a probe orbiting venus exerts a gravitational force of 2.58×10³ n on venus. venus has a mass of 4.87×10²⁴…

Explanation:

Step1: Recall gravitational - force formula

The gravitational - force formula is $F = G\frac{Mm}{r^{2}}$, where $F$ is the gravitational force, $G$ is the gravitational constant, $M$ is the mass of Venus, $m$ is the mass of the probe, and $r$ is the distance between the center of Venus and the probe. We need to solve for $r$. First, we can re - arrange the formula for $r$: $r^{2}=G\frac{Mm}{F}$, so $r=\sqrt{G\frac{Mm}{F}}$.

Step2: Substitute the given values

Given $F = 2.58\times10^{3}\ N$, $G = 6.67\times10^{-11}\ N\cdot m^{2}/kg^{2}$, $M = 4.87\times10^{24}\ kg$, and $m = 655\ kg$. Substitute these values into the formula: [ \begin{align*} r&=\sqrt{\frac{6.67\times 10^{-11}\times4.87\times 10^{24}\times655}{2.58\times 10^{3}}}\ &=\sqrt{\frac{6.67\times4.87\times655\times10^{-11 + 24}}{2.58\times 10^{3}}}\ &=\sqrt{\frac{6.67\times4.87\times655\times10^{13}}{2.58\times 10^{3}}}\ &=\sqrt{\frac{6.67\times4.87\times655}{2.58}\times10^{13 - 3}}\ &=\sqrt{\frac{6.67\times4.87\times655}{2.58}\times10^{10}} \end{align*} ] Calculate $\frac{6.67\times4.87\times655}{2.58}$: $6.67\times4.87 = 32.5829$, $32.5829\times655 = 21341.8995$, $\frac{21341.8995}{2.58}\approx8272$. So $r=\sqrt{8272\times10^{10}}=\sqrt{8272}\times10^{5}\approx91\times10^{5}=9.10\times 10^{6}\ m$.

Answer:

$9.10$