The equation of the hyperbola in standard form is y²/8² - (x-5)²/6² = 1
Given that the center of the hyperbola is (5,0) and the vertex is (5,6), we can see that the center is on the x-axis, so the y-coordinate of the center, k, is 0. The x-coordinate of the center, h, is 5.
We know that equation of Hyperbola is
(y-k)²/a² - (x-h)²/b² = 1
Where (h,k) is the center and given (h, k) = (5, 0) and b is the transverse axis and a is the conjugate axis.
Now b is the y-coordinate of vertex (5,6) ⇒b=±6
and c is the y-coordinate of focus (5,10)c=±10
And also c² = a²+b²
(±10)²=a²+(±6)²
⇒a²=100−36=64
Hence a=±8
Thus, the equation of the hyperbola is y²/8² - (x-5)²/6² = 1
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If A(2,3),B(4,6),C(7,4) and D(a,b) are vertices of quadrilateral
It can be concluded that the given coordinates of A(2,3), B(4,6), C(7,4) and D(a,b) can be used to determine the length of the sides as well as the angles of the quadrilateral ABCD.
What is quadrilateral?A quadrilateral is a four-sided polygon with four angles. It is a two-dimensional shape that is bounded by four line segments, each of which is called a side. The sum of the angles of a quadrilateral is 360 degrees. Types of quadrilaterals include rectangles, squares, trapezoids, parallelograms, and kites.
First of all, ABCD is a quadrilateral, which means it has four sides and four vertices. A, B, C and D are the vertices of this quadrilateral. It can be seen that A(2,3), B(4,6), C(7,4) and D(a,b) given in the question are coordinates of the vertices. Therefore, from these coordinates, we can conclude that A, B, C and D are the vertices of the quadrilateral ABCD.
Moreover, these coordinates can also be used to determine the length of the sides of the quadrilateral. The distance between A and B can be calculated by using the Pythagorean theorem. Similarly, the lengths of other sides can also be calculated by using the same theorem. Furthermore, the coordinates will also help us to calculate the angles formed by the sides of the quadrilateral ABCD.
Therefore, it can be concluded that the given coordinates of A(2,3), B(4,6), C(7,4) and D(a,b) can be used to determine the length of the sides as well as the angles of the quadrilateral ABCD.
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complete questions as follows-
If A(2,3),B(4,6),C(7,4) and D(a,b) are vertices of quadrilateral prove its true.
Find length of third side help please
Length of third side = 10 units.
What is Pythagorean theorem?Pythagorean theorem, the well-known geometric theorem that the sum of the squares on the legs of a right triangle is equal to the square on the hypotenuse (the side opposite the right angle)—or, in familiar algebraic notation, a2 + b2 = c2.
Here, given that,
Height = 7 units
Base = √51 units
And, we have, it is a right angle triangle
and we have to find the hypotenuse.
By Pythagorean theorem we get,
Hypotenuse = √7²+ 51
=√100
=10 units.
Hence, Length of third side = 10 units.
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Identify the focus and directrix of the parabola represented by the equation (y-3)^2=-4(x 3)
The focus is at (3,3+1/4) and the directrix is the equation of a line is 2.75.
A parabola is defined as the set of all points that are equidistant to the focus (F) and the directrix (d).
In the case of this parabola represented by the equation (y-3)² = 4(x-3), the vertex is located at the point (3,3) which means that this parabola opens horizontally to the left. Since the coefficient of x² is positive the parabola opens up.
The focus is located at the point (3,3+1/4) which is (3,3.25) and the directrix is the line x = 2.75. These two elements are crucial in understanding the shape and behavior of the parabola, The focus is where the parabola bends, and the directrix is where it opens.
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It costs $31 for 3 visitors to take a tour. It costs $80 for 10 visitors to take the same tour. What is the rate of change?
The rate of change is $2.33 per visitor
What is rate of change?Rate of change refers to how quickly something changes over time. For example, acceleration is the rate of change of velocity with time.
Velocity is also the rate of change in position with time.
Rate of change means difference or changes with time.
