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24

?
Here is a different solid sphere and a different solid cone.
All measurements are in cm.
The surface area of the sphere is equal to the total surface area of the cone.
(b) Find r:h
Give your answer in the form 1: √n
where n is an integer.

24 ? Here is a different solid sphere and a different solid cone. All measurements-example-1
User Samrat
by
8.3k points

2 Answers

5 votes

Answer:


1 : √(8)

Explanation:

The surface area of a sphere is given by the formula:


\boxed{S.A._(\sf sphere)=4\pi r^2}

where r is the radius of the sphere.

The surface area of a cone is the sum of the area of its circular base and the curved area. Therefore:


\boxed{S.A._(\sf cone)=\pi r^2 + \pi r l}

where r is the radius of the base of the cone and
l is the slant height.

As we need to find the ratio of the radius (r) to the perpendicular height (h) of the cone, we need to rewrite
l in terms of r and h. To do this, we can use Pythagoras Theorem, since r and h are the legs of a right triangle with
l as the hypotenuse.


r^2+h^2=l^2


l=√(r^2+h^2)

Substitute the expression for
l into the formula for the equation for the surface area of a cone:


\boxed{S.A._(\sf cone)=\pi r^2 + \pi r √(h^2+r^2)}

where r is the radius and h is the perpendicular height of the cone.

If the total surface area of the sphere is equal to the total surface area of the cone, then:


4\pi r^2=\pi r^2 + \pi r √(h^2+r^2)

Subtract πr² from both sides of the equation:


3\pi r^2=\pi r √(h^2+r^2)

Divide both sides of the equation by πr:


3r=√(h^2+r^2)

Square both sides of the equation:


9r^2=h^2+r^2

Subtract r² from both sides:


8r^2=h^2

Square root both sides:


√(8)\;r=h

Divide both sides by √8 h:


(r)/(h)=(1)/(√(8))

Therefore, the ratio of r : h is:


\boxed{r : h = 1 : √(8)}

User Uwe Honekamp
by
7.7k points
4 votes

Answer:

Let's denote the radius of the sphere by "r" and the height of the cone by "h".

The surface area of the sphere is given by 4πr² and the total surface area of the cone is given by πr√(r² + h²) + πr². We are given that these two are equal, so we can set them equal to each other and solve for r:h.

4πr² = πr√(r² + h²) + πr²

4πr² - πr² = πr√(r² + h²)

3πr² = πr√(r² + h²)

9r⁴ = r²(r² + h²) (squaring both sides)

9r² = r² + h²

8r² = h²

r:h = 1 : √8 = 1 : 2√2 (simplifying the ratio)

Explanation:

User Buck Calabro
by
8.4k points

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