Which equation represents a parabola with a focus at (0,-2) and a directrix of y=6?
Answers
[tex]F(0,-2)[/tex]
Given that the directrix is parallel to the x-axis, then the ordinary equation is given by:
[tex](x-h)^{2} = 4p(y-k)[/tex]
We need to find V(h,k), being V the vertex.
We know that these distances are always the same, namely:
│FV│ = │VD│
Being D the directrix. Given that the focus F is on the y-axis and the directrix is parallel to the x-axis, then the vertex V will also be on this axis, so h = 0.
As │FV│ = │VD│, then:
[tex]k = \frac{6-2}{2}[/tex], that is the middle point of the segment FD, so:
V(0,2)
Now │FV│= │p│= │2-(-2)│=4
Given that the vertex and focus are below the directrix, then the parabola open down, therefore: [tex]p\ \textless \ 0[/tex]
Lastly, the equation is:
[tex]x^{2} = -4(4)(y-2) = -16y+32[/tex]
[tex]y = -\frac{ x^{2} }{16} + 2[/tex]
The equation of a parabola with a focus at (0,-2) and a directrix of y=6 is x^2 = -16(y - 2).
To find the equation of a parabola with a focus at (0,-2) and a directrix of y=6, you need to use the standard form of the equation of a parabola that opens upwards or downwards. The general form of this type of parabola is (x - h)^2 = 4p(y - k), where (h,k) is the vertex of the parabola, and p is the distance from the vertex to the focus (if the parabola opens upwards or downward) or to the directrix (if the parabola opens sideway).
Given that the focus is at (0,-2) and the directrix is at y=6, the vertex of the parabola will be located midway between them. The distance between the focus and directrix is 8 units, so the vertex will be 4 units from each, which puts the vertex at (0, 2). Therefore, h=0 and k=2.
Since the focus is below the directrix, our parabola opens downward, and the value of p is negative. The distance p is half the distance between the focus and directrix, so p=-4. Plugging these values into the general form, we get (x - 0)^2 = 4(-4)(y - 2), which simplifies to x^2 = -16(y - 2).
This is the equation that represents the desired parabola.
Related Questions
Trigonometry Unit Circle question (see photo)
Answers
More specifically x = cos(theta) and y = sin(theta) make up the point (x,y) which is where point P is located. Because we have a unit circle, the equation of the circle is x^2+y^2 = 1. This leads to cos^2(theta) + sin^2(theta) = 1 which is the pythagorean trig identity.
The altitude to the hypotenuse of a right triangle has a length of 12. what could be the lengths of the two segments of the hypotenuse ?
Answers
we know that
The length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of the two segments of the hypotenuse
h²=x*y
where
h is the length of the altitude to the hypotenuse
h=12
x and y are the two segments of the hypotenuse
so
12²=x*y----------> 144=x*y
the lengths of the two segments of the hypotenuse could be
x=6
y=24
x*y=144
the answer is
6 and 24
A rectangle has a perimeter of 34 cm and an area of 52 cm2. its length is 5 more than twice its width. write and solve a system of equations to find the dimensions of the rectangle
Answers
length=13cm
Larry's lemons is a street vendor business that sells lemonade and lemon bars. A cup of lemonade sells for $2 and a lemon bar sells for $1.50. When all related business expenses are included, a cup of lemonade costs $0.25 to prepare and a lemon bar costs $0.20 to prepare. Last Monday, one of the vendors selling Larry's Lemons sold at least $500 worth of lemonade and lemon bars and its expenses were no more than $100. At least 150 cups of lemonade were sold. Let x be the number of cups of lemonade sold last Monday and y be the number of lemon bars sold last Monday. Which ordered pair representing a combination of cups of lemonade and lemon bars could have been sold last Monday and make sense in the context of the situation?
Answers
The one possible ordered pair that satisfies all conditions is (x, y) = (150, 134).
