Completely bemused, do not know how to get the answer.

a)8π
b)7√2
c)27√π
d)7√×7√×1

13- Determine the exact total surface area of a sphere with radius 2√2 metres.

a) An inverted cone with side length 4 metres is placed on top of the sphere so that the centre of its base is 0.5 metres above the centre of the sphere.

b) Find the radius of the cone exactly.

c) Find the area of the curved surface of the cone exactly.

d) What are the exact dimensions of a box that could precisely fit the cone connected to the sphere?

And I am equally bemused by your comment.

What do you mean by "the answer" ?
You have simply written down 4 numbers.
What about them ?

13.
SA of sphere is 4πr^2
= 4π(2√2)^2
= 32π m^2

Still trying to visualize your cone somehow wobbling on top of , or hovering over the sphere.
Does the cone have the same radius as the sphere?

The first 4 numbers are the answers

To determine the exact total surface area of a sphere with radius 2√2 meters, we can follow the given steps:

a) An inverted cone with side length 4 meters is placed on top of the sphere so that the center of its base is 0.5 meters above the center of the sphere.

The cone is inverted, which means its tip is pointing downwards and it rests on top of the sphere. The side length of the cone is given as 4 meters, which refers to the slant height or the distance from the tip to the base along the curved surface. The center of the cone's base is positioned 0.5 meters above the center of the sphere.

b) To find the radius of the cone exactly, we can use the Pythagorean theorem.

The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

In the given scenario, the radius of the sphere forms the hypotenuse of a right-angled triangle with the radius of the cone as one side and the slant height as the other side.

Let's denote the radius of the cone as r. The slant height of the cone is given as 4 meters, and the radius of the sphere is 2√2 meters.

Using the Pythagorean theorem, we can express the relationship as follows:

(r^2) + (2√2)^2 = (4^2)

Simplifying the equation:

r^2 + 8 = 16

r^2 = 16 - 8

r^2 = 8

Taking the square root of both sides:

r = √8

Simplifying further:

r = √(4 * 2)

r = 2√2

Therefore, the radius of the cone is 2√2 meters.

c) To find the area of the curved surface of the cone exactly, we can use the formula for the lateral surface area of a cone:

Lateral Surface Area = π * r * s

where r is the radius of the cone and s is the slant height.

In this case, the radius of the cone is 2√2 meters and the slant height is 4 meters. Substituting these values into the formula:

Lateral Surface Area = π * 2√2 * 4

Lateral Surface Area = 8π√2

Therefore, the area of the curved surface of the cone is 8π√2 square meters.

d) Finally, to determine the exact dimensions of a box that could precisely fit the cone connected to the sphere, we can consider the dimensions of the cone and the sphere.

Since the base of the inverted cone rests on the top of the sphere, the box would need to accommodate the cone's dimensions as well as the sphere's dimensions.

The dimensions of the box would be the length, width, and height required to enclose the cone and the sphere.

The length and width would be equal to the diameter of the base of the cone (2 * radius). The height would be the sum of the cone's height (0.5 meters) and the sphere's diameter (2 * radius).

Therefore, the exact dimensions of the box that could precisely fit the cone connected to the sphere would be:

Length = Width = Diameter of the base of the cone = 2 * radius = 2 * (2√2) = 4√2 meters

Height = Cone's height + Sphere's diameter = 0.5 + 2 * radius = 0.5 + 2 * (2√2) = 0.5 + 4√2 meters

Hence, the exact dimensions of the box would be 4√2 meters (length and width) and 0.5 + 4√2 meters (height).