The equation of the ellipse with foci at (39,0) and (-39,0) and a constant difference of 30 is given as follows:
x²/39² + y²/9² = 1.
How to obtain the equation of the elipse?The equation of an ellipse of center (h,k) is given by the rule presented as follows:
(x - h)²/a² + (y - k)²/b² = 1.
In which:
a is the distance from the center to the focus.b is the distance from the center to the vertex.The coordinates of the foci are given as follows:
(39,0) and (-39,0).
The center is the midpoint of these two coordinates, hence:
h = (39 - 39)/2 = 0.k = (0 + 0)/2 = 0.The coefficient a is given as follows:
a = |39 - 0| = |0 - 39| = 0.
Considering the constant difference of 30, the constant b is given as follows:
b = a - 30 = 39 - 30 = 9.
Thus the equation of the ellipse is given as follows:
x²/39² + y²/9² = 1.
Missing InformationThe problem asks for the equation of the ellipse with the given features.
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a rectangular field will be bounded by a wall on one side and by fencing on the other three sides. what is the largest area you can enclose with 120 feet of fencing?
The largest area you can enclose with 120 feet of fencing is 900 square feet.
To find the maximum area you can enclose with 120 feet of fencing, you can use the formula for the area of a rectangle, which is A = lw, where A is the area, l is the length, and w is the width. Since the length of the rectangle is bounded by a wall and cannot be changed, the width will be the variable we seek to maximize.
Since the perimeter of the rectangle is fixed at 120 feet, we can set up the following equation:
2l + 2w = 120
This equation represents the total length of the fence needed to enclose the rectangular field. We can solve for w in terms of l by rearranging the equation as follows:
2w = 120 - 2l
w = (120 - 2l)/2
To maximize the area of the rectangle, we want to maximize the value of w. We can do this by making l as small as possible, since this will make w as large as possible. Therefore, we can set l equal to the minimum possible value, which is the length of the wall.
Let's say the length of the wall is x feet. Then, we can substitute x for l in the equation for w to find the maximum possible width:
w = (120 - 2x)/2
To find the maximum area, we can then substitute this value for w back into the formula for the area of a rectangle:
A = lw
= x((120 - 2x)/2)
= (120x - 2x^2)/2
This is a quadratic function, and we can use the techniques for maximizing quadratic functions to find the maximum value of A. To do this, we take the derivative of the function and set it equal to zero:
A' = (120 - 4x)/2 = 0
120 - 4x = 0
x = 30
Substituting this value back into the equation for A, we find that the maximum possible area is:
A = (120 * 30 - 2 * 30^2)/2
= (3600 - 1800)/2
= 1800/2
= 900 square feet
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What is the process of forecasting future expenses and savings called?
The process of forecasting future expenses and savings is called financial planning.
What is forecasting?Forecasting is the process of predicting future events or outcomes based on past data, trends, and other information. It is an important tool for businesses, allowing them to plan ahead and make informed decisions. Forecasting can help organizations anticipate changes in the market, anticipate customer demand, plan investments, and more. If done properly, it can help organizations achieve their goals and maximize profits.
Financial planning is the process of creating a budget, setting goals, and developing strategies to help manage money and reach financial goals. It includes estimating future income and expenses, assessing current financial resources, and creating a plan for spending, saving, and investing. Financial planning can help you to stay on track towards meeting your financial goals and create a secure financial future.
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Find an exponential function in the form y = a b with a y-
intercept of (0,5) and a multipler of 3
[tex]y = 5(3) {}^{x} [/tex]
Why is the power of 2 called square?
The power of 2 is called a square due to the definition of the area of a square.
What are a square and Power?
A square is a quadrilateral with four equal sides and four equal angles that is a regular quadrilateral. The square's angles are at a straight angle or 90 degrees. The square's diagonals are also equal and split at a 90-degree angle.
The base number is the factor that is multiplied by itself, and the exponent is the number of times the same base number has been multiplied. Power is defined as a base number raised to the exponent.
The power of two is called a square.
The square function's name derives from the fact that a square with sides of length [tex]l[/tex] equals [tex]l^2[/tex] has a significant role to play in the definition of area.
The area is inversely proportional to size: the area of a form that is [tex]n[/tex] times larger is [tex]n^2[/tex] times larger.
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How to find all pairs in array of integers whose sum is equal to given number?
To find all pairs in an array of integers whose sum is equal to a given number, iterate through the array and for each element, check if there is another element in the array that, when added to the current element, is equal to the given number, and if such an element is found, add the pair (current element, other element) to a list of pairs; repeat this process until all elements in the array have been checked.
