A cone has a diameter of 12 feet and a slant height of 20 feet, if you triple both dimensions will it triple the surface area?? PLEASE ANSWER QUICKLY!!!
No. Areas are proportional to the square of linear dimensions, for similar figures. In your case, the area increases by a factor of 9.
To determine if tripling both dimensions of a cone will triple its surface area, let's first calculate the surface area of the original cone.
The formula for the surface area of a cone is:
Surface Area = π * r * (r + slant height)
Given that the diameter of the original cone is 12 feet, the radius (r) will be half of the diameter, which is 12/2 = 6 feet.
Substituting the radius (r = 6 feet) and the slant height (20 feet) into the formula, we get:
Surface Area = π * 6 * (6 + 20)
Surface Area = π * 6 * 26
Surface Area = 156π
Now, let's calculate the new surface area when both dimensions (diameter and slant height) are tripled.
Since we are tripling the dimensions, the new diameter will be 12 feet * 3 = 36 feet, and the new slant height will be 20 feet * 3 = 60 feet.
Again, we can calculate the new surface area using the formula:
Surface Area = π * r * (r + slant height)
For the new cone, the radius (r) will be 36/2 = 18 feet.
Substituting the radius (r = 18 feet) and the slant height (60 feet) into the formula, we get:
Surface Area = π * 18 * (18 + 60)
Surface Area = π * 18 * 78
Surface Area = 1404π
Comparing the two surface areas, we find that
Surface Area (Original Cone) = 156π
Surface Area (New Cone) = 1404π
So, tripling both dimensions does not result in tripling the surface area of the cone. The surface area is significantly larger in the new cone.
To determine whether tripling both the diameter and slant height of a cone will triple its surface area, we need to use the formula for the surface area of a cone.
The formula for the surface area of a cone is:
Surface Area = π * r * (r + l)
Where:
- r is the radius of the base (half of the diameter)
- l is the slant height
Given that the diameter of the cone is 12 feet, the radius (r) would be half of the diameter, which is 6 feet. The slant height (l) is given as 20 feet.
Now, let's calculate the surface area using these values:
Surface Area = π * (6 feet) * (6 feet + 20 feet)
Surface Area = π * 6 feet * 26 feet
Surface Area = 156π square feet
Now, if we triple both the diameter and slant height, the new dimensions would be:
- Diameter = 3 * 12 feet = 36 feet
- Slant Height = 3 * 20 feet = 60 feet
Let's calculate the new surface area using these new dimensions:
Surface Area (new) = π * (18 feet) * (18 feet + 60 feet)
Surface Area (new) = π * 18 feet * 78 feet
Surface Area (new) = 1404π square feet
Comparing the original surface area (156π square feet) with the new surface area (1404π square feet), we can see that the new surface area is approximately 9 times larger than the original surface area.
Therefore, tripling both the diameter and slant height of a cone will increase the surface area by a factor of 9, not 3.