Answer :
Sure, let's break down each part of the problem step by step:
### (i) How long is its radius?
Given:
- The circumference of the circular base is 66 inches.
To find the radius, we use the formula for the circumference of a circle:
[tex]\[ C = 2\pi r \][/tex]
Where:
- [tex]\( C \)[/tex] is the circumference.
- [tex]\( r \)[/tex] is the radius.
Rearranging the formula to solve for the radius [tex]\( r \)[/tex]:
[tex]\[ r = \frac{C}{2\pi} \][/tex]
Plugging in the given circumference:
[tex]\[ r = \frac{66}{2\pi} \][/tex]
Upon calculation:
[tex]\[ r \approx 10.504 \text{ inches} \][/tex]
The radius of the hat is approximately 10.504 inches.
### (ii) How many square inches of material is required to make the hat?
To find the surface area of the cone, we need to calculate both the area of the base and the lateral (side) surface area.
1. Area of the base (A_base):
The area [tex]\( A \)[/tex] of a circle is given by:
[tex]\[ A_\text{base} = \pi r^2 \][/tex]
Using the radius found previously:
[tex]\[ A_\text{base} = \pi \times (10.504)^2 \][/tex]
2. Lateral surface area (A_lateral):
The lateral surface area of a cone can be found using:
[tex]\[ A_\text{lateral} = \pi r l \][/tex]
Where:
- [tex]\( l \)[/tex] is the slant height (13.7 inches).
Using the radius:
[tex]\[ A_\text{lateral} = \pi \times 10.504 \times 13.7 \][/tex]
Adding both areas to find the total surface area:
[tex]\[ \text{Total Surface Area} = A_\text{base} + A_\text{lateral} \][/tex]
Upon calculation:
[tex]\[ \text{Total Surface Area} \approx 798.739 \text{ square inches} \][/tex]
The material required to make the hat is approximately 798.739 square inches.
### (iii) How much cubic inch of the air does it hold?
To find the volume of the cone, we use the formula for the volume of a cone:
[tex]\[ V = \frac{1}{3} \pi r^2 h \][/tex]
Where:
- [tex]\( V \)[/tex] is the volume.
- [tex]\( r \)[/tex] is the radius.
- [tex]\( h \)[/tex] is the height of the cone.
First, we need to determine the height [tex]\( h \)[/tex]. Using the Pythagorean theorem for the right triangle formed by the radius, height, and slant height:
[tex]\[ l^2 = r^2 + h^2 \][/tex]
[tex]\[ h^2 = l^2 - r^2 \][/tex]
[tex]\[ h = \sqrt{l^2 - r^2} \][/tex]
Using the known values:
[tex]\[ h = \sqrt{(13.7)^2 - (10.504)^2} \][/tex]
Upon calculation, we can then use the height to find the volume:
[tex]\[ V \approx \frac{1}{3} \pi \times (10.504)^2 \times h \][/tex]
Upon final calculation:
[tex]\[ V \approx 1016.226 \text{ cubic inches} \][/tex]
The volume of air the hat holds is approximately 1016.226 cubic inches.
### (i) How long is its radius?
Given:
- The circumference of the circular base is 66 inches.
To find the radius, we use the formula for the circumference of a circle:
[tex]\[ C = 2\pi r \][/tex]
Where:
- [tex]\( C \)[/tex] is the circumference.
- [tex]\( r \)[/tex] is the radius.
Rearranging the formula to solve for the radius [tex]\( r \)[/tex]:
[tex]\[ r = \frac{C}{2\pi} \][/tex]
Plugging in the given circumference:
[tex]\[ r = \frac{66}{2\pi} \][/tex]
Upon calculation:
[tex]\[ r \approx 10.504 \text{ inches} \][/tex]
The radius of the hat is approximately 10.504 inches.
### (ii) How many square inches of material is required to make the hat?
To find the surface area of the cone, we need to calculate both the area of the base and the lateral (side) surface area.
1. Area of the base (A_base):
The area [tex]\( A \)[/tex] of a circle is given by:
[tex]\[ A_\text{base} = \pi r^2 \][/tex]
Using the radius found previously:
[tex]\[ A_\text{base} = \pi \times (10.504)^2 \][/tex]
2. Lateral surface area (A_lateral):
The lateral surface area of a cone can be found using:
[tex]\[ A_\text{lateral} = \pi r l \][/tex]
Where:
- [tex]\( l \)[/tex] is the slant height (13.7 inches).
Using the radius:
[tex]\[ A_\text{lateral} = \pi \times 10.504 \times 13.7 \][/tex]
Adding both areas to find the total surface area:
[tex]\[ \text{Total Surface Area} = A_\text{base} + A_\text{lateral} \][/tex]
Upon calculation:
[tex]\[ \text{Total Surface Area} \approx 798.739 \text{ square inches} \][/tex]
The material required to make the hat is approximately 798.739 square inches.
### (iii) How much cubic inch of the air does it hold?
To find the volume of the cone, we use the formula for the volume of a cone:
[tex]\[ V = \frac{1}{3} \pi r^2 h \][/tex]
Where:
- [tex]\( V \)[/tex] is the volume.
- [tex]\( r \)[/tex] is the radius.
- [tex]\( h \)[/tex] is the height of the cone.
First, we need to determine the height [tex]\( h \)[/tex]. Using the Pythagorean theorem for the right triangle formed by the radius, height, and slant height:
[tex]\[ l^2 = r^2 + h^2 \][/tex]
[tex]\[ h^2 = l^2 - r^2 \][/tex]
[tex]\[ h = \sqrt{l^2 - r^2} \][/tex]
Using the known values:
[tex]\[ h = \sqrt{(13.7)^2 - (10.504)^2} \][/tex]
Upon calculation, we can then use the height to find the volume:
[tex]\[ V \approx \frac{1}{3} \pi \times (10.504)^2 \times h \][/tex]
Upon final calculation:
[tex]\[ V \approx 1016.226 \text{ cubic inches} \][/tex]
The volume of air the hat holds is approximately 1016.226 cubic inches.
