Question

√R²- 4.5² + √R² - 10.5²=64

Answer 1

√R²-4,5²+√R²-10,5²=64

R-20,25+R-110,25-64=0

2R-194,5=0

2R=194,5

R=97,25

или если условие такое, то решение

√(R²-4,5²)+√(R²-10,5)=64

(R-4,5)+(R-10,5)=64

2R=64+4,5+10,5

2R=79

R=39,5

Answer 2

To solve the equation √(R² - 4.5²) + √(R² - 10.5²) = 64, you can follow these steps:

1. Start by isolating one of the square roots on one side of the equation.

   √(R² - 4.5²) = 64 - √(R² - 10.5²)

2. Square both sides of the equation to eliminate the square roots.

   (R² - 4.5²) = (64 - √(R² - 10.5²))²

3. Expand the right side of the equation.

   R² - 4.5² = 64² - 2 * 64 * √(R² - 10.5²) + (R² - 10.5²)

4. Now, isolate the term with the square root:

   2 * 64 * √(R² - 10.5²) = 64² - R² + 10.5² - R² + 4.5²

5. Square both sides of this equation to eliminate the square root:

   4 * (64²) * (R² - 10.5²) = (64² - R² + 10.5² - R² + 4.5²)²

6. Simplify both sides:

   4 * 64⁴ * (R² - 10.5²) = (64² - R² + 10.5² - R² + 4.5²)²

7. Divide both sides by 4 * 64⁴:

   R² - 10.5² = [(64² - R² + 10.5² - R² + 4.5²)²] / (4 * 64⁴)

8. Now, simplify the right side:

   R² - 110.25 = [(64² - 2R² + 10.5² + 4.5²)²] / (4 * 64⁴)

9. Continue simplifying:

   R² - 110.25 = [(64² - 2R² + 115.5²)²] / (4 * 64⁴)

10. Expand the square on the right side:

    R² - 110.25 = [4096² - 2 * 64² * R² + 2 * 115.5² * 64² - 2 * 115.5² * 64² + 2R² * 115.5² - 4R⁴] / (4 * 64⁴)

11. Simplify further:

    R² - 110.25 = [4096² - 2 * 64² * R² - 2 * 115.5² * 64² + 2R² * 115.5² - 4R⁴] / (4 * 64⁴)

12. Combine like terms:

    R² - 110.25 = [4096² - 2 * 64² * R² - 2 * 115.5² * 64² - 4R⁴] / (4 * 64⁴)

13. Now, isolate the R² terms on one side:

    R² - R² - 110.25 = [4096² - 2 * 64² * R² - 2 * 115.5² * 64² - 4R⁴] / (4 * 64⁴)

14. Simplify:

    -110.25 = [4096² - 2 * 64² * R² - 2 * 115.5² * 64² - 4R⁴] / (4 * 64⁴)

15. Multiply both sides by 4 * 64⁴ to isolate the right side:

    -110.25 * 4 * 64⁴ = 4096² - 2 * 64² * R² - 2 * 115.5² * 64² - 4R⁴

16. Calculate the left side:

    -110.25 * 4 * 64⁴ ≈ -51273728

17. Rewrite the equation:

    -51273728 = 4096² - 2 * 64² * R² - 2 * 115.5² * 64² - 4R⁴

18. Move everything to one side:

    4R⁴ - 2 * 64² * R² - 2 * 115.5² * 64² - 4096² + 51273728 = 0

19. Now, you have a quartic equation in terms of R. This equation can be challenging to solve analytically. You may need to use numerical methods or a calculator to find its approximate solutions.

Please note that the solutions to this equation might involve complex numbers, as quartic equations can have complex roots. Numerical methods like the Newton-Raphson method or using a graphing calculator/computer software would be useful for finding approximate solutions.