

Estimated Time:1m 30s

Question 1366Question →

A solar power monitoring station at Site K, located at longitude $24^\circ\text{W}$ , records local solar noon ( $12:00\text{ PM}$ ). At the exact same instant, a remote telecommunication hub at Site M records its local solar time as $05:20\text{ PM}$ . What is the longitude of Site M in degrees East?

Show answer & explanation

Answer: 56

Answer

The longitude of Site M is

56^\circ\text{E}

.

The time difference between Site K (

12:00\text{ PM}

) and Site M (

05:20\text{ PM}

) is

5\text{ hours } 20\text{ minutes}

. Converting time to longitude (

15^\circ

per hour and

1^\circ

per

4\text{ minutes}

) yields an angular distance of

80^\circ

. Since Site M is ahead in time, it is located east of Site K. Subtracting

24^\circ

from

80^\circ

accounts for the distance to the Prime Meridian (

0^\circ

), leaving

56^\circ

in the Eastern Hemisphere (

56^\circ\text{E}

).

Step-by-Step Solution

1

Calculate the time difference between the two locations.

Time difference =

17:20 - 12:00 = 5\text{ hours } 20\text{ minutes}

(

5.333\text{ hours}

).

Differences in local solar time directly reflect differences in longitude.

2

Convert the time difference into angular degrees of longitude using the rate of

15^\circ

per hour (

1^\circ

per

4\text{ minutes}

).

Angular distance =

(5\text{ hours} \times 15^\circ/\text{hr}) + (20\text{ min} \div 4\text{ min}/^\circ) = 75^\circ + 5^\circ = 80^\circ

.

The Earth completes a

360^\circ

rotation in 24 hours.

3

Determine the direction of displacement and find the longitude of Site M.

Longitude of Site M =

80^\circ - 24^\circ\text{W} = 56^\circ\text{E}

.

Site M is ahead in time, so it lies to the east of Site K (

24^\circ\text{W}

). Traversing

80^\circ

eastward takes

24^\circ

to reach the Prime Meridian (

0^\circ

) and the remaining

56^\circ

into the Eastern Hemisphere.

Key Concept

Calculating longitude across meridians from local solar time differences

Question 1367Question →

At a weather station in Port Harcourt, Nigeria, the maximum thermometer recorded a temperature of $33.0^\circ\text{C}$ during the day, and the minimum thermometer recorded $21.0^\circ\text{C}$ at night. What is the diurnal temperature range in degrees Celsius ( $^\circ\text{C}$ ) recorded for that day?

Show answer & explanation

Answer: 12

Answer

The diurnal temperature range recorded for the day is

12.0^\circ\text{C}

.

The diurnal temperature range is defined as the difference between the maximum and minimum temperatures recorded within a single 24-hour day. Subtracting

21.0^\circ\text{C}

from

33.0^\circ\text{C}

yields

12.0^\circ\text{C}

.

Step-by-Step Solution

1

Extract the given maximum and minimum temperature values from the problem statement.

Maximum temperature =

33.0^\circ\text{C}

, Minimum temperature =

21.0^\circ\text{C}

.

Diurnal range requires the daily maximum and minimum readings.

2

Apply the diurnal temperature range formula.

\text{Diurnal Range} = \text{Maximum Temperature} - \text{Minimum Temperature}

.

The diurnal range represents the total temperature variation experienced within a 24-hour period.

3

Perform the subtraction.

33.0^\circ\text{C} - 21.0^\circ\text{C} = 12.0^\circ\text{C}

.

Subtracting the minimum temperature from the maximum temperature gives the numeric difference.

Key Concept

Diurnal Temperature Range Calculation

Question 1368Question →

A regional transport planning board mapped a single continuous railway network ( $p = 1$ ) connecting $15$ key industrial towns ( $v = 15$ ) with $22$ direct rail tracks ( $e = 22$ ). What is the cyclomatic number ( $C$ ) of this transport network?

