Mathematics High School

If the telescope is 1 m deep and 8 m wide, how far is the focus from the vertex?
1
2
4
16
Find the vertex, focus, directrix, and focal width of the parabola.

negative 1 divided by 12 times x squared = y

Answers

Answer 1
First, let me introduce the general equation of the parabola:

(x-h)^2 = +/- 4a(y-k) or (y-k)^2=+/- 4a(x-h), where

(h,k) are the coordinates of the vertex
a is the distance of the vertex to the focus
4a = length of lactus rectum or the focal width

If the equation contains (x-h)^2, then the parabola passes the x-axis twice. Similarly, (y-k)^2 passes the y-axis twice. If the sign is (-), it opens to the left(if y-axis) or downward (if x-axis). If the sign is (+), it opens to the right(if y-axis) or upward (if x-axis). 

The equation of the parabola is -1/12 x^2 = y. Rearranging to the general form:

x^2 = -12y
Therefore, 

-4a = -12
4a = 12
a = 3, and the parabola is facing downwards.

The vertex is (0,0) at the origin.
The focus is (0,-3). Since it is negative, the focus is situated downwards, hence -3.
The directrix is the mirror image of the focus. Hence, it is a line passing +3 on the y-axis. y=3
Focal width is 4a which is equal to 12 units.

Related Questions

Identify the values of a, b, and c that would be used in the quadratic formula to solve the equation 3x2− 5x + 5 = 0. A. a = 3, b = 5, c = 5 B. a = 3, b = −5, c = 5 C. a = 5, b = −5, c = 0 D. a = −3, b = 5, c = −5

Answers

3x2− 5x + 5 = 0.
a=3 b=-5 c=5

A. a = 3, b = 5, c = 5
B. a = 3, b = −5, c = 5
C. a = 5, b = −5, c = 0
D. a = −3, b = 5, c = −5

answer is B


How many years will it take $1,000 invested at 7% interest to earn $280?

Answers

7% of 1,000 is 70.
280/70 = 4
It will take 4 years.

A cell phone company orders 500 new phones from a manufacturer. If the probability of a phone being defective is 2.6%, predict how many of the phones are likely to be defective. Round to the nearest whole number.
A.)16 phones
B.)11 phones
C.)13 phones
D.)130 phones

Answers

2.6% of 500 =
0.026(500) = 13 <==

Why is it difficult to measure circumference accurately? answers?

Answers

One of either the circumference or distance across is not rational. If you have a bit of string precisely 1 inch long and you make it into an idealize circle, the distance across of that circle will be an irrational value. You could try to measure it, however regardless of how precisely you measured it, it would in any case not be very sufficiently precise. So, you can never know precisely what the diameter is just by measuring.

 

What's more, on the off chance that you took that string and made a width with it, the circumference would be irrational and you could never have the capacity to measure the circumference precisely enough. Whatever you quantified would be close however not correct.


Find the area of the figure. PLEASE HELP

Answers

find area of larger figure

then subtract area of cut out

 larger figure 40 *36 = 1440 square yards

cut out = 20 *10 = 200 square yards

1440-200 = 1240 square yards total

A = 40*36 - 20*10 = 1440 - 200 = 1240 yd²

Are theses right ? Or are some wrong please let me know wrong and right ones

Answers

I can't find any mistakes (wolframmed them). Make sure you do use parenthesis (you missed one in 10). Without the parenthesis, officially they would be incorrect.

Describe the relationship between the segments made when secant lines intersect outside a circle. Use the following image as reference.

Answers

Let the extensions of secants AD and BC meet at point P.

Let the measure of ar DC be 2a, then m(DAC)=m(DBC)=a, since angles DAC and DBC are both inscribed angles intercepting arc DC.

m(P)=b.

