Three circles with different radii have their centers on a line. The two smaller circles are inside the largest circle, and each circle is tangent to the other two. The radius of the largest circle is 10 meters. Together the area of the two smaller circles is 68% of the area of the largest circle. Find the product of the radii of the smaller circles.

Let the radii of the two smaller circles be x and a0-x

pi x^2 + pi(10-x)^2 = .68 * pi * 10^2
x = 2,8

2*8 = 16

16

Let's assume the radius of the first smaller circle is r1 and the radius of the second smaller circle is r2.

Given that the radius of the largest circle is 10 meters, we know that the area of the largest circle is π * (10^2) = 100π square meters.

The area of each smaller circle can be calculated using the formula A = π * r^2, where A is the area and r is the radius.

The area of the first smaller circle is π * (r1^2) square meters.
The area of the second smaller circle is π * (r2^2) square meters.

According to the given information, the sum of the areas of the two smaller circles is 68% of the area of the largest circle. In mathematical terms, this can be expressed as:

π * (r1^2) + π * (r2^2) = 0.68 * (100π)
r1^2 + r2^2 = 68

Since the circles are tangent to each other, the distance between their centers along the line is equal to the sum of their radii. Therefore, we can express this as:

r1 + r2 = 10

We now have a system of two equations:

r1^2 + r2^2 = 68
r1 + r2 = 10

To solve this system, we can use a variety of methods. One way to proceed is to solve the second equation for r1 and substitute it into the first equation:

r1 = 10 - r2

Substituting this into the first equation:

(10 - r2)^2 + r2^2 = 68

Expanding and simplifying:

100 - 20r2 + r2^2 + r2^2 = 68
2r2^2 - 20r2 + 32 = 0

Dividing through by 2 gives:

r2^2 - 10r2 + 16 = 0

We can solve this quadratic equation using factoring or the quadratic formula:

(r2 - 2)(r2 - 8) = 0

This gives two possible solutions for r2: r2 = 2 or r2 = 8.

If we substitute r2 = 2 into the equation r1 + r2 = 10, we find r1 = 8.

Alternatively, if we substitute r2 = 8 into the equation r1 + r2 = 10, we find r1 = 2.

Since we are asked to find the product of the radii of the smaller circles, we can calculate:

r1 * r2 = 8 * 2 = 16

Therefore, the product of the radii of the smaller circles is 16.

To solve this problem, let's denote the radii of the two smaller circles as r1 and r2. We need to find the product of r1 and r2.

Given that the radius of the largest circle is 10 meters, we know that its area is π * (10^2) = 100π square meters.

Now, we are given that the combined area of the two smaller circles is 68% of the area of the largest circle. Therefore, the combined area of the two smaller circles is 0.68 * 100π = 68π square meters.

Since the circles are tangent to each other, the combined area of the two smaller circles can be calculated by subtracting the overlap area from the sum of their individual areas.

The area of the overlap can be found using the Pythagorean Theorem. When considering the centers of the circles and the radii, we can form two right triangles. In both triangles, the hypotenuse is the distance between the centers of the smaller circles, which is equal to the sum of their radii, r1 + r2.

By applying the Pythagorean Theorem, we have:
(r1 + r2)^2 = (r1^2) + (r2^2)

Expanding this equation, we get:
r1^2 + 2r1r2 + r2^2 = r1^2 + r2^2

Simplifying, we find:
2r1r2 = 0

Since the radii of the smaller circles cannot be 0 (as they must be greater than 0), we conclude that r1r2 = 0.

This implies that the two smaller circles do not overlap, and their combined area is the sum of their individual areas.

Now, we can set up an equation using the areas of the two smaller circles:
π(r1^2) + π(r2^2) = 68π

Canceling out π from both sides, we get:
r1^2 + r2^2 = 68

We are looking for the product of r1 and r2, which is (r1 * r2).

We can use the identity (a + b)^2 = a^2 + b^2 + 2ab to rewrite the equation as follows:
(r1 + r2)^2 - 2(r1 * r2) = 68

Since we know that (r1 + r2) = 10 (given that the radius of the largest circle is 10 meters), we can substitute this value into the equation:
10^2 - 2(r1 * r2) = 68

Simplifying further, we get:
100 - 2(r1 * r2) = 68

Now, let's solve for (r1 * r2):
2(r1 * r2) = 100 - 68
2(r1 * r2) = 32
r1 * r2 = 16

Therefore, the product of the radii of the smaller circles is 16.