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svetlana [45]
3 years ago
15

The triangle T has vertices at (-2, 1), (2, 1) and (0,-1). (It might be an idea to

Mathematics
1 answer:
Firdavs [7]3 years ago
6 0

Rewrite the boundary lines <em>y</em> = -1 - <em>x</em> and <em>y</em> = <em>x</em> - 1 as functions of <em>y </em>:

<em>y</em> = -1 - <em>x</em>  ==>  <em>x</em> = -1 - <em>y</em>

<em>y</em> = <em>x</em> - 1  ==>  <em>x</em> = 1 + <em>y</em>

So if we let <em>x</em> range between these two lines, we need to let <em>y</em> vary between the point where these lines intersect, and the line <em>y</em> = 1.

This means the area is given by the integral,

\displaystyle\iint_T\mathrm dA=\int_{-1}^1\int_{-1-y}^{1+y}\mathrm dx\,\mathrm dy

The integral with respect to <em>x</em> is trivial:

\displaystyle\int_{-1}^1\int_{-1-y}^{1+y}\mathrm dx\,\mathrm dy=\int_{-1}^1x\bigg|_{-1-y}^{1+y}\,\mathrm dy=\int_{-1}^1(1+y)-(-1-y)\,\mathrm dy=2\int_{-1}^1(1+y)\,\mathrm dy

For the remaining integral, integrate term-by-term to get

\displaystyle2\int_{-1}^1(1+y)\,\mathrm dy=2\left(y+\frac{y^2}2\right)\bigg|_{-1}^1=2\left(1+\frac12\right)-2\left(-1+\frac12\right)=\boxed{4}

Alternatively, the triangle can be said to have a base of length 4 (the distance from (-2, 1) to (2, 1)) and a height of length 2 (the distance from the line <em>y</em> = 1 and (0, -1)), so its area is 1/2*4*2 = 4.

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<h3>How to draw the line plot</h3>

From the question, we have the following frequency table

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4\frac 18                       2

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Next, we plot the number line.

Such that the lengths of the scrap woods are on the x-axis while their frequencies are on the y-axis

See attachment for the line plot

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brainly.com/question/15853311

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Divide using long division. check your answers. 1. (x 3 – 13x 2 + 7x – 48) ÷ (x 2 + 3) 3. (x 3 + 5x 2 – 3x – 1) ÷ (x – 1) 5. (x
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5. We can conclude that the result of the division is x+1 with a remainder of 5, or in other notation: x+1+ \frac{5}{x-4}. Check the steps in the first image.

7. To check if x+4 is a factor of x^{3}+3x^{2}-10x-24, we are going to perform long division. If the result of the long division has no remainder, x+4 is a factor of x^{3}+3x^{2}-10x-24.
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9. To check if x-3 is a factor of x^{3}+3x^{2}-10x-24, we are going to perform long division. If the result of the long division has no remainder, x-4 is a factor of x^{3}+3x^{2}-10x-24.
Since the result of the long division has no remainder, we can conclude that x-4 is a factor of x^{3}+3x^{2}-10x-24. Check the steps in the third image.

10. We can conclude that the result of the synthetic division is -2x^{2}+9x+5. Check the steps in the fourth image. 

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