Then triangles APC and BPD are similar, because they have 2 pairs of common angles.

from the similarity of APC and BPD, we can write the ratios:

[tex] \frac{AP}{BP} = \frac{PC}{PD} = \frac{AC}{BD} [/tex]

What would you do to isolate the variable in the equation below, using only one step? x + 5 = -12 Select one: a. Add 5 to both sides of the equation. b. Subtract 5 from both sides of the equation. c. Add 12 to both sides of the equation. d. Subtract 12 from both sides of the equation.

Answers

subtract 5 from both sides then it would look like x=-17   
Final answer:

To isolate the variable in x + 5 = -12, subtract 5 from both sides of the equation. This leads to the equation x = -12 - 5.

Explanation:

In order to isolate the variable x in the equation x + 5 = -12, you should subtract 5 from both sides of the equation. This gives us the equation in a simplified form that allows us to determine the value of x. The idea here is to cancel out the +5 on the left side by subtracting 5 from both sides, resulting in x = -12 - 5. Hence, the answer is option b.

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Which is equivalent to 5√1,215^x?

A)243x
B)1,215^1/5x
C)1,215^1/5x
D)243^1/x

Answers

Answer:

Option C is correct

The expression [tex](1215)^{\frac{1}{5}x}[/tex]  is equivalent to [tex]\sqrt[5]{1215^x}[/tex]

Step-by-step explanation:

Simplify the radical Expression: [tex]\sqrt[5]{1215^x}[/tex]

A radical expression can be written by using the exponents and also using the following procedure:

[tex]\sqrt[n]{x} = x^{\frac{1}{n}}[/tex]

when x be non-negative then n can be any index

when x is negative, then n can be odd.

The following radical expression can be written as:

[tex]\sqrt[5]{1215^x} = (1215^x)^{\frac{1}{5}}[/tex] = [tex](1215)^{\frac{1}{5}x}[/tex]

Therefore, the expression [tex](1215)^{\frac{1}{5}x}[/tex]  is equivalent to [tex]\sqrt[5]{1215^x}[/tex]





The table shows data from a survey about the number of times families eat at restaurants during a week. The families are either from Rome, Italy or New York, New York:

           High   Low   Q1     Q3    IQR   Median    Mean        σ
Rome 18         1        3       7       4         6.5       6.4           4.3
NY      14        1       4.5    8.5      4        5.5        6.1            3.2

Which of the choices below best describes how to measure the center of this data?

- Both centers are best described with the mean.

-Both centers are best described with the median.

- The Rome data center is best described by the mean. The New York data center is best described by the median.

-The Rome data center is best described by the median. The New York data center is best described by the mean.

Answers

Final answer:

For Rome, both the mean and the median are almost identical, suggesting a well-balanced data set and either can be used to describe the center. For New York, the median might be a better measure if the data are skewed or contain outliers, but more information would be needed to make a definitive judgment.

Explanation:

When considering the best measure to describe the center of a data set, we should look at the presence of outliers and the distribution of the data. The mean is the numerical average and is sensitive to extreme values, while the median is the middle value that splits the data set in half and is less affected by outliers.

For Rome, the mean is 6.4 and the median is 6.5, which are very close to each other, indicating that the data is likely symmetric without extreme outliers. Thus, in Rome's case, either the mean or the median could serve as a good measure of center. On the other hand, New York's mean is 6.1 and the median is 5.5, suggesting slightly skewed data or the presence of outliers. However, without more data points or visual representation, it's difficult to determine which is more representative solely based on these summary statistics.

Considering the provided values and common statistical practice, the centers can potentially be described as follows:

The Rome data center can be best described by either the mean or the median as they are similar.

The New York data center might be best described by the median if outliers or skewness are present.

For one photography session, Dexter earns no less than $50, but no more than $100. Which inequality can be used to represent his earnings, e?

e ≥ 50 or e ≤ 100
e > 50 or e < 100
50 < e < 100
50 ≤ e ≤ 100

Answers

d is the answer
50 ≤ e ≤ 100

Answer:

The answer is D.

Step-by-step explanation:

We know this because it is an "and" problem because it is numbers in between, not outside.

