The radius and height of a cylinder are in the ratio 5:7 and its volume is 550 cm3. Find its radius.

Let radius (r) = 5x and height (h) = 7x.
Now. Volume = 550

rightwards double arrow space 22 over 7 cross times space 5 straight x space cross times space 5 straight x space cross times space 7 straight x space equals space 550
rightwards double arrow space space space straight x cubed equals fraction numerator 550 cross times 7 over denominator 22 cross times 25 cross times 7 end fraction
x = 1
∴ The required radius = 5x = 5 x 1 = 5cm.

1498 Views

A wooden article was made by scooping out a hemisphere from each end of a solid cylinder. If the height of the cylinder is 20 cm and radius of the base is 3.5 cm, find the total surface area of the article.


Let r cm be the radius, h cm be the height of the cylinder thenr = 3.

Let r cm be the radius, h cm be the height of the cylinder then
r = 3.5 cm
and    h = 20 cm.
Let r1 be the radius of the hemisphere, then
r1 = 3.5 cm
Now,
Total surface area of the area of the article
= curved surface area of the cylinder
+ 2 (curved surface area of hemisphere)
= 2πrh + 2 (2 πr12)
= 2 πrh + 4 πr2    [∵ r = r1]
= 2 π r(h + 2r)

equals space 2 space straight x space 22 over 7 straight x space 3.5 space left parenthesis 20 space plus 2 space straight x space 3.5 right parenthesis
equals space 2 space straight x space 22 over 7 space straight x space 3.5 space straight x space 27 space equals space 594 space cm squared

1148 Views

A toy is in the form of a cone mounted on a hemisphere of common base radius 7cm. The total height of the toy is 31 cm. Find the total surface area of the toy. 

Let r cm be the radius and h cm the height of the cone, then r = 7 cm, and h = 24 cm. Let l cm be the slant height of am cone, then

l italic space italic equals italic space square root of straight r squared plus straight h squared end root
space space equals space square root of left parenthesis 7 right parenthesis squared plus left parenthesis 24 right parenthesis squared end root
space space equals space square root of 49 plus 576 end root space equals space square root of 625 equals 25 space cm

Let r cm be the radius and h cm the height of the cone, then r = 7 cm

Let r1 cm be the radius of hemisphere then
r1 = 7 cm
Now, Total surface area of toy
= C.S.A of cone + C.S.A. of hemisphere

equals space πrl space plus space 2 πr subscript 1 squared equals πrl plus 2 πl squared space left square bracket because space straight r subscript 1 equals space rl space right square bracket
equals space 22 over 7 straight x 7 space left parenthesis 25 space plus space 2 space straight x space 7 right parenthesis
equals space 22 space left parenthesis 25 space plus space 14 right parenthesis
equals space left parenthesis 22 space straight x space 39 right parenthesis space cm squared space equals space 858 space cm squared



Tips: -

open parentheses bold Use bold space bold pi bold space bold equals bold space bold 22 over bold 7 close parentheses
384 Views

The decorative block shown in fig. is made of two solids - a cube and a hemisphere. The base of the block is a cube with edge 5 cm, and the hemisphere fixed on the top has a diameter of 4.2 cm. Find the total surface area of the block.



Let the side (edge) of the cube be a cm, then a= 5 cm.Let radius of t

Let the side (edge) of the cube be a cm, then a
= 5 cm.
Let radius of the hemisphere be r, then
r = 2.1 cm
Now,
The total surface area of the block
= T.S.A of cube - base area of hemisphere + C.S.A of hemisphere
= 6 (side)2 – straight pir2 + 2rstraight pi2
= 150 – straight pir2 + 2 straight pir2
= 150 + straight pi2

equals space open parentheses 150 plus 22 over 7 straight X space 2.1 space straight X space 2.1 close parentheses space cm squared
equals space left parenthesis 150 space plus space 13.86 right parenthesis space cm squared
equals space 163.86 space cm squared.

Tips: -

open parentheses bold Use bold space bold pi bold space bold equals bold space bold 22 over bold 7 close parentheses
673 Views

The interior of building is in the form of a right circular cylinder of radius 7 m and height 6 m, surmounted by a right circular cone of same radius and of vertical angle 60°. Find the cost of painting the building from inside at the rate of Rs. 30/m2

We have, r1 = Radius of the base of the cylinder = 7m
r2 = Radius of the base of the cone = 7 m
h1 = Height of the cylinder = 6m


We have, r1 = Radius of the base of the cylinder = 7mr2 = Radius of

In right triangle EOC, we have

sin space 30 degree space equals space OC over CE
rightwards double arrow space space space 1 half equals 7 over CE
rightwards double arrow space space CE space equals space 14 space straight m

Thus, slant height (CE) = 14 m
Now,Surface area of building = Curved surface area of cylinder + curved surface area of cone

equals space 2 πr subscript 1 straight h subscript 1 plus πr subscript 1 straight l equals πr subscript 1 left parenthesis 2 straight h subscript 1 plus l right parenthesis
equals space 22 over 7 space straight x space 7 space left parenthesis 2 space straight x space 6 space plus 14 right parenthesis equals open parentheses 22 over 7 space straight x space 7 space straight x space 26 close parentheses space straight m squared equals 572 space straight m squared

Cost of painting the building from inside at the rate of Rs. 30/m2
= Rs. (572 x 30) = Rs. 17160.  



3277 Views

A solid is composed of a cylinder with hemispherical ends. If the whole length of the solid is 100 cm and the diameter of the hemispherical ends is 28 cm, find the cost of polishing the surface of the solid at the rate of 5 paise per sq. cm.

We have,
r = radius of the cylinder
= radius of hemispherical ends = 14 cm
h = height of cylinder = 72 cm
∴ Total surface area = curved surface area of the cylinder + surface areas of hemispherical ends
= (2straight pirh +2 x 2straight pir2) cm2
= 2straight pir (h + 2r) cm2


We have,r = radius of the cylinder= radius of hemispherical ends = 14
equals space 2 space straight x space 22 over 7 straight x space 14 space left parenthesis 72 space plus space 2 space straight x space 14 right parenthesis space cm squared
equals space open parentheses 2 space straight x space 22 over 7 straight x space 14 space straight x space 100 close parentheses space cm squared
equals space 8800 space cm squared
Now, cost of polishing the surface of the solid at the rate of 5 paise per sq. m

equals open parentheses 8800 space straight x space 5 over 100 close parentheses space equals Rs. space 440



1223 Views