Respuesta :
The Halloween conical hat, with given height, circular base and brim
extension has the following calculated parameters;
Part a. The slant height is 18.2 inches
Part b. The volume of the cone is [tex]37\frac{1}{2} \cdot \pi[/tex] in.³
Part c. The area of the brim, A = 36·π in.²
Part d. The area of the brim is found by subtracting the area of the base of the cone from the area covered by the perimeter of the brim
Reasons:
Known parameters;
Height of the conical portion, h = 18 inches
Base circumference, C = 5·π inches
Part a. Slant height of the conical portion; Required
Solution:
The circumference of a circle, C = 2·π·r
Therefore;
[tex]r = \dfrac{C}{2 \cdot \pi}[/tex]
Which gives;
[tex]r = \dfrac{5 \cdot \pi}{2 \cdot \pi} = \dfrac{5}{2} = 2.5[/tex]
Radius, r = 2.5 inches
According to Pythagoras's theorem, we have; s² = r² + h²
Where;
s = The slant height of the cone
s² = 2.5² + 18² = 330.25
s = √(330.25) ≈ 18.2
- The slant height, s ≈ 18.2 inches
Part b. The measure in cubic inches of candy that exactly fills the conical portion of the hat is the volume of the cone.
[tex]Volume \ of \ a \ cone = \dfrac{1}{3} \cdot \pi \cdot r^2 \cdot h[/tex]
Therefore;
[tex]V = \dfrac{1}{3} \times \pi \times 2.5^2 \times 18 = 37\frac{1}{2} \cdot \pi[/tex]
- The volume of the cone, V = [tex]37\frac{1}{2}[/tex]·π in.³
Part c. The extension of the brim from the base of the cone = 4 inches
The radius of the brim, R = Radius of the base of the cone + 4 inches
∴ R = 2.5 inches + 4 inches = 6.5 inches
Area of the brim, A = Area of the 6.5 inch circle - Area of the circular base of the cone
∴ A = π × 6.5² - π × 2.5² = 36·π
- The area of the brim, A = 36·π in.²
Part d. The procedure for solving the question in part c, is described as follows;
- The area of the brim can be found by finding the entire area of the circle formed by the perimeter of the brim, then subtracting the area of the base of the cone from that area.
Learn more here:
https://brainly.com/question/17023854