The ordered pair (x, y) that satisfies the following conditions:
1. The total revenue from selling x cups of lemonade and y lemon bars is at least $500. This gives us the inequality:
[tex]\[ 2x + 1.50y \geq 500 \][/tex]
2. The total expenses for preparing x cups of lemonade and y lemon bars is no more than $100. This gives us the inequality:
[tex]\[ 0.25x + 0.20y \leq 100 \][/tex]
3. At least 150 cups of lemonade were sold, which gives us the inequality:
[tex]\[ x \geq 150 \][/tex]
To find a combination of x and y that satisfies all these conditions, we can start by considering the minimum number of lemonade cups sold, which is 150. We can then calculate the revenue and expenses for this minimum number and see how many lemon bars would be needed to meet the revenue requirement while keeping the expenses within the limit.
Let's start with the minimum number of lemonade cups:
[tex]\[ x = 150 \][/tex]
The revenue from lemonade alone would be:
[tex]\[ 2 \times 150 = \$300 \][/tex]
The expenses for lemonade alone would be:
[tex]\[ 0.25 \times 150 = \$37.50 \][/tex]
Now, we need to find out how many lemon bars (y) would be needed to make up the remaining revenue to at least $500 while keeping the total expenses at or below $100.
Let's denote the remaining revenue needed as R and calculate it:
[tex]\[ R = 500 - 300 = \$200 \][/tex]
The revenue from selling y lemon bars is $1.50y, so we have:
[tex]\[ 1.50y \geq 200 \][/tex]
[tex]\[ y \geq \frac{200}{1.50} \][/tex]
[tex]\[ y \geq 133.33 \][/tex]
Since we cannot sell a fraction of a lemon bar, we round up to the nearest whole number, so at least 134 lemon bars must be sold.
Now let's check the expenses for the lemon bars. We have $100 - $37.50 = $62.50 left for expenses. The cost to prepare each lemon bar is $0.20, so we calculate the maximum number of lemon bars (y) that can be prepared with the remaining expenses:
[tex]\[ 0.20y \leq 62.50 \][/tex]
[tex]\[ y \leq \frac{62.50}{0.20} \][/tex]
[tex]\[ y \leq 312.5 \][/tex]
Combining the two conditions for y, we find that y must be at least 134 but no more than 312.
Therefore, one possible ordered pair that satisfies all conditions is (x, y) = (150, 134). This means that at least 150 cups of lemonade and at least 134 lemon bars were sold last Monday to meet the revenue and expense conditions.
How many solutions can be found for the equation −4x − 11 = 2(x − 3x) + 13? (4 points) none or one or two or infinitrly many
Answers
Answer:
None
Step-by-step explanation:
There are no solutions to this equation.
BRAINLIESTTTT ASAP!!
please answer :)
Answers
3z - 4 ≤ 17
add 4 to both sides:
3z-4 + 4 ≤ 17 + 4
simplify:
3z ≤ 21
divide both sides by 3 :
3z / 3 ≤ 21 / 3
z ≤ 21 / 3
z ≤ 7
hope this helps!
The tiles below represent the polynomial 2x2 + 5x + 3.
What is the factorization of 2x2 + 5x + 3?
A. (2x + 3)(x + 1)
B. (x + 3)(x + 3)
C. (x + 3)(x + 1)
D. (2x + 3)(x + 3)
Answers
2x^2 + 5x + 3 factors into (2x + 3)(x + 1).
Answer:
Factorization of 2x²+5x+3 is:
A. (2x + 3)(x + 1)
Step-by-step explanation:
We have to factorize the expression:
2x²+5x+3
We will solve this expression by splitting the middle term method
2x²+5x+3
=2x²+2x+3x+3
=2x(x+1)+3(x+1)
=(2x+3)(x+1)
Hence, factorization of 2x²+5x+3 is:
A. (2x + 3)(x + 1)
express the product of 2x2 + 6z - 8 and x + 3 in standard form
Answers
Answer:
[tex]2x^3+12x^2+10x-8[/tex]
Step-by-step explanation:
The product is found using the distributive property.
[tex](x+3)(2x^2+6x-8)\\x*2x^2 + x*6x+x*(-8)=2x^3+6x^2-8x\\\\and\\\\3*2x^2+3*6x+3*(-8)=6x^2+18x-8[/tex]
Now combine the two products by adding like terms together.
