Verified answer

Answer:

given below

Step-by-step explanation:

Let the radius of the cone be 5x, and the height be 6x, where x is a constant. The volume (V) of a cone is given by V = (1/3)πr²h.

Given that V = 1256 cm³, we have:

\[\frac{1}{3} \times 3.14 \times (5x)^2 \times 6x = 1256\]

Solving this equation for x:

\[157 \times 25x³ = 1256\]

\[x³ = \frac{1256}{3925}\]

\[x = \frac{\sqrt[3]{1256}}{\sqrt[3]{3925}}\]

Now, the radius (r) and height (h) can be expressed in terms of x:

\[r = 5x = 5 \times \frac{\sqrt[3]{1256}}{\sqrt[3]{3925}}\]

\[h = 6x = 6 \times \frac{\sqrt[3]{1256}}{\sqrt[3]{3925}}\]

To find the total surface area (A) of the cone, we use the formula \(A = πr(r + l)\), where \(l\) is the slant height.

The slant height \(l\) can be found using the Pythagorean theorem: \(l = \sqrt{r^2 + h^2}\).

Substitute the expressions for \(r\) and \(h\) into the formula for \(l\) and then into the formula for \(A\). After some simplification, you should get the answer.

After performing the calculations, the correct option appears to be:

**C: 31.4[10 + 2 sqrt(61)] sq. cm**

Answer:

Let radius of cone =5x

Height of cone = 12x

l

2

=

√

169

x

2

=

13

x

It is given that volume =

314

c

m

2

1

3

π

(

5

x

)

2

(

12

x

)

=

314

1

3

×

3.14

×

25

×

12

×

x

3

=

314

x

3

=

314

×

3

3.14

×

25

×

12

=

1

x=1cm

Radius r=5

×

1=5cm

Height h=12

×

1=12 cm

and slant height l=13

×

l=13 cm

Now total surface area of cone

=

π

r

(

l

+

r

)

=

3.14

×

5

(

13

+

5

)

=

3.14

×

5

×

8

=

282.60

c

m

2

Hence total surface area of cone is 282.60

c

m

2

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