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lawyer [7]
3 years ago
13

A man wants to take a goat, a bag of cabbage, and a wolf over to an island. His boat will only hold him and one animal or item.

If the goat is left with the cabbage, he’ll eat it. If the wolf is left with the goat, he’ll eat it. How can the man transport all three to the island without anything being eaten?
Mathematics
1 answer:
Paul [167]3 years ago
5 0

Answer:

The wolf does not eat cabbage, so the crossing can start with the goat.

The man leaves the goat and returns, puts the cabbage in the boat and

takes it across. On the other bank, he leaves the cabbage but takes the

goat.He leaves the goat on the first bank and takes the wolf across. He leaves the cabbage with the wolf and rows back alone.He takes the goat across.

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If <img src="https://tex.z-dn.net/?f=%5Cmathrm%20%7By%20%3D%20%28x%20%2B%20%5Csqrt%7B1%2Bx%5E%7B2%7D%7D%29%5E%7Bm%7D%7D" id="Tex
Harman [31]

Answer:

See below for proof.

Step-by-step explanation:

<u>Given</u>:

y=\left(x+\sqrt{1+x^2}\right)^m

<u>First derivative</u>

\boxed{\begin{minipage}{5.4 cm}\underline{Chain Rule for Differentiation}\\\\If  $f(g(x))$ then:\\\\$\dfrac{\text{d}y}{\text{d}x}=f'(g(x))\:g'(x)$\\\end{minipage}}

<u />

<u />\boxed{\begin{minipage}{5 cm}\underline{Differentiating $x^n$}\\\\If  $y=x^n$, then $\dfrac{\text{d}y}{\text{d}x}=xn^{n-1}$\\\end{minipage}}

<u />

\begin{aligned} y_1=\dfrac{\text{d}y}{\text{d}x} & =m\left(x+\sqrt{1+x^2}\right)^{m-1} \cdot \left(1+\dfrac{2x}{2\sqrt{1+x^2}} \right)\\\\ & =m\left(x+\sqrt{1+x^2}\right)^{m-1} \cdot \left(1+\dfrac{x}{\sqrt{1+x^2}} \right) \\\\ & =m\left(x+\sqrt{1+x^2}\right)^{m-1} \cdot \left(\dfrac{x+\sqrt{1+x^2}}{\sqrt{1+x^2}} \right)\\\\ & = \dfrac{m}{\sqrt{1+x^2}} \cdot \left(x+\sqrt{1+x^2}\right)^{m-1}  \cdot \left(x+\sqrt{1+x^2}\right)\\\\ & = \dfrac{m}{\sqrt{1+x^2}}\left(x+\sqrt{1+x^2}\right)^m\end{aligned}

<u>Second derivative</u>

<u />

\boxed{\begin{minipage}{5.5 cm}\underline{Product Rule for Differentiation}\\\\If  $y=uv$  then:\\\\$\dfrac{\text{d}y}{\text{d}x}=u\dfrac{\text{d}v}{\text{d}x}+v\dfrac{\text{d}u}{\text{d}x}$\\\end{minipage}}

\textsf{Let }u=\dfrac{m}{\sqrt{1+x^2}}

\implies \dfrac{\text{d}u}{\text{d}x}=-\dfrac{mx}{\left(1+x^2\right)^\frac{3}{2}}

\textsf{Let }v=\left(x+\sqrt{1+x^2}\right)^m

\implies \dfrac{\text{d}v}{\text{d}x}=\dfrac{m}{\sqrt{1+x^2}} \cdot \left(x+\sqrt{1+x^2}\right)^m

\begin{aligned}y_2=\dfrac{\text{d}^2y}{\text{d}x^2}&=\dfrac{m}{\sqrt{1+x^2}}\cdot\dfrac{m}{\sqrt{1+x^2}}\cdot\left(x+\sqrt{1+x^2}\right)^m+\left(x+\sqrt{1+x^2}\right)^m\cdot-\dfrac{mx}{\left(1+x^2\right)^\frac{3}{2}}\\\\&=\dfrac{m^2}{1+x^2}\cdot\left(x+\sqrt{1+x^2}\right)^m+\left(x+\sqrt{1+x^2}\right)^m\cdot-\dfrac{mx}{\left(1+x^2\right)\sqrt{1+x^2}}\\\\ &=\left(x+\sqrt{1+x^2}\right)^m\left(\dfrac{m^2}{1+x^2}-\dfrac{mx}{\left(1+x^2\right)\sqrt{1+x^2}}\right)\\\\\end{aligned}

              = \dfrac{\left(x+\sqrt{1+x^2}\right)^m}{1+x^2}\right)\left(m^2-\dfrac{mx}{\sqrt{1+x^2}}\right)

<u>Proof</u>

  (x^2+1)y_2+xy_1-m^2y

= (x^2+1) \dfrac{\left(x+\sqrt{1+x^2}\right)^m}{1+x^2}\left(m^2-\dfrac{mx}{\sqrt{1+x^2}}\right)+\dfrac{mx}{\sqrt{1+x^2}}\left(x+\sqrt{1+x^2}\right)^m-m^2\left(x+\sqrt{1+x^2\right)^m

= \left(x+\sqrt{1+x^2}\right)^m\left(m^2-\dfrac{mx}{\sqrt{1+x^2}}\right)+\dfrac{mx}{\sqrt{1+x^2}}\left(x+\sqrt{1+x^2}\right)^m-m^2\left(x+\sqrt{1+x^2\right)^m

= \left(x+\sqrt{1+x^2}\right)^m\left[m^2-\dfrac{mx}{\sqrt{1+x^2}}+\dfrac{mx}{\sqrt{1+x^2}}-m^2\right]

= \left(x+\sqrt{1+x^2}\right)^m\left[0]

= 0

8 0
1 year ago
M is the midpoint of AB. A is at (-5, 1) and M is (-1,4). Find the coordinates of B. Select one: O a (-3.2.5) O b.(-1.4) O c. (2
coldgirl [10]

Answer:

A (-3, 2.5)

Step-by-step explanation:

Midpoint formula is m=(x1 + x2/2), (y1 + y2/2)

So add the xs: -5 + -1 = -6

And the ys: 1 + 4 = 5

Then divide them by 2

(-3, 2.5)

4 0
3 years ago
Help me pleaseeeessss
wlad13 [49]

Answer:

28/53

Step-by-step explanation:

Hey There!

So they want us to find the sine of angle S

well remember is sohcahtoa

S - sine

O - opposite

H - hypotenuse

meaning that

Sin=\frac{opposite}{hypotenuse}

The opposite side of angle S is 28 and the hypotenuse is 53

so sinS=\frac{28}{53}

5 0
2 years ago
A dartboard has a diameter of 18 inches.What are its radius and circumference?
allsm [11]
Diameter of the dartboard as given in the question = 18 inches
Then
Radius of the dartboard will be = 18/2 inches
                                                  = 9 inches
We already know that
Circumference of a circle = <span>2π</span>r
We also know that the value of <span>π = 3.14159
Then
Circumference of the dartboard = 2 * 3.14159 * 9 inches
                                                    = 56.548 inches
                                                    = 56.55 inches
So the radius of the dartboard is 9 inches and the circumference of the dartboard is 56.55 inches. I hope the procedure is absolutely clear to you.
</span>
3 0
3 years ago
from a group of 7 people, you randomly select 2 of them. what is the probability that they are the 2 of the oldest people in the
spayn [35]

Answer:

1/21

Step-by-step explanation:

3 0
3 years ago
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