[tex]2x^3+6x^2-8x + 6x^2+18x-8\\2x^3+12x^2+10x-8[/tex]
In the diagram, SR = 4sqrt 2 and QR = sqrt 10. What is the perimeter of parallelogram PQRS?
A. sqrt 10 units
B. 8 sqrt 2 + sqrt 10 units
C. 16 sqrt 2 units
D. 8 sqrt 2 + 8 units
Answers
Answer: [tex]B.\ 8\sqrt2+\sqrt{10}[/tex]
Answer: B
Step-by-step explanation:
A rectangular photograph is mounted on a poster and has a two inch border on each side. The poster itself is mounted on a frame whose sides are the same length as the sides of the poster. The frame cost $2 per inch and the cost of the frame was $160. If the area of the photograph is 231 inches squared. What are the dimensions of the frame?
Answers
w: width of the photograph
l: length of the photograph
The cost of the box is:
2 * (2 * (w + 2)) + 2 * (2 * (l + 2)) = 160
The area is:
(w + 2) * (l + 2) = 231
Solving the system of equations we have:
w = 5 inches
l = 31 inches
Then, the dimensions of the frame are:
w + 2 = 7 inches
l + 2 = 33 inches
Answer:
the dimensions of the frame are:
7 inches * 33 inches
In a right rectangles pyramid with base edges a= 18 cm and b= 10 cm the slant height toward a is k= 13 cm while the slant height towards b is l= 15 cm. What is the surface area of the pyramid
Answers
Base : 18 x 10 = 180 cm²
1st set of lateral faces:
A = 1/2 x b x h
1/2 x 13 x 18
A = 117 cm² x 2
234 cm²
2nd set of lateral faces:
1/2 x 10 x 15
A = 75 cm² x 2
150 cm²
150 cm² +234 cm² +180 cm²
The surface area of the pyramid is 564 cm².
Find the total area of the regular pyramid.
Answers
notice, the base of the "square" pyramid, is a square, and it has 4 triangular faces with a base of 2, and a height of √(10).
so the total surface area is the area of the base plus all 4 triangular faces' areas.
[tex]\bf \stackrel{\textit{squarish base}}{(2\cdot 2)}~~~~+~~~~\stackrel{\textit{4 triangular faces}}{4\left[ \cfrac{1}{2}(2)(\sqrt{10}) \right]}[/tex]
Answer:
Area of the regular pyramid = 16.64 square units.
Step-by-step explanation:
Given : Regular pyramid .
To find: Find the total area of the regular pyramid.
Solution : We have given that regular pyramid.
Area = 4 ( area of triangle ) + area of base .
Area of the regular pyramid = 4 ( [tex]\frac{1}{2} * base * height + side * side[/tex].
Area of the regular pyramid = [tex]4(\frac{1}{2}* 2* \sqrt{10} + 2*2[/tex]
Area of the regular pyramid = [tex]4( \sqrt{10} ) + 4[/tex]
Area of the regular pyramid = (4 [tex]( \sqrt{10} ) \+\ 1[/tex].
Area of the regular pyramid = 16.64 square units.
Therefore, Area of the regular pyramid = 16.64 square units.
6 is what percent of 8?
Answers
BRAINLIEST AND 20 POINTS ANSWER ASAP PLZ
Anyone have answers for Geometry B Unit 6 Lesson 10 test?? Surface area and volume? 31 questions.. my first question is..
1. use euler’s formula to find the missing number
Vertices-13
Edges-28
Faces-?
A.17 B.16 C.18 D.20
and the last one is
31. Whats the maximum vol. of a square pyramid that can fit inside a cube with a side length of 24 cm?
A.2,304 B.4,608 C.6,912 D.13,824
Answers
r = 28 - 13 + 2
r = 15 + 2
r = 17, so the first answer is correct.
7. The surface area of a cone is A = pi*r*sqrt(r^2 + H^2)
A = pi*(7)(sqrt(49 + 1849)
A = pi*(7)(43.57)
A = pi*305 = 959 m^2, so the first answer is correct.
13. The volume of the slab is V = HLW
V = (5 yards)(5 yards)(1/12 yards)
V = 25/12 cubic yards
So it costs $46.00*(25/12) = $95.83 of total concrete. The third answer is correct.
