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Question

The diameter of Mars is about 6800 km. Approximate the surface area of Mars using the formula for the surface area of a sphere, S=4πr2S=4 \pi r^{2}. Write your answer in scientific notation.

Solution

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Using the given formula, S=4πr2,S=4\pi r^2 , with a diamter of 6.8×1036.8\times10^3 kmkm, then

S=4π(6.8×1032)2( r a d i u s = d i a m e t e r 2)=4π(3.4×103)2=4π(3.41(2)×103(2))( u s e (ambn)p=ampbnp)=4π(3.42×106)=4π(11.56×106)145.27×1061.4527×102×1061.4527×102+61.4527×108.\begin{align*} S&=4\pi \qty(\dfrac{6.8\times10^3}{2})^2 &\qty(\text{radius }=\dfrac{\text{diameter}}{2}) \\\\&= 4\pi \qty(3.4\times10^3)^2 \\&= 4\pi \qty(3.4^{1(2)}\times10^{3(2)}) &\qty(\text{use }\qty(a^mb^n)^p=a^{mp}b^{np}) \\&= 4\pi \qty(3.4^{2}\times10^{6}) \\&= 4\pi \qty(11.56\times10^{6}) \\&\approx 145.27\times10^{6} \\&\approx 1.4527\times10^2\times10^{6} \\&\approx 1.4527\times10^{2+6} \\&\approx 1.4527\times10^{8} .\end{align*}

Hence, the surface area, S,S, of Mars is approximately 1.4527×1081.4527\times10^{8} square kilometers.

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