We also know it is "no less" or "no more" than 50 or 100. so we use a less than and equal too. Not just a less than sign.

Really hoped this helped, kind of in a test right now so just giving a brief way of how to figure this stuff out.

Continue and have fun learning!

Which of the following is a solution of x2 + 5x = −2?

5 plus or minus the square root of 33 divided by two.
5 plus or minus the square root of 17 divided by two
negative 5 plus or minus the square root of 33 divided by two.
negative 5 plus or minus the square root of 17 divided by two.

Answers

x^2 + 5x = -2
x^2 + 5x + 2 = 0

x = -b (+-) sqrt (b^2 - 4ac) / 2a
a = 1, b = 5, and c = 2

x = -5 (+-) sqrt (5^2 - 4(1)(2)) / 2(1)
x = -5 (+-) sqrt (25 - 8) / 2
x = -5 (+-) sqrt (17) / 2

answer is : negative 5 plus or minus the square root of 17 divided by 2

Answer:

negative 5 plus or minus the square root of 17 divided by two: [tex]x=\frac{-5\pm\sqrt{17}}{2}[/tex].

Step-by-step explanation:

We have the equation

[tex]x^2+5x=-2[/tex]

[tex]x^2+5x+2=0[/tex].

To solve it, we are going to use the general formula. This formula is used to find the solutions of quadratic equations.

[tex]x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}[/tex]

where a is the coefficient of [tex]x^2[/tex], in this case a=1, b is the coefficient of x, in this case b=5 and c is the independent term, in this case c=2.

[tex]x=\frac{-5\pm\sqrt{5^2-4(1)(2)}}{2(1)}[/tex]

[tex]x=\frac{-5\pm\sqrt{25-8}}{2}[/tex]

[tex]x=\frac{-5\pm\sqrt{17}}{2}[/tex].

what is the length of a circle that has a central angle of 98 degrees and 6 cm rounded to the nearest 100th. using 3.14

Answers

Arc length = 2[tex] \pi [/tex]r * Θ/360 = [tex] \pi [/tex]d * Θ/360
so...
If we take pi to be 3.14,
Arc length = 2(3.14)(6) * 98/360
= 37.68 * 98/360 = 10.26 cm

Which graph shows a triangle and its reflection image over the x-axis?
A. first picture
B. 2nd picture
C. picture 3
D. picture 4

A. is wrong

Answers

Assuming the choices are A,B,C,D in that order from left to right (according to your attachments), then it would be the fourth one. 

To make sure we're on the same page, I'm attaching your fourth drawing below. It's the result of taking a triangle and reflecting it over the horizontal x axis. Think of it like mirroring an image.

Answer-

In picture 4, the triangle is reflected over x axis.

Solution-

While reflecting a point over x axis, (x, y) → (x, -y) rule is followed.

Only the sign of y-coordinate is changed, keeping x-coordinate intact.

In the first picture, though the image is reflected, but it is reflected over y axis.

In the second picture, the image is first reflected over y axis, then it is translated 1 unit down.

We rotating a point 90° clockwise, (x, y) → (y. -x) rule is followed.

In the third picture, the triangle is rotated 90° clockwise and then translated 1 unit to right.

But, in case of the fourth picture, the figure is reflected over x axis. Only the y coordinates sign are changed, all the x coordinates are same as before.


Find the missing value. Show your work
(GEOMETRY, PLEASE HELP!!)

Answers

sin51=y/12

y=12sin51 units

y≈9.33 units (to nearest hundredth of a unit)

...

tanα=12/5

α=arctan2.4°

α≈67.38°  (to nearest hundredth of a degree)

...

tan13=x/24

x=24tan13 units

x≈5.54 units (to nearest hundredth of a unit)

...

sin20=10/x

x=10/sin20 units

x≈29.24 units (to nearest hundredth of a unit)

the area of Juan's garden after he adds a border x feet wide is equal to (3+2x)(6+2x) which polynomial represents the area of Juan's garden?

