A wooden stool is in the form of a frustrum of a cone with slant edge 40cm,top diameter 30 cm and bottom diameter 50cm
(@) calculate the perpendicular height of the stool
(B) calculate the total surface area of the stool in terms of m squared
(C) calculate the volume of wood used to make the stool in terms of m squared
a) bottom radius = 25
top radius = 15
difference = 10
so h^2 + 100 = 1600
h = 38.73
area add top, bottom and side
top = pi (225)
bottom = pi(625)
side = pi (25+15)(40)
Draw a side view
h^2 + 10^2 = 40^2
there are handy formulas for the surface area, or you can think of it as the bottom part of a complete cone, so just subtract the area that was cut off
unless you know the thickness of the wood, it's impossible to know its volume.
To solve these questions, we will need to use the formulae for the volume and surface area of a frustrum of a cone.
(A) To calculate the perpendicular height of the stool:
We are given the slant edge and the top and bottom diameters. The slant edge of the frustrum of the cone is equivalent to the slant height of the cone.
Using the Pythagorean theorem, we can find the perpendicular height. The slant height (l) is the hypotenuse, the top radius (r1) is the shorter side, and the bottom radius (r2) is the longer side of a right triangle.
Using the formula:
l^2 = h^2 + (r2 - r1)^2
Substituting the given values, we have:
40^2 = h^2 + (25 - 15)^2
Simplifying the equation:
1600 = h^2 + 10^2
1600 - 100 = h^2
1500 = h^2
Taking the square root on both sides:
h = √1500 ≈ 38.73 cm
Therefore, the perpendicular height of the stool is approximately 38.73 cm.
(B) To calculate the total surface area of the stool in terms of square meters:
The total surface area (A) of the frustrum of a cone is given by the formula:
A = π(r1^2 + r2^2 + r1r2) + πl(r1 + r2)
Substituting the given values, we have:
A = π(15^2 + 25^2 + 15 * 25) + π * 40 * (15 + 25)
Simplifying the equation:
A = π(225 + 625 + 375) + π * 40 * 40
A = π(1225) + π * 1600
A = 1225π + 1600π
Therefore, the total surface area of the stool in terms of square meters is 2825π square centimeters.
(C) To calculate the volume of wood used to make the stool in terms of cubic meters:
The volume (V) of the frustrum of a cone is given by the formula:
V = (1/3)πh(r1^2 + r2^2 + r1r2)
Substituting the given values, we have:
V = (1/3) * π * 38.73 * (15^2 + 25^2 + 15 * 25)
V = (1/3) * π * 38.73 * (225 + 625 + 375)
V = (1/3) * π * 38.73 * 1225
Therefore, the volume of wood used to make the stool in terms of cubic meters is (1/3) * 38.73 * 1225π cubic centimeters.