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Mumz [18]
2 years ago
15

PLEASE HELP

Mathematics
2 answers:
Sholpan [36]2 years ago
7 0
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 formx= \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
NNADVOKAT [17]2 years ago
5 0

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

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Hi there! Sorry to bother you, but can someone help me out with this?
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We can rule out choices B through D because they are valid items to use in any proof. A definition is a statement (or set of statements) set up in a logical fashion that is very clear and unambiguous. This means there cannot be any contradiction to the definition. An example of a definition is a line is defined by 2 points (aka a line goes through 2 points).

A postulate is a term that refers to a basic concept that doesn't need much proof to see why it's true. An example would be the segment addition postulate which says we can break up a segment into smaller pieces only to glue those pieces back together and get the original segment back.

A theorem is more rigorous involving items B and C to make a chain of statements leading from a hypothesis to a conclusion. You usually would see theorems in the form "if this, then that". Where "this" and "that" are logical statements of some kind. One theorem example is the SSS congruence theorem that says "if two triangles have three pairs of congruent corresponding sides, then the triangles are congruent". Chaining previously proven/established theorems is often done to form new theorems. So math builds on itself.

A conjecture is basically a guess. You cannot just blindly guess and have it be valid in a proof. You can have a hypothesis and have it lead to a conclusion (whether true or false) but simply blindly guessing isn't going to cut it. So that's why conjectures aren't a good idea in a proof.

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Of course, they may not realize they are lying but it's always a good idea to verify any claim no matter how trivial. Math tries to be as impartial as possible to have every theorem require proof. Some proofs are a few lines long (we consider these trivial) while others take up many pages, if not an entire book, depending on the complexity of the theorem.

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If you wanted to go for a statement that doesn't require proof, then you'd go for an axiom or postulate. Another example of such would be something like "if two straight lines intersect, then they intersect at exactly one point".

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