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3 years ago

A line has the equation y = 3x + 4.

Mathematics

1 answer:

oksano4ka [1.4K]3 years ago

7 0

Answer:

b. 3/1 c. -1/3

Step-by-step explanation:

When a line is parallel to another they don't intersect and instead go in the same direction. Therefore they have the same slope

b. 3/1

When a line is perpendicular it has to be the negative reciprocal. This means that it you have to flip the numbers in the fraction and make it negative (unless it is already negative, then it becomes positive)

c. -1/3

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Converting to polar coordinates will help make this computation easier. Set

$\begin{cases}x=\rho\cos\theta\sin\varphi\\y=\rho\sin\theta\sin\varphi\\z=\rho\cos\var\phi\end{cases}\implies\mathrm dx\,\mathrm dy\,\mathrm dz=\rho^2\sin\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi$

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$\displaystyle\iiint_R\mathrm dV=\int_0^{2\pi}\int_0^{\arctan\frac{\sqrt{r^2-h^2}}h}\int_{h\sec\varphi}^r\rho^2\sin\varphi\,\mathrm d\rho\,\mathrm d\varphi\,\mathrm d\theta$

You should get

$\dfrac{2\pi}3\left(r^3\arctan\dfrac{\sqrt{r^2-h^2}}h-\dfrac{h^3}2\left(\dfrac{r\sqrt{r^2-h^2}}{h^2}+\ln\dfrac{r+\sqrt{r^2-h^2}}h\right)\right)$

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