Tüm alıştırma soruları

Soru 421Soru →

City P is located at longitude $38^\circ\text{E}$ where the local time is 4:20 p.m. At the exact same moment, the local time at City Q is 9:40 a.m. on the same day. What is the longitude of City Q in degrees West of the Greenwich Meridian?

Cevabı ve açıklamayı göster

Cevap: 62

Cevap

City Q is located at longitude

62^\circ\text{W}

.

The calculation demonstrates that a time difference of 6 hours and 40 minutes equals

100^\circ

of longitude (

400 \text{ minutes} \div 4 \text{ minutes/degree}

). Because City Q is earlier in time (9:40 a.m.) than City P (4:20 p.m.), City Q is located

100^\circ

to the west of

38^\circ\text{E}

. Subtracting

38^\circ

to reach

0^\circ

leaves

62^\circ

in the Western Hemisphere, giving a final position of

62^\circ\text{W}

.

Adım Adım Çözüm

1

Find the local time difference between City P and City Q

Local time difference is 6 hours and 40 minutes (400 minutes)

Converting both times to 24-hour format (16:20 and 09:40) allows direct subtraction:

16:20 - 09:40 = 6\text{ h } 40\text{ min}

.

2

Convert time difference into angular longitude difference

Longitude difference is

100^\circ

Earth rotates

1^\circ

every 4 minutes. Dividing 400 minutes by 4 yields

100^\circ

of total longitude separation.

3

Determine the relative direction of City Q from City P

City Q is located

100^\circ

west of City P

Places with earlier local times lie to the west because the Earth rotates from west to east.

4

Calculate the final longitude position west of Greenwich (

0^\circ

)

Longitude of City Q is

62^\circ\text{W}

Moving

100^\circ

west from

38^\circ\text{E}

covers

38^\circ

to reach

0^\circ

, and the remaining

62^\circ

extends into the Western Hemisphere (

100^\circ - 38^\circ = 62^\circ

).

Anahtar Kavram

Calculation of longitude position using local time differences across the Prime Meridian

Soru 422Soru →

Zenith Freight Services extracted the following ledger balances at the end of its financial year:

- Capital: ₦150,000
- Premises: ₦120,000
- Cash at Bank: ₦35,000
- Motor Vehicles: ₦60,000
- Creditors: ₦25,000
- Sales: ₦180,000
- Purchases: ₦110,000
- Salaries Expense: ₦30,000

Calculate the total debit balance column of the trial balance in Naira (₦).

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Cevap: 355000

Cevap

The total debit column of the trial balance is ₦355,000.

The trial balance debit total is obtained by summing all debit balance accounts: Premises (₦120,000), Cash at Bank (₦35,000), Motor Vehicles (₦60,000), Purchases (₦110,000), and Salaries Expense (₦30,000), giving a total of ₦355,000. This equals the credit column total (Capital ₦150,000 + Creditors ₦25,000 + Sales ₦180,000 = ₦355,000).

Adım Adım Çözüm

1

Classify ledger balances into debit and credit entries based on standard accounting rules.

Debit items: Premises (Asset), Cash at Bank (Asset), Motor Vehicles (Asset), Purchases (Cost), Salaries Expense (Expense). Credit items: Capital (Equity), Creditors (Liability), Sales (Revenue).

Assets and expenses carry normal debit balances, while liabilities, equity, and revenue carry normal credit balances.

2

Sum all debit entries to find the total of the trial balance debit column.

₦120,000 + ₦35,000 + ₦60,000 + ₦110,000 + ₦30,000 = ₦355,000.

Extracting the trial balance requires totaling the debit side to verify double-entry arithmetic equality with the credit side.

Anahtar Kavram

Classification of Ledger Balances into Trial Balance Debit and Credit Columns

Soru 423Soru →

A marine navigational radar system emits electromagnetic pulses with a wavelength of $3.0\text{ cm}$ . Given that the speed of electromagnetic waves in air is $3.0 \times 10^8\text{ m/s}$ , what is the frequency of the radar signal in gigahertz ( $\text{GHz}$ )?

Cevabı ve açıklamayı göster

Cevap: 10

Cevap

The frequency of the radar signal is

10\text{ GHz}

.

Converting

3.0\text{ cm}

to meters gives

\lambda = 0.03\text{ m}

. Using

f = c / \lambda

,

f = (3.0 \times 10^8\text{ m/s}) / 0.03\text{ m} = 1.0 \times 10^{10}\text{ Hz}

. Dividing by

10^9\text{ Hz/GHz}

yields

10\text{ GHz}

.

Adım Adım Çözüm

1

Convert the given wavelength into fundamental SI units (meters).

