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n200080 [17]
3 years ago
10

Darien drew a quadrilateral on a coordinate grid. He rotated the figure 180 degrees and then translated it left 4 units. What ar

e the coordinates of the image of point P?

Mathematics
1 answer:
VikaD [51]3 years ago
6 0
If we take a point (x,y) and rotate it 180 degrees about the origin, the two coordinates change sign. That is, the point (x,y) becomes (or moves to) (-x,-y).

If you then take that point which is now (-x,-y) and translate it (slide it over) 4 units left you are taking the x-coordinate and moving it left 4 which is taking the x-coordinate and subtracting 4. The point (-x,-4) when you subtract 4 from the -x becomes (-x-4,-y)

So the two transformations (rotation of 180 degrees followed by a translation 4 units left) take a point (x,y) and move it to its image (-x-4,-y)

Now we are asked for the image of point P. The coordinates of P can be found from the diagram. If you start at the origin (where the x and y axis cross or the center of the plane you are given) to travel to P you move 5 units right and 2 units up. That is, the coordinates of P are (5,2)

So for P (x,y) = (5,2)

The image of P is (-x-4,-y) which is (-5-4, -2). That is, the image of P is (-9,-2)
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<h3>How do you find the maximum volume of the cylinder?</h3>

The formula for the volume of the cylinder v = \pir^{2}h, To find the maximum volume of the cylinder we apply the condition of maxima  \frac{\mathrm{d} v}{\mathrm{d} r} = 0.

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                                                             h =   \frac{5000-2\pi r^{2} }{2\pi rh} .............(1)

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Substitute the value of h in the above equation

\Rightarrow                                       = \pi r^{2} × \left [ \frac{5000-2\pi  r^{2}}{2\pi r} \right ]

\Rightarrow                                      = \frac{r}{2} ×  (5000-2\pi r^{2} )

\Rightarrow                                    v = 2500r-\pi r^{3} ..............(2)

Now, for the maximum volume of the cylinder  \frac{\mathrm{d} v}{\mathrm{d} r} = 0

\Rightarrow                                                            \frac{\mathrm{d} (2500r-\pi r^{3})}{\mathrm{d} r} = 0

\Rightarrow                                                                      3\pi r^{2} = 2500

\Rightarrow                                                                          r^{2} = \frac{2500}{3\pi }

\Rightarrow                                                                          r = \frac{50}{\sqrt{3\pi } }

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Then, v = 2500r-\pi r^{3}

\Rightarrow              =  2500 \frac{50}{\sqrt{3\pi } } -\pi (\frac{50}{\sqrt{3\pi } } )^{3}                              

\Rightarrow             = \frac{125000}{\sqrt{3\pi } } - \frac{125000}{3\sqrt{3\pi } }

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\Rightarrow             = \frac{250000}{3\sqrt{3\pi } }                                                              

\Rightarrow          v = 27147.355

Hence, The maximum volume of the cylinder is 27147.355 at the maximum point  r = \frac{50}{\sqrt{3} } .                                                

To learn more about total surface area and volume of the cylinder from the given link

brainly.com/question/16095729

#SPJ4                                                                                   

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