Question

(1,000 cm³ = 1 l)

1. traffic cone design

a traffic cone has a base radius of 10 cm and a height of 24 cm.
(a) calculate the slant height of the cone.
(b) determine the lateral surface area of the cone if it is to be painted with reflective coating.

1. a metal funnel consists of a conical part with a height of 12 cm and radius of 5 cm.

(a) find the slant height.
(b) calculate the lateral surface area of the funnel (excluding the circular base).

Explanation:

Step1: Calculate the slant height of the cone (for 7(a) and 8(a))

Use the Pythagorean theorem \(l=\sqrt{r^{2}+h^{2}}\), where \(r\) is the radius and \(h\) is the height.
For 7(a): \(r = 10\space cm\), \(h=24\space cm\)
\(l=\sqrt{10^{2}+24^{2}}=\sqrt{100 + 576}=\sqrt{676}=26\space cm\)
For 8(a): \(r = 5\space cm\), \(h = 12\space cm\)
\(l=\sqrt{5^{2}+12^{2}}=\sqrt{25+144}=\sqrt{169}=13\space cm\)

Step2: Calculate the lateral surface area of the cone (for 7(b) and 8(b))

Use the formula \(S=\pi rl\), where \(r\) is the radius and \(l\) is the slant height.
For 7(b): \(r = 10\space cm\), \(l = 26\space cm\)
\(S=\pi\times10\times26=260\pi\space cm^{2}\approx260\times3.14 = 816.4\space cm^{2}\)
For 8(b): \(r = 5\space cm\), \(l = 13\space cm\)
\(S=\pi\times5\times13=65\pi\space cm^{2}\approx65\times3.14=204.1\space cm^{2}\)

Answer:

7(a) The slant height is \(26\space cm\).
7(b) The lateral surface area is approximately \(816.4\space cm^{2}\).
8(a) The slant height is \(13\space cm\).
8(b) The lateral surface area is approximately \(204.1\space cm^{2}\).