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The Equation of parabola with focus (3,1) and directrix y = 3 is [tex]10x^2 + 9y^2-60x -14y+91 =0[/tex]

A parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves.

Here, focus (3,1) and directrix y=3

Let the locus of parabola be (x, y)

We know that, in Parabola

Distance between locus to focus = Distance between locus to directrix line

[tex]\sqrt{(x_{2}-x_{1})^2+(y_{2}-y_{1})^2 }[/tex] = [tex]\frac{Ax_{1}+By_{1}+C }{\sqrt{A^2 + B^2 + C^2} }[/tex]

[tex]\sqrt{(x-3)^2+(y-1)^2} = \frac{0.x+1.y-3}{\sqrt{0^2+1^2+(-3)^2} }[/tex]

On squaring both sides, we get

[tex](x-3)^2 + (y-1)^2 = (\frac{y-3}{\sqrt{10} } )^2[/tex]

[tex]x^2 -6x + 9 + y^2 -2y + 1 = \frac{y^2 -6y +9}{10}[/tex]

[tex]10x^2 + 9y^2-60x -14y+91 =0[/tex]

Thus, The Equation of parabola with focus (3,1) and directrix y = 3 is [tex]10x^2 + 9y^2-60x -14y+91 =0[/tex]

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Answer:

The value of y is 4.

Step-by-step explanation:

Note the system of equation:

x + y = 24

5y = x

First, plug in 5y for x in the first equation:

x + y = 24

(5y) + y = 24

Simplify. Combine like terms. Like terms are terms that have the same amount of the same variable:

(5y + y) = 24

(6y) = 24

Isolate the variable, y. Note the equal sign, what you do to one side, you do to the other. Divide 6 from both sides of the equation:

(6y)/6 = (24)/6

y = 24/6 = 4

Now, plug in 4 for y in one of the given equations in the system of equation:

x + y = 24

x + (4) = 24

Isolate the variable, x. Note the equal sign, what you do to one side, you do to the other. Subtract 4 from both sides of the answer:

x + 4 = 24

x + 4 (-4) = 24 (-4)

x = 24 - 4

x = 20

x = 20 , y = 4

~

Check. Use the other equation. Plug in 4 for y, and 20 for x:

5y = x

5(4) = (20)

5 * 4 = 20

20 = 20 (True)

~

The value of y is 4.

Answer:

C.

Step-by-step explanation:

Find 5% of 19:

19 x 0.05 = 0.95

0.95 + 19 = 19.95

answer is C. 19.95

(22 + 26) / 8 = x

(/ this means division)

The product of the expressions 2 × 10⁵⁸ and 2 × 10⁻²³ in the scientific notation will be 2 × 10³⁵.

It is also known as the product. If the object n is given to m times then we just simply multiply them.

The expressions are given below.

2 × 10⁵⁸ and 2 × 10⁻²³

Then the product of the expressions will be

(2 × 10⁵⁸) × (2 × 10⁻²³) = 2 × 10⁵⁸⁻²³

(2 × 10⁵⁸) × (2 × 10⁻²³) = 2 × 10³⁵

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Answer:

The third measurement is 62°

Step-by-step explanation:

Hey there! Please take a look at the explanation for clear understanding to the answer and also let me know in comment if you have any doubts!

The theorem regarding it states that, sum of all 3 interior angles must equal to 180°.

That means, we will have this equation:

[tex]\displaystyle \large{28^{\circ} + 90^{\circ} + x=180^{\circ}}[/tex] because we know two interior angles and one missing angle - sum of them equals to 180 degrees.

Then we solve for x-term to find the third measurement!

[tex]\displaystyle \large{118^{\circ}+x=180^{\circ}}[/tex]

Subtract both sides by 118 degrees.

[tex]\displaystyle \large{118^{\circ}-118^{\circ}+x=180^{\circ}-118^{\circ}}\\\\\displaystyle \large{x=62^{\circ}}[/tex]

Henceforth, the third measurement is 62°

Answer:

$7.43

Step-by-step explanation:

f (fries) = 1.01

d (drinks) = 1.07

1.05f + 1.07d = ?

To solve this problem, you need to replace the variable with the amount of each item.