21. First, find the volume of the rectangular prism. V = HLW
V = (15 cm)(5 cm)(7 cm)
V = 525 cm^3
Next, find the volume of the pyramid. V = 1/3(BH), where H is the height of the pyramid and B is the area of the base of the pyramid. Note that B = (15 cm)(5 cm) = 75 cm^2
V = (1/3)(75 cm^2)(13 cm)
V = 325 cm^3
Add the two volumes together, the total volume is 850 cm^3. The fourth answer is correct.
22. The volume of a square pyramid is V = 1/3(S^2)(H), where S is the side length and H is the height.
V = (1/3)(20^2 in^2)(21 in)
V = 2800 in^3
Now that we know the volume of this pyramid, and that both pyramids have equal volume, we plugin our V to the equation for the volume.
2800 = (1/3)(84)(S^2)
2800 = 28S^2
100 = S^2
10 in = S, so we have a side length of 10 in, and the first answer is correct.
The missing number using Euler's formula is: Option A. 17
The maximum volume of a square pyramid is: Option B. 4,608
What is Euler's formula?"It is a geometrical formula. V − E + F = 2, where V represents number of vertices, E represents number of edges and F represents number of faces."
What is square pyramid?"Square pyramid is a three dimensional geometrical figure where four triangular sides are associated to square base."
What is cube?"A cube is a three-dimensional geometric structure with six congruent square face."
Formula for volume of a square pyramid:[tex]V=\frac{1}{3}a^{2}h[/tex]
where [tex]a[/tex] represents the length of square base and [tex]h[/tex] represents the height of the pyramid.
Consider the first question,
number of vertices (V) = 13
number of edges (E) = 28
So, using Euler's formula:
[tex]13-28+F=2[/tex]
⇒ [tex]-15+F=2[/tex]
⇒ [tex]F=2+15[/tex]
⇒ [tex]F=17[/tex]
So, the number of faces are 17.
Hence, the correct answer is option A. 17
Consider last question,
the side length of a cube = 24 cm
As the square pyramid fit inside a cube.
⇒ the length of the square base of a pyramid [tex]b[/tex] = 24 cm
and the height of a square pyramid [tex]h[/tex] = 24 cm
So, the volume of a square pyramid is,
[tex]V=\frac{1}{3} a^{2} h[/tex]
⇒ [tex]V=\frac{1}{3}[/tex] × [tex]24^{2}[/tex] × [tex]24[/tex]
⇒ [tex]V= 4608[/tex] [tex]cm^{3}[/tex]
Therefore, the maximum volume of a square pyramid that can fit inside a cube with a side length of 24 cm is [tex]4608[/tex] [tex]cm^{3}[/tex].
And the correct answer is option B. 4,608
Learn more about Euler's formula here,
https://brainly.com/question/22069428
Learn more about volume of a square pyramid here:
https://brainly.com/question/2501401
#SPJ2
If four times a number plus 3 is 11, what is the number? A. 16 B. 5 C. 2 D. 4
Answers
PLEASE HELP I REALLY NEED IT
What are the x-intercepts of the quadratic function?
f(x)=x2−3x−10
Enter your answers in the boxes.
_______ and ________
Answers
the x-intercepts of the quadratic function f(x)=x²−3x−10 is when
f(x)=0
x²−3x−10=0
Group terms that contain the same variable, and move the constant to the opposite side of the equation
(x²−3x)=10
Complete the square. Remember to balance the equation by adding the same constants to each side.
(x²−3x+2.25)=10+2.25
Rewrite as perfect squares
(x-1.5)²=12.25----------> (+/-)[x-1.5]=3.5
(-)[x-1.5]=3.5-----> x2=-2
the answer is
the x-intercepts of the quadratic function are
x=5
x=-2
Answer:
5 and -2 :)
explanation:
Select from the drop-down menu to correctly compare the numbers. 4.5872...[ ] 14−−√
>
<
=
Answers
> is ur answer just took the test
hope this helps............(:
4.5872 > [tex]\sqrt{14}[/tex]
We have two numbers.
