\( \left. \begin{array} { l } { \frac { 9 \times 10 ^ { 9 } \times 3 \times 10 ^ { - 12 } } { ( 0,15 - D ) ^ { 2 } } = \frac { 9 \times 10 ^ { 9 } \times 7 \times 10 ^ { - 12 } } { D ^ { 2 } } } \\ { = 0,091 m \checkmark } \end{array} \right. \)

To solve the equation given in the problem, we start with the two expressions set equal to each other: \[ \frac{9 \times 10^9 \times 3 \times 10^{-12}}{(0.15 - D)^2} = \frac{9 \times 10^9 \times 7 \times 10^{-12}}{D^2} \] We can simplify this equation by canceling out the common factors on both sides. The \(9 \times 10^9\) and \(10^{-12}\) terms can be canceled: \[ \frac{3}{(0.15 - D)^2} = \frac{7}{D^2} \] Next, we cross-multiply to eliminate the fractions: \[ 3D^2 = 7(0.15 - D)^2 \] Now, we expand the right side: \[ 3D^2 = 7(0.0225 - 0.3D + D^2) \] Distributing the 7 gives: \[ 3D^2 = 0.1575 - 2.1D + 7D^2 \] Now, we can rearrange the equation to bring all terms to one side: \[ 3D^2 - 7D^2 + 2.1D - 0.1575 = 0 \] This simplifies to: \[ -4D^2 + 2.1D - 0.1575 = 0 \] Multiplying through by -1 to make the leading coefficient positive: \[ 4D^2 - 2.1D + 0.1575 = 0 \] Now we can use the quadratic formula \(D = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 4\), \(b = -2.1\), and \(c = 0.1575\). Calculating the discriminant: \[ b^2 - 4ac = (-2.1)^2 - 4 \cdot 4 \cdot 0.1575 \] \[ = 4.41 - 2.52 = 1.89 \] Now we can find \(D\): \[ D = \frac{-(-2.1) \pm \sqrt{1.89}}{2 \cdot 4} \] \[ = \frac{2.1 \pm 1.375}{8} \] Calculating the two possible values for \(D\): 1. \(D = \frac{2.1 + 1.375}{8} = \frac{3.475}{8} \approx 0.434375\) 2. \(D = \frac{2.1 - 1.375}{8} = \frac{0.725}{8} \approx 0.090625\) Since the problem states that \(D = 0.091 m\) is a solution, we can conclude that: \[ D \approx 0.091 \, \text{m} \checkmark \] Thus, the solution is confirmed.