The diagram shows a solid circular cone and a solid sphere. The cone has radius 5x cm and height 12x cm. The sphere has radius r cm. The cone has the same total surface area as the... The diagram shows a solid circular cone and a solid sphere. The cone has radius 5x cm and height 12x cm. The sphere has radius r cm. The cone has the same total surface area as the sphere. Show that \(r^2 = \frac{45}{2}x^2\). [The curved surface area, A, of a cone with radius r and slant height l is A = \(\pi rl\).] [The surface area, A, of a sphere with radius r is A = \(4\pi r^2\).] Read more

Steps to Solve

1. Calculate the slant height of the cone The slant height, $l$, can be found using the Pythagorean theorem: $l = \sqrt{h^2 + r^2}$, where $h$ is the height and $r$ is the radius of the cone. In this case, $h = 12x$ and the radius is $5x$.

$$ l = \sqrt{(12x)^2 + (5x)^2} = \sqrt{144x^2 + 25x^2} = \sqrt{169x^2} = 13x $$

2. Calculate the total surface area of the cone The total surface area of the cone is the sum of the curved surface area and the area of the circular base. Curved surface area $= \pi rl = \pi (5x)(13x) = 65\pi x^2$. Area of the circular base $= \pi r^2 = \pi (5x)^2 = 25\pi x^2$. Total surface area of the cone $= 65\pi x^2 + 25\pi x^2 = 90\pi x^2$.

3. Calculate the surface area of the sphere The surface area of the sphere is given by $4\pi r^2$.

4. Equate the surface areas and solve for (r^2) Since the cone and sphere have the same total surface area: $4\pi r^2 = 90\pi x^2$. Divide both sides by $4\pi$: $r^2 = \frac{90}{4}x^2 = \frac{45}{2}x^2$.