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 probe and the center of Venus. We need to solve for $r$. First, we can re - arrange the formula for $r$: $r^{2}=G\frac{Mm}{F}$, then $r=\sqrt{G\frac{Mm}{F}}$.

Step2: Substitute the given values

We are given that $F = 2.58\times10^{3}\text{ N}$, $G = 6.67\times10^{-11}\text{ N}\cdot\text{m}^{2}/\text{kg}^{2}$, $M = 4.87\times10^{24}\text{ kg}$, and $m = 655\text{ kg}$. Substitute these values into the formula: [ \begin{align*} r&=\sqrt{\frac{6.67\times 10^{-11}\text{ N}\cdot\text{m}^{2}/\text{kg}^{2}\times4.87\times 10^{24}\text{ kg}\times655\text{ kg}}{2.58\times 10^{3}\text{ N}}}\ &=\sqrt{\frac{6.67\times4.87\times655\times10^{-11 + 24}}{2.58\times 10^{3}}\text{ m}^{2}}\ &=\sqrt{\frac{6.67\times4.87\times655\times10^{13}}{2.58\times 10^{3}}\text{ m}^{2}}\ &=\sqrt{\frac{6.67\times4.87\times655}{2.58}\times10^{10}\text{ m}^{2}} \end{align*} ] Calculate $\frac{6.67\times4.87\times655}{2.58}$: $6.67\times4.87 = 32.4829$, $32.4829\times655 = 21276.3995$, $\frac{21276.3995}{2.58}=8246.6665$. So $r=\sqrt{8246.6665\times10^{10}\text{ m}^{2}}\approx9.08\times 10^{6}\text{ m}$

Answer:

$9.08\times 10^{6}$