A paper cone has a base diameter of 8cm and a height of 3cm(a) use Pythagoras theorem to calculate it's slant height (b)if the cone is cut and opened out into the sector of a circle what is the angel of the sector
calculate the curved surface area of the cone in terms of pi
Make a sketch to see that r = 5, h = 3
then s^2 = 5^2 + 3^2 = 34
slant height = √34 cm
circumference of base of cone
= 2π(5) = 10π cm
This becomes the arc-length in a circle of circumference of 2π√34
let the angle of the sector be Ø
arc = rØ
10π = √34Ø
Ø = 10π/√34 radians
= appr 5.388 radians or appr 308.7°
a is 5cm b is 288 degree
Abeg i no understand how una tey do this one ooo pls expantiate better o ti daru momi loju nsin biko na beg i de beg o mathematician
To find the slant height of a cone using Pythagoras' theorem, we can consider the cone as a right triangle. According to Pythagoras' theorem, the square of the hypotenuse (slant height) is equal to the sum of the squares of the other two sides.
Let's label the sides of the right triangle formed by the cone:
- The height (h) of the cone is one of the sides of the right triangle.
- The radius of the base (r) is also a side of the right triangle.
Given that the base diameter is 8 cm, we can calculate the radius by dividing the diameter by 2. Therefore, the radius, r = 8 cm / 2 = 4 cm.
Using the Pythagorean theorem, we have:
(hypotenuse)^2 = (height)^2 + (radius)^2
Let's substitute the values we know:
(slant height)^2 = (3 cm)^2 + (4 cm)^2
Simplifying:
(slant height)^2 = 9 cm^2 + 16 cm^2
(slant height)^2 = 25 cm^2
To find the slant height, we need to take the square root of both sides:
slant height = √(25 cm^2)
slant height = 5 cm
Therefore, the slant height of the cone is 5 cm.
To find the angle of the sector when the cone is cut and opened out, we need to calculate the circumference of the base.
The circumference of a circle is given by the formula: C = 2πr, where r is the radius.
Given that the base radius is 4 cm, the circumference of the base is:
C = 2π(4 cm)
C = 8π cm
The sector formed by cutting and opening the cone will have the same arc length as the circumference of the base. The entire circumference of a circle corresponds to an angle of 360 degrees.
Therefore, to find the angle of the sector, we need to find what fraction of the total circumference the base covers.
Let's calculate this fraction:
Arc length / Total circumference = Angle / 360 degrees
Substituting the values we know:
(8π cm) / (8π cm) = Angle / 360 degrees
Simplifying:
1 = Angle / 360 degrees
To isolate the angle, we multiply both sides by 360 degrees:
Angle = 1 * 360 degrees
Angle = 360 degrees
Therefore, the angle of the sector formed when the cone is cut and opened out is 360 degrees.