Today our topic of discussion is Rectangular solid.
Rectangular solid
Solids
Rectangular solid
The region surrounded by three pairs of parallel rectangular planes or surfaces is known as rectangular solid.
Let, ABCDEFGH is a rectangular solid, whose length AB = a , and breadth
BC = b and height AH = c
- Determining the diagonal: AF is the diagonal of the rectangular solid ABCDEFGH.
In ΔABC , BC ⊥ AB and AC is the hypotenuse.
∴A * C² = A * B² + B * C² = a² + b²
Again, in triangle ABC . FC perp AC and AF is hypotenuse.
∴A * F² = A * C² + C * F²= a ²+ b ² + c ²
∴ AF = √(a² + b² + c²)
∴the diagonal of the rectangular solid = sqrt(a² + b² + c²)
- Determination of area of the whole surface: There are 6 surfaces of the rectangular solid where the opposite surfaces are of equal dimensions.
Area of the whole surface of the rectangular solid
=2 (area of the surface of ABCD + area of the surface of ABGH + area of the surface ofBCFG)
= 2(ABAD + ABAH + BCBG)
= 2(ab + ac + bc) = 2(ab + bc + ca)
- Volume of the rectangular solid = length x width x height = abc
Example 28. The length, width and height of a rectangular solid are 25 cm, 20 cm and 15 cm respectively. Determine its area of the whole surface, volume and the length of the diagonal.
Solution: Let, the length of the rectangular solid is a = 25 cm, width b = 20 cm and height c = 15 cm
∴ Area of the whole surface of the rectangular solid = 2(ab + bc + ca) = 2(25 * 20 + 20 * 15 + 15 * 25) = 2350 square cm and Volume = abc = 25 x 20 x 15 = 7500 cube cm and length of its diagonal = √(a² + b² + c ²)
= √(25² + 20² + 15²) = √(624 + 400 + 225) = √(1250) = 35.363 cm (approx.)
The required area of the whole surface is 2350 square cm, volume 7500 cubic cm and length of its diagonal 35.363 cm (approx.).
Work: Determine the volume, area of the whole surface and the length of the diagonal of your mathematics book after measuring its length, width and height.
Cube
If the length, width and height of a rectangular solid are equal, it is called a cube. Let, ABCDEFGH is a cube.
Its length = width = height = a units
- The length of diagonal of the cube = sqrt(a² + a² + a²) = √(3a ²) = √(3) * a
- The area of the whole surface of the cube = 2(aa + aa + aa) = 2(a² + a² + a²) = 6a²
- The volume of the cube = aaa = a³
Example 29. The area of the whole surface of a cube is 96 square metre. Determine the length of its diagonal.
Solution: Let, the sides of the cube is a.
∴ The area of its whole surface = 6a² and length of its diagonal = √3a
As per question, 6a296 or, a2 = 16..a=4
∴ The length of the diagonal = √3.4 6.928 metre (approx.).
The required length of the diagonal 6.928 metre (approx.).
Work: The sides of 3 metal cubes are 3 cm, 4 cm and 5 cm respectively. A new cube is formed by melting these 3 cubes. Determine the area of the whole surface and the length of the diagonal of the new cube.
Cylinder
The solid formed by a complete revolution of any rectangle about one of its sides as axis is called a cylinder or a right circular cylinder. The two ends of a right circular cylinder are circular surfaces. The curved face is called curved surface and the total plane is called whole surface. The side of the rectangle which is parallel to the axis and revolves about the axis is called the generator line of the cylinder.
The figure above is a right circular cylinder, whose radius is r and height h.
- Area of the base = πr²
- Area of the curved surface = perimeter of the base x height=2rh
- Area of the whole surface = (πr² + 2πrh + πr²) = 2πr(r + h)
- Volume Area of the base x height = r²h
Example 30. If the height of a right circular cylinder is 10 cm and radius of the base is 7 cm, determine its volume and the area of the whole surface.
