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Ghella [55]
1 year ago
6

Find the slope. A. y= -1/2x - 19/2.

Mathematics
1 answer:
Dmitry [639]1 year ago
5 0

The equation of the line follows the following general structure:

y=mx+b

Where m is the slope of the line and b is the y intercept.

Find the corresponding values in the given formula, this way:

In the given equation, m has a value of -1/2, it means the slope is -1/2.

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A particle moves along line segments from the origin to the points (2, 0, 0), (2, 4, 1), (0, 4, 1), and back to the origin under
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The work is equal to the line integral of \vec F over each line segment.

Parameterize the paths

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  • from (2, 0, 0) to (2, 4, 1) by \vec r_2(t)=2\,\vec\imath+4t\,\vec\jmath+t\,\vec k with 0\le t\le1,
  • from (2, 4, 1) to (0, 4, 1) by \vec r_3(t)=(2-t)\,\vec\imath+4\,\vec\jmath+\vec k with 0\le t\le2, and
  • from (0, 4, 1) to (0, 0, 0) by \vec r_4(t)=(4-4t)\,\vec\jmath+(1-t)\,\vec k with 0\le t\le1

The work done by \vec F over each segment (call them C_1,\ldots,C_4) is

\displaystyle\int_{C_1}\vec F\cdot\mathrm d\vec r_1=\int_0^2\vec0\cdot\vec\imath\,\mathrm dt=0

\displaystyle\int_{C_2}\vec F\cdot\mathrm d\vec r_2=\int_0^1(t^2\,\vec\imath+24t\,\vec\jmath+32t^2\,\vec k)\cdot(4\,\vec\jmath+\vec k)\,\mathrm dt=\int_0^1(96t+32t^2)\,\mathrm dt=\frac{176}3

\displaystyle\int_{C_3}\vec F\cdot\mathrm d\vec r_3=\int_0^2(\vec\imath+(24-12t)\,\vec\jmath+32\,\vec k)\cdot(-\vec\imath)\,\mathrm dr=-\int_0^2\mathrm dt=-2

\displaystyle\int_{C_4}\vec F\cdot\mathrm d\vec r_4=\int_0^1((1-t)^2\,\vec\imath+2(4-4t)^2\,\vec k)\cdot(-4\,\vec\jmath-\vec k)\,\mathrm dt=-2\int_0^1(4-4t)^2\,\mathrm dt=-\frac{32}3

Then the total work done by \vec F over the particle's path is 46.

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