What is an array?A group of objects, pictures, or numbers organized in rows and columns is known as an array. Arrays provide as concrete examples of multiplication concepts.
Why do we need arrays?Arrays are needed in cases where you need to store and manipulate large amounts of data efficiently.
Some of the benefits of using arrays include:
Arrays allow you to store a large number of items in a single data structure, which makes it easier to work with large datasets. Arrays allow you to access and modify the items in the array quickly, using the index of the item.There are numerous techniques for locating all pairs in an integer array whose total is equal to a specific value. Here is one strategy you may employ:
In order to store the pairs you find, create a blank list.Check if there are any other elements in the array that, when combined with the current element, equal the specified number as you iterate over the array.Add the pair (current element, other element) to the list of pairs if you come across such an element.Once you have tested each entry in the array, repeat this step.To learn more about arrays visit:
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The graph represents a quadratic function. Write an equation of the function in standard form.
The equation for the quadratic function in standard form is f(x) = x² - 11x + 24
Writing Quadratic equationFrom the question, we are to write an equation for the quadratic function in standard form.
From the given quadratic graph,
The roots of the function are 3 and 8.
That is,
x = 3 and x = 8
Then,
The factors of the quadratic function are (x -3) and (x - 8)
Thus,
We can write that
(x - 3)(x - 8) = 0
x² - 8x - 3x + 24 = 0
Simplify
x² - 11x + 24 = 0
Hence, the equation is f(x) = x² - 11x + 24
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How do you know if there is a limit or not?
By examining the function's graph, there are four alternative techniques to decide if a limit is there.
1. if the graph has a hole in the line with a point for that value of x on a different value of y, then the limit DOES exist.
2. The limit still exists if there is a gap in the graph at the value that x is nearing and no other point for a different value of the function.
3. The limit does not exist if the graph exhibits a vertical asymptote, which is a pair of lines that approach the value of the limit and continue upward or downward without constraints.
4. The limit does not exist as x approaches a particular integer if the graph is approaching two different numbers from two different directions. There can't be two distinct numbers.
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What are the 3 types of forecasting?
The three main types of forecasting are qualitative, quantitative, and time series.
What is forecasting?Forecasting is a predictive tool used to make decisions and plan for the future. It involves analyzing past trends and data to predict future events and outcomes. Forecasting helps businesses and organizations prepare for potential market changes, plan ahead for budgeting, and make informed decisions. It is also an invaluable tool for understanding consumer behavior and market trends.
Qualitative forecasting is a subjective approach that relies on expert opinion, intuition, and past experiences. Quantitative forecasting is a more structured approach that uses statistical techniques and data-driven models to make predictions. Time series forecasting is a form of quantitative forecasting that uses historical data to predict future values. It is especially useful for predicting cyclical trends and seasonal variations. All three approaches can be used together to create a comprehensive forecasting strategy for a business.
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Francisco is working two summer jobs, making $15 per hour life guarding and making $8 per hour clearing tables. In a given week, he can work at most 8 total hours and must earn no less than $80. If x represents the number of hours lifeguarding and y represents the number of hours clearing tables, write and solve a system of inequalities graphically and determine one possible solution.
The system of inequalities is:
x + y ≤ 8 (1)
5x +8y ≥ 80 (2)
The graphical solution is attached and one possible solution is (-30,30).
What is inequality?
The relation between two unequal expressions is defined as inequality.
Given that, the person makes $15 per hour of lifeguarding and $8 per hour clearing tables.
It is also given that x represents the number of hours of lifeguarding and y represents the number of hours of clearing tables.
The person has at most 8 hours a week, therefore, it follows:
x + y ≤ 8 (1)
The earnings of the person in a week are:
15x + 8y
Since the earnings must be no less than $80, it follows:
15x +8y ≥ 80 (2)
Hence, the system of inequalities is:
x + y ≤ 8 (1)
5x +8y ≥ 80 (2)
The graphical solution is attached and one possible solution is (-30,30).
I need steps for this....
The discriminant is 60 and the equation has two real roots with parabola intersecting x - axis at two points.
We are given that the equation is [tex]-3x^{2} +6x +2 = 0[/tex]
The formula for discriminant is given by
D = [tex]{b^{2} -4ac}[/tex] where the equation is of the form [tex]ax^{2} + bx + c = 0[/tex]
Therefore,
a = -3
b = 6
c = 2
Putting these values in formula of discriminant, we get
D = [tex]{6^{2} - 4(-3)(2) }[/tex]
D = 36 +24
D = 60
For the quadratic equation [tex]ax^{2} + bx + c = 0[/tex] , the expression D = [tex]b^{2} - 4ac[/tex] is called the discriminant.