1.05(3) + 1.07(4) = ?

Now we can solve by multiplying and adding:

3.15 + 4.28 = ?

7.43

Answer:

Step-by-step explanation:

Q1

Q2

Q3

Q4

Q5

Using the z-distribution, it is found that the margin of error for the true proportion of times this die would land on five dots is of 0.1338.

A confidence interval of proportions is given by:

[tex]\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]

In which:

The margin of error is given by:

[tex]M = z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]

In this problem, we have a 95% confidence level, hence[tex]\alpha = 0.95[/tex], z is the value of Z that has a p-value of [tex]\frac{1+0.95}{2} = 0.975[/tex], so the critical value is z = 1.96.

The estimate and the sample size are given as follows:

[tex]\pi = \frac{25}{30} = 0.8333, n = 30[/tex].

Hence the margin of error is given by:

[tex]M = z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]

[tex]M = 1.96\sqrt{\frac{0.8333(0.1667)}{30}}[/tex]

M = 0.1338.

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Answer:

Step-by-step explanation:

To convert degrees to radians: [tex]n(\frac{\pi }{180})[/tex] where n is degrees.

For n = 13

[tex]13(\frac{\pi}{180} ) = 0.229 radians[/tex] when rounded to the nearest thousandth

The area of the rectangle ACEG is the amount of space on it, and the area of square HJFG is 9 square units

The area of a rectangle is:

Area = Length * Width

The area of a square is:

Area = Length * Length

From the question, the given parameters are:

AB =AH = x ;

BC=5;

HG=3

This means that the area of square HJFG is:

Area = HG * HG

Area = 3 * 3

Area = 9

Note that the other areas cannot be calculated because the question is incomplete

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Answer:

18.4 years

Step-by-step explanation:

The compound interest formula:

[tex]A=P(1+\frac{r}{n} )^{nt}[/tex]

For this problem we have

[tex]P= 2000\\r= 0.06\\n= 4\\A= 2000(3)= 6000[/tex]

We plug our numbers...

[tex]6,000= 2,000(1+\frac{0.06}{4} )^{4t} \\[/tex]

Then divide

[tex]6,000/2,000=3[/tex]

[tex]3= (1.015)^{4t}[/tex]

Apply log both sides

[tex]3log=(1.015) 4tlog[/tex]

Solve for t

[tex]t= \frac{3log}{(4log(1.015))} =18.4 years[/tex]

Answer:

-4 < x < -2

Step-by-step explanation:

Taking the left part of the inequality :

Taking the right part of the inequality :

Solution :

Answer:

Total ounces should be 12

and hence

x + y = 12

and total money to spend is 50, hence

3.5x + 4.5y = 50

Therefore, these are the two ✌️ equations you should use

x + y = 12

3.5x + 4.5y = 50

Answer:

(5, -3)

Step-by-step explanation:

Solve for x and solve for y by making each set of brackets equal to 0

Answer: 48 degrees

Step-by-step explanation:

Dilations preserve angle measure.

Answer:

a. <B = 53°

b. BC = 14

c. x = 14

Step-by-step explanation:

a. <B = ?

Use Sine.

Sine(<B) = 12 /15

<B = Sin^-1(12/15)

<B = 53.13

<B = 53°

b. BC = ?

Either use Sine(<A) = BC/16 or Cosine(<C) = BC/16

Will be using Cosine.

Cosine(<C) = BC/16

Cosine(30°) = BC/16

BC = 16 * Cos(30°)

BC = 13.856...

BC = 14

c. x = ?

Looking at the triangle. The triangle is isosceles which mean two side angle or side length are the same.

x = 14

Answer:

A: 53.13

B:13.86

C: 14

Step-by-step explanation:

A can be found by using the trigonometric ratio Sin.

Sin=

[tex] \frac{opp}{hyp} [/tex]

And we have our opposite side and hypotenuse. so insert it:

Sin=12/15

Therefore use sin inverse and input that fraction

[tex]sin { }^{ - 1} ( \frac{12}{15} )[/tex]

=53.13010235

Rounded off to two decimals

=53. 13

B can be found using sin again. Since we know that Sin is opp/hyp and we have the hyp, we are now looking for the opp.