\lambda = 3.0\text{ cm} = 3.0 \times 10^{-2}\text{ m}

The speed of light

c

is given in meters per second, so the wavelength must be expressed in meters to keep units consistent.

2

Rearrange the electromagnetic wave speed formula

c = f \lambda

to solve for frequency

f

.

f = \frac{c}{\lambda} = \frac{3.0 \times 10^8\text{ m/s}}{3.0 \times 10^{-2}\text{ m}} = 1.0 \times 10^{10}\text{ Hz}

Frequency is the ratio of wave propagation speed to wavelength.

3

Convert the frequency from hertz to gigahertz.

f = \frac{1.0 \times 10^{10}\text{ Hz}}{10^9\text{ Hz/GHz}} = 10\text{ GHz}

The question explicitly asks for the answer in gigahertz (

\text{GHz}

).

Anahtar Kavram

Electromagnetic wave propagation equation (

c = f \lambda

) and unit metric prefixes.

Soru 424Soru →

The following information was extracted from the incomplete records of Chukwuma Traders for the year ended 31 December 2025:

Transaction / Account Details	Amount ( $\text{NGN}$ )
Trade debtors balance (1 January 2025)	$12,000$
Cash received from trade debtors	$45,000$
Discount allowed to debtors	$1,200$
Returns inwards	$800$
Bad debts written off	$500$
Trade debtors balance (31 December 2025)	$15,500$
Cash sales made during the year	$18,000$

Calculate the total sales for the year ended 31 December 2025 in Naira ( $\text{NGN}$ ).

Cevabı ve açıklamayı göster

Cevap: 69000

Cevap

The total sales for the year ended 31 December 2025 is NGN 69,000.

Total sales is computed by summing credit sales derived from the Total Debtors Control Account (

\text{NGN } 51,000

) and cash sales (

\text{NGN } 18,000

), yielding

\text{NGN } 69,000

.

Adım Adım Çözüm

1

Determine Credit Sales using the Total Debtors Control Account equation

Credit Sales =

\text{NGN } 51,000

The Total Debtors Account credits all items that reduce trade debtors (cash received, discounts allowed, returns inwards, bad debts, and closing balance) and debits opening debtors and credit sales. Rearranging gives:

\text{Credit Sales} = (45,000 + 1,200 + 800 + 500 + 15,500) - 12,000 = 63,000 - 12,000 = 51,000

.

2

Add Cash Sales to Credit Sales to find Total Sales

Total Sales =

\text{NGN } 69,000

Total sales comprises both cash sales and credit sales (

\text{Total Sales} = \text{Cash Sales} + \text{Credit Sales} = 18,000 + 51,000 = 69,000

).

Anahtar Kavram

Reconstruction of Total Sales from Incomplete Records

Tahmini Süre:2m 0s

Soru 425Soru →

On 30th June 2026, the adjusted cash book of Emeka Enterprises showed a debit balance of $\text{NGN } 45,000$ . A comparison with the bank statement revealed unpresented cheques totaling $\text{NGN } 12,500$ and uncredited lodgements of $\text{NGN } 8,200$ . What is the balance as per the bank statement as at 30th June 2026?

Cevabı ve açıklamayı göster

Cevap: 49300

Cevap

The balance as per the bank statement as at 30th June 2026 is NGN 49,300.

Starting from a favourable (debit) adjusted cash book balance of NGN 45,000, unpresented cheques (NGN 12,500) are added because they represent payments recorded in the cash book that have not yet been processed by the bank. Uncredited lodgements (NGN 8,200) are subtracted because they represent deposits recorded in the cash book that have not yet been credited by the bank. This gives a final bank statement credit balance of NGN 49,300.

Adım Adım Çözüm

1

Identify starting balance from the adjusted cash book

Adjusted Cash Book balance (Debit) = NGN 45,000

Reconciliation starts from the corrected cash book figure to arrive at the bank statement balance.

2

Add unpresented cheques

45,000 + 12,500 = NGN 57,500

Unpresented cheques have been debited/subtracted in the cash book but not yet cleared by the bank, so the bank balance is higher by this amount.

3

Deduct uncredited lodgements

57,500 - 8,200 = NGN 49,300

Uncredited lodgements have been added in the cash book but not yet credited on the bank statement, so the bank balance is lower by this amount.

Anahtar Kavram

Reconciling the Adjusted Cash Book Balance to the Bank Statement Balance

Soru 426Soru →

A solar observation station located at longitude $23^\circ\text{W}$ records local solar noon ( $12:00\text{ PM}$ ) at a specific instant. At that exact moment, a research vessel at sea notes its local solar time as $7:16\text{ PM}$ ( $19:16$ ) on the same day. What is the longitude of the research vessel in degrees East?