We have to compare these two numbers.
The square root of a number is always less than ?The square root of a number is always less then the number itself.
According to the question, we have -
A = 4.5872
B = [tex]\sqrt{14}[/tex]
Now -
The value of B = 3.472.
Clearly, A > B
Hence, 4.5872 > [tex]\sqrt{14}[/tex].
To solve more questions on comparing numbers, visit the link below-
https://brainly.com/question/15451569
#SPJ2
Consider the system of inequalities and its graph. y ≤ –0.75x y ≤ 3x – 2 In which section of the graph does the actual solution to the system lie?
A. 1
B. 2
C. 3
D. 4
Answers
The actual solution lie in region 4.
EXPLANATION
The given system of inequalities are
[tex]y \leqslant - 0.75x \: and \: y \leqslant 3x - 2[/tex]
Choose a point from each region and substitute in to the system of inequalities, the one that satisfies the system simultaneously is the actual solution.
Region 2: We test for
[tex](0,2)[/tex]
Thus,
[tex]2 \leqslant - 0.75(0)\: and \: 2 \leqslant 3(0)- 2[/tex]
[tex]2 \leqslant 0\: and \: 2 \leqslant- 2[/tex]
The above statement is false.
For region 1: We test for
[tex](-2,0)[/tex]
[tex]0 \leqslant - 0.75( - 2)\: and \: 0 \leqslant 3( - 2) - 2[/tex]
[tex]0 \leqslant 1.5\: and \: 0 \leqslant - 8[/tex]
The first statement is true but the second statement is false, so the whole thing is false.
For Region 3: We test
[tex](2,0)[/tex]
[tex]0 \leqslant - 0.75(2) \: and \: 0 \leqslant 3(2)- 2[/tex]
[tex]0 \leqslant - 1.5 \: and \: 0 \leqslant 4[/tex]
The first statement is false and the second is true so the whole statement is not true.
For region 4: We test
[tex](0,-4)[/tex]
[tex]0 \leqslant - 0.75( - 4)\: and \: - 4 \leqslant 3( 0) - 2[/tex]
[tex]0 \leqslant 3\: and \: - 4 \leqslant - 2[/tex]
Both statements are true, region 4 represents the actual solution to the system.
Answer:
4
Step-by-step explanation:
trust me
Need answer asap evaluate the limit or state that the limit does not exist 7n-8n/2n
Answers
write the product in its simplest form:
8y^7*6y^7
Answers
find the circumference of the circle r=4 Ft
Answers
circumference= pi x diameter(8ft)
Answer: 25.12 feet
Step by step explanation:
The number of boys varies directly as the number of girls and inversely as the number of teachers when there are 24 boys and 16 girls there are 2 teachers how many girls are there when there are 73 boys and 1 teacher?
Answers
thus:
b=kg/t
where k is the constant of proportionality
k=bt/g
b=24, when g=16 and t=2
k=(24*2)/16
k=3
hence:
b=(3g)/2
thus the number of girls when:
b=73 and t=1 will be:
73=(3*g)/1
73=3g
g=73/3
g=24.333~ 24 girls
Which value makes g true (x-3)(x+5)=x^2+gx-15
Answers
You bought a new car for $15,000 and know that it loses 1 5 of its value every year. The equation that models the value of your car is y = 15000( 4 5 )x. Which is the MOST reasonable domain for this function?
Answers
y=15000(1/5)^x
the graph will be represented as follows:
From the graph we see that the domain is all real numbers.
Answer:
0 < X < 20
Step-by-step explanation:
Lincoln Technical College has 856 students. They expect 700 guests for a special speaker. The custodian has set up 1,500 chairs. How many more chairs are needed if everyone is to have a seat? 3. Draw a picture or a chart that shows the information and the question
Answers
57 chairs are needed if everyone is to have a seat
To calculate the number of additional chairs needed, we first need to determine the total number of seats required. This is the sum of the students, guests, and speaker. Then, we subtract the number of chairs already set up to find out the additional chairs needed.