A. 4x^4+30x+18
B. 4x^2+22x+27
C. 4x^2+18x+18
D. 4x^2+6x+18

Answers

 (3+2x)(6+2x) =
18 + 6x + 12x + 4x² = 
4x² + 18x + 18

Can someone please answer this for me I'm begging

Answers

if the length of JK is 16.4 miles, it will be diameter of this circle, the radius will be JK/2 = 8.2 miles

As we know the length of circle is 2π*r  in this case r = 8.2

The arc JM is 1/4 length of the whole circle so arc JM = (2π*8.2)*(1/4) = 2π*8.2/4 = π*8.2/2 = π*4.1

if we assume that π=3.14 the answer will be 3.14 * 4.1 = 12.874.... And as the answer needs to be rounded to the nearest tenth it will be 12.874≈12.9

So the answer will be 12.9

Identify all the sets to which 0.41567 belongs. Choose from rational,irrational,whole,and integer

Answers

0.41567 is:
 
... rational because it can be expressed as a fraction (e.g., 41567/100000)
... not irrational for the same reason
... not whole (obviously)
... not integer (obviously)

Answer:

1.) D

2.) C

3.) C

4.) B

5.) A

6.) D

7.) A

8.) D

9.) A

10.) D

Step-by-step explanation:

If at least one independent variable is between-groups and at least one is within-groups, the design is referred to as:

Answers

A variable is an object that you are trying to measure. It has two types; independent and dependent variable. An independent variable is a variable that is not affected by other variables. A dependent variable on the other hand is a variable that depends on the other variables. Suppose you want to know the quality of sleep of the students in their academic performance. The quality of sleep is the independent variable and the academic performance is the dependent variable. There are also controlled variables in an experiment. A controlled experiment is an experimental setup designed to test the hypotheses. It has one or more conditions (independent variables) and measures (dependent variables). It is necessary to change more than one variable, but all of the experimental conditions will be controlled so that only the variables being examined change and the amount or way of change is measured.

The price of a community pool membership has a one-time sign-up fee and a monthly fee. The price can be modeled by the function y = 20x + 50, where x is the number of months.

What is the slope, and what does it represent?

Answers

the slope in this equation is 20. This represents the monthly fee. Every month u go, u pay 20 bucks

Answer:

The answer is 20 and it represents the monthly fee.

Step-by-step explanation:


Boating across a river, the current moves you downstream. Instead of going directly across the river, the boat moves at a 10 degree angle. If the river is 420 feet across, how far downstream will you end up?

Answers

To solve this problem, we use the tan function. The distance across the river is the adjacent side, the distance drownstream is the opposite side while the direction of the boat is the hypotenuse.

tan θ = opposite side / adjacent side

tan 10 = opposite side / 420 ft

opposite side = 74.06 ft

Therefore the boat will move about 74 ft downstream.

A pyramid has a regular hexagonal base with side lengths of 4 and a slant height of 6 find the toal area of the pyramid

Answers

now, if you check the picture below, the top-side has the base alone, the hexagon, notice, if you divide a circle the 360° of a circle in 6 even angles, like the hexagon does, you end up with 6 angles of 60° each, and 6 equilateral triangles made by them, now, if you run an angle bisector, like we did on the bottom side of the hexagon, you end up splitting the angle in equal halves.

so.. then we end up with the a 30, 60, 90 angles triangle, and thus we apply the 30-60-90 rule, to get the apothem.

now the area of a regular polygon is just (1/2) (apothem)(perimeter).

now, we know the apothem, the perimeter of the hexagon is just 4+4+4+4+4+4.  So, just get those values, plug them in to get the area of the hexagonal base.

if you look at the picture bottom-side, the slant-height of the pyramid, is just the altitude/height of the triangular face.  So, you really have 6 triangles whose base is 4 and height is 6, and you know how to get the area of a triangle.

so, just get the area of the base, plus the area of the triangular faces, sum them up, and that's the total area of the hexagonal pyramid.