The value of the discriminant shows how many roots does the quadratic equation has:
If D > 0 then the quadratic equation has two distinct real roots.If D = 0 then the quadratic equation has one repeated real root.If D < 0 then the quadratic equation has no real roots.Now, since D = 60 > 0, we can say that the equation has two real roots and therefore, the parabola is intersecting the x-axis at two places.
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Which of the quantities below represents a velocity?
A. 25 m/s
B. 10 km/min
C. 15 mi/h eastward
D. 3 mi/h
Step-by-step explanation:
Velocity is a measure of the rate at which an object moves through space. It is a vector quantity, meaning that it has both magnitude (size) and direction. The standard unit for velocity is meters per second (m/s).
Of the quantities listed above, the following represent velocities:
A. 25 m/s: This represents a velocity of 25 m/s, because it gives both the speed (magnitude) and direction of the object's movement.
C. 15 mi/h eastward: This represents a velocity of 15 miles per hour (mi/h) in an eastward direction, because it gives both the speed and direction of the object's movement.
D. 3 mi/h: This represents a velocity of 3 mi/h, because it gives both the speed and direction (implied to be eastward or westward) of the object's movement.
B. 10 km/min: This does not represent a velocity, because it gives the distance traveled (10 km) but not the time over which the distance was traveled. Without the time, it is not possible to determine the speed or rate of movement.
Please help me with this question
many thanks
answer :
d÷d = 0
then 10 remain
What is an example of a whole number?
Answer:
52,81,8,926 anything that is not a decimal, fraction,or negative number
Step-by-step explanation:
How do you add rational expressions examples?
The Example of addition of Rational Expressions is "Add 2/3 and 6/7" .
The Rational Expression is defined as the expression , that is represented in the form of p/q , where q ≠ 0 .
Example of Addition of Rational Expression is
Add 2/3 + 6/7 .
In order to add the above two rational expression , we first make the denominator of both the terms same by taking LCM ,
the LCM of denominators 3 and 7 is 21 .
So , the expression becomes ⇒ 14/21 + 18/21 .
⇒ (14 + 18)/21
Simplifying further ,
we get ;
⇒ 32/21
Therefore , the example of addition of rational expressions is explained above .
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The equation of a parabola is y=5x^2+15x–25. Write the equation in vertex form.
Write any numbers as integers or simplified proper or improper fractions
The equation of the parabola y = 5 · x² + 15 · x - 25 in vertex form is y + 145 / 4 = 5 · (x + 3 / 2)².
How to find the vertex form of a quadratic equation
Herein we find the equation of a parabola in standard form, that is, an equation of the form:
y = a · x² + b · x + c
Where:
a, b, c - Real coefficientsx - Independent variabley - Dependent variableFrom this equation we must derive the vertex form, whose definition is shown below:
y - k = C · (x - h)²
Where:
C - Vertex constant(h, k) - Coordinates of the center.The vertex form is found by algebra properties. First, write the equation of the parabola:
y = 5 · x² + 15 · x - 25
Second, use algebra properties until the vertex form is found:
y + 25 = 5 · x² + 15 · x
y + 25 = 5 · (x² + 3 · x)
y + 25 + 45 / 4 = 5 · (x² + 3 · x + 9 / 4)
y + 145 / 4 = 5 · (x + 3 / 2)²
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what is the largest integer that always divides $(n 1)(n 3)(n 5)(n 7)(n 9)$ for all positive even integers $n$?
The largest integer that always divides (n+1)(n+3)(n+5)(n+7)(n+9) for all positive even integers n is 15.
What is positive integer?Positive integers are numbers such as 1, 2, 3, etc. They are also called countable or natural numbers. A positive number is any number greater than zero.
Let P(n) = (n+1)(n+3)(n+5)(n+7)(n+9) be a polynomial for all positive even integers, n . We have to calculate the largest integer that always divides the P(n) .
Let "l" be the largest integer that divides P(n) for each n∈N . Then, l must divide both P(1) and P(2)
i.e, P(1) = (1+1)(1+3)(1+5)(1+7)(1+9) = 2×4×6×8×10
= 2^8×5×3
P(2) = 3×5×7×9×11 , hence , l divides 15.
Since, n + 5 ≡ (n+2) mod(3), one (and exactly one) of the numbers n+1 , n+3 , n+5 must be multiple of 3 . Thus, 3/P(n) for each n∈N.
