A paper cone has a base diameter of 8 and a height of 3cm make a sketch of the cone and hence use Pythagoras theorem to calculate the slant height
Cannot sketch on these posts.
radius = 4
4^2 + 3^2 = x^2
Solve for x.
Radius of base = 4 cm, height = 3 cm
Volume = (1/3)π(4^2)(3) = 16π cm^3
slant height --- s
s^2 = 4^2 + 3^2
s = 5 , (did your recognize the standard 3-4-5 right angled triangle?)
So the radius of the sector is 5 and the arclength is the circumference of the base of the cone.
Circumference of base = arc of sector = 8π cm
circumference of circle containing our sector = 10π cm, so the area of the sector is 4/5 the area of the big circle
area of sector = (4/5)π(5^2) = 20π cm^2
Of course I could have just used the formula
lateral area of cone = πrl, where r is the radius of the cone and l is the slant height
= π(4)(5) = 20π
for the sector angle:
sector-angle/360 = 8π/10π = 4/5
sector angle = 288°
To make a sketch of the cone, follow these steps:
1. Draw a horizontal line to represent the table or surface on which the cone stands.
2. At the left endpoint of the line, draw a vertical line to represent the height of the cone (3cm).
3. Then draw a curved line that starts at the top of the vertical line and connects to the right endpoint of the horizontal line. Make sure the curved line is slanted and gradually widens as it goes down to form a cone shape.
Now, let's use Pythagoras' theorem to calculate the slant height of the cone. Pythagoras' theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In this case, the slant height is the hypotenuse (l) of a right-angled triangle formed by the slant height, the radius (r), and the height (h) of the cone.
Using Pythagoras' theorem, we have:
l^2 = r^2 + h^2
In this case, the radius (r) is half the base diameter, so r = 8/2 = 4.
Plugging in the values we know, we get:
l^2 = 4^2 + 3^2
l^2 = 16 + 9
l^2 = 25
To find the slant height (l), we take the square root of both sides:
l = √25
l = 5 cm
Therefore, the slant height of the cone is 5cm.