I still can't figure out sig figs... how to deal with volume?

Question

"Jupiter is approximately a sphere of radius $6.99 \times 10^7 \text{ m}$. (c) What is its volume in cubic kilometers? "

I have $V = \frac{4}{3}\pi r^3$. Then, $\frac{4}{3}\pi (6.99 \times 10^7\text{ m})^3$. Then, $2.93 \times 10^{22}\text{ m}^3$. Then, $10^{13}\text{ km}^3$.

But the software says the answer is "1.43e+15." I don't understand.

Answer 1

I've managed to detect your mistake. What you have done when you've inputted it in your calculator is done this: $$\frac{4}{3}\pi\times 6.99\times (10^7)^3\approx 2.9279644\times 10^{22}$$ Instead of the correct one: $$\frac{4}{3}\pi\times (6.99\times 10^7)^3\approx 1.4306063\times 10^{24}$$

Be sure to put parentheses where necessary. If you still cannot get this result on your calculator, please let me know.

Answer 2

One of the steps requires you to calculate $(6.99 \times 10^7)^3$. It appears that you calculated $6.99 \times (10^7)^3$ instead, which is very different.

Answer 3

This time I will try to break down the problem a little bit more clearly than with the area example, to hopefully help you understand this. I would highly recommend checking out some simple manipulation examples, for starters, Khan Academy on DA, Khan Academy on Sig. Figs, and Khan Academy on Scientific Notation. I think you need to work on your baseline understanding before attempting these results.

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\begin{align} \frac{4}{3} \pi r^3 &= \frac{4}{3} \cdot \pi \cdot (6.99 \times 10^7 m )^3 \\ &= \frac{4}{3}\pi \cdot (6.99)^3 \times (10^7)^3 \times (m)^3 \\ &= \frac{4}{3}\pi \cdot (6.99 \cdot 6.99 \cdot 6.99) \times (10^{7 \cdot 3} )\times (m \cdot m \cdot m )\\ &= 1430 \times (10^{21}) \times (m\cdot m\cdot m) \\ &= (1.43 \times 10^3) \times 10^{21} \times (m\cdot m\cdot m) \\ &= 1.43 \times 10^{3 + 21}\times (m \cdot m\cdot m)\\ &= 1.43 \times 10^{24} \times m\cdot m \cdot m\\ &= 1.43 \times 10^{24} \times m\cdot m \cdot m \times (\frac{1000m}{1000m}) \\ &= 1.43 \times 10^{24} \times m\cdot m \cdot m \times (\frac{1km}{1000m}) \\ &= 1.43 \times 10^{24} \times m\cdot m \cdot m \times (\frac{1km}{1000m} \cdot \frac{1km}{1000m} \cdot \frac{1km}{1000m}) \\ &= 1.43 \times 10^{24} \times (\frac{1km \cdot m}{1000m} \cdot \frac{1km\cdot m}{1000m} \cdot \frac{1km\cdot m}{1000m}) \\ &= 1.43 \times 10^{24} \times (\frac{1km}{1000} \cdot \frac{1km}{1000} \cdot \frac{1km}{1000}) \\ &= 1.43 \times 10^{24} \times (\frac{1km}{10^3} \cdot \frac{1km}{10^3} \cdot \frac{1km}{10^3}) \\ &= 1.43 \times 10^{24} \times \frac{km^3}{(10^3)^3} \\ &= 1.43 \times \frac{10^{24}}{(10^3)^3} \times km^3 \\ &= 1.43 \times \frac{10^{24}}{10^9} \times km^3 \\ &= 1.43 \times 10^{24 - 9} \times km^3 \\ &= 1.43 \times 10^{15} km^3 \end{align}