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English Senior High

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We are,(to a remarkable degree, the right distance from the right sort of star, one e 5 of ten billion and we wouldn't be here now./ We are also fortunate to orbit where we that is big enough to radiate lots of energy, but not so big as to burn itself out swiftly t 1s a curiosity bf physics that the larger a stor the more rapidly it burns. Had our sun Ocen ten times as massive、it would have evhonsted itself after ten million years instead of do. 1o0 much nearer and evervthing on Farth would have boiled away. Much rarther away and everything would have frozen. の14 m 1978, an astrophysicist named Micheel Hart made some calculations and Concluded that Earth would have been uninhabitable had it been just 1 percent rartner That's not much, and in fact it wasn't enough. percent 10 from or 5.percent closer to the Sun. The figures have since been refined and made a little more generous 5 nearer and I5 percent farther are thought to be more accurate assessments 1oI om zone of habitability - but that is still a narrow belt. To appreciate just how narrow, you have only to look at Venus. Venus 1s only ©10 15 twenty-five million miles closer to the Sun than we are. The Sun's warmth reaches it just two minutes before it touches us. In size and composition, Venus is very like Earth, but the small difference in orbital distance made all the difference to (3)how it turned out. It appears that during the early years of the solar system Venus was only slightly warmer than Earth and probably had oceans. But those few degrees of extra 20 warmth meant that Venus could not hold on to its surface water, with disastrous consequences for its climate. As its water evaporated, the hydrogen atoms escaped into space, and the oxygen atoms combined with carbon to form a dense atmosphere of the greenhouse gas CO2. Venus became stifling. Although people of my age will recall a time when astrononmers hoped that Venus might harbor life beneath its padded 25 clouds, possibly even a kind of tropical vegetation, we now know that it is much too fierce an environment for any kind of life that we can reasonably conceive of. Its surface temperature is a roasting 470 degrees centigrade (roughly 900 degrees Fahrenheit), which is hot enough to melt lead, and the atmospheric pressure at the surface is ninety times that of Earth, or more than any human body could withstand We lack the technology to make suits or even spaceships that would allow us to visit Our knowledge of Venus's surface is based on distant radar imagery and som。 disturbing noise from an unmanned Soviet probe that was dropped hopefully into the

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Mathematics Senior High

f(x)のところはfじゃなきゃダメですか? P(x)で置いても大丈夫ですか?

100 基本例題63 解から係数決定(虚数解) OOO00 3次方程式 x°+ax+bx+10=0 の1つの解が x=2+i であるとき、史 の定数 a, bの値と他の解を求めよ。 (山梨学院大) p.94 基本事項2, 基本 62 AOIRUNI CHART OSOLUTION x=α がf(x)30 の解 → f(α)=0 代入する解は1個 (x=D2+i) で, 求める値は2個 (aとb)であるが, 複素数の相等 A, Bが実数のとき A+Bi=0 A30 かつ B=0 により, a, bに関する方程式は2つできるから, a, bの値を求めることができる。 また,実数を係数とする n次方程式が虚数解 α をもつとき, 共役な複素数αも 解であることを用いて, 次のように解いてもよい。 別解1,2 αとαが解であるから,方程式の左辺は(x-α)(x-α) すなわち x°-(α+a)x+aa で割り切れることを利用する。 3つ目の解をんとして, 3次方程式の解と係数の関係を利用する。 別解3 解答 |inf. x-2=iと変形して 両辺を2乗すると x°-4x+5=0 x=2+i がこの方程式の解であるから (2+)°+a(2+)?+6(2+i)+10=0 ここで,(2+i)=2°+3·2°%+3·2ポ+パ=2+11i, 81=D6 これを利用して (2+)°=2°+2.2i+ぴ=3+4i であるから ー +ax+bx+10 の次数を 2+11i+a(3+4i)+6(2+)+10=0 ( ( 下げる方法(別解1の3行 0+x1-(目以降と同じ)もある。 8 とすると、 他方 iについて整理すると (b.89 基本例題56 参照) 3a+26+12+(4a+b+11)i=0 3a+26+12, 4a+b+11 は実数であるから 全この断り書きは重要。 A, Bが実数のとき 3a+26+12=0, 4a+b+11=0 0ヶ預の a=-2, b=-3 x°-2x°-3x+10=0 A+Bi=0 これを解いて ゆえに,方程式は f(x)=x°-2x?2_3x+10 とすると C-x)(1-3)- → A=0 かつ B=0 こる 開題国 f(-2)=(-2)°-2·(-2)?-3·(-2)+10=0 =-IS よって,f(x)は x+2 を因数にもつから s-ー )-合益立除法 f(x)=(x+2)(x?2_4x+5) 10 -2 1 -2 -3 8-=o 81=d -2 8 -10 したがって,方程式は (x+2)(x°-4x+5)=0 x+2=0 または x°-4x+5=0 ゆえに 1 -4 5 0 x2-4x+5=0 を解くと x=2±i よって, 他の解は x=-2, 2-i

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