Two solid right circular cones and radii r1 and r2 and have the same height h, they are method and react into a cylinder of same height. Show that the radius of the base of cylinder is √[r1]°2+[r2]°2^3.

Check for typos please.

If you mean merged, same total volume, then with the same height, the base areas add up to the original.
pi R^2 = pi r1^2 + pi r2^2
R2 = sqrt( r1^2 + r2^2)

To show that the radius of the base of the cylinder is √(r1^2 + r2^2), we can use the properties of similar triangles.

Let's start by visualizing the problem. We have two cones with radii r1 and r2 and the same height h. When we melt and pour these cones into a cylinder of the same height, the shape becomes a cylinder.

The ratio of the radius of a cone to its height is constant. Therefore, we can express the ratio of r1 to h as a fraction, say x.

So, we have r1/h = x. Rearranging this equation, we get r1 = xh.

Similarly, r2/h = x, which means r2 = xh.

Now, let's consider the cylinder. The cylinder has the same height h as the cones, and we need to find the radius of its base.

The base of the cylinder is formed by combining the bases of the two cones, which gives us a circle with a radius equal to the sum of the radii of the cones.

The radius of the base of the cylinder, let's call it R, is given by R = r1 + r2.

Substituting the values of r1 and r2, we get R = xh + xh = 2xh.

Now, we need to find the value of x. To do that, we can use the property of similar triangles.

Consider a right triangle formed by the height of the cone h, the radius of the cone r1, and the slant height of the cone, which is the hypotenuse. This triangle is similar to the right triangle formed by the height of the cylinder h, the radius of the cylinder R, and the slant height of the cylinder.

Using the Pythagorean theorem, we can write:

(r1^2) + (h^2) = (slant height of the cone)^2

(R^2) + (h^2) = (slant height of the cylinder)^2

Since the slant height of the cone is the same as the slant height of the cylinder, we can equate these two equations:

(r1^2) + (h^2) = (R^2) + (h^2)

Simplifying this equation, we get:

r1^2 = R^2

Substituting the value of R (2xh), we get:

(xh)^2 = (2xh)^2

Simplifying further, we have:

x^2h^2 = 4x^2h^2

Dividing both sides by h^2, we get:

x^2 = 4x^2

Simplifying this equation, we get:

3x^2 = 0

Since x cannot be zero, we must have:

3x^2 = 0

Dividing both sides by 3, we get:

x^2 = 0

Taking the square root of both sides, we find:

x = 0

Therefore, the radius of the base of the cylinder, R = 2xh, simplifies to:

R = 2(0)h

R = 0

Hence, the radius of the base of the cylinder is zero.

Apologies, I made a mistake in my explanation. It seems there was an error during the solution steps. Let's try again to find the correct answer.

Given that r1 and r2 are the radii of two cones, and they are combined to form a cylinder with the same height.

The formula to calculate the volume of a cone is V = (1/3)πr^2h, where r is the radius and h is the height.

For the first cone: V1 = (1/3)π*r1^2h
For the second cone: V2 = (1/3)π*r2^2h

Since the cones are melted and formed into a cylinder, their volumes should be equal to the volume of the cylinder.

Thus, V1 + V2 = πR^2h, where R is the radius of the cylinder.

Substituting the volumes of the cones into the equation, we have:

(1/3)π*r1^2h + (1/3)π*r2^2h = πR^2h

Canceling out the common terms and rearranging, we get:

r1^2 + r2^2 = 3R^2

Now, we need to find the radius R of the cylinder in terms of r1 and r2.

R^2 = (r1^2 + r2^2) / 3

Taking the square root of both sides, we get:

R = √[(r1^2 + r2^2) / 3]

Now, let's simplify this further.

R = √[(r1^2 + r2^2) / 3]

Multiplying both the numerator and denominator by 3, we have:

R = √[3(r1^2 + r2^2) / 9]

Since the square root of a fraction is equal to the square root of the numerator divided by the square root of the denominator, we get:

R = (√(3(r1^2 + r2^2))) / (√9)

Simplifying the denominator (√9 = 3), we have:

R = (√(3(r1^2 + r2^2))) / 3

Therefore, the radius of the base of the cylinder is given by:

R = (√(3(r1^2 + r2^2))) / 3

Hence, the correct answer is R = (√(3(r1^2 + r2^2))) / 3.