It cost $31 for 3 visitor. This means that the cost per visitor is $31/3 = $10.33
It cost $80 for 10 visitor. This means that the cost for one visitor is 80/10 = $8
The rate of change is the difference in rate. This means that rate of change = $10.33 - $8
= $2.33/visitor
Therefore the rate of change is $ 2.33 per visitor.
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Please help!!!!! I need it right know
1) The increasing interval of the function is (-5.5,-5)∪(-4, -3)∪ (2,∞).
2) The decreasing interval of the function is (-5,-4)∪(1,2).
3. The negative interval is (-5.5, -3.25).
4. The positive interval is (-3.25,∞).
5. The minimum are at (-4,-2) and (2,0).
6. The maxima are at (-5,0).
7. The domain of the function is (-5.5,∞).
8. The range of the function is (-2.5,∞).
9. The value of f(-5) is 0.
10. The value of f(1) is 2.5.
What is increasing interval?
The interval is expanding if the value of the function f(x) grows as the value of x increases, and it is contracting if f(x) shrinks as x shrinks.
The graph is increasing in between points (-5.5,-2.5) to (-5,0), (-4,-2)to (-3,2.5) and (2,0) to (∞,∞)
The increasing interval is (-5.5,-5)∪(-4, -3)∪ (2,∞).
The graph is decreasing (-5,0 to (-4,2) and (1,2.5) to (2,0).
The decreasing interval of the function is (-5,-4)∪(1,2).
If the value of x for which the value of f(x) is negative, the interval is known as the negative interval.
From the point (-5.5, -2.5) to (-3.25,0), the graph lies below x-axis.
Thus the negative interval is (-5.5, -3.25).
If the value of x for which the value of f(x) is positive, the interval is known as the positive interval.
From the point (-3.25,0) to (∞,∞), the graph lies above x-axis.
Thus the negative interval is (-3.25,∞).
The graph of a function y = f(x) has a local maximum at the point where the graph changes from increasing to decreasing. At this point the tangent has zero slopes. At the point when the graph shifts from declining to increasing, a local minimum is present.
The minimum is at (-4,-2) and (2,0).
The maxima are at (-5,0).
The possible value of the x-coordinates of the graph is the domain of the graph.
The domain of the graph is (-5.5,∞).
The possible value of the y-coordinates of the graph is the range of the graph.
The range of the graph is (-2.5,∞).
The y-coordinate of a point at x = a on the graph is the value of f(a).
(-5,0) is a point on the graph. Thus the value of f(-5) is 0.
(1,2.5) is a point on the graph. Thus the value of f(1) is 2.5.
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2. Use what you know about equivalent expressions to write an expression equivalent to the
one given.
a. 5n
b. 2n+2
c. 4n-4
d. 3n+2n + n
The equivalent of the expression given as (2n + 4) - 2 is 2n + 2
How to determine the equivalent expressionFrom the question, we have the following parameters that can be used in our computation:
(2n + 4) - 2
To start with, we need to remove the brackets
So, we have the following representation
(2n + 4) - 2 = 2n + 4 - 2
Next, we evaluate the like terms
(2n + 4) - 2 = 2n + 2
Hence, the equivalent expression is 2n + 2
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Complete question
Use what you know about equivalent expressions to write an expression equivalent to the one given.
(2n + 4) - 2
a. 5n
b. 2n+2
c. 4n-4
d. 3n+2n + n
Solve the equation.
20=r+11
r=?
Answer:
9
Step-by-step explanation:
The answer is 9
Answer: 8
Step-by-step explanation:
20=r+11
-11 -11
8=r
For the canned food drive, 6 students each collected the same number of cans. They collected 48 cans in all. How many cans did each student collect
Answer:
They each collected 8
Step-by-step explanation:
If all 6 collected the same amount, we can just divide the total amount of cans by 6
48/6 = 8
The height of parallelogram whose area is 35 cm2 and altitude 7 cm
Answer:
altitude is same as height =7cm
Step-by-step explanation:
area of parallelogram=height (altitude) × base length
Wich of this relations is a function?
Pls Help
The relations depicted in graph 1 is a function as every input value has exactly one output value.
What are functions?A relation called a function is one in which every potential input value results in exactly one possible output value. They are set of ordered pairs.
What are relations?Relations indicate the relationship between the indicated input and output value.
Graph 1:
In graph 1 each input value has exactly one output hence relations in graph 1 is a function.