Show answer & explanation

Answer: 8

Answer

The cyclomatic number of the transport network is 8.

The cyclomatic number (

C

) measures structural redundancy and closed loops within a graph network using the standard topological equation

C = e - v + p

. Substituting

e = 22

,

v = 15

, and

p = 1

gives

22 - 15 + 1 = 8

, representing

8

independent loops in the network.

Step-by-Step Solution

1

Identify given network variables from the problem stem

e = 22

(edges/tracks),

v = 15

(vertices/towns),

p = 1

(connected component)

Topological indices require accurate counting of fundamental structural elements.

2

Apply the formula for the cyclomatic number

C = e - v + p

The cyclomatic number measures the number of fundamental circuits or redundant loops in a transport graph.

3

Compute the quantitative value

C = 22 - 15 + 1 = 8

Subtracting vertices from edges and adding the component count yields the exact number of independent loops.

Key Concept

Cyclomatic Number in Transport Network Analysis

Question 1369Question →

An agricultural province in West Africa covers a total land area of $80,000\text{ km}^2$ . Arable land suitable for crop cultivation accounts for $25\%$ of this total land area. If the physiological population density of the province is $200\text{ persons per km}^2$ of arable land, what is the total population of the province?

Show answer & explanation

Answer: 4000000

Answer

The total population of the province is

4,000,000

people.

Physiological density measures the population relative to the amount of arable land available (

200\text{ persons per km}^2

). First, finding

25\%

of the total

80,000\text{ km}^2

gives

20,000\text{ km}^2

of arable land. Multiplying this arable area by the physiological density of

200\text{ persons/km}^2

yields a total population of

4,000,000

people.

Step-by-Step Solution

1

Calculate the total area of arable land in the province

Arable land area =

20,000\text{ km}^2

Physiological density is calculated strictly based on arable (cultivable) land area, not total surface area.

25\%

of

80,000\text{ km}^2

equals

20,000\text{ km}^2

.

2

Apply the physiological density equation to calculate the total population

Total population =

4,000,000

people

Since

\text{Physiological Density} = \frac{\text{Total Population}}{\text{Arable Land Area}}

, multiplying the physiological density (

200\text{ persons/km}^2

) by the arable land area (

20,000\text{ km}^2

) gives the total population (

200 \times 20,000 = 4,000,000

).

Key Concept

Physiological Population Density

Question 1370Question →

A mining concession covers an area of $40\text{ cm}^2$ on a topographical map drawn to a scale of $1 : 50,000$ . If this map is reduced to a scale of $1 : 200,000$ , what is the area of the concession on the reduced map in $\text{cm}^2$ ?

Show answer & explanation

Answer: 2.5

Answer

The area of the mining concession on the reduced map is

2.5\text{ cm}^2

.

When a map is reduced from a scale of

1 : 50,000

to

1 : 200,000

, the linear scale is reduced by a factor of

4

(since

200,000 / 50,000 = 4

). Because area changes proportionally to the square of the linear change, the area scale factor is

(1/4)^2 = 1/16

. Multiplying the original area of

40\text{ cm}^2

by

1/16

gives

2.5\text{ cm}^2

.

Step-by-Step Solution

1

Calculate the linear reduction factor.

Linear scale factor

k = \frac{50,000}{200,000} = \frac{1}{4} = 0.25

The linear change ratio is determined by comparing the old scale denominator to the new scale denominator.

2

Calculate the area scale factor.

Area factor

k^2 = (0.25)^2 = \frac{1}{16} = 0.0625

Area scale varies with the square of the linear scale factor.

3

Determine the area on the reduced map.

New Map Area

= 40\text{ cm}^2 \times 0.0625 = 2.5\text{ cm}^2

Applying the area reduction factor to the original map area yields the new map area.

Key Concept

Relationship between linear scale factor and area scale factor during map reduction

Question 1371Question →

A wildlife sanctuary covers an area of $24\text{ cm}^2$ on Map P, which is drawn to a scale of $1 : 60,000$ . If Map P is enlarged to produce Map Q with a scale of $1 : 20,000$ , what is the area of the sanctuary on Map Q in $\text{cm}^2$ ?

