Whakaoti i te 20^2+left(1.5tright)^2-left(12+1.5tright)^2/60t=frac{34^2+{left(1.5tright)}^2-{left(30+1.5tright)}^2}{-102t}

$$\frac{ { 20 }^{ 2 } + { \left(1.5t \right) }^{ 2 } - { \left(12+1.5t \right) }^{ 2 } }{ 60t } = \frac{ { 34 }^{ 2 } + { \left(1.5t \right) }^{ 2 } - { \left(30+1.5t \right) }^{ 2 } }{ -102t }$$

Whakaoti mō t

$$t = \frac{32}{7} = 4\frac{4}{7} \approx 4.571428571$$

Ngā Upane mō te Whakaoti Whārite Rārangi

Tē taea kia ōrite te tāupe t ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 1020t, arā, te tauraro pātahi he tino iti rawa te kitea o 60t,-102t.

$$17\left(20^{2}+\left(1.5t\right)^{2}-\left(12+1.5t\right)^{2}\right)=-10\left(34^{2}+\left(1.5t\right)^{2}-\left(30+1.5t\right)^{2}\right)$$

Tātaihia te 20 mā te pū o 2, kia riro ko 400.

$$17\left(400+\left(1.5t\right)^{2}-\left(12+1.5t\right)^{2}\right)=-10\left(34^{2}+\left(1.5t\right)^{2}-\left(30+1.5t\right)^{2}\right)$$

Whakarohaina te \left(1.5t\right)^{2}.

$$17\left(400+1.5^{2}t^{2}-\left(12+1.5t\right)^{2}\right)=-10\left(34^{2}+\left(1.5t\right)^{2}-\left(30+1.5t\right)^{2}\right)$$

Tātaihia te 1.5 mā te pū o 2, kia riro ko 2.25.

$$17\left(400+2.25t^{2}-\left(12+1.5t\right)^{2}\right)=-10\left(34^{2}+\left(1.5t\right)^{2}-\left(30+1.5t\right)^{2}\right)$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(12+1.5t\right)^{2}.

$$17\left(400+2.25t^{2}-\left(144+36t+2.25t^{2}\right)\right)=-10\left(34^{2}+\left(1.5t\right)^{2}-\left(30+1.5t\right)^{2}\right)$$

Hei kimi i te tauaro o 144+36t+2.25t^{2}, kimihia te tauaro o ia taurangi.

$$17\left(400+2.25t^{2}-144-36t-2.25t^{2}\right)=-10\left(34^{2}+\left(1.5t\right)^{2}-\left(30+1.5t\right)^{2}\right)$$

Tangohia te 144 i te 400, ka 256.

$$17\left(256+2.25t^{2}-36t-2.25t^{2}\right)=-10\left(34^{2}+\left(1.5t\right)^{2}-\left(30+1.5t\right)^{2}\right)$$

Pahekotia te 2.25t^{2} me -2.25t^{2}, ka 0.

$$17\left(256-36t\right)=-10\left(34^{2}+\left(1.5t\right)^{2}-\left(30+1.5t\right)^{2}\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 17 ki te 256-36t.

$$4352-612t=-10\left(34^{2}+\left(1.5t\right)^{2}-\left(30+1.5t\right)^{2}\right)$$

Tātaihia te 34 mā te pū o 2, kia riro ko 1156.

$$4352-612t=-10\left(1156+\left(1.5t\right)^{2}-\left(30+1.5t\right)^{2}\right)$$

Whakarohaina te \left(1.5t\right)^{2}.

$$4352-612t=-10\left(1156+1.5^{2}t^{2}-\left(30+1.5t\right)^{2}\right)$$

Tātaihia te 1.5 mā te pū o 2, kia riro ko 2.25.

$$4352-612t=-10\left(1156+2.25t^{2}-\left(30+1.5t\right)^{2}\right)$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(30+1.5t\right)^{2}.

$$4352-612t=-10\left(1156+2.25t^{2}-\left(900+90t+2.25t^{2}\right)\right)$$

Hei kimi i te tauaro o 900+90t+2.25t^{2}, kimihia te tauaro o ia taurangi.

$$4352-612t=-10\left(1156+2.25t^{2}-900-90t-2.25t^{2}\right)$$

Tangohia te 900 i te 1156, ka 256.

$$4352-612t=-10\left(256+2.25t^{2}-90t-2.25t^{2}\right)$$

Pahekotia te 2.25t^{2} me -2.25t^{2}, ka 0.

$$4352-612t=-10\left(256-90t\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te -10 ki te 256-90t.

$$4352-612t=-2560+900t$$

Tangohia te 900t mai i ngā taha e rua.

$$4352-612t-900t=-2560$$

Pahekotia te -612t me -900t, ka -1512t.

$$4352-1512t=-2560$$

Tangohia te 4352 mai i ngā taha e rua.

$$-1512t=-2560-4352$$

Tangohia te 4352 i te -2560, ka -6912.

$$-1512t=-6912$$

Whakawehea ngā taha e rua ki te -1512.

$$t=\frac{-6912}{-1512}$$

Whakahekea te hautanga \frac{-6912}{-1512} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te -216.

$$t=\frac{32}{7}$$