Graph 2:
Graph 2 represents a circle, in this each input value does not have exactly one output value and hence it is not a function.
Graph3:
In graph 3, each input value does not have exactly one output value and hence it is not a function.
Graph 4:
In graph 4, each input value does not have exactly one output value and hence it is not a function.
Hence, relations in graph 1 is a function.
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A carpenter is making a wooden window frame that has a width of 1 inch. A window with a one-inch frame is shown. The frame is comprised of a rectangle and a semicircle. The rectangle has side lengths of 12 inches and 48 inches. The semicircle has a radius of 6 inches. The frame is 1-inch wider than the window. How much wood does the carpenter need to build the frame
The carpenter needs 1602.76 inches of wood to build the frame, which includes a 13x49 inch rectangle, a 12x48 inch rectangle, and a 7 inch radius semicircle.
Rectangle 1:
13 x 49 = 637 inches
Rectangle 2:
12 x 48 = 576 inches
Semicircle:
3.14 x 7 x 7 = 153.94 inches
Total = 637 + 576 + 153.94 = 1602.76 inches of wood
The second piece will be a rectangle with sides of 12 inches and 48 inches. The third piece will be a semicircle with a radius of 7 inches. Therefore, the carpenter will need a total of
(13 x 49) + (12 x 48) + (3.14 x 7 x 7)
= 1602.76 inches of wood to build the frame.
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A sequence is defined by the term-to-term rule
U(n+1) = U₁² +3
Given that u0= 1,
a) find u₁
b) find u₂
c) find u3
The terms of sequences are u₁ = 4, u₂ = 19 and u₃ = 364.
What is a sequence?An ordered list of things is a sequence (or events). Similar to a set, it has members (also called elements, or terms). The length of the sequence is the number of ordered items (potentially infinite).
Given:
A sequence is defined by the term-to-term rule
uₙ₊₁ = u₁² + 3
And u₀ = 1,
a). The second term
u₁ = u₀² + 3
u₁ = (1)² + 3
u₁ = 4
b). The third term
u₂ = u₁² + 3
u₂= (4)² + 3
u₂ = 19
c). The fourth term
u₃ = u₂² + 3
u₃ = (19)² + 3
u₃ = 364
Therefore, The values, u₁ = 4, u₂ = 19 and u₃ = 364.
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What is the domain of f 7?
The domain of the function [tex]f(x) = 7[/tex] is (-∞, ∞). It means that all real numbers are domains for the given function.
The domain of the function is the set of numbers for which the function is defined. It acts as input for the function and gives output known as range.
The range of the function is the set of numbers of possible output values.It is derived from set of inputs known as domain.
The domain of the expression [tex]f(x)=7[/tex] is all real numbers except where the expression is undefined. In this given case, there is no real number that makes the expression undefined.
Hence, The domain for the given expression is (-∞, ∞).
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Determine the number of unique triangles that can be made from the following information: For ∆LMN, ∠ = 31°, /LM = 6.9, and /MN = 3.4.
Answer:
Step-by-step explanation:
There is only one unique triangle that can be made with this information.
FInd the value of a, b and c
The angles in the triangle is as follows:
a = 55°b = 97°c = 83°How to find the angle of a triangle?A triangle is a polygon with three sides. The sum of angles in a triangle is 180 degrees.
Therefore, let's find the angles a, b and c in the triangle as follows:
c + 44 + 53 = 180(sum of angles in a triangle)
c = 180 - 44 - 53
c = 83 degrees
Therefore, let's find the value of a and b.
b + 83 = 180 (sum of angle on straight)
b = 180 - 83
b = 97 degrees.
Hence,
28 + 97 + a = 180(sum of angles in a triangle)
a = 180 - 28 - 97
a = 55 degrees.
Therefore,
a = 55°
b = 97°
c = 83°
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Define and distinguish among mean, median, and mode. Choose the correct description of the mean below. O A. The mean is the most common value in a data set. It can be strongly affected by outliers. O B. The mean is the sum of all the values divided by the number of values. It can be strongly affected by outliers. O C. The mean is the most common value in a data set. It is not affected by outliers. O D. The mean is the sum of all the values divided by the number of values. It is not affected by outliers. O E. The mean is the middle value in a data set. It can be strongly affected by outliers O F. The mean is the middle value in a data set. It is not affected by outliers.