Since, n+7 ≡ (n+2)mod(5) and n+9 ≡ (n+4)mod(5),
one (and exactly one) of the numbers n+1 , n+3 , n+5 , n+7 and n+9 , must be multiple of 5.
Thus, 5/P(n) for each n∈N.
Therefore, 15/p(n) for each n∈N , so that l = 15 .
So, required value of n is 15.
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i need help please.
Answer: ( x + 5, y + 11)
Step-by-step explanation:
Let's pick point S at ( -9, -9); it changes to point S' at (-4, 2)
We see the x increase by 5 and the y increase by 11, so the answer is
( x + 5, y + 11)
Is Y =- 4 a vertical line?
Answer: No, it is a horizontal line across -4 y
Step-by-step explanation:
Your friend hides a dollar bill under one of three plastic cups. Under the other
two cups is a paper clip. You get to keep whatever is under the cup you
choose, but you don't know which contains the dollar bill. After you choose,
your friend lifts one of the other cups to reveal a paper clip and then gives you
the option of changing your choice.
Should you switch to the other cup?
Answer:
Yes, you should
Step-by-step explanation:
Because the probability of winning the dollar increases from \frac{1}{3} 3 1 to \frac{2}{3} 3 2 by switching.
E-loan, an online lending service, recently offered 36-month auto loans at 6% compounded monthly to applicants with good credit ratings. If you have a good credit rating and can afford monthly payments of $539, how much can you borrow from E-loan? What is the total interest you will pay for this loan?
I can borrow $17717.48from E-Loan.
The interest I will pay for the loan is $1,686.52
Now, According to the question:
We know,
The formula of present value is
PV = PMT [tex](\frac{ 1-(1+i)^-^n}{i} )[/tex]
PMT = The monthly payment = $539
i= Rate of interest per period = 6%/12 = 0.005
n = The number of month = 36 months
Therefore,
PV = 539[tex](\frac{ 1-(1+0.005)^-^3^6}{0.005} )[/tex]
≈$17717.48
Therefore I can borrow $17717.48 from E-Loan.
The interest = Paid amount - Loan amount
=$[(539×36)-17717.48]
= $1,686.52
The interest I will pay for the loan is $1,686.52
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or
Last year, the average monthly revenue at the Cedarburg Times newspaper was $66,490. This year, the average monthly revenue so far has been $59,841. What is the percent of decrease in the average monthly revenue?
The percent of the decrease in the average monthly revenue is 10%.
How to calculate the percent of the decrease?We can calculate the percent of the decrease by using the following formula:
The percent of the decrease = (previous revenue - current revenue) / previous revenue × 100%
Last year, the average monthly revenue at the Cedarburg Times newspaper was $66,490.
This year, the average monthly revenue so far has been $59,841.
The percent of the decrease = (previous revenue - current revenue) / previous revenue × 100%
previous revenue = $66,490.
current revenue = $59,841
The percent of the decrease = ($66,490 - $59,841) / $66,490 × 100%
The percent of the decrease = 6,649 / $66,490 × 100%
The percent of the decrease = 0.1 × 100%
Apply the multiplication operation, and we get
The percent of the decrease = 10%
Thus, the drop in the average monthly revenue is 10%.
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Solve the equation:5(3+6a)=-15+6(5+5a)
Answer:
0.5Step-by-step explanation:
To solve the equation 5(3+6a) = -15+6(5+5a), we need to use the rules of algebra to isolate the variable "a" on one side of the equation and simplify the other side.
First, we can distribute the 5 on the left side of the equation to obtain 53 + 56a = -15+6(5+5a). This gives us 15 + 30a = -15+6(5+5a).
Next, we can distribute the 6 on the right side of the equation to obtain 15 + 30a = -15+65 + 65a. This gives us 15 + 30a = -15+30 + 30a.
Now we can combine like terms on both sides of the equation to obtain 30a = -15 + 30, which simplifies to 30a = 15. Finally, we can divide both sides of the equation by 30 to obtain a = 0.5.
Therefore, the solution to the equation 5(3+6a) = -15+6(5+5a) is a = 0.5
How do you identify the type of similarity that two triangles might be?
To easily recognize comparable triangles, you can apply three theorems that are peculiar to triangles.
Two triangles can be compared using one of three theorems: Angle-Angle (AA), Side-Angle-Side (SAS), and Angle-Angle-Side (SS-AS) (SSS).
The symbol representing two triangles with the same angles on all three sides is SSS, which stands for "side, side, side." If two triangles' third sides match those of another triangle, the two triangles are said to be congruent.
The SAS regulation states as much. If the two sides and included angle of one triangle match the two sides and included angle of another triangle, the triangles are congruent. Any angle formed by two specified sides is considered to be an included angle.
