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3 years ago

The difference of twelve times a number and nine times a number

Mathematics

2 answers:

inna [77]3 years ago

6 0

Since the problem states "a number" and again "a number", I assume it;'s the same number. Let the number be x.

12x - 9x = 3x

The difference of twelve times a number and nine times a number is three times the number.

Crank3 years ago

6 0

12x-9y

Difference means subtraction, so the difference between 12 times a number is known as 12x because we do not know what is being multiplied by 12. Same with 9 times a number. Since 12 and 9 have two numbers that we don't know the value of, we have to change one of them to a different variable because we do not know if it is the same number or not. So it would be 12x-9y, or any letters you choose.

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MATH HELP!!!!! If y = 15 when x = 2, what is the value of x when y = 7?

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To answer this item, we have 4 as the speed of the kayaker in still water and the speed of current be y.

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and consider a plane

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$\displaystyle\iiint_R\mathrm dV=\int_{-\sqrt{r^2-h^2}}^{\sqrt{r^2-h^2}}\int_{-\sqrt{r^2-h^2-x^2}}^{\sqrt{r^2-h^2-x^2}}\int_h^{\sqrt{r^2-x^2-y^2}}\mathrm dz\,\mathrm dy\,\mathrm dx$

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

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

Now, the volume can be computed with the integral

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

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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3 years ago