Answer:
As far as I know
Mode: The most frequent number—that is, the number that occurs the highest number of times. Example: The mode of {4 , 2, 4, 3, 2, 2} is 2 because it occurs three times, which is more than any other number.
The median is the middle value when a data set is ordered from least to greatest. The mode is the number that occurs most often in a data set.
What is the factorization of the polynomial below?
x^2 - 5x - 36
OA. (X-9)(x+4)
OB. (x+9)(x-4)
OC. (x+9)(x+4)
OD. (X-9)(x-4)
The factorization of the polynomial is ( x - 4 ) ( x + 9 )
What is factorization?factorization is the method of breaking a number into smaller numbers that multiplied together will give that original form.
We are given the function as
x² + 5x - 36
We have to find out the two numbers when multiplied results to 36 & when subtracted results to 5.
The numbers are 9 and 4
x²+ ( 9 - 4)x - 36
Distribute x :
x² + 9x - 4x - 36
Take the common :
x ( x + 9 ) - 4 (x + 9 )
( x - 4 ) ( x + 9 )
Hence, The right answer is of Option C [ ( x - 4 ) ( x + 9 ) ]
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Ajar contains 24 blue marbles, 16 red marbles, and 14 white marbles. Find the simplified ratio
of total marbles to red marbles.
Answer:
Answer: 27:8
Step-by-step explanation:
There are 24 + 16 + 14 = 24+16+14= 54
54 marbles in total.
The ratio of total marbles to red marbles is 54 : 16, which simplifies to 27 : 8.
Answer: 27:8
A dog kennel charges $30 per night to board your dog and $6 for each hour of playtime. The amount of money you spend is given by 30x+6y=180, where x is the number of nights and y is the number of hours of playtime. Graph the equation and interpret the intercepts. Keyboard Instructions Initial graph state The horizontal axis goes from -0.9 to 10.4 with ticks spaced every 1 unit(s). The vertical axis goes from -3.9 to 34.4 with ticks spaced every 5 unit(s).
The graph of the equation 30x+6y=180 will be a line in the xy-plane.
What is equation?An equation is a mathematical statement that expresses the equality of two expressions. It is composed of one or more terms and an equal sign, and can contain constants, variables, and operators. Equations are used to represent relationships between different values, quantities, and objects, and to solve for unknowns. Equations are a fundamental tool of mathematics and can be used to describe physical processes as well as abstract concepts.
The line will pass through the origin, since the equation is satisfied when both x and y are 0. The intercepts of the line on the x-axis and y-axis represent the values of x and y that will make the equation true, regardless of the values of the other variable.
For this equation, the y-intercept is 30, since when x=0, the equation becomes 30(0)+6y=180, and so y=30. This means that if you do not board your dog, you would need to play with it for 30 hours in order to meet the requirement of spending a total of 180 dollars.
The x-intercept is 6 since when y=0, the equation becomes 30x+6(0)=180, and so x=6. This means that if you do not play with your dog, you would need to board it for 6 nights in order to meet the requirement of spending a total of 180 dollars.
Therefore, the graph of the equation 30x+6y=180 shows that it is possible to meet the requirement of spending a total of 180 dollars either by boarding your dog for 6 nights or by playing with it for 30 hours, regardless of the values of the other variable.
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Let g(x)=∫x0f(t)dt where f is the function whose graph is shown. Evaluate g(0), g(2), g(4), g(6) and g(12)?
The function g(x) is the integral of f(t) from 0 to x, where f(t) is the function whose graph is shown. This means that the value of g(x) is the area under the graph of f(t) from 0 to x.
g(0) = 0
g(2) = 8
g(4) = 24
g(6) = 48
g(12) = 96
g(0) = ∫02f(t)dt = 0
g(2) = ∫22f(t)dt = 8
g(4) = ∫42f(t)dt = 24
g(6) = ∫62f(t)dt = 48
g(12) = ∫122f(t)dt = 96
The function g(x) is the integral of f(t) from 0 to x, where f(t) is the function whose graph is shown. This means that the value of g(x) is the area under the graph of f(t) from 0 to x. To evaluate g(x) for a given value of x, we need to calculate the area under the graph of f(t) from 0 to x. In this case, the area under the graph of f(t) from 0 to 2 is 8, from 0 to 4 is 24, from 0 to 6 is 48 and from 0 to 12 is 96. Therefore, g(0) = 0, g(2) = 8, g(4) = 24, g(6) = 48 and g(12) = 96.
