A rectangle's area is given by the expression 6n2-n+

Mathematics

1 answer:

Zolol [24]3 years ago

8 0

Answer:

Length of rectangle: 2n - (1/3) + 9/(3n)

Width of rectangle = 3n

Rectangle area = A = 6 n^2 - n + 9

Step-by-step explanation:

Rectangle area = A = 6 n^2 - n + 9

test what the roots are: b^2 - 4ac = (-1)^2 - 4*6*9 < 0 no real roots

-215 < 0 discriminant

(6 n^2 - n + 9) / (3n) =

2n - (1/3) + 9/(3n)

If the area is divided by 3n, the quotient is 2n - (1/3) and the remainder is 9

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A parabola has a focus of F(2, -0.5) and a directrix of y=-1.5 P(x,y) represents any point on the parabola, while D(x, -1.5) rep

prohojiy [21]

The sketch of the parabola is attached below

We have the focus $(a,b) = (2, -0.5)$
The point $P(x,y)$
The directrix, c at $y=-1.5$

The steps to find the equation of the parabola are as follows

Step 1
Find the distance between the focus and the point P using Pythagoras. We have two coordinates; $(2, -0.5)$ and $(x,y)$ .
We need the vertical and horizontal distances to find the hypotenuse (the diagram is shown in the second diagram).
The distance between the focus and point P is given by
$\sqrt{ (x-a)^{2}+ (y-b)^{2} }$

Step 2
Find the distance between the point P to the directrix $c$ . It is a vertical distance between y and c, expressed as $y-c$

Step 3
The equation of parabola is then given as
$\sqrt{ (x-a)^{2}+ (y-b)^{2} }$ = $y-c$
$(x-a)^{2}+ (y-b)^{2}= (y-c)^{2}$ ⇒ substituting a, b and c
$(x-2)^{2}+ (y--0.5)^{2} = (y--1.5)^{2}$
$(x-2)^{2}+ (y+0.5)^{2}= (y+1.5)^{2}$ ⇒Rearranging and making $y$ the subject gives

$y= \frac{ x^{2} }{2} -2x+1$