Answer:
h = 16.0 cm
Step-by-step explanation:
Note: I am assuming that the surface area of 258 is the total surface area which includes the Lateral Area (L) and Base Area(B)
Total Surface Area of a right cone
= Lateral Area(L) + Base Area(B)
All numbers in intermediate calculations are rounded to tenth of centimeter i.e. 1 decimal point
Base Area is the area of the circle with radius 4 = [tex]\displaystyle \pi \cdot16 = 50.3[/tex] cm²
Total Area = 258 cm²
So Lateral Area = 258 - 50.3 = 207.7 cm²
Lateral Area is given by the formula
[tex]\displaystyle L = \pi r s \textrm{ where s is the slant height }[/tex]
So we get
[tex]\displaystyle \rm pi \cdot 4 \cdot s = 207.7\\\\\textrm {This gives } s = \dfrac{207.7}{4\pi} = 16.5\\\\[/tex] cm
Slant height is given by the formula:
[tex]s = \sqrt{r^2 + h^2} \textrm{ where r is the radius and h the height}[/tex]
Plugging in values we get
[tex]16.5 = \sqrt{4^2 + h^2} = \sqrt{16 + h^2}[/tex]
Square both sides
[tex]16.5^2 = 16 + h^2[/tex]
==> [tex]h^2 = 16.5^2 - 16 = 256.25\\\\h = \sqrt{256.25} = 16.003 = 16.0[/tex] rounded to 1 decimal place
So final answers is h = 16 cm