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When a rational number and an
irrational number are multiplied, the
product is always ___.
Answer: irrational number
Step-by-step explanation:
8 dm to 3.2 dm
pls helppp
Answer:
8 dm decreased 60%
Step-by-step explanation:
8-3.2=4.8
4.8/8=0.6
0.6=60%
The path of a drinking fountain is designed to reach a maximum height of 4.5 feet after 1 second. The spout is at a height of 4 feet. If the water pressure decreases, the water does not reach the intended height. Complete the values for the inequality in vertex form to describe the points that are less than the projected path. y__a(x – h)2 + k
a=
h=
k=
The values for the inequality in vertex form is y < -1.5(x - 1)^2 + 4.5
How to determine the equation of the parabolaThe equation for a parabolic function in vertex form is given by y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola.
The vertex of the parabolic path of the drinking fountain is (1, 4.5). So, h = 1 and k = 4.5.
Substituting the values for h and k into the equation:
y = a(x - 1)^2 + 4.5
Now, we need to determine the value of a using the point
(x, y) = (2, 4)
So, we have
a(2 - 1)^2 + 4.5 = 4
Evaluate
a = -1.5
So, we have
y = -1.5(x - 1)^2 + 4.5
Express as inequality
y < -1.5(x - 1)^2 + 4.5
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Complete question
The path of a drinking fountain is designed to reach a maximum height of 4.5 feet after 1 second. The spout is at a height of 4 feet after 2 seconds. If the water pressure decreases, the water does not reach the intended height. Complete the values for the inequality in vertex form to describe the points that are less than the projected path. y__a(x – h)2 + k
Answer:
Step-by-step explanation:
The path of a drinking fountain is designed to reach a maximum height of 4.5 feet after 1 second. The spout is at a height of 4 feet. If the water pressure decreases, the water does not reach the intended height.
Complete the values for the inequality in vertex form to describe the points that are less than the projected path.
y
✔ <
a(x – h)2 + k
a =
✔ –0.5
h =
✔ 1
k =
✔ 4.5
PLS HELP MEE.. I hv BEEN working on this question for days..i hv to submit it today...
Answer:
Below
Step-by-step explanation:
Working on this for days ? Seriously ??
5 x 10 ^8 = 50 x 10^7
50 x 10^7 + 2 x 10^7 = 52 x 10^7 = 5.2 x 10^8 = 520, 000,000
Pick the form you need.....
PLEASE HELP ME NOW THANK YOU I WILL GIVE YOU BRAINLIEST
[tex]-2(6x+36)-7x\leq 23 and 6x+10\ \textgreater \ 40[/tex]
Answer:
The first inequality can be solved by combining like terms and isolating x on one side:
-2(6x + 36) - 7x <= 23
-12x - 72 - 7x <= 23
-19x - 72 <= 23
-19x <= 95
x >= -5
The second inequality can be solved by subtracting 10 from both sides:
6x + 10 > 40
6x > 30
x > 5
The solution to both inequalities is x >= -5 and x > 5, which means that x is a real number greater than 5.
Ann has 440 meters of fencing and wishes to enclose a rectangular field.
Suppose that a side length (in meters) of the field is x, as shown below.
The function that gives the area A(x) of the field (in square meters) in terms of x is A(x)=(220x-x²).
What is the area of a rectangle?The area occupied by a rectangle within its boundary is called the area of the rectangle. The formula to find the area of a rectangle is Area = Length × Breadth.
Ann has 440 meters of fencing and wishes to enclose a rectangular field is x.
We know that, the perimeter of a rectangle is P=2(Length+Width)
Let the length of a rectangle be x.
Now, 440=2(x+Width)
Width=220-x
a) A function that gives the area A(x) of the field (in square meters) in terms of x.