PLEASE HELP!!!
How can one thirdx − 2 = one fourthx + 11 be set up as a system of equations?

3y − x = −6
4y − x = 44

3y + x = −6
4y + x = 44

3y − 3x = −6
4y − 4x = 44

3y + 3x = −6
4y + 4x = 44

Answers

1/3x - 2 = 1/4x + 11....it looks to me like one y was subbed in for another...

ur 2 equations are :
y = 1/3x - 2 and y = 1/4x + 11...but we need them in standard form...

y = 1/3x - 2
-1/3x + y = -2 ..multiply by 3
-x + 3y = -6.....or 3y - x = -6 <==

y = 1/4x + 11
-1/4x + y = 11 ...multiply by 4
-x + 4y = 44 or 4y - x = 44 <==

The system of equations are 3y − x = −6 and 4y − x = 4

What is Equation?

Two or more expressions with an Equal sign is called as Equation.

One third x − 2 = one fourth x + 11

1/3x - 2 = 1/4x + 11

Now let us take y as 1/3x - 2

y= 1/3x - 2

3y=x-6

Subtract x on both sides

3y-x=-6

and Let us consider the right side equation

y=1/4x + 11

4y=x+44

Subtract x on both sides

4y-x=44

Hence, the system of equations are 3y − x = −6 and 4y − x = 44

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WILL GIVE A BRAINLIEST IF UR RIGHT!!

Evaluate f(–2) in the given function below.

f(x)= 3x-3 if -5 ≤ x < 3 x+4 if 3 ≤ x ≤ 12


A.
–6
B.
–3
C.
2
D.
–9

Answers

F(x)=3x-3

f(-2)=3(-2)-3

f(-2)=-6-3=-9

so D is correct

What is the peremeter of this polygon? (With picture)

Answers

check the picture below.

so.. simply, use the distance formula, to get their length an add them up, and that's the perimeter of the polygon.


[tex]\bf \textit{distance between 2 points}\\ \quad \\ \begin{array}{lllll} &x_1&y_1&x_2&y_2\\ % (a,b) &({{ -1}}\quad ,&{{ 2}})\quad % (c,d) &({{ 2}}\quad ,&{{ 4}})\\ &({{ 2}}\quad ,&{{ 4}})\quad % (c,d) &({{ 3}}\quad ,&{{ -2}})\\ &({{ 3}}\quad ,&{{ -2}})\quad % (c,d) &({{ -2}}\quad ,&{{ -3}})\\ &({{ -2}}\quad ,&{{ -3}})\quad % (c,d) &({{ -1}}\quad ,&{{ 2}}) \end{array}\qquad % distance value d = \sqrt{({{ x_2}}-{{ x_1}})^2 + ({{ y_2}}-{{ y_1}})^2}[/tex]

[tex]\bf -------------------------------\\\\ d=\sqrt{[2-(-1)]^2+(4-2)^2}\implies d=\sqrt{(2+1)^2+(2)^2} \\\\\\ d=\sqrt{3^2+2^2}\implies \boxed{d=\sqrt{13}}\\\\ -------------------------------\\\\ d=\sqrt{(3-2)^2+(-2-4)^2}\implies d=\sqrt{1^2+(-6)^2}\implies \boxed{d=\sqrt{37}}\\\\ -------------------------------\\\\ d=\sqrt{(-2-3)^2+[-3-(-2)]^2}\implies d=\sqrt{(-5)^2+(-3+2)^2} \\\\\\ d=\sqrt{(-5)^2+(-1)^2}\implies \boxed{d=\sqrt{26}}[/tex]

[tex]\\\\ -------------------------------\\\\ d=\sqrt{[-1-(-2)]^2+[2-(-3)]^2}\implies d=\sqrt{(-1+2)^2+(2+3)^2} \\\\\\ d=\sqrt{(1)^2+(5)^2}\implies \boxed{d=\sqrt{26}}[/tex]

so, those are their lengths, sum them all up, that's the polygon's perimeter.

