Question

PLEASE HELP

Answer 1

1) We have that the equation is $x^2=20y$ , hence y=x^2/20. The standard equation of such an equation is y=$\frac{1}{4p} x^2$. Hence, p=5 in this case. The focus is at (0,5) and the directrix is at y=-5 (a tip is that the directrix is always "opposite" the focus point of a parabola; if the directrix is at x=-7 for example, the focus is at (7,0)).
2) Similarly, we have that the equation is $x=3y^2 \\ \frac{1}{4p} =3$. Thus, p=1/12. In this case, the parabola opens along the x-axis and the focus is at (1/12, 0). Also, the directrix is at x=-1/12. Hence the correct answer is B.
3) We are given that the parabola has a p of 9. Also, the focus lies along the y-axis, hence the parabola is opening along the y-axis. Finally, the focus is on the positive half, so the parabola is opening upwards. The equation for this case is y=$y=\frac{1}{4p} x^2= \frac{1}{36 } x^2$.
4) Similarly as above. The directrix is superfluous, we only need the p-value. THe same comments about the parabola apply and if we substitute p=8 in the formula: $y= \frac{1}{4p} x^2$ we get y=$\frac{1}{32} x^2$.
5) This is somewhat different, even though we do not need the directrix again. The focus lies on the x-axis, thus the parabola opens in this direction. The focus lies on the positive part of the axis, thus the parabola opens to the right. We also are given p=7. Hence, the equation we need is of the form$x= \frac{1}{4p} y^2$. Substituting p=7, we get $x= \frac{1}{28} y^2$.
6) The equation of a prabola with a vertex at (0,0) is of the form y=-ax^2. The minus sign is needed since the parabola is downwards. Since we are given anothe point, we can calculate a. We have to take y=-74 and x=14 feet (since left to right is 28, we need to take half). $-a= \frac{y}{x^2} = \frac{-74}{14^2} =-0.378$. Thus a=0.378. Hence the correct expressions is y=-0.378*$x^2$

Answer 2

Answer:

1.

Given the parabolic equation:

$x^2=20y$

The equation of parabola is given by:

$(x-h)^2 =4p(y-k)$                       .....[A]

where,

|4p| represents the focal width of the parabola

Focus = (h, k+p)

Vertex = (h, k)

Directrix (y) = k -p

On comparing given equation with equation [A] we have;

we have;

4p = 20

Divide both sides by 4 we have;

p = 5

Vertex =(0,0)

Focus = (0, 0+5) = (0, 5)

Focal width = 20

Directrix:

y = k-p = 0-5 = -5

⇒y = -5

Therefore, only option A is correct

2.

Given the parabolic equation:

$x = 3y^2$

Divide both sides by 3 we have;

$y^2 = \frac{1}{3}x$

The equation of parabola is given by:

$(y-k)^2 =4p(x-h)$             ....[B]

Vertex = (h, k)

Focus = (h+p, k)

directrix: x = k -p

Focal width = 4p

Comparing given equation with equation [B] we have;

$4p = \frac{1}{3}$

Divide both sides by 4 we have;

$p = \frac{1}{12}$

Focal width = $\frac{1}{3} = 0.33..$

Vertex = (0, 0)

Focus = $(0+\frac{1}{12}, 0) =(\frac{1}{12}, 0)$

directrix:

$x = 0-\frac{1}{12}=-\frac{1}{12}$

⇒$x = -\frac{1}{12}$

Therefore, option B is correct.

3.

The equation of parabola that opens upward is:

$x^2 = 4py$

For the given problem:

Axis of symmetry:

x = 0

Distance from a focus to the vertex on the axis of the symmetry:

p = 9

then;

4p = 36

⇒$x^2 = 36y$

Divide both sides by 36 we have;

$y = \frac{1}{36}x^2$

Therefore, the only option A is correct.

4.

The equation of parabola that opens upward is:

$x^2 = 4py$

Given that: Focus = (0, 8) and directrix: y = -8

Distance from a focus to the vertex and vertex to directrix is same:

i,e

|p| = 8

Then,

4p = 32

⇒$x^2 = 32y$

Divide both sides by 32 we have;

$y = \frac{1}{32}x^2$

Therefore, the only option A is correct.

5.

The equation of parabola that opens right is:

$y^2 = 4px$

Given that:

Focus: (7, 0) and directrix: x = -7

Distance from a focus to the vertex and vertex to directrix is same:

i,e

|p| = 7

then

4p = 28

⇒$y^2 =28x$

Divide both sides by 32 we have;

$x= \frac{1}{28}y^2$

Therefore, the option B is correct.

6.

As per the statement:

A building has an entry the shape of a parabolic arch 74 ft high and 28 ft wide at the base as shown below.

The equation of parabola is given by:

$x^2 = -4py$            .....[C]

Substitute the point (14, -74) we have;

Put x = 14 and y = -74

then;

$(14)^2 = -4p \cdot (-74)$

⇒$196 = 4p \cdot 74$

Divide both sides by 74 we have;

$4p = \frac{196}{74} = \frac{98}{37}$

Substitute in the equation [C] we have;

$x^2 = -\frac{98}{37}y$

or

$y = -\frac{37}{98}x^2$

Therefore,  an equation for the parabola if the vertex is put at the origin of the coordinate system is, $y = -\frac{37}{98}x^2$