Tropical, mild and arctic

How many percent of the Earth's surface lies in the tropical, mild, and arctic range? The border between the ranges is the parallel 23°27' and 66°33'.

Correct answer:

p₁ =  8.2592 %
p₂ =  51.9459 %
p₃ =  39.7949 %

Step-by-step explanation:

$$\begin{gathered} A=23+27\mathrm{/}60={\displaystyle \frac{469}{20}}=23.45\text{ ° } \\ B=66+33\mathrm{/}60={\displaystyle \frac{1331}{20}}=66.55\text{ ° } \\ r=6371\text{ km } \\ {h}_1=r-r\cdot \cos (A\mathrm{°}\to \text{ rad})=r-r\cdot \cos (A\mathrm{°}\cdot {\displaystyle \frac{\pi }{180}})=6371-6371\cdot \cos (23.45\mathrm{°}\cdot {\displaystyle \frac{3.1415926}{180}})=526.19555\text{ km } \\ {h}_2=r-r\cdot \cos (B\mathrm{°}\to \text{ rad})=r-r\cdot \cos (B\mathrm{°}\cdot {\displaystyle \frac{\pi }{180}})=6371-6371\cdot \cos (66.55\mathrm{°}\cdot {\displaystyle \frac{3.1415926}{180}})=3835.66927\text{ km } \\ S=4\pi \cdot r^2\mathrm{/}2=4\cdot 3.1416\cdot 6371^2\mathrm{/}2\doteq 255032235.955{\text{km}}^2 \\ {S}_1=2\pi \cdot r\cdot h_1=2\cdot 3.1416\cdot 6371\cdot 526.1955\doteq 21063699.1055{\text{km}}^2 \\ {S}_2=2\pi \cdot r\cdot h_2=2\cdot 3.1416\cdot 6371\cdot 3835.6693\doteq 153542506.717{\text{km}}^2 \\ {p}_1=100\cdot {\displaystyle \frac{S_1}{S}}=100\cdot {\displaystyle \frac{21063699.1055}{255032235.955}}=8.2592\mathrm{\%} \end{gathered}$$

$$\begin{gathered} S_3=S_2-S_1=153542506.717-21063699.1055=132478807.611{\text{km}}^2 \\ {p}_2=100\cdot {\displaystyle \frac{S_3}{S}}=100\cdot {\displaystyle \frac{132478807.611}{255032235.955}}=51.9459\mathrm{\%} \end{gathered}$$

$p_3=100-(p_1+p_2)=100-(8.2592+51.9459)=39.7949\mathrm{\%}$

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