Can someone please help me

Answers

V = Volume of cylinder
B = area of base
h = height of cylinder

It turns out that
V = B*h
In a similar way to that of a cube

If we know that V = 84pi and h = 7, then we can plug those values into the formula above, and solve for B

V = B*h
84pi = B*7
84pi/7 = B*7/7 ... divide both sides by 7
12pi = B
B = 12pi

Therefore the base is 12pi square units

what is the average rate of change for this quadratic function please help

Answers

Answer: Choice A) 2

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

Explanation:

When x = 0, the value of y is y = -6. You can see this as you draw a vertical line through x = 0. This vertical line is the y axis. The parabola crosses the y axis at -6. This point is the y intercept.

When x = 3, the value of y is y = 0. This is one of the two x intercepts. An x intercept is a location where the graph crosses the x axis.

So we have the two points (0,-6) and (3,0). Let's find the slope of the line through these two points

(x1,y1) = (0,-6)
(x2,y2) = (3,0)

Slope:
m = (y2-y1)/(x2-x1)
m = (0-(-6))/(3-0)
m = (0+6)/(3-0)
m = 6/3
m = 2

The slope of the line through those two point is m = 2. 

Consider the trinomial x2 – 9x + 18. Which pair of numbers has a product of ac and a sum of b? What is the factored form of the trinomial?

Answers

x^2 - 9x + 18
AC = 18
Factors of AC: 1 * 18, 2 * 9, 3 * 6, -1 * -18, -2 * -9, -3 * -6
Sum to -9: 19, 11, 9, -19, -11, -9

(x - 3)(x - 6)

The required pair of number are -3 and -6

The factored form of the trinomial is (x-3) and (x-6)

The standard form of a quadratic equation is expressed as:

[tex]ax^2+bx+c=0[/tex]

Given the expression [tex]x^2 - 9x + 18.[/tex]

[tex]a=1\\b=-9\\c=18[/tex]

We are to get two values that we must add to get b, and that we will multiply that will give c.

The given values are -6 and -3

Check:

[tex]-6+(-3)=-6-3=-9\\-6(-3) = 18[/tex]

Factorize the trinomial

[tex]x^2-9x+18\\x^2-6x-3x+18\\x(x-6)-3(x-6)\\(x-3)(x-6)[/tex]

Hence the factored form of the trinomial is (x-3) and (x-6).

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a scale model of a school building is 5 inches tall. If the building is 100 feet tall, find the scale of the model

Answers

It's 20 feet to one inch or 10 feet to half an inch.

Answer with Step-by-step explanation:

A scale model of a school building is 5 inches tall.

Actual size of the school building is 100 feet.

1 feet=12 inches

100 feet=12×100 inches

             = 1200 inches

5 inch is representing 1200 inches

1 inch represents  1200/5 inches

1 inch represents  240 inches

or  1 inch represents  12×20 inch

or 1 inch represents 20 feet

Scale of the model is:

1 inch represents  240 inches

1 inch represents 20 feet

The top of a cliff overlooking the ocean is 250 feet above sea level. The sea floor at the foot of the cliff is 40 feet below sea level. A rock falls off the cliff and drops toe the sea floor. What number represents the change in elevation of the rock.

Answers

Final answer:

The change in elevation of the rock is 290 feet.

Explanation:

The change in elevation of the rock can be calculated by finding the difference in height between the top of the cliff and the sea floor.

Given that the top of the cliff is 250 feet above sea level and the sea floor is 40 feet below sea level, we can calculate the change in elevation by subtracting the elevation of the sea floor from the elevation of the top of the cliff.

The change in elevation is therefore 250 feet - (-40 feet), which simplifies to 250 feet + 40 feet = 290 feet. So, the number that represents the change in elevation of the rock is 290 feet.

Other Questions
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