case study
CONE
Christmas tree-the evergreen fir tree has traditionally been used to celebrate Winter festivals for thousands of years. It is one of the best examples for conical Shape
Rita and Sam are siblings. Both of them decorated their house for Christmas celebration. Their father gifted them a Christmas tree.
a) Both the trees are with same height but the base radius of Rita’s tree is twice that of Sam’s. What is the ratio of their volumes?
b) Sam placed his tree with a support of cylindrical iron rod, which is fixed to the base of the tree, the height of the rod is 35cm and volume is 5390cm3. Find the radius of the rod.
Answers
Answer:
a)1:2
b)Height of cylindrical bucket(h
1
)=32 cm
Radius of the base of the bucket (r
1
)=18 cm
∴Volume of the sand in the cylindrical bucket=πr
1
2
h
1
Height of conical heap (h
2
)=24 cm
let the radius of the conical heap=r
2
∴Volume of the sand in conical heap=
3
1
πr
2
2
h
2
According to the question
The volume of the sand in the cylindrical bucket=Volume of the sand in the conical shape
πr
1
2
h
1
=
3
1
πr
2
2
h
2
⇒π×(18)
2
×32=
3
1
π×r
2
2
×24
⇒r
2
2
=
24
3×18
2
×32
⇒r
2
2
=18
2
×4
⇒r
2
=18×2=36cm
Slant height of heap=
r
2
2
+h
2
2
⇒
36
2
+24
2
⇒
1296+576
⇒
1872
⇒
144×13
⇒12
13
cm.
a) The ratio of their volume is 1x : 2x (where x is the unknown volume)
b)Given that :
Volume of the cylinder = 5390 cm³
Height of the cylinder = 35 cm
Formula used:
The volume of the cylinder = πr²h
Where,
r = radius of the cylinder
h = height of the cylinder
V = volume of the cylinder
D = diameter of the cylinder
Calculation:
Volume = πr²h
⇒ 5390 = (22/7) × r² × 35
⇒ (5390 × 7)/(22 × 35) = r²
⇒ 49 = r²
⇒ r = 7 cm
therefore the radius is 7cm.