# FIND RADIUS OF CONE WHEN GIVEN SURFACE AREA

To find lateral surface area and total surface area of cone, we use the formulas given below.

Lateral surface area =  πrl

Total surface area =  πrl +  πr2

πr(l + r)

Here r = radius, l = slant height

Find length of radius of the following cones given below.

Problem 1 :

Solution :

Total surface area πr(l + r)

πr(l + r) = 36π

radius(r) = x and slant height (l) = 5 cm

xπ(5 + x) = 36π

5x + x2 = 36

x2 + 5x - 36 = 0

(x + 9) (x - 4) = 0

x = -9 and x = 4

So, the required radius is 4 cm.

Problem 2 :

Solution :

Total surface area = πr(l + r)

πr(l + r) = 9600π

radius(r) = x and slant height (l) = 1 m = 100 cm

xπ(100 + x) = 9600π

100x + x2 = 9600

x2 + 100x  - 9600 = 0

(x + 160)(x - 60) = 0

x = -160 and x = 60

So, the radius is 60 cm.

Problem 3 :

The cylinder and cone has the same surface area. Express L in terms of x.

Solution :

Surface area of cylinder = 2πr(h + r)

Surface area of cone = πr(l + r)

height of cylinder = 3x and slant height of cone = l

2πx(3x + x) = πx(l + x)

2x(4x) = x(l + x)

8x2 = lx + x2

Subtracting x2 on both sides.

7x2 = lx

l = 7x

Problem 4 :

A cone and cylinder are joined to make a solid

Find the total surface area of the solid.

Solution :

Total surface area of the figure given above =

lateral surface area of cylinder + lateral surface area of cone

= 2πrh + πrl

πr(2h + l)

= πr(2(r+6) + (3r/2))

= πr(2r + 12 + (3r/2))

= πr((4r + 24 + 3r)/2)

= πr((7r + 24)/2)

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