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3 years ago

This circle is centered at the origin, and the length of its radius is 4. What is

Mathematics

1 answer:

Sonbull [250]3 years ago

6 0

Answer:

D. x² + y² = 4²

Step-by-step explanation:

The general equation of a circle is given in the attached graphic.

In this case, we can see that the center of the circle is at the origin i.e (0,0) and that the radius is 4 units.

the values for h, k and r in the general equation are as follows:

h = 0, k=0, r = 4

putting this into the general equation:

(x-h)² + (y-k)² = r²

(x-0)² + (y-0)² = 4²

x² + y² = 4²

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3 years ago

Find the average value of f over region

SVETLANKA909090 [29]

$D$ is a right triangle with base length 1 and height 8, so the area of $D$ is $\dfrac12(1)(8)=4$ .

The average value of $f(x,y)$ over $D$ is given by the ratio

$\dfrac{\displaystyle\iint_Df(x,y)\,\mathrm dA}{\displaystyle\iint_D\mathrm dA}$

The denominator is just the area of $D$ , which we already know. The average value is then simplified to

$\displaystyle\frac74\iint_Dxy\,\mathrm dA$

In the $x,y$ -plane, we can describe the region $D$ as all points $(x,y)$ that lie between the lines $y=0$ and $y=8x$ (the lines which coincide with the triangle's base and hypotenuse, respectively), taking $0\le x\le1$ . So, the integral is given by, and evaluates to,

$\displaystyle\frac74\int_{x=0}^{x=1}\int_{y=0}^{y=8x}xy\,\mathrm dy\,\mathrm dx=\frac78\int_{x=0}^{x=1}xy^2\bigg|_{y=0}^{y=8x}\,\mathrm dx$
$=\displaystyle56\int_{x=0}^{x=1}x^3\,\mathrm dx$
$=14x^4\bigg|_{x=0}^{x=1}$
$=14$

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3 years ago