Answer:
[tex]\sf h=\sqrt{153}\;in[/tex]
Step-by-step explanation:
A cone has a circular base with an apex directly above its center. When a vertical plane passes through the apex and intersects the cone, the resulting cross-section forms an isosceles triangle. The base of this triangle corresponds to the diameter of the cone's circular base.
In an isosceles triangle, the altitude (height) from the apex to the base bisects the base and divides the triangle into two congruent right triangles. Therefore:
- The hypotenuse of each right triangle corresponds to the slant height of the cone.
- The base of each right right triangle corresponds to the radius of the circular base of the cone.
- The height of each right triangle corresponds to the height of the cone.
Therefore, to find h, we can use the Pythagorean Theorem which states that in a right triangle, the square of the hypotenuse (longest side) is equal to the sum of the squares of its legs.
In this case, one leg is labelled "h", the other leg measures 4 inches, and the hypotenuse is 13 inches. Therefore:
[tex]\sf 4^2+h^2=13^2 \\\\16+h^2=169 \\\\h^2=169-16 \\\\ h^2=153 \\\\h=\sqrt{153}\; in[/tex]
So, the height of the cone is:
[tex]\Large\boxed{\boxed{ \sf h=\sqrt{153}\;in}}[/tex]