Cevabı ve açıklamayı göster

Cevap: 86

Cevap

The research vessel is located at

86^\circ\text{E}

.

The time difference between 12:00 PM and 7:16 PM is 7 hours and 16 minutes (436 minutes). Since 4 minutes correspond to 1° of longitude, the angular difference is 436 ÷ 4 = 109°. Because the vessel's solar time is later than the station's time, the vessel lies to the East. Moving 109° East from 23°W requires 23° to reach the 0° Greenwich Meridian and an additional 86° into the Eastern Hemisphere, placing the vessel at 86°E.

Adım Adım Çözüm

1

Determine the time difference between the solar observation station and the research vessel.

Time difference = 19:16 - 12:00 = 7 hours and 16 minutes = 436 minutes.

Difference in local solar time corresponds to longitudinal separation.

2

Convert the time difference into longitudinal degrees using the rate of Earth's rotation (

1^\circ = 4\text{ minutes}

).

Longitudinal difference = 436 ÷ 4 = 109°.

The Earth rotates 360° in 24 hours, which equals 1° for every 4 minutes of time difference.

3

Determine the direction of the vessel relative to the station.

The vessel is East of the station.

Local solar time at the vessel (7:16 PM) is ahead of the station (12:00 PM), meaning the vessel lies further East.

4

Calculate the absolute longitude in the Eastern Hemisphere.

Vessel longitude = 109° - 23° = 86°E.

Traversing 109° East starting from 23°W uses 23° to reach the Greenwich Meridian (0°) and the remaining 86° extends into the Eastern Hemisphere.

Anahtar Kavram

Calculating longitude from local solar time difference across meridians

Soru 427Soru →

A progressive transverse wave traveling through a primary medium is governed by the mathematical wave equation $y = 0.05 \sin\left(100\pi t - 2.5\pi x\right)$ , where $x$ and $y$ are measured in meters and $t$ is in seconds. Upon entering a secondary medium, the wave undergoes refraction such that its wavelength decreases by $20\%$ . What is the speed of the wave in the secondary medium in $\text{m/s}$ ?

Cevabı ve açıklamayı göster

Cevap: 32

Cevap

The speed of the wave in the secondary medium is

32\text{ m/s}

.

Comparing

y = 0.05 \sin\left(100\pi t - 2.5\pi x\right)

with

y = A \sin(\omega t - kx)

yields

\omega = 100\pi\text{ rad/s}

and

k = 2.5\pi\text{ rad/m}

. This gives a source frequency

f = \frac{\omega}{2\pi} = 50\text{ Hz}

and initial wavelength

\lambda_1 = \frac{2\pi}{k} = 0.8\text{ m}

. Because wave frequency is invariant across boundaries,

f

remains

50\text{ Hz}

in the second medium. The new wavelength is

\lambda_2 = 0.8\text{ m} \times 0.80 = 0.64\text{ m}

. Consequently, the wave speed in the second medium is

v_2 = 50\text{ Hz} \times 0.64\text{ m} = 32\text{ m/s}

.

Adım Adım Çözüm

1

Extract angular frequency and wave number from the wave equation

\omega = 100\pi\text{ rad/s}

and

k = 2.5\pi\text{ rad/m}

Matching coefficients in

y = A \sin(\omega t - kx)

allows determination of temporal and spatial characteristics.

2

Determine wave frequency and original wavelength

f = 50\text{ Hz}

and

\lambda_1 = 0.8\text{ m}

Using fundamental relationships

f = \frac{\omega}{2\pi}

and

\lambda = \frac{2\pi}{k}

.

3

Calculate the refracted wavelength under constant frequency

\lambda_2 = 0.64\text{ m}

while

f

remains

50\text{ Hz}

Wave frequency depends solely on the wave source and remains unchanged during medium transitions.

4

Compute wave propagation speed in the new medium

v_2 = 32\text{ m/s}

Applying the wave equation

v = f\lambda

with the updated wavelength.

Anahtar Kavram

Wave equation parameter extraction and frequency invariance during refraction

Tahmini Süre:2m 0s

Soru 428Soru →

On a topographical map drawn to a scale of $1 : 25,000$ , two hilltop stations, Station A and Station B, are separated by a map distance of $16\text{ cm}$ . If Station A is situated at an elevation of $580\text{ m}$ above sea level and Station B is at an elevation of $180\text{ m}$ , what is the gradient of the slope between the two stations expressed as the value $N$ in the ratio $1 : N$ ?