Show answer & explanation

Answer: 216

Answer

The area of the sanctuary on Map Q is

216\text{ cm}^2

.

When a map is enlarged from a scale of

1 : 60,000

to

1 : 20,000

, the linear dimension increases by a factor of

\frac{60,000}{20,000} = 3

. Because area is a two-dimensional measurement, the area scale factor is the square of the linear scale factor (

3^2 = 9

). Thus, the new area on Map Q is

24\text{ cm}^2 \times 9 = 216\text{ cm}^2

.

Step-by-Step Solution

1

Find the linear scale enlargement factor (

k

)

k = \frac{\text{Old Scale Denominator}}{\text{New Scale Denominator}} = \frac{60,000}{20,000} = 3

Enlarging a map scale from

1 : 60,000

to

1 : 20,000

increases linear dimensions by a factor of 3.

2

Determine the area scale multiplier

\text{Area Scale Factor} = k^2 = 3^2 = 9

Area changes according to the square of the linear scale change ratio.

3

Calculate the new map area

24\text{ cm}^2 \times 9 = 216\text{ cm}^2

Multiplying the original area on Map P by the area scale factor yields the enlarged area on Map Q.

Key Concept

Map Enlargement Area Calculation

Question 1372Question →

In a municipality located in West Africa with an estimated mid-year population of $250{}000$ , a demographic survey recorded $6{}250$ live births and $2{}500$ deaths within a single year. What is the Rate of Natural Increase (RNI) of this population expressed as a percentage?

Show answer & explanation

Answer: 1.5

Answer

The Rate of Natural Increase (RNI) of the population is

1.5\%

.

The Rate of Natural Increase (RNI) measures the surplus of births over deaths in a population over a specific period. Subtracting annual deaths (

2{}500

) from annual live births (

6{}250

) yields a natural increase of

3{}750

. Dividing this increase by the total population (

250{}000

) gives

0.015

, which equals

1.5\%

when multiplied by

100

.

Step-by-Step Solution

1

Calculate the net natural increase in population.

Net natural increase =

3{}750

individuals.

Natural increase measures population growth resulting solely from natural demographic events (live births minus deaths).

2

Express the natural increase as a percentage of the mid-year population.

RNI =

1.5\%

.

Dividing the natural increase (

3{}750

) by the total mid-year population (

250{}000

) and multiplying by

100

converts the absolute increase into an annual growth percentage.

Key Concept

Rate of Natural Increase (RNI)

Question 1373Question →

In a demographic study of an agricultural region in West Africa covering a total surface area of $150{}000\text{ km}^2$ ( $20\%$ of which consists of non-arable mountainous terrain and water bodies), the population at the beginning of a 10-year census period was $9{}000{}000$ . Over the 10-year period, official demographic records indicate $2{}700{}000$ live births, $1{}050{}000$ deaths, $600{}000$ inbound immigrants, and $150{}000$ outbound emigrants. What is the physiological population density of the region in persons per square kilometer of arable land at the end of the 10-year period?

Show answer & explanation

Answer: 92.5

Answer

92.5 persons per square kilometer of arable land

To calculate the physiological population density, the final population must be divided strictly by the area of arable land. The total ending population is obtained by taking the initial population (

9{}000{}000

), adding natural increase (

2{}700{}000 - 1{}050{}000 = 1{}650{}000

), and adding net migration (

600{}000 - 150{}000 = 450{}000

), which gives

11{}100{}000

persons. The arable land area is

80\%

of

150{}000\text{ km}^2 = 120{}000\text{ km}^2

. Dividing

11{}100{}000

by

120{}000

yields

92.5\text{ persons/km}^2

.

Step-by-Step Solution

1

Calculate the area of arable land

Arable land area =

150{}000\text{ km}^2 \times (1 - 0.20) = 120{}000\text{ km}^2

Physiological population density measures population relative to arable (cultivable) land only, excluding non-arable terrain.

