Find all cube roots of 8 Solve for x and y 3x2 15y2 15 x2

Find all cube roots of 8. Solve for x and y: 3x^2 + 15y^2 = 15 x^2 + 4y^2 = 4 Suppose you are at a river resort and rent a boat for 5 hours starting at 7 am. You are told that the boat will travel 8 mph upstream and 12 mph returning. You decide that you would like to go as far up the river as far as you can and still be back at noon. At what time should you turn back, and how far from the resort will you be at that time?

Solution

17)let the cubeth root of 8 is x

x³ =8

x³=8+0i

x³=8(1+0i)

x³=8(cos0+isin0)

x=(8(cos0+isin0))^1/3,x=(8(cos(0+2)+isin(0+2)))^1/3,x=(8(cos(0+4)+isin(0+4)))^1/3

x=(8(cos0+isin0))^1/3,x=(8(cos(2)+isin(2)))^1/3,x=(8(cos(4)+isin(4)))^1/3

x=2(cos0+isin0)^1/3,x=2(cos(2)+isin(2))^1/3,x=2(cos(4)+isin(4))^1/3

x=2(cos(0_/3)+isin(0_/3)),x=2(cos(2_/3)+isin(2_/3)),x=2(cos(4_/3)+isin(4_/3))

x=2(1+i0),x=2(-(1_/2)+i(3 _/2)),x=2(-(1_/2)-i(3 _/2))

x=2,x=-1+i3 ,x=-1-i3

therefore all cube roots are 2,-1+i3 ,-1-i3

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18) given 3x²+15y²=15---------------->(1)

,x²+4y²=4. multiply by 3 on both sides

=>3x²+12y²=12----------------->(2)

3x²+15y²-3x²-12y²=15-12

3y²=3

y²=1

y =-1 , y =1

x²+4y²=4

x²+4*1=4

x²=0

x=0

so x =0 , y =-1, y=1