Cevabı ve açıklamayı göster

Cevap: 10

Cevap

The denominator

N

in the slope gradient ratio

1 : N

is

10

(representing a gradient ratio of

1 : 10

).

To calculate the gradient expressed as

1 : N

, determine the Vertical Interval (VI

= 580\text{ m} - 180\text{ m} = 400\text{ m}

) and the Horizontal Equivalent (HE

= 16\text{ cm} \times 25,000 = 400,000\text{ cm} = 4,000\text{ m}

). Dividing VI by HE gives

\frac{400\text{ m}}{4,000\text{ m}} = \frac{1}{10}

, which corresponds to a gradient of

1 : 10

, making

N = 10

.

Adım Adım Çözüm

1

Determine the Vertical Interval (VI)

\text{VI} = 580\text{ m} - 180\text{ m} = 400\text{ m}

The Vertical Interval is the vertical height difference between the two specified elevations.

2

Calculate the Horizontal Equivalent (HE) on the ground

\text{HE} = 16\text{ cm} \times 25,000 = 400,000\text{ cm} = 4,000\text{ m}

Multiply the map distance by the scale denominator to find the ground distance, then divide by 100 to convert centimeters to meters.

3

Calculate the gradient ratio

\text{Gradient} = \frac{\text{VI}}{\text{HE}} = \frac{400\text{ m}}{4,000\text{ m}} = \frac{1}{10} = 1 : 10

Gradient is calculated as Vertical Interval divided by Horizontal Equivalent, simplified to a fraction with a numerator of 1.

Anahtar Kavram

Calculation of slope gradient using vertical interval, map distance, and representative fraction scale

Tahmini Süre:1m 30s

Soru 429Soru →

The following figures were extracted from the books of Ade & Sons Manufacturing Enterprises for the financial year ended 31st December 2025:

- Direct materials used: ₦145,000
- Direct labor cost: ₦65,000
- Factory overhead expenses: ₦38,000
- Opening work-in-progress (1st Jan 2025): ₦22,000
- Closing work-in-progress (31st Dec 2025): ₦27,000

What is the total cost of production for the year?

Cevabı ve açıklamayı göster

Cevap: 243000

Cevap

The total cost of production for the year is ₦243,000.

To calculate the total cost of production, add direct materials consumed (₦145,000) and direct labor (₦65,000) to find the prime cost of ₦210,000. Add factory overhead expenses (₦38,000) to obtain the gross cost of production of ₦248,000. Finally, add opening work-in-progress (₦22,000) and subtract closing work-in-progress (₦27,000) to arrive at the net cost of production of ₦243,000.

Adım Adım Çözüm

1

Calculate Prime Cost

₦210,000

Prime cost is the sum of all direct manufacturing costs, including direct materials consumed and direct labor.

2

Add Factory Overheads

₦248,000

Factory overheads represent indirect factory costs incurred during the manufacturing process.

3

Adjust for Opening and Closing Work-in-Progress

₦243,000

Opening WIP is added because it represents partially finished goods from the previous period completed in the current period. Closing WIP is deducted because it represents partially finished goods that are not yet complete at year-end.

Anahtar Kavram

Valuation and Adjustment for Work-in-Progress (WIP) in Manufacturing Accounts

Soru 430Soru →

City A is located at longitude $25^\circ\text{W}$ and City B is located at longitude $65^\circ\text{E}$ . Calculate the difference in local solar time between the two cities in hours.

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

The time difference between City A and City B is 6 hours.

Because City A (

25^\circ\text{W}

) and City B (

65^\circ\text{E}

) lie in opposite hemispheres relative to the Greenwich Meridian (

0^\circ

), the total longitudinal difference between them is the sum of their absolute longitudes (

25^\circ + 65^\circ = 90^\circ

). Given that Earth rotates

15^\circ

of longitude every hour (

360^\circ / 24\text{ hours}

), dividing

90^\circ

by

15^\circ\text{/hour}

gives a total time difference of 6 hours.

Adım Adım Çözüm

1

Calculate the total angular distance between the two longitudes.

25^\circ\text{W} + 65^\circ\text{E} = 90^\circ

Because the two locations are in different hemispheres (West and East), their longitudinal values must be added to find the total separation across the Prime Meridian.

2

Convert longitudinal degrees into time difference.

90^\circ / 15^\circ\text{ per hour} = 6\text{ hours}

Earth completes one full rotation of

360^\circ

in 24 hours, which corresponds to an angular velocity of

15^\circ

per hour.

Anahtar Kavram

Calculating time difference from longitudinal distance across hemispheres.

