A journey around a right angle triangle, pocket money shared among five classmates, the surface area and volume of a sphere, and distributing Rs. 4,320 in parts

Understanding the topic

This question combines geometry, ratios and mensuration. Draw the relationships before substituting values. Part (b) is grammatically awkward, so its stated relationships must be translated explicitly. The coherent interpretation is that Shahbaz receives an amount equal to the sum of Nasir’s and Ali’s amounts.

(a) A man travels over the path of a right-angle triangle having base and hypotenuse 4 and 5 kilometers, respectively. After a complete round he continues in the same direction for 6 km and then turns at 90 degree and continues for another 8 km. How long he has travelled and how far he is from his starting point?

The right-angled triangle has base 4 km and hypotenuse 5 km. Let its third side be p.

By the Pythagorean theorem:

p² + 4² = 5²

p² = 25 - 16 = 9

p = 3 km

One complete round of the triangle therefore covers:

3 + 4 + 5 = 12 km

He then travels another 6 km, turns through 90°, and travels 8 km:

Total distance = 12 + 6 + 8 = 26 km

After completing the triangle, he has returned to his starting point. His last two movements are perpendicular displacements of 6 km and 8 km. His straight-line distance from the start is:

Displacement = √(6² + 8²)

= √100 = 10 km

Answer: He travels 26 km in total and finishes 10 km from his starting point.

(b) Hassan, Ali, Akbar, Nasir and Shahbaz are classmates having different pocket money. Hassan's pocket money is one third as much as of Ali and Ali has five times as much as Akbar. Akbar has thrice as much as Nasir and Shahbaz gets equal to Nasir and that of Ali. If they get Rs. 8000 then find the pocket money of each.

The last relationship is awkwardly printed. The coherent intended reading is:

Shahbaz = Nasir + Ali

Let Nasir receive x.

• Nasir = x
• Akbar = 3x
• Ali = 5(3x) = 15x
• Hassan = (1/3)(15x) = 5x
• Shahbaz = x + 15x = 16x

Their total is Rs. 8,000:

x + 3x + 15x + 5x + 16x = 8000

40x = 8000

x = 200

Therefore:

• Nasir = Rs. 200
• Akbar = Rs. 600
• Hassan = Rs. 1,000
• Ali = Rs. 3,000
• Shahbaz = Rs. 3,200

Check: 200 + 600 + 1,000 + 3,000 + 3,200 = 8,000.

(c) What will be the surface area and volume of a sphere if it has a radius of 7 m?

Given radius:

r = 7 m

Using π = 22/7, the surface area is:

Surface area = 4πr²

= 4 × 22/7 × 7²

= 616 m²

The volume is:

Volume = (4/3)πr³

= 4/3 × 22/7 × 7³

= 4312/3 m³

= 1,437 1/3 m³ ≈ 1,437.33 m³

Answer: Surface area = 616 m²; volume = 4,312/3 m³, approximately 1,437.33 m³.

(d) Distribute Rs. 4320 among Zain, Aslam and Ashraf in such a way that if zain gets 2 parts then Aslam gets three parts, whereas Ashraf gets seven parts.

The required ratio is:

Zain : Aslam : Ashraf = 2 : 3 : 7

Total parts:

2 + 3 + 7 = 12

Value of one part:

Rs. 4,320 ÷ 12 = Rs. 360

Therefore:

• Zain = 2 × 360 = Rs. 720
• Aslam = 3 × 360 = Rs. 1,080
• Ashraf = 7 × 360 = Rs. 2,520

Check: 720 + 1,080 + 2,520 = 4,320.