Question

1. a wooden stool is in the form of a frustum of a cone with slant edge 40cm, top diameter 30cm and bottom diameter 50cm.

a) calculate the perpendicular height of the stool
(3mk)
b) calculate the total surface area of the stool
(3mk)
c)calculate the volume of wood used to make the stool in terms of
(3mk)
d)given that the density of the wood used to make the stool is 0.8g/cm³, calculate the mass of the stool in kg
(1mk)

Explanation:

Step1: Find radii

The top diameter \(d_1 = 30\mathrm{cm}\), so the top radius \(r_1=\frac{30}{2}=15\mathrm{cm}\). The bottom diameter \(d_2 = 50\mathrm{cm}\), so the bottom radius \(r_2=\frac{50}{2}=25\mathrm{cm}\). Let the slant height \(l = 40\mathrm{cm}\). Using the Pythagorean theorem for the frustum of a cone \(h=\sqrt{l^{2}-(r_2 - r_1)^{2}}\).

Step2: Calculate height

Substitute \(r_1 = 15\mathrm{cm}\), \(r_2=25\mathrm{cm}\) and \(l = 40\mathrm{cm}\) into the formula: \((r_2 - r_1)=25 - 15=10\mathrm{cm}\). Then \(h=\sqrt{40^{2}-10^{2}}=\sqrt{(40 + 10)(40-10)}=\sqrt{50\times30}=\sqrt{1500}=10\sqrt{15}\approx38.73\mathrm{cm}\)

Step3: Calculate total surface area

The total surface area of the frustum of a cone \(A=\pi(r_1^{2}+r_2^{2}+l(r_1 + r_2))\). Substitute \(r_1 = 15\mathrm{cm}\), \(r_2=25\mathrm{cm}\) and \(l = 40\mathrm{cm}\): \(A=\pi(15^{2}+25^{2}+40\times(15 + 25))=\pi(225+625+40\times40)=\pi(225 + 625+1600)=\pi(2450)=2450\pi\mathrm{cm}^{2}\approx7696.9\mathrm{cm}^{2}\)

Step4: Calculate volume

The volume of the frustum of a cone \(V=\frac{1}{3}\pi h(r_1^{2}+r_2^{2}+r_1r_2)\). Substitute \(r_1 = 15\mathrm{cm}\), \(r_2=25\mathrm{cm}\) and \(h = 10\sqrt{15}\mathrm{cm}\): \(V=\frac{1}{3}\pi\times10\sqrt{15}(15^{2}+25^{2}+15\times25)=\frac{10\sqrt{15}\pi}{3}(225 + 625+375)=\frac{10\sqrt{15}\pi}{3}(1225)=\frac{12250\sqrt{15}\pi}{3}\mathrm{cm}^{3}\approx157080\mathrm{cm}^{3}\)

Step5: Calculate mass

Given density \(
ho=0.8\mathrm{g/cm}^{3}\). Using the formula \(m=
ho V\). First, \(V=\frac{12250\sqrt{15}\pi}{3}\mathrm{cm}^{3}\approx157080\mathrm{cm}^{3}\). Then \(m = 0.8\times V\). \(m=0.8\times157080 = 125664\mathrm{g}\). Convert to kg: \(m=\frac{125664}{1000}=125.664\mathrm{kg}\)

Answer:

a) The perpendicular height of the stool is approximately \(38.73\mathrm{cm}\)
b) The total surface area of the stool is approximately \(7696.9\mathrm{cm}^{2}\)
c) The volume of wood used is approximately \(157080\mathrm{cm}^{3}\)
d) The mass of the stool is \(125.664\mathrm{kg}\)