Soru 431Soru →

A ship captain at sea observes local solar noon ( $12:00\text{ PM}$ ) on Monday. At that precise moment, a radio time signal from Greenwich ( $0^\circ$ ) indicates that Greenwich Mean Time (GMT) is $05:20\text{ PM}$ on Monday. What is the longitude of the ship in degrees West of the Prime Meridian?

Cevabı ve açıklamayı göster

Cevap: 80

Cevap

The longitude of the ship is

80^\circ\text{W}

(80 degrees West).

The time difference between local solar noon (

12:00\text{ PM}

) and GMT (

05:20\text{ PM}

) is 5 hours and 20 minutes, or

5\frac{1}{3}

hours. Multiplying

5.333\text{ hours}

by

15^\circ

per hour yields

80^\circ

. Since local solar time is behind GMT, the ship is situated

80^\circ

West of the Prime Meridian.

Adım Adım Çözüm

1

Calculate the time difference between local solar time and Greenwich Mean Time (GMT)

05:20\text{ PM} - 12:00\text{ PM} = 5\text{ hours and } 20\text{ minutes} = 5.333\text{ hours}

The distance in longitude is directly proportional to the difference between local time and GMT.

2

Convert the time difference into angular distance in degrees of longitude

5\frac{1}{3}\text{ hours} \times 15^\circ/\text{hour} = 80^\circ

Earth rotates

360^\circ

in 24 hours, which corresponds to

15^\circ

of longitude for every 1 hour of time difference.

3

Determine hemisphere direction based on whether local time is ahead or behind GMT

Western Hemisphere (

80^\circ\text{W}

)

Local time (

12:00\text{ PM}

) is earlier than GMT (

05:20\text{ PM}

), placing the ship west of the Prime Meridian.

Anahtar Kavram

Calculating longitude using local time and Greenwich Mean Time (GMT)

Soru 432Soru →

An open storage container is designed in the shape of a frustum of a right circular cone. The top radius of the container is $4\text{ m}$ , the bottom radius is $1\text{ m}$ , and its vertical height is $7\text{ m}$ . Taking $\pi = \frac{22}{7}$ , what is the volume of the container in cubic metres ( $\text{m}^3$ )?

Cevabı ve açıklamayı göster

Cevap: 154

Cevap

The volume of the container is

154\text{ m}^3

.

Using the volume formula for a frustum of a cone

V = \frac{1}{3}\pi h (R^2 + Rr + r^2)

with

R = 4\text{ m}

,

r = 1\text{ m}

,

h = 7\text{ m}

, and

\pi = \frac{22}{7}

yields

V = \frac{1}{3} \times \frac{22}{7} \times 7 \times (16 + 4 + 1) = 22 \times 7 = 154\text{ m}^3

.

Adım Adım Çözüm

1

Identify the given dimensions of the frustum of the cone.

Top radius

R = 4\text{ m}

, bottom radius

r = 1\text{ m}

, height

h = 7\text{ m}

, and

\pi = \frac{22}{7}

.

Establishing known parameter values for the frustum volume formula.

2

Apply the standard formula for the volume of a frustum of a circular cone.

V = \frac{1}{3}\pi h (R^2 + Rr + r^2)

This formula accounts for the non-uniform cross-sectional area of a truncated cone.

3

Calculate the sum of the squares and the product of the radii.

R^2 + Rr + r^2 = 4^2 + (4)(1) + 1^2 = 16 + 4 + 1 = 21

Evaluating the quadratic radius factor in the frustum equation.

4

Substitute all numeric values and simplify.

V = \frac{1}{3} \times \frac{22}{7} \times 7 \times 21 = 22 \times 7 = 154\text{ m}^3

Simplifying by cancelling

7

in the numerator and denominator, then dividing

21

by

3

yields

22 \times 7 = 154

.

Anahtar Kavram

Volume of a Frustum of a Cone

Soru 433Soru →

Given the matrices $A = \begin{pmatrix} 4 & x \\ -1 & 3 \end{pmatrix}$ and $B = \begin{pmatrix} 2 & 1 \\ 3 & 4 \end{pmatrix}$ , if the determinant of the product matrix $AB$ is $95$ , what is the value of $x$ ?

Cevabı ve açıklamayı göster

Cevap: 7

Cevap

The value of

x

is

7

.

Using the property

\det(AB) = \det(A)\det(B)

, we find

\det(B) = (2)(4) - (1)(3) = 5

and

\det(A) = (4)(3) - (x)(-1) = 12 + x

. Setting

5(12 + x) = 95

yields

60 + 5x = 95

, which solves to

x = 7

.

Adım Adım Çözüm

1

Calculate the determinant of matrix

B

\det(B) = (2)(4) - (1)(3) = 5

The determinant of a

2 \times 2

matrix

\begin{pmatrix} a & b \\ c & d \end{pmatrix}

is

ad - bc

.

