Respuesta :
Given:
The circumference of the base of a cone is 10π cm.
The circumference of the base of a second similar cone is 20π.
To find:
The ratio of the surface area of the first cone to that of the second cone.
Solution:
Let the radii of base of two cones are [tex]r_1[/tex] and [tex]r_2[/tex] respectively.
Circumference of the circular base is [tex]2\pi r[/tex], where, r is radius.
We have,
[tex]2\pi r_1=10\pi\text{ cm}[/tex]
[tex]r_1=\dfrac{10\pi}{2\pi}\text{ cm}[/tex]
[tex]r_1=5\text{ cm}[/tex]
And,
[tex]2\pi r_2=20\pi\text{ cm}[/tex]
[tex]r_2=\dfrac{20\pi}{2\pi}\text{ cm}[/tex]
[tex]r_2=10\text{ cm}[/tex]
It two cons are similar, then ratio of there areas is equal to square of the ratio of there corresponding dimensions, i.e., radius or heights.
[tex]\dfrac{A_1}{A_2}=\left(\dfrac{r_1}{r_2}\right)^2[/tex]
[tex]\dfrac{A_1}{A_2}=\left(\dfrac{5}{10}\right)^2[/tex]
[tex]\dfrac{A_1}{A_2}=\left(\dfrac{1}{2}\right)^2[/tex]
[tex]\dfrac{A_1}{A_2}=\dfrac{1}{4}[/tex]
The ratio form is
[tex]V_1:V_2=1:4[/tex]
Therefore, the ratio of the surface area of the first cone to that of the second cone is 1:4.
The ratio of the surface area of first cone to that of the second cone is; 1:4
Surface Area
Formula for the circumference of the base of a cone is;
C = 2πr
Where;
C is circumference
r is radius
Thus;
- For the first cone;
10π = 2πr
r = 10π/2π
r = 5
- For the second cone;
20π = 2πr
r = 20π/π
r = 10
Now, formula for surface area of the base is;
A = πr²
Thus, ratio of both area of first cone to second cone is;
A1/A2 = 5²/10²
A1/A2 = 25/100
A1/A2 = 1/4
Thus, ratio of first cone to that of the second cone is; 1:4
Read more on surface area of cone at; https://brainly.com/question/6613758
