Question

75 points. Will give certified if work is shown and the answer is correct.

Determine the difference between the volumes of two dwarf planets. Where one planet has a radius of 832 mi; while the other has a radius of 829 mi (express your volumes and final answers in terms of
`\pi` )

Answer 1

Volume of the first dwarf planet (r₁ = 832 mi):


`V_1=(4)/(3)\cdot\pi\cdot r_1^3=(4)/(3)\cdot\pi\cdot 832^3=(2303721472)/(3)\pi\approx7.679\cdot10^8\pi\,\text{mi}^3`

Volume of the second dwarf planet (r₂ = 829 mi):


`V_2=(4)/(3)\cdot\pi\cdot r_2^3=(4)/(3)\cdot\pi\cdot 829^3=(2278891156)/(3)\pi\approx7.5963\cdot10^8\pi\,\text{mi}^3`

So difference between the volumes is:


`V_1-V_2\approx7.679\cdot10^8\pi-7.5963\cdot10^8\pi=0.0827\cdot10^8\pi=\boxed{8270000\pi\,\text{mi}^3}`

or if we want exact value (we use (a³-b³) = (a-b)(a²+ab+b²) ):


`V_1-V_2=(4)/(3)\cdot\pi\cdot r_1^3-(4)/(3)\cdot\pi\cdot r_2^3=(4)/(3)\pi(r_1^3-r_2^3)=(4)/(3)\pi(832^3-829^3)=\\\\\\=(4)/(3)\pi(832-829)(832^2+832\cdot829+829^2)=\\\\\\=(4)/(3)\pi\cdot3(692224+689728+687241)=4\pi\cdot2069193=\boxed{8276772\pi\,\text{mi}^3}`