Question

a nasa deep space probe has an observed deviation from it’s predicted path of $5.5936 \times 10^{-10} \frac{m}{s^2}$ while the pioneer 10 and 11 spacecrafts have an observed deviation of $8.74 \times 10^{-10} \frac{m}{s^2}$. compared to the pioneer spacecraft’s deviation, how many times greater are the deep space probe deviations?
the deep space probe deviations are \boxed{} times more than the pioneer spacecraft’s.
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Explanation:

Step1: Identify the values

Let \( D = 5.5936\times10^{- 10}\frac{m}{s^{2}} \) (deep space probe deviation) and \( P=8.74\times10^{-10}\frac{m}{s^{2}} \) (Pioneer deviation). Wait, no, wait. Wait, the question is "how many times greater are the deep space probe deviations compared to Pioneer?". Wait, no, wait, maybe I mixed up. Wait, the deep space probe's deviation is \( 5.5936\times 10^{-10}\) and Pioneer's is \( 8.74\times 10^{-10}\)? Wait, no, that can't be. Wait, maybe I read it wrong. Wait, the problem says: deep space probe has \( 5.5936\times 10^{-10}\frac{m}{s^{2}} \), Pioneer has \( 8.74\times 10^{-10}\frac{m}{s^{2}} \). Wait, but if we want to find how many times greater deep space probe is than Pioneer, we divide deep space by Pioneer. Wait, but \( 5.5936 < 8.74 \), that would be less. Wait, maybe I mixed up the numbers. Wait, no, let's check again.

Wait, the problem: "how many times greater are the deep space probe deviations compared to the Pioneer spacecraft’s deviation". So we need to compute \( \frac{\text{Deep Space Probe Deviation}}{\text{Pioneer Deviation}} \)

So \( \text{Deep Space Probe Deviation} = 5.5936\times 10^{-10}\frac{m}{s^{2}} \)

\( \text{Pioneer Deviation} = 8.74\times 10^{-10}\frac{m}{s^{2}} \)

Wait, but that would be \( \frac{5.5936\times 10^{-10}}{8.74\times 10^{-10}} \). The \( 10^{-10} \) cancels out. So \( \frac{5.5936}{8.74} \)

Wait, let's calculate that. \( 5.5936\div8.74 \). Let's do the division:

\( 5.5936\div8.74 \). Let's multiply numerator and denominator by 10000 to eliminate decimals: \( 55936\div87400 \). Let's divide numerator and denominator by 4: \( 13984\div21850 \). Divide by 2: \( 6992\div10925 \approx 0.64 \)? Wait, that can't be. Wait, maybe I mixed up the numbers. Wait, maybe the deep space probe is \( 8.74\times 10^{-10} \) and Pioneer is \( 5.5936\times 10^{-10} \)? No, the problem says: "deep space probe has an observed deviation... \( 5.5936\times 10^{-10}\frac{m}{s^{2}} \) while the Pioneer 10 and 11 spacecrafts have an observed deviation of \( 8.74\times 10^{-10}\frac{m}{s^{2}} \)". Wait, that would mean Pioneer's deviation is larger. So the question is "how many times greater are the deep space probe deviations compared to the Pioneer spacecraft’s deviation" – maybe it's a typo, or maybe I misread. Wait, no, maybe the deep space probe is \( 8.74\times 10^{-10} \) and Pioneer is \( 5.5936\times 10^{-10} \)? Wait, let's check the numbers again.

Wait, the user's problem: "A NASA deep space probe has an observed deviation from it’s predicted path of \( 5.5936\times 10^{-10}\frac{m}{s^{2}} \) while the Pioneer 10 and 11 spacecrafts have an observed deviation of \( 8.74\times 10^{-10}\frac{m}{s^{2}} \). Compared to the Pioneer spacecraft’s deviation, how many times greater are the deep space probe deviations?"

Wait, that would be \( \frac{5.5936\times 10^{-10}}{8.74\times 10^{-10}} = \frac{5.5936}{8.74} \). Let's compute that:

\( 5.5936 \div 8.74 \). Let's do this division:

\( 8.74 \times 0.64 = 8.74 \times 0.6 + 8.74 \times 0.04 = 5.244 + 0.3496 = 5.5936 \). Oh! So \( 5.5936\div8.74 = 0.64 \)? Wait, but that would mean deep space probe's deviation is 0.64 times that of Pioneer, which is smaller. But the question says "how many times greater", which is confusing. Maybe the numbers are reversed? Maybe deep space probe is \( 8.74\times 10^{-10} \) and Pioneer is \( 5.5936\times 10^{-10} \)? Let's check: \( 8.74\div5.5936 \approx 1.56 \)? No, wait, \( 5.5936\times1.56 \approx 8.72 \), close to 8.74. Wait, but according to the problem, deep space is 5.5936 and Pioneer is 8.74. So the calculation is \( \frac{5.5936}{8.74} = 0.64 \). Wait, but maybe the question is "how many times smaller", but it says "greater". Maybe a mistake in the problem, but let's follow the numbers.

So Step 1: Divide the deep space probe's deviation by Pioneer's deviation.

\( \text{Ratio} = \frac{5.5936\times 10^{-10}}{8.74\times 10^{-10}} \)

Step 2: Simplify the \( 10^{-10} \) terms: \( \frac{5.5936}{8.74} \times \frac{10^{-10}}{10^{-10}} = \frac{5.5936}{8.74} \times 1 = \frac{5.5936}{8.74} \)

Step 3: Calculate the division: \( 5.5936 \div 8.74 = 0.64 \)

Wait, but that's less than 1. So maybe the problem has the numbers reversed. But according to the given numbers, that's the result.

Answer:

\( 0.64 \)