Let's break it down step by step:
1. Total number of seats required:
- Students: 856
- Guests: 700
- Speaker: 1 (assuming the speaker needs a seat)
Total seats required = 856 (students) + 700 (guests) + 1 (speaker) = 1,557 seats
2. Number of chairs already set up: 1,500
3. Additional chairs needed:
Total seats required - Chairs already set up = 1,557 - 1,500 = 57
So, Lincoln Technical College needs 57 more chairs to accommodate everyone if all students, guests, and the speaker are to have a seat.
Lincoln Technical College has 856 students and expects 700 guests for a special speaker, making the total number of attendees 1,557. The custodian has already set up 1,500 chairs. To find out how many more chairs are needed, we subtract the number of chairs already set up from the total seats required. This calculation reveals that 57 additional chairs are needed to ensure that everyone has a seat. It's essential to consider all attendees, including students, guests, and the speaker, to accurately determine the total seating requirements.
Complete question:
Lincoln Technical College has 856 students. They expect 700 guests for a special speaker. The custodian has set up 1,500 chairs. How many more chairs are needed if everyone is to have a seat?
three times the perimeter of a triangle is the same as 75 decreased by twice the perimeter. what is the perimeter of the triangle?
Answers
3x = 75 - 2x
Add 2x from both sides:
5x = 75
Divide both sides by 5:
x = 15
Answer: The perimeter is 15 units.
10 POINTS!! WILL MARK BRAINLIEST!!
Circle O is shown below. The diagram is not drawn to scale.
If m∡R = 28°, what is m∡ O?
Please Explain.
Answers
Answer:
Measure of angle O is 56 degrees.
Step-by-step explanation:
We can see from diagram that NQ is minor arc and NRQ is major arc. Our angle R (inscribed angle) and O (central angle) are corresponding to minor arc NQ. We will use inscribed angle theorem which states that measure of inscribed angle is one-half the measure of central angle.
We are given that measure of inscribed angle R is 28 degrees. To find measure of our central angle O we will multiply 28 by 2.
[tex]28*2=56[/tex]
Therefore, measure of angle O will be 56 degrees.
The gallup poll interviews 1600 people. of these, 18% say that they jog regularly. the news report adds: "the poll had a margin of error of plus or minus three percentage points." you can safely conclude that
a. 95% of all gallup poll samples like this one give answers within ±3% of the true population value.
b. the percent of the population who jog is certain to be between 15% and 21%.
c. 95% of the population jog between 15% and 21% of the time.
d. we can be 3% confident that the sample result is true.
e. if gallup took many samples, 95% of them would find that exactly 18% of the people in the sample jog.
Answers
Final answer:
The ±3 percent represents the margin of error in the Gallup poll, indicating the potential variation in the poll results due to sampling. The percentage of people who jog regularly could be as low as 15% or as high as 21%.
Explanation:
The ±3 percent represents the margin of error in the Gallup poll. The margin of error is a measure of the uncertainty or potential variation in the poll results due to the sampling process. In this case, it means that the percentage of people who say they jog regularly could be as low as 15% or as high as 21%.
Solve this problem: –282 – (+1,017) = ? A. 1,299 B. 735 C. –735 D. –1,299
Answers
Happy to help! :)
imagine the ( - ) has a one behind it like so
-282 -1 (+1,017)
you need to distribute the 1, so 1,017 become a negative
subtracting two negative numbers is like adding positive numbers, except the negative still exists, as a result
-282 - 1,017
= - 1,299
To find the height of a pole, a surveyor moves 120 feet away from the base of the pole and then, with a transit 8 feet tall, measures the angle of elevation to the top of the pole to be 26. to the nearest foot, what is the height of the pole
Answers
Let AB be the height of the pole; CD be the height of the transit; BC is the distance from the base of the pole to the transit.
Triangle ADE is a right triangle with angle D measuring 26 degrees. Using the tangent function, we have
[tex]tan\:26^{\circ} =\frac{AE}{120}[/tex]
So,
[tex]AE=120\:tan\:26^{\circ} [/tex]
[tex]AE=58.53\:[/tex]
Therefore, the overall height of the pole is
[tex]AB=58.53+8=66.53\:feet[/tex]
The height of the pole is 66.53 feet.