Three cubes with sides in the ration 3 : 4 : 5 are melted to form a single cube whose diagonal is 12? 3 cm. The sides of the cubes are?

Let h be the height,
Volume of the cone = V
and curved surface area of the cone = c
? 1/3 ?r²h = V, ?rl = c
i.e., ?r?r² + h² = c
?²r²(r² + h²) = c²
Consider 3?Vh³ - c²h² + 9V²
= 3? x 1/3?r²h x h³ - ?²r²(r² + h²)h + 9 x 1/9 x ?²r⁴ h²
= ?²r²h⁴ - ?²r⁴h² - ?²r²h⁴ + ?²r⁴ = 0

Height of the cylinder = 10 x Diameter of each ball
= 10 x 10 = 100 cm
? Required empty space
= ? (5)² (100) - 4/3 ? (5)³ x 10
= ? (5)² x 10 [10 - 20/3]
= ? (5)² x 10 [30 - 20/3] = 2500/3 ? cm³

Given that, Height = 1 m
Diameter = 140 cm or 140/100 m = 1.40 m
? r = 1.40/2 = 0.7 m

Now,
Required sheet = Total surface area = 2?r( h + r)
= (22/7) x 1.4 x (1 + 0.7) = 7.48 sq m
? Sheet required = 7.48 sq m

Given that, the height and radius of a right circular ,etal cone (solid) are 8 cm and 2 cm, respectively.
i.e., h = 8 cm and r = 2 cm
Let the radius of the sphere is R
Then, by condition,
1/3 ? r² h = 4/3 ? R³ h
? 4 x 8 = 4 R³
? R³ = (2)³ = ? R = 2
? Radius odf the sphere = 2 cm

Let the breadth and height of room be m and h m, respectively.
Then, according to the question, l x b = n x Area occupied by one patient
? 14 x b = 56 x 22
? b = (56 x 22)/14 = 8.8 m³
? h x Area of occupied by one patient = 8.8
h = 8.8/2.2
h = 4 m

External radius = 2.5 cm,
length = 100 cm
? External volume = [? x (2.25)² x 100] cm²
internal radius = 1.5 cm
? internal volume = [? x (1.5)² x 100]cm³
Volume of metal
= [? x (2.5)² x 100 - ? x (1.5)² x 100] cm³
= ? x 100 x [(2.5)² - (1.5)²] cm³
= (22/7 x 100 x 4 x 1 x 21/1000) kg
= 26.4 kg.

Curved surface area of cylinder = 2?rh cm²
Total surface area of cylinder = 2?r(h + r) cm²
Now, According to the question
2?rh/2?r(h + r) = 1/2
? h/(h + r) = 1/2
? 2h = h + r
? Now, Total surface area = 616 cm²
? 2? (h + r) = 616
? 2?r (r + r) = 616 (? h = r)
? 2?r(2r) = 616
? 4?r² = 616
? r = ?style="text-decoration:overline">616/4?
= ?616 x 7/4 x 22
= ?7 x 7 =
? r = h = 7 cm
? Volume = ? r² h
= (22/7) x (7 x 7) x 7 = 1078 cm³

If 10 circular plates each of thickness 3 cm, each are placed one above the other, then it forms a cylinder with height (3 x 10 = 30 cm) and a hemisphere of radius 6 cm is placed on the top just to cover the cylinder that means its radius is 6 cm also
? Radius of hemisphere (R) = 6 cm
Radius of cylinder (r) = 6 cm
and height of cylinder (h) = 30 cm
? Volume of the solid = Volume of cylinder + Volume of hemisphere
= ?r²h + (2/3)?R³
= ?(6)² x 30 + (2/3)?(6)³
= ? x 36 x 30 + (2/3)? x 216
= 1080? + 2? x 72 = 1080? + 144?
= 1224? cm³
Which is the required volume of solid.

Given series is a consecutive prime number series. Therefore, the next term will be 23.

Series Pattern
+6, +9, +12, +15....
? Missing term = 6 + 6 = 12