2

Express the determinant of matrix

A

in terms of

x

\det(A) = (4)(3) - (x)(-1) = 12 + x

Applying the determinant formula to matrix

A

gives

12 - (-x) = 12 + x

.

3

Apply the determinant product property

\det(AB) = \det(A) \cdot \det(B)

5(12 + x) = 95

The determinant of the product of two square matrices is equal to the product of their individual determinants.

4

Solve the linear equation for

x

60 + 5x = 95 \implies 5x = 35 \implies x = 7

Subtract

60

from both sides and divide by

5

to isolate

x

.

Anahtar Kavram

Determinant of Matrix Product Property

Soru 434Soru →

A stone is thrown vertically downwards from the top of a $60\text{ m}$ high tower with an initial speed of $5\text{ m/s}$ . Taking acceleration due to gravity $g = 10\text{ m/s}^2$ , calculate the time taken, in seconds, for the stone to reach the ground.

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

The stone takes

3\text{ s}

to reach the ground.

Applying the equation

h = ut + \frac{1}{2}gt^2

with

h = 60\text{ m}

,

u = 5\text{ m/s}

, and

g = 10\text{ m/s}^2

yields

60 = 5t + 5t^2

. Dividing by

5

gives

t^2 + t - 12 = 0

, which factorizes into

(t + 4)(t - 3) = 0

. Rejecting

t = -4\text{ s}

leaves the correct time of

3\text{ s}

.

Adım Adım Çözüm

1

Set up the vertical motion equation

Using

h = ut + \frac{1}{2}gt^2

, substitute

h = 60\text{ m}

,

u = 5\text{ m/s}

, and

g = 10\text{ m/s}^2

.

This equation directly relates displacement, initial speed, acceleration, and time.

2

Form and solve the quadratic equation for time

60 = 5t + 5t^2 \Rightarrow t^2 + t - 12 = 0 \Rightarrow (t - 3)(t + 4) = 0

, giving

t = 3\text{ s}

.

Time must be positive, so the physically meaningful solution is

t = 3\text{ s}

.

Anahtar Kavram

Vertical motion under gravity with non-zero initial downward velocity

Soru 435Soru →

A copper rod has an initial length of $100\text{ cm}$ at $25^\circ\text{C}$ . When its temperature is increased to $75^\circ\text{C}$ , its length becomes $100.085\text{ cm}$ . What is the area expansivity of copper in $10^{-5}\text{ K}^{-1}$ ?

Cevabı ve açıklamayı göster

Cevap: 3.4

Cevap

The area expansivity of copper is

3.4 \times 10^{-5}\text{ K}^{-1}

, giving a numerical value of

3.4

in units of

10^{-5}\text{ K}^{-1}

.

Linear expansivity is obtained as

\alpha = \frac{\Delta L}{L_0 \Delta T} = \frac{0.085\text{ cm}}{100\text{ cm} \times 50\text{ K}} = 1.7 \times 10^{-5}\text{ K}^{-1}

. Since area expansivity

\beta

is related to linear expansivity by

\beta = 2\alpha

, multiplying by

2

yields

3.4 \times 10^{-5}\text{ K}^{-1}

, or

3.4

in units of

10^{-5}\text{ K}^{-1}

.

Adım Adım Çözüm

1

Calculate the temperature change

\Delta T = 75^\circ\text{C} - 25^\circ\text{C} = 50\text{ K}

Thermal expansion depends directly on the change in temperature.

2

Find the change in length

\Delta L = 100.085\text{ cm} - 100\text{ cm} = 0.085\text{ cm}

The expansion is the difference between the final length and original length.

3

Determine linear expansivity (

\alpha

)

\alpha = \frac{\Delta L}{L_0 \Delta T} = \frac{0.085}{100 \times 50} = 1.7 \times 10^{-5}\text{ K}^{-1}

Linear expansivity defines fractional change in length per degree temperature change.

4

Compute area expansivity (

\beta

)

\beta = 2\alpha = 2 \times (1.7 \times 10^{-5}) = 3.4 \times 10^{-5}\text{ K}^{-1}

Area (superficial) expansivity is equal to twice the linear expansivity.

Anahtar Kavram

Relationship between linear expansivity (

\alpha

) and area expansivity (

\beta = 2\alpha

).

Soru 436Soru →

A galvanometer with an internal resistance of $5\text{ }\Omega$ produces a full-scale deflection when a current of $10\text{ mA}$ flows through it. Calculate the shunt resistance, in ohms, required to convert this galvanometer into an ammeter capable of measuring currents up to $50\text{ mA}$ .

