Question

Знайдіть радіуси двох кругів, які мають зовнішній дотик, якщо їх площі відносяться як 4 : 25, а відстань між центрами дорівнює 14 см.​

Answer 1

Ответ:

4см і 10см.

Объяснение:

S=pi×r²

x - коефіцієнт пропорційності

S1/S2

(pi×r1²)/(pi×r2²)=4/25

r1²/r2²=4/25

r1/r2=2/5

2×r2=5×r1

r2=2,5r1

оскільки дотик зовнішній, то радіуси додаються

r1+r2=14см

r1+2,5r1=14см

3,5r1=14см

r1=4см

r2=2,5×4=10см.

Answer 2

Calculating the Radii of Circles with Tangent and Given Area Ratio

To find the radii of two circles that have an external tangent and a given ratio of areas, we can use the following approach:

1. Understand the Given Information: - The ratio of the areas of the two circles is given as 4:25. - The distance between the centers of the circles is 14 cm.

2. Calculate the Radii: - Let's assume the radii of the circles are r1 and r2. - The ratio of the areas of the circles is equal to the square of the ratio of their radii. This means: \[ \frac{{\text{Area of circle 1}}}{{\text{Area of circle 2}}} = \frac{{\pi \times r1^2}}{{\pi \times r2^2}} = \frac{4}{25} \] - From the given ratio, we can express the radius of the second circle in terms of the radius of the first circle: \[ r2 = \sqrt{\frac{25}{4}} \times r1 = \frac{5}{2} \times r1 \]

3. Use the Distance Between Centers: - The distance between the centers of the circles is 14 cm. Using this information, we can form an equation involving the radii: \[ r1 + r2 = 14 \]

4. Solve for the Radii: - Substituting the expression for \( r2 \) from the area ratio into the equation involving the distance between centers, we get: \[ r1 + \frac{5}{2} \times r1 = 14 \] \[ \frac{7}{2} \times r1 = 14 \] \[ r1 = \frac{2}{7} \times 14 \] \[ r1 = 4 \, \text{cm} \] - Using the relationship between \( r1 \) and \( r2 \), we can find \( r2 \): \[ r2 = \frac{5}{2} \times r1 = \frac{5}{2} \times 4 = 10 \, \text{cm} \]

Answer:

The radius of the first circle is 4 cm, and the radius of the second circle is 10 cm.