So use Sin(60)=BC/16

Multiply 16 on both sides to get: 16Sin(30)=BC

13.85640646=BC

rounded off to two decimals:

13.86

C can be found using Tan. Tan is opp/adjacent

We have the adjacent and we are looking for the opposite.

So use:

tan(45)=X/14

Multiply 14 on both sides

14tan(45)=X

14=X

Hope that helps!

Answer:

Step-by-step explanation:

Because a square must be positive no matter what the sign of x is.

(-2)^2 = 4

2^2 = 4

Answer & Step-by-step explanation:

USDA's Dietary Guidelines for Americans provide percent ranges for each macronutrient. For adults, the organization recommends consuming 10%–35% of calories from protein, 45%–65% from carbohydrates, and 20%–35% of calories from fat. They also suggest consuming less than 10% of total calories from saturated fat.

Answers:

x = 16

y = 41

z = 123

==========================================================

Explanation:

Focus on the large triangle on the left.

The three interior angles of any triangle add to 180 degrees.

57+82+y = 180

y+139 = 180

y = 180-139

y = 41

-------------------

Now focus on the large triangle on the right.

We'll use the same idea as before.

x+102+62 = 180

x+164 = 180

x = 180-164

x = 16

-------------------

The last triangle to focus on is the small one in the middle.

Use the values of x and y to find z.

x+y+z = 180

16+41+z = 180

57+z = 180

z = 180-57

z = 123

Answer:

The angles inside of a triangle always add up to 360 therefore we can easily determine what the angle of x and y is because they are a part of two triangles that have two angles each.  Let's first determine x.

[tex]x = 180 - (102+ 62)[/tex]

[tex]x = 180 - 164[/tex]

[tex]x = 16[/tex]

The second step that we need to do is to determine y using 82 and 57.

[tex]y = 180 - (82 + 57)[/tex]

[tex]y = 180 - 139[/tex]

[tex]y = 41[/tex]

The final step is to determine z using x and y

[tex]z = 180 - (x + y)[/tex]

[tex]z = 180 - (16 + 41)[/tex]

[tex]z = 180 - 57[/tex]

[tex]z = 123[/tex]

Therefore, x = 15 and y = 41 and z = 123

Hope this helps!

Step-by-step explanation:

= 3x - 2x + 24x

= 3 × 4 - 2 × 4 + 24 × 4

= 12 - 8 + 96

= 12 + 96 - 8

= 108 - 8

= 100

Answer:

- a frequency table is how much a number occurs, for example there are 4 10's in data list given so the frequency is 4

- the prompt asked for a width of 6 so 1 - 6, there was no numbers of 1, 2, 3, 4, 5, 6 in the list making the frequency 0

Answer:

2

Step-by-step explanation:

Answer:

B. Unlikely

Step-by-step explanation:

Generally speaking, a probability of 0.11 to 0.40 is unlikely. 1/5 is 0.2, which falls in this range.

Answer:

 (a) isosceles

 (b), (c) scalene

Step-by-step explanation:

The lengths of the sides can be found from the distance formula:

 d = √((x2 -x1)^2 +(y2 -y1)^2)

Here, there are 9 lengths to be computed, so it is useful to let a spreadsheet or graphing calculator do it.

We find triangle (a) has two sides the same length, so it is isosceles.

The other two triangles have three different side lengths, so they are scalene.

_____

Additional comment

It is impossible to specify an equilateral triangle using rational coordinates.

Answer: Circle d

Step-by-step explanation:

y=1/2(0.25)^0-1. +2

y=1/2(0.25)^-1 +2

y=1/2*4+2

y=2+2

y=4

Step-by-step explanation:

set x to 0 to find y intercept

Find some points

x=1

x=2

As you did the graph already so I haven't attached it .

The Y intercept

Put x=0

Y intercept is 4

Answer:

[tex]\frac{-2}{x^{2} } -2[/tex]

Step-by-step explanation:

[tex]x^{2} + \frac{x-2}{x^{3} } -x^{2} +2x-2[/tex]

= [tex]\frac{-2}{x^{2} } -2[/tex]

Answer: see photo#2

Step-by-step explanation: