the slant height of a frustum of a cone is 4 cm and the perimeter of its circular ends are 18 cm and 6 cm. find the curved surface area of the frustum and the volume of frustum in terms of given measurement. Are the curved surface area and the volume of the frustum numerically equal? explain it truthfully.
Answers
Answer:
Given,
Perimeter of top circle having radius (r) is
2πr=6 cm
Perimeter of bottom circle having radius (R) is
=18 cm
Slant height of frustum of cone l=4 cm
Hence,
Radius of top circle r=
2π
6
=
π
3
cm
Radius of bottom circle R=
2π
18
=
π
9
cm
Hight of frustum of cone is
h=
l
2
−(R−r)
2
h=
4
2
−(
π
9
−
π
3
)
2
cm
h=
16−(
π
6
)
2
cm
h=
16−
π
2
36
cm
h=
12.35244
cm
h=3.5146 cm
curved surface area of frustum of cone =π(R+r)l
=((R+r)l)π
=((
π
9
+
π
3
)×4)π cm
2
=((
π
12
)×4)π cm
2
=(
π
48
)π cm
2
=48 cm
2
Total surface area of the frustum of cone=48 cm
2
Given,
Slant height of frustum of cone (l) = 4 cm
Let ratio of the top and bottom circles be r1 and r2
And given perimeters of its circular ends as 18 cm and 6 cm
⟹ 2πr1 = 18 cm; 2πr2 = 6 cm
⟹ πr1= 9 cm and πr2 = 3 cm
We know that,
Curved surface area of frustum of a cone = π(r1 + r2)l
= (πr1+πr2)l = (9 + 3) × 4
= (12) × 4 = 48 cm2
Therefore, the curved surface area of the frustum = 48 cm2.