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2 years ago

Find two consecutive whole numbers that /76 lies between.

Mathematics

2 answers:

kozerog [31]2 years ago

7 0

35 and 41 are two consecutive numbers

liraira [26]2 years ago

6 0

76 lies between ,< 35 and 41 >

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What is the value of x in the solution to the system of linear equations? {y=3x+2 y=x-4 a –7 b –3 c 1 d 5

lyudmila [28]

Since y must equal both 3x+2 and x-4, we deduce that 3x+2 and x-4 must equal each other. So, we have

$3x+2=x-4 \iff 2x+2=-4 \iff 2x=-6 \iff x=-3$

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2 years ago

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2 years ago

A sphere of radius r is cut by a plane h units above the equator, where

Anika [276]

Consider the top half of a sphere centered at the origin with radius $r$ , which can be described by the equation

$z=\sqrt{r^2-x^2-y^2}$

and consider a plane

$z=h$

with $0$ . Call the region between the two surfaces $R$ . The volume of $R$ is given by the triple integral

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

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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2 years ago

Lauren made $45 babysitting. She put 1/3 in her savings and spent 3/5 of the remainder. How much money did she spend? (OR YOU CA

earnstyle [38]

Lauren spent $18. She put 1/3 of $45 ($15) in a savings account, then spent 3/5 of the remaining $30 ($18).

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2 years ago