2

Calculate the natural population increase

Natural Increase =

2{}700{}000\text{ births} - 1{}050{}000\text{ deaths} = 1{}650{}000\text{ persons}

Natural population change is the net difference between live births and deaths.

3

Calculate net migration

Net Migration =

600{}000\text{ immigrants} - 150{}000\text{ emigrants} = 450{}000\text{ persons}

Net migration accounts for population change caused by movement into and out of the region.

4

Determine final total population

Final Population =

9{}000{}000 + 1{}650{}000 + 450{}000 = 11{}100{}000\text{ persons}

Total population at the end of the period equals initial population plus natural increase plus net migration.

5

Calculate physiological population density

Physiological Density =

\frac{11{}100{}000\text{ persons}}{120{}000\text{ km}^2} = 92.5\text{ persons/km}^2

Physiological density is computed by dividing the total ending population by the arable land area.

Key Concept

Physiological Population Density and Net Population Change

Question 1374Question →

A hydrologist conducts a morphometric analysis on a river basin using a topographic map drawn at a scale of $1:25,000$ . Stream ordering reveals $28$ first-order streams, $7$ second-order streams, and $1$ third-order stream. If the total basin area measured on the map is $32\text{ cm}^2$ , what is the stream frequency ( $F_s$ ) of the basin in $\text{streams per km}^2$ ?

Show answer & explanation

Answer: 18

Answer

The stream frequency of the river basin is

18\text{ streams per km}^2

.

To determine stream frequency (

F_s

), first calculate the total stream count (

N = 28 + 7 + 1 = 36

). Next, convert the map area of

32\text{ cm}^2

to actual ground area using the scale

1:25,000

. Since

1\text{ cm}

on the map represents

0.25\text{ km}

,

1\text{ cm}^2

represents

0.0625\text{ km}^2

. Multiplying

32\text{ cm}^2

by

0.0625\text{ km}^2/\text{cm}^2

gives a basin area of

2.0\text{ km}^2

. Finally, divide the total number of streams by the basin area (

36 / 2.0 = 18\text{ streams per km}^2

).

Step-by-Step Solution

1

Sum the number of streams of each order to find the total stream count (

N

).

N = 28 + 7 + 1 = 36\text{ streams}

.

Stream frequency considers the entire stream network comprising all stream orders in the basin.

2

Convert map area to ground area using the map scale (

1:25,000

).

1\text{ cm} = 0.25\text{ km} \implies 1\text{ cm}^2 = 0.0625\text{ km}^2

. Actual Area

A = 32 \times 0.0625 = 2.0\text{ km}^2

.

Stream frequency must be expressed per unit of actual ground area in

\text{km}^2

rather than map area.

3

Divide total number of streams by actual basin area.

F_s = \frac{36}{2.0} = 18\text{ streams/km}^2

.

Stream frequency (

F_s

) is defined as the ratio of total number of streams to the drainage basin area.

Key Concept

Stream Frequency (

F_s

) Calculation in River Basin Morphometry

Estimated Time:2m 0s

Question 1375Question →

A cartographer constructs a quantitative dot map to represent the distribution of livestock across a region. On the map, a total of $64$ dots represent a total population of $160,000$ sheep. If Province X has a sheep population of $35,000$ , how many dots should be plotted to represent Province X on this map?

Show answer & explanation

Answer: 14

Answer

14 dots are required to represent Province X.

To find the dot count for Province X, first determine the scale value of a single dot:

\frac{160,000\text{ sheep}}{64\text{ dots}} = 2,500\text{ sheep per dot}

. Next, divide the population of Province X by the dot value:

\frac{35,000\text{ sheep}}{2,500\text{ sheep/dot}} = 14\text{ dots}

.

Step-by-Step Solution

1

Determine the quantity represented by one dot.

Each dot represents

2,500

sheep.