Cevabı ve açıklamayı göster

Cevap: 1.25

Cevap

The required shunt resistance is

1.25\text{ }\Omega

.

To extend the range of a galvanometer, a shunt resistor

S

is placed in parallel with it. The potential difference across the galvanometer equals the potential difference across the shunt:

I_g R_g = (I - I_g) S

. Substituting the given values

R_g = 5\text{ }\Omega

,

I_g = 10\text{ mA}

, and maximum current

I = 50\text{ mA}

yields

10\text{ mA} \times 5\text{ }\Omega = (50\text{ mA} - 10\text{ mA}) \times S

, solving to

S = \frac{50}{40} = 1.25\text{ }\Omega

.

Adım Adım Çözüm

1

Find the current passing through the parallel shunt resistor

I_s = 40\text{ mA}

By Kirchhoff's current law, the total maximum current divides into the galvanometer current and the shunt current (

I = I_g + I_s

).

2

Calculate the required shunt resistance

S

S = 1.25\text{ }\Omega

Because the galvanometer and shunt resistor are connected in parallel, the potential difference across both branches is equal (

I_g R_g = I_s S

).

Anahtar Kavram

Conversion of a galvanometer to an ammeter using a low-resistance shunt in parallel

Soru 437Soru →

A coin lies at the bottom of a vessel filled with a liquid to a depth of $14.0\text{ cm}$ . If the refractive index of the liquid relative to air is $1.40$ , calculate the apparent upward displacement of the coin in centimeters when viewed vertically from directly above.

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

The apparent upward displacement of the coin is

4.0\text{ cm}

.

Refraction at the liquid-air boundary makes an object at real depth

h = 14.0\text{ cm}

appear at an apparent depth

h' = \frac{h}{n} = \frac{14.0}{1.40} = 10.0\text{ cm}

. The apparent upward displacement is the difference between real depth and apparent depth:

d = 14.0\text{ cm} - 10.0\text{ cm} = 4.0\text{ cm}

.

Adım Adım Çözüm

1

Identify the given values and formula for refractive index in terms of depth.

Real depth

h = 14.0\text{ cm}

, refractive index

n = 1.40

. Formula:

n = \frac{\text{Real depth}}{\text{Apparent depth}} = \frac{h}{h'}

.

Light rays bending away from the normal upon leaving the denser liquid cause the coin to appear closer to the surface.

2

Calculate the apparent depth (

h'

).

h' = \frac{14.0\text{ cm}}{1.40} = 10.0\text{ cm}

.

Rearranging the refractive index formula gives

h' = \frac{h}{n}

.

3

Calculate the apparent upward displacement (

d

).

d = h - h' = 14.0\text{ cm} - 10.0\text{ cm} = 4.0\text{ cm}

.

The displacement is the distance between the actual position at the bottom and the virtual image position.

Anahtar Kavram

Real depth, apparent depth, and apparent displacement

Soru 438Soru →

A copper container with an initial volume of $800 \text{ cm}^3$ at $25^\circ\text{C}$ is completely filled with oil. The real cubic expansivity of the oil is $6.5 \times 10^{-4} \text{ K}^{-1}$ and the linear expansivity of copper is $1.5 \times 10^{-5} \text{ K}^{-1}$ . What volume of oil (in $\text{cm}^3$ ) will overflow when the temperature of the system is raised to $75^\circ\text{C}$ ?

Cevabı ve açıklamayı göster

Cevap: 24.2

Cevap

The volume of oil that overflows is

24.2 \text{ cm}^3

.

When a container completely filled with liquid is heated, both liquid and container expand. The overflow volume equals the apparent volume expansion of the liquid

\Delta V_a = V_0 \gamma_a \Delta T

. The apparent cubic expansivity

\gamma_a

is obtained by subtracting the container's volume expansivity (

\gamma_v = 3\alpha = 4.5 \times 10^{-5} \text{ K}^{-1}

) from the liquid's real cubic expansivity (

\gamma_r = 6.5 \times 10^{-4} \text{ K}^{-1}

), giving

\gamma_a = 6.05 \times 10^{-4} \text{ K}^{-1}

. Multiplying by

V_0 = 800 \text{ cm}^3

and

\Delta T = 50 \text{ K}

gives

24.2 \text{ cm}^3

.

Adım Adım Çözüm

1

Calculate the cubic expansivity of the copper container

\gamma_v = 3 \alpha = 3 \times (1.5 \times 10^{-5} \text{ K}^{-1}) = 4.5 \times 10^{-5} \text{ K}^{-1} = 0.45 \times 10^{-4} \text{ K}^{-1}

The volumetric expansion coefficient of a solid container is three times its linear expansivity.