A triangle is comparable to another if its two matching angles are congruent, according to the AA postulate of Euclidean geometry. The AA postulate is derived from the observation that the total internal angle of a triangle is always equal to 180°.
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Which expression is equivalent to the following complex fraction (- 2 x 5 y )/( 3 y 2 x?
The expression expressed as a single fraction is -2y(yx-2)/3x(yx+5)
How to simplify the expression as a single fraction?You should know that simplification involves making an expression look more simpler than it was
The given expression is 2/(x+5/y)÷3/(y-2/x)
First of all let us express the denominator as a single fraction
x+5/y
⇒(yx +5)/y
In the same way express the expression y-2/x = (yx-2)÷x
By substitution we have
(-2)/yx+5)/(y/3)÷(yx-2/x)
⇒(-2y)/(yx+5) ÷ (3x/ (yx -2)
Simplifying this further to get
(-2y)/(yx +5) * (yx -2)/(3x)
⇒-2y(yx-2)/3x(yx +5)
In conclusion, the expression expressed as a single fraction is -2y(yx-2)/3x(yx+5)
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Correct question:
Which expression is equivalent to the following complex fraction? -2/x+5/y/3/y-2/x
Answer:the answer is
Step-by-step explanation:
just did the test
What is a linear graph formula?
y = MX + b is the required linear graph formula.
What is the linear graph formula?Linear equations with two variables have straight-line graphs.
There are numerous ways to express linear equations: Y = mx+ b is the slope-intercept form.
The slope is m and the y-intercept is the point in an equation of the type y = MX + b, such as y = 2x 3. (0, b).
Write the equation in the form y = MX + b to determine the slope m and the y-intercept.
This will allow you to graph the equation using the slope and y-intercept (0,
b).
Plot the y-intercept next.
3) Depending on whether the slope is positive or negative, go up, down, or left or right from the y-intercept.
Therefore, y = MX + b is the required linear graph formula.
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find an equation of the tangent line to the curve at the given point. y = 8ex cos(x), (0, 8)
The equation of the line tangent to the curve at the points , y = 8ex cos(x), (0, 8) as calculated is y' = 0.
We will differentiate y = 4ex cos(x) so as to find the required equation.
we get the differential as ,
y' = - 8eˣsin(x)
given the value of x as 0 and y as 8
y' = -8e⁰sin(0)
Now as we know that the value of sin( 0 ) is 0 , therefore the equation becomes,
y'= 0
Differentiation is a branch of calculus which is used to calculate the rate of change in a given object in a specified time. The other branch of calculus is integration.
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What is a parallel class 6?
In geometry, parallel lines are the lines which maintain the same distance apart from each other and never intersect at any point.
What is parallel class 6?
In geometry, parallel lines are the lines which maintain the same distance apart from each other and never intersect at any point.
All rungs are perpendicular to the left post is a line perpendicular to one of a set of parallel lines, it is perpendicular to all parallel lines.
The rungs are parallel
lines that are perpendicular to the same line are parallel.
The rungs are perpendicular to both posts and the posts are parallel to each other.
if a line is perpendicular to one of a set of parallel lines, it is perpendicular to all the parallel lines.
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What are the 3 different options for horizontal asymptotes?
The 3 different options for horizontal asymptotes are:
If the degree of the numerator < the degree of the denominator.If the degree of the numerator = degree of the denominator. If the degree of the numerator > the degree of the denominator.What are horizontal asymptotes?This refers to a horizontal line y = b where the graph approaches the line as the inputs approach infinity (∞) or negative infinity (–∞).
The 3 different options for horizontal asymptotes
If the degree of the numerator < the degree of the denominator, y = 0 (x-axis), and If n<m, the x-axis, y=0 is the horizontal asymptote.If the degree of the numerator = degree of the denominator. If n=m, then y=an / bm . That is, the ratio of the leading coefficients.It is the horizontal asymptote. If the degree of the numerator > the degree of the denominator. If n>m, there is no horizontal asymptote.Note: If n=m+1, there is an oblique or slant asymptote.
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Which two functions are inverses of each other?
The two functions are inverse to each other if the two graphs are symmetric with respect to the line y = x (mirror images over y = x ), then they are inverse functions.
Functions
A function is defined as a relation between a set of inputs having one output each. In simple words, a function is a relationship between inputs where each input is related to exactly one output.
inverse functions
the inverse function of a function f is a function that undoes the operation of f. The inverse of f exists if and only if f is bijective, and if it exists, is denoted by F^-1.
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