Dividing the overall sheep population (

160,000

) by the total number of dots (

64

) establishes the dot value.

2

Calculate the number of dots for Province X.

14 dots.

Dividing the provincial population (

35,000

) by the single-dot value (

2,500

) yields the exact number of dots required for visual representation.

Key Concept

Dot Map Scale and Quantifier Calculation

Question 1376Question →

At a coastal weather station situated at sea level ( $0\text{ m}$ ), the recorded dry-bulb air temperature is $31.2^\circ\text{C}$ . A secondary meteorological station is established on an adjacent highlands plateau at an altitude of $2,400\text{ m}$ . Assuming a standard environmental lapse rate of $6.5^\circ\text{C}$ per $1,000\text{ m}$ of ascent in the troposphere, what is the calculated air temperature at the highlands station in degrees Celsius ( $^\circ\text{C}$ )?

Show answer & explanation

Answer: 15.6

Answer

The expected air temperature at the highlands station is

15.6^\circ\text{C}

.

Air temperature in the lowest layer of the atmosphere (troposphere) decreases with altitude at the standard environmental lapse rate of

6.5^\circ\text{C}

per

1,000\text{ m}

(

0.65^\circ\text{C}

per

100\text{ m}

). For an elevation increase of

2,400\text{ m}

, the total temperature drop equals

2.4 \times 6.5^\circ\text{C} = 15.6^\circ\text{C}

. Subtracting this reduction from the baseline sea-level reading of

31.2^\circ\text{C}

gives

15.6^\circ\text{C}

.

Step-by-Step Solution

1

Determine altitude difference in thousands of meters

2.4\text{ units of } 1,000\text{ m}

The environmental lapse rate is specified per

1,000\text{ m}

elevation gain.

2

Compute total atmospheric temperature drop

15.6^\circ\text{C}

drop

Multiply the elevation change in thousands of meters (

2.4

) by the lapse rate (

6.5^\circ\text{C}

).

3

Calculate final temperature at plateau elevation

15.6^\circ\text{C}

Subtract the total calculated temperature drop from the sea-level temperature (

31.2^\circ\text{C} - 15.6^\circ\text{C}

).

Key Concept

Environmental Lapse Rate and Vertical Temperature Variation

Question 1377Question →

On a topographical map representing a mountainous terrain, a lower index contour line is marked at $500\text{ m}$ above sea level and a higher index contour line is marked at $900\text{ m}$ above sea level. If there are $4$ equal contour intervals between these two index contours, what is the elevation in metres of the third contour line above the $500\text{ m}$ index contour?

Show answer & explanation

Answer: 800

Answer

The elevation of the third contour line above the

500\text{ m}

index contour is

800\text{ m}

.

The vertical difference between the

500\text{ m}

and

900\text{ m}

index contours is

400\text{ m}

. Dividing this difference by the

4

contour intervals gives a uniform vertical interval of

100\text{ m}

per contour line. Therefore, three contour lines above the

500\text{ m}

line correspond to an elevation of

500\text{ m} + 3(100\text{ m}) = 800\text{ m}

.

Step-by-Step Solution

1

Find the total elevation difference between the two known index contours

900\text{ m} - 500\text{ m} = 400\text{ m}

This establishes the cumulative elevation change across the four intervals.

2

Calculate the Vertical Interval (V.I.) between adjacent contour lines

\frac{400\text{ m}}{4} = 100\text{ m}

The contour interval is uniform across the map, so dividing the vertical change by the number of spaces yields the elevation change per line.

3

Calculate the elevation at the third contour line above the base index contour

500\text{ m} + (3 \times 100\text{ m}) = 800\text{ m}

Moving up three contour lines from

500\text{ m}

increases the elevation by three times the vertical interval.

Key Concept

Vertical Interval and Contour Line Elevation Determination

Question 1378Question →

A demographic survey carried out in an agricultural district in Nigeria recorded the following age structure: $24,300$ children under 15 years of age, $45,000$ adults in the economically active age range of 15–64 years, and $6,300$ elderly residents aged 65 years and above. What is the dependency ratio of this district, expressed as a percentage?

