The surface area of a cone is 38 in^2. The radius of a similar cone is triple the radiusof the orginal cone. What is the surface area of the new cone?
Similar cones have area proportional to the square of the linear dimensions.
So if the new cone has three times the radius, the area is 3²=9 time that of the original cone.
To find the surface area of the new cone, we need to know the relationship between the surface area and the radius of a cone.
The formula for the surface area of a cone is given by:
SA = π * r * (r + l)
Where:
SA = Surface area of the cone
π = Pi (approximately 3.14)
r = radius of the base of the cone
l = slant height of the cone
Let's start by calculating the radius of the original cone.
Given that the surface area of the original cone is 38 in^2, we can set up the equation:
38 = π * r * (r + l)
However, we don't know the slant height, so it's not possible to solve for the radius directly.
Now, let's move on to the new cone. We are told that the radius of the new cone is triple the radius of the original cone. So, let's denote the radius of the original cone as "r" and the radius of the new cone as "3r".
The formula for the surface area of the new cone will be:
SA_new = π * (3r) * (3r + l_new)
Now, we need to find the slant height of the new cone, denoted as "l_new". Since the two cones are similar, we know that the ratio of the slant heights is the same as the ratio of the radii:
l_new / sl = 3r / r
Simplifying this equation, we get:
l_new = 3 * sl
Now, substitute the value of "l_new" back into the surface area formula for the new cone:
SA_new = π * (3r) * (3r + 3sl)
We know that 38 is the surface area of the original cone, so we can substitute the values into the equation:
38 = π * r * (r + sl)
Now, we can rearrange this equation to solve for "sl":
sl = (38 / (π * r)) - r
Substitute the value of "sl" into the equation for "l_new":
l_new = 3 * [(38 / (π * r)) - r]
Finally, substitute the value of "l_new" and "r" into the surface area equation for the new cone:
SA_new = π * (3r) * (3r + 3 * [(38 / (π * r)) - r])