2

Determine the apparent cubic expansivity of the oil

\gamma_a = \gamma_r - \gamma_v = 6.5 \times 10^{-4} \text{ K}^{-1} - 0.45 \times 10^{-4} \text{ K}^{-1} = 6.05 \times 10^{-4} \text{ K}^{-1}

The apparent expansion of a liquid accounts for the concurrent thermal expansion of the containing vessel.

3

Compute the temperature increase

\Delta T = 75^\circ\text{C} - 25^\circ\text{C} = 50 \text{ K}

Temperature change is the final temperature minus the initial temperature.

4

Calculate the volume of oil that overflows

\Delta V_a = V_0 \gamma_a \Delta T = 800 \text{ cm}^3 \times (6.05 \times 10^{-4} \text{ K}^{-1}) \times 50 \text{ K} = 24.2 \text{ cm}^3

The overflow volume is equal to the apparent increase in volume of the liquid.

Anahtar Kavram

Apparent and Real Expansion of Liquids

Soru 439Soru →

In a agricultural survey of 120 farmers in a community, 65 grow maize, 50 grow yam, and 42 grow cassava. Furthermore, 24 grow both maize and yam, 18 grow both maize and cassava, and 15 grow both yam and cassava. If 8 farmers grow none of these three crops, find the number of farmers who grow all three crops.

Cevabı ve açıklamayı göster

Cevap: 12

Cevap

12 farmers grow all three crops.

Using the 3-set inclusion-exclusion principle, the total number of farmers growing at least one crop is

120 - 8 = 112

. Expanding

|M \cup Y \cup C| = |M| + |Y| + |C| - (|M \cap Y| + |M \cap C| + |Y \cap C|) + |M \cap Y \cap C|

gives

112 = 65 + 50 + 42 - 24 - 18 - 15 + x

. Simplifying yields

112 = 100 + x

, which gives

x = 12

.

Adım Adım Çözüm

1

Determine the cardinality of the union of all three sets.

|M \cup Y \cup C| = 112

Subtract the farmers who grow none of the crops from the universal set size (

120 - 8 = 112

).

2

Set up the Inclusion-Exclusion equation for three sets.

112 = 65 + 50 + 42 - (24 + 18 + 15) + x

Inclusion-exclusion states that

|A \cup B \cup C| = n(A) + n(B) + n(C) - n(A \cap B) - n(A \cap C) - n(B \cap C) + n(A \cap B \cap C)

.

3

Solve for the unknown value

x

representing the intersection of all three sets.

x = 12

Simplifying gives

112 = 100 + x

, which leads directly to

x = 12

.

Anahtar Kavram

Principle of Inclusion-Exclusion for 3 Sets

Soru 440Soru →

A ray of light traveling in air strikes the flat surface of a transparent glass slab at an angle of incidence of $60^\circ$ . If the refractive index of the glass slab relative to air is $\sqrt{3}$ , what is the angle of refraction inside the glass slab in degrees?

Cevabı ve açıklamayı göster

Cevap: 30

Cevap

The angle of refraction inside the glass slab is

30^\circ

.

According to Snell's Law,

n = \frac{\sin i}{\sin r}

. Substituting

n = \sqrt{3}

and

i = 60^\circ

gives

\sqrt{3} = \frac{\sin 60^\circ}{\sin r}

. Since

\sin 60^\circ = \frac{\sqrt{3}}{2}

, rearranging gives

\sin r = \frac{\sqrt{3}/2}{\sqrt{3}} = 0.5

. Taking the inverse sine of

0.5

gives

r = 30^\circ

.

Adım Adım Çözüm

1

Identify Snell's law formula relating the angle of incidence and angle of refraction.

n = \frac{\sin i}{\sin r}

Snell's law describes how light bends when crossing the boundary between two optical media.

2

Substitute the given values

i = 60^\circ

and

n = \sqrt{3}

into the equation.

\sqrt{3} = \frac{\sin 60^\circ}{\sin r}

Plugging in the known parameters allows us to isolate the unknown sine of the angle of refraction.

3

Substitute

\sin 60^\circ = \frac{\sqrt{3}}{2}

and rearrange for

\sin r

.

\sin r = \frac{\sqrt{3}/2}{\sqrt{3}} = 0.5

Canceling

\sqrt{3}

from both sides yields a simple numerical value for

\sin r

.

4

Take the inverse sine of

0.5

to find

r

.

r = \arcsin(0.5) = 30^\circ

The angle whose sine is

0.5

is

30^\circ

.

Anahtar Kavram

Snell's Law of Refraction

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