the hat of a clown is in the form of a cone. the slant height of the cone is 15cm and it makes the angle of 45 degree with the height of the hat. what is the diameter of clown's head ?
To determine the diameter of the clown's head, we first need to find the height of the cone.
Given the slant height (s) of the cone is 15 cm, and the angle it makes with the height (h) of the hat is 45 degrees, we can use trigonometry to solve for h.
Since the sine of an angle is defined as the ratio of the length of the opposite side to the hypotenuse, we can write:
sin(45 degrees) = h / 15 cm
Simplifying the equation, we have:
h = 15 cm * sin(45 degrees)
h = 15 cm * (√2 / 2)
h = 15 cm * (√2) / 2
h ≈ 10.61 cm
Now, to find the diameter of the clown's head, we can use the fact that the slant height of the cone is equal to the height of the cone plus the radius of the base. Since the slant height is 15 cm and the height is 10.61 cm, we can solve for the radius.
s = h + r
15 cm = 10.61 cm + r
Simplifying the equation, we have:
r = 15 cm - 10.61 cm
r ≈ 4.39 cm
Finally, the diameter (d) is equal to twice the radius (d = 2r), so:
d = 2 * 4.39 cm
d ≈ 8.78 cm
Therefore, the diameter of the clown's head is approximately 8.78 cm.
Draw a diagram. If the hat's radius is r, then
r^2+r^2 = 15^2
and the diameter is just 2r