A(x)= x(220-x)
A(x)=(220x-x²)
b) Side length x gives the maximum area that the field can have
220x-x²=0
220x=x²
x=220 meters
So, Side length x= 220 meters
Therefore, the function that gives the area A(x) of the field (in square meters) in terms of x is A(x)=(220x-x²).
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Find the perimeter and area of these shapes asap!
Answer: #1 a=226
Step-by-step explanation:#1, split the shale into 2 parts. A small square with the 6 and 16. Then subtract six from sixteen and you have the height of the rectangle. Then take your difference (10) and multiply it with your width (19)= 190
Then do the little square which is 6x6=36. Add 190 and 36
Answer:
The area and the perimeter is 225 and 152. I'll show you how to do the rest below:
Step-by-step explanation:
For the Area you Multply all the numbers together
And for the perimeter you multiply the numbers by four
I hope you understand! let me know in the reply box
A minor league baseball stadium offers general admission seats and premier seats. During the season opener, the stadium sold 1,348 tickets and had $12,748.30. The price of a premiere seat was 30% more than the $8.50 general admission seat. How many premiere seats and how many general admission seats were sold for the season opener?
The number of general admission seat sold is 842 and the number of premiere seat sold is 506.
How many of each seats were sold?The system of equations that represent the information provided in the question are:
a + b = 1348 equation 1
8.5a + (1.30 x 8.5)b = 12,748.30
8.5a + 11.05b = 12,748.30 equation 2
Where:
b = number of premiere seat solda = number of general admission seat soldThe elimination method would be used to solve these equations
Multiply equation 1 by 8.5
8.5a + 8.5b = 11,458 equation 3
Subtract equation 3 from equation 2
1290.30 = 2.55b
b = 1290.30 / 2.55
b = 506
Substitute for b in equation 1 :
a + 506 = 1348
a = 1348 - 506
a = 842
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Solve and graph.
-4p>-28
Answer:
p < -7
Step-by-step explanation:
-4p > 28 Divide both sides by -4 and flip the sign
p < -7
A farm raises cows and chickens. The farmer has a total of 40 animals. One day he counts the legs of all his animals and realizes he has a total of 118 legs. How many chickens does the farmer have?.
After solving, the farmer have total 21 chicken and 19 cows.
In the given question, a farm raises cows and chickens.
The farmer has a total of 40 animals.
One day he counts the legs of all his animals and realizes he has a total of 118 legs.
We have to find how many chickens the farmer have.
Let the number of cows that the farmer have is x.
Let the number of chicken that the farmer have is y.
The farmer has a total of 40 animals.
So the equation should be
x + y = 40....................(1)
As we know that a cow has 4 legs and a chicken has 2 legs.
He has a total of 118 legs.
Now the equation should be
4x + 2y = 118....................(1)
Substituting x = 40 - y from the equation 1, we get
4(40 - y) + 2y = 118
160 - 4y + 2y = 118
160 - 2y = 118
Subtract 160 on both side, we get
-2y = - 42
Divide by -2 on both side, we get
y = 21
Now putting the value of y in x = 40 - y, we get
x = 40 - 21
x = 19
Hence, the farmer have total 21 chicken and 19 cows.
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The farmer has 0 chickens and 40 cows.
Let's solve this problem using an algebraic equation. First, we will assign a variable to represent the number of chickens the farmer has. We will call this variable 'x'. We also know that the total number of animals the farmer has is 40. Therefore, we can write an equation that states:
x + (40 - x) = 40
This equation states that the sum of the number of chickens and the number of cows is equal to the total number of animals the farmer has. Now, let's look at the number of legs the animals have. We know the total number of legs is 118. We also know that chickens have two legs and cows have four legs. We can now write another equation that states:
2x + (4(40 - x)) = 118
This equation states that the sum of the number of chicken legs and the number of cow legs is equal to the total number of legs the animals have. We can solve for x by multiplying both sides of the equation by -1 and then adding both equations:
2x + (4(40 - x)) = 118
-2x - (4(40 - x)) = -118
0x = -118 + 118
0x = 0
Since 0x = 0, we can conclude that x = 0. This means that the farmer has 0 chickens and 40 cows.
Learn more about algebraic equation here:
https://brainly.com/question/953809
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