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3 years ago

HELP ME THIS IS THE LAST ONE

Mathematics

1 answer:

anzhelika [568]3 years ago

4 0

Answer:

100,000m

Step-by-step explanation:

find the square root of 100 squared +1,000squared

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If x = a cosθ and y = b sinθ , find second derivative

Olin [163]

I'm guessing the second derivative is for <em>y</em> with respect to <em>x</em>, i.e.

$\dfrac{\mathrm d^2y}{\mathrm dx^2}$

Compute the first derivative. By the chain rule,

$\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{\mathrm dy}{\mathrm d\theta}\dfrac{\mathrm d\theta}{\mathrm dx}=\dfrac{\frac{\mathrm dy}{\mathrm d\theta}}{\frac{\mathrm dx}{\mathrm d\theta}}$

We have

$y=b\sin\theta\implies\dfrac{\mathrm dy}{\mathrm d\theta}=b\cos\theta$

$x=a\cos\theta\implies\dfrac{\mathrm dx}{\mathrm d\theta}=-a\sin\theta$

and so

$\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{b\cos\theta}{-a\sin\theta}=-\dfrac ba\cot\theta$

Now compute the second derivative. Notice that $\frac{\mathrm dy}{\mathrm dx}$ is a function of $\theta$ ; so denote it by $f(\theta)$ . Then

$\dfrac{\mathrm d^2y}{\mathrm dx^2}=\dfrac{\mathrm df}{\mathrm dx}$

By the chain rule,

$\dfrac{\mathrm d^2y}{\mathrm dx^2}=\dfrac{\mathrm df}{\mathrm d\theta}\dfrac{\mathrm d\theta}{\mathrm dx}=\dfrac{\frac{\mathrm df}{\mathrm d\theta}}{\frac{\mathrm dx}{\mathrm d\theta}}$

We have

$f=-\dfrac ba\cot\theta\implies\dfrac{\mathrm df}{\mathrm d\theta}=\dfrac ba\csc^2\theta$

and so the second derivative is

$\dfrac{\mathrm d^2y}{\mathrm dx^2}=\dfrac{\frac ba\csc^2\theta}{-a\sin\theta}=-\dfrac b{a^2}\csc^3\theta$

4 0

3 years ago

Solve for x. 8^2x −3 = ( 1 16 )^x − 2 plz help

azamat

Answer:

$\large\boxed{x=\dfrac{17}{10}=1.7}$

Step-by-step explanation:

$8=2^3\\\\\dfrac{1}{16}=\dfrac{1}{2^4}=2^{-4}\qquad\text{used}\ a^{-n}=\dfrac{1}{a^n}\\\\8^{2x-3}=\left(\dfrac{1}{16}\right)^{x-2}\\\\(2^3)^{2x-3}=(2^{-4})^{x-2}\qquad\text{use}\ (a^n)^m=a^{nm}\\\\2^{3(2x-3)}=2^{-4(x-2)}\iff3(2x-3)=-4(x-2)\\\\\text{use the distributive property}\ a(b+c)=ab+ac\\\\(3)(2x)+(3)(-3)=(-4)(x)+(-4)(-2)\\\\6x-9=-4x+8\qquad\text{add 9 to both sides}\\\\6x=-4x+17\qquad\text{add 4x to both sides}\\\\10x=17\qquad\text{divide both sides by 10}\\\\x=1.7$

6 0

3 years ago