Show answer & explanation

Answer: 68

Answer

68%

The dependency ratio expresses the relationship between the dependent segment of the population (those under 15 and those 65 and above) and the productive segment (ages 15–64). Summing the dependents (

24,300 + 6,300

) gives

30,600

. Dividing

30,600

by the working-age population of

45,000

and multiplying by

100

yields

68\%

.

Step-by-Step Solution

1

Determine the size of the dependent population.

Total dependents =

24,300 + 6,300 = 30,600

.

Demographic dependency includes individuals under 15 years and those aged 65 years and older.

2

Calculate the dependency ratio.

\text{Dependency Ratio} = \left(\frac{30,600}{45,000}\right) \times 100 = 68\%

.

The dependency ratio measures the number of economic dependents supported by every 100 working-age individuals.

Key Concept

Dependency Ratio Calculation

Question 1379Question →

In triangle $PQR$ , the side lengths are $p = 8\text{ cm}$ and $q = 15\text{ cm}$ , and the included angle $\angle R = 60^\circ$ . What is the length of side $r$ in centimeters?

Show answer & explanation

Answer: 13

Answer

The length of side

r

is

13\text{ cm}

.

Using the Cosine Rule formula

r^2 = p^2 + q^2 - 2pq \cos R

with

p=8

,

q=15

, and

R=60^\circ

, we calculate

r^2 = 64 + 225 - 240(0.5) = 169

. Taking the square root gives

r = 13\text{ cm}

.

Step-by-Step Solution

1

Identify the given values and appropriate trigonometric rule

Givens:

p = 8\text{ cm}

,

q = 15\text{ cm}

,

\angle R = 60^\circ

. Since two sides and the included angle (SAS) are given, use the Cosine Rule:

r^2 = p^2 + q^2 - 2pq \cos R

.

The Cosine Rule is required to find the third side when two sides and their included angle are known.

2

Substitute the values into the Cosine Rule formula

r^2 = 8^2 + 15^2 - 2(8)(15) \cos 60^\circ

Replacing variables with their numerical equivalents sets up the algebraic calculation.

3

Calculate the terms and evaluate

r^2

r^2 = 64 + 225 - 240 \times 0.5 = 289 - 120 = 169

Since

\cos 60^\circ = 0.5

, simplify the arithmetic operations.

4

Solve for side length

r

r = \sqrt{169} = 13\text{ cm}

Take the square root of both sides to obtain the length of side

r

.

Key Concept

Applying the Cosine Rule to find an unknown side given two sides and the included angle (SAS)

Question 1380Question →

If the determinant of the matrix $A = \begin{pmatrix} k & 1 & 2 \\ 3 & -1 & 0 \\ 2 & 4 & 1 \end{pmatrix}$ is equal to $17$ , what is the value of $k$ ?

Show answer & explanation

Answer: 8

Answer

The value of

k

is

8

.

Expanding the matrix determinant along the first row gives

\det(A) = k(-1) - 1(3) + 2(14) = -k + 25

. Setting

-k + 25 = 17

leads to

-k = -8

, so

k = 8

.

Step-by-Step Solution

1

Perform cofactor expansion along the first row of matrix

A

.

\det(A) = k((-1)(1) - (0)(4)) - 1((3)(1) - (0)(2)) + 2((3)(4) - (-1)(2))

Expanding along the first row uses the formula \det(A) = a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13}.

2

Evaluate the products and simplify the algebraic expression for the determinant.

\det(A) = -k - 3 + 28 = -k + 25

Simplifying each sub-determinant term yields a linear expression in

k

.

3

Equate the expression to

17

and solve for

k

.

-k + 25 = 17 \implies k = 8

Subtracting 25 from both sides gives

-k = -8

, which simplifies to

k = 8

.

Key Concept

Determinant of a 3x3 Matrix

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