Solution: Let, the height of the right circular cylinder is h = 10 cm and radius
of the base is r.
∴ Its volume = r²h
3.1416 x 72 x 10 1539.38 cube cm (approx.)
and the area of the whole surface = 2πr(r + h)
= 2 * 3.1416 * 7(7 + 10) = 747.7 square metre (approx.)
Work: Make a right circular cylinder using a rectangular paper. Determine the area of its whole surface and the volume.
Example 31. The outer measurements of a box with its top are 10 cm, 9 cm and 7 cm respectively and the area of the whole inner surface is 262 square metre and the thickness of its wall is uniform on all sides.
1) Find the volume of the box.
2) Find the thickness of its wall.
3) If the length of a diagonal of a rhombus having sides equal to the largest length of the box is 16 cm, then find its area.
Solution:
1) The outer measurements of the box with top are 10 cm, 9 cm and 7 cm. .. The outer volume of the box = 10 * 9 * 7 = 630 cube cm
2) Let, the thickness of the box is z. The outer measurements of the box with top are 10 cm, 9 cm 7 cm.
∴The inside measurements of the box are respectively a = (10 – 2x) (9 – 2x) and c = (7 – 2x) cm. b =
The area of the whole surface of the inner side of the box = 2(ab + bc + ca)
As per question, 2(ab + bc + ca) = 262
o*r_{1} (10 – 2x)(9 – 2x) + (9 – 2x)(7 – 2x) + (7 – 2x)(10 – 2x) = 131
or, 90 – 38x + 4x² + 63 – 32x + 4x² + 70 – 34x + 4x² – 131 = 0
or, 12x² – 104x + 92 = 0
or, 3x² – 26x + 23 = 0
3x² – 3x – 23x + 23 = 0
3x(x – 1) – 23(x – 1) = 0
or, (x – 1)(3x – 23) = 0
or ,x=1
or, 3x – 23 = 0
or,x= 23/3 =7.67( approx .)
But the thickness of a box cannot be greater than or equal to any of the sides.
The required thickness of the box is 1 cm
3) Let, length of each side of ABCD is 10 cm and the diagonals intersect each other at the point 0.
We know, diagonals of rhombus bisect each other at right angle.
OA = OC, OB = OD
In the right-angled triangle AAOB, hypotenuse
AB = 10
Here, A * B² = 10² = 100 = 36 + 64
= 6² + 8² = O * B² + O * A² [according to the figure]
∴ OB = 6 OA = 8
∴ diagonal AC = 2 * 8 = 16 cm and diagonal BD = 2 * 6 = 12 cm
∴ area of the rhombus ABCD= 1/2 × AC x BD =1/2 x 16 x 1296 square
Example 32. If the length of diagonal of the surface of a cube is 8sqrt(2) cm, determine the length of its diagonal and volume.
Solution:
Let, the side of the cube is a.
∴ The length of diagonal of the surface = √2a . Length of diagonal = √2a and volume = a³
As per question, √2a = 8√2 or, a = 8
∴The length of the cube’s diagonal = √3 * 8 = 13.856 cm (approx.) and volume = 8³ = 512 cube cm.
The required length of the diagonal is 13.856 cm (approx.) and volume 512 cubic cm.
Example 33. The length of a rectangle is 12 cm and width 5 cm. Determine the area of its whole surface and the volume of the solid that is formed by revolving the rectangle around its greater side.
Solution: Given that, the length of a rectangle is 12 cm. and width 5 cm. If it is revolved around the greater side, a right circular cylinder is formed with height h = 12 cm and radius of the base r = 5cm
The whole surface of the produced solid = 2πr(r+h)
= 2 * 3.1416 * 5(5 + 12) = 534.071 square cm (approx.) and volume = r²h
= 3.1416 * 5² * 12 = 942.48 cube cm (approx.)
The required area of whole surface is 534.071 square cm (approx.) and volume 942.48 cube cm (approx.)
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