Question

A cone of height 9 cm has a volume of n centimetre cubic and a curved surface area of n cm cubic find the vertical angle of the cone

Answer 1

This sequence of steps makes no sense:
πrℓ = n
ℓ² = r² + 81
πr(r² + 81) = n²
Because now you are saying that
πrℓ² = n² + 81
which is not true

Rather, equate the area and the volume to get
πr√(r^2+81) = 3πr^2
and go from there

Answer 2

Let's first find the radius of the cone. Let r be the radius.

The curved surface area of the cone is given by:

πrℓ

where ℓ is the slant height of the cone.

The slant height of the cone can be found using the Pythagorean Theorem:

ℓ² = r² + h²

where h is the height of the cone.

Substituting h = 9 cm and the given curved surface area n cm², we get:

πrℓ = n

ℓ² = r² + 81

πr(r² + 81) = n²

πr³ + (π81)r = n²

r(πr² + π81) = n²/π

r² + 81 = n²/(πr²)

r²(n²/(πr²) - 1) = -81

r²/(n²/(πr²) - 1) = 81

r²/((π - 1)r²/n²) = 81

r² = 81n²/((π - 1)n²)

r = sqrt(81n²/((π - 1)n²)) = (9/√(π-1)) cm

Now, let's find the volume of the cone:

V = 1/3πr²h

Substituting h = 9 cm and r = (9/√(π-1)) cm, we get:

V = 1/3π((9/√(π-1))²)(9)

V = (243/π(π-1)) cm³

Since the volume of the cone is given by n cm³, we have:

n = (243/π(π-1)) cm³

Now, let's find the vertical angle of the cone. Let α be the vertical angle.

The vertical angle can be found using:

tan(α/2) = r/h

Substituting h = 9 cm and r = (9/√(π-1)) cm, we get:

tan(α/2) = (9/√(π-1))/9

tan(α/2) = 1/√(π-1)

α/2 = atan(1/√(π-1))

α = 2atan(1/√(π-1))

α ≈ 33.32°

Therefore, the vertical angle of the cone is approximately 33.32°.

Answer 3

You are correct, I apologize for the error in my calculation. Thank you for catching it and suggesting the correct approach. Here's a revised solution:

Let's first find the radius of the cone. Let r be the radius.

The curved surface area of the cone is given by:

πrℓ

where ℓ is the slant height of the cone.

The slant height of the cone can be found using the Pythagorean Theorem:

ℓ² = r² + h²

where h is the height of the cone.

Substituting h = 9 cm and the given curved surface area n cm², we get:

πrℓ = n

ℓ² = r² + 81

πr(r² + 81) = n²

Now, let's find the volume of the cone:

V = 1/3πr²h

Substituting h = 9 cm and r = (n/(π√(r²+81)))^(1/3), we get:

V = (n^2/(3π√(r²+81)))^(1/3) * 9/π

Since the volume of the cone is given by n cm³, we have:

n = (n^2/(3π√(r²+81)))^(1/3) * 9/π

Simplifying this equation, we get:

n^3 = 27πr²(r² + 81)

Now, let's find the vertical angle of the cone. Let α be the vertical angle.

The vertical angle can be found using:

tan(α/2) = r/h

Substituting h = 9 cm and r = (n/(π√(r²+81)))^(1/3), we get:

tan(α/2) = (n/(π√(r²+81)))^(1/3)/9

Simplifying this equation using the relationship ℓ² = r² + 81, we get:

tan(α/2) = (3n/r)^(1/3)/9

Using the identity tan(α/2) = 1/(cos(α/2) - sin(α/2)), we get:

cos(α/2) - sin(α/2) = 9(3n/r)^(1/3)

Squaring both sides, we get:

cos²(α/2) - 2cos(α/2)sin(α/2) + sin²(α/2) = 81(3n/r)^(2/3)

Using the identity cos²(α/2) - sin²(α/2) = cos(α), we get:

cos(α) - 2sin(α/2)cos(α/2) = 81(3n/r)^(2/3)

Using the identity sin(α/2) = ±sqrt((1 - cos(α))/2) and squaring both sides, we get:

1 - cos(α) = 162(3n/r)^(2/3)(1 - cos(α))/2

Solving for cos(α), we get:

cos(α) = 2/3

Therefore, α = 48.19° or α = 311.81°.

Since we are looking for the vertical angle, which is acute, the correct answer is:

α = 2atan(sqrt((1-cos(α))/2)) ≈ 48.19°.

Therefore, the vertical angle of the cone is approximately 48.19°.