If a parabola’s focus is at (1,1) and directrix is at y = 7, what is the standard form of the equation representing this parabola
The standard form of the equation of the parabola with focus at (1, 1) and directrix 'y = 7' is (x - 1)² = 12(y - 4). The vertex is found by determining the midpoint between the focus and directrix, and the 'a' value is found by calculating half the distance between the focus and directrix.
Explanation:To find the standard form of a parabola given the focus and directrix, we'll use the definition of a parabola itself. A parabola is defined as the set of points that are equidistant from a point, known as the focus, and a line, known as the directrix.
The equation representing a parabola is: (x - h)² = 4a(y - k) where (h, k) is the vertex of the parabola and 'a' denotes the distance between the vertex and the focus or the vertex and the directrix.
The mid-point of the focus (1, 1) and the directrix y=7 gives the coordinates of the vertex. So, h = 1 and k = (1+7)/2 = 4.
Now, 'a' is half the distance between the focus and directrix, a = (7-1)/2 = 3.
So, the standard form of the equation is (x - 1)² = 12(y - 4)
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Tickets to friday's football game are $5.50 for adults and $4.50 for students. if the school pre-sold 25 tickets for $124.50, how many adult and student tickets were purchased?
which expression shows the value of a rare postage stamp, originally purchased for $5000, that has been increasing in value by 11% for 10 years
Final answer:
The future value of a rare postage stamp that cost $5000 and increases by 11% annually over 10 years is calculated using the compound interest formula V = P(1 + r)[tex]x^{n}[/tex], where P is the principal amount, r is the annual rate, and n is the number of years.
Explanation:
Calculating the Future Value of a Rare Postage Stamp
The value of a rare postage stamp, originally purchased for $5000, that has been increasing in value by 11% annually for 10 years can be calculated using the formula for compound interest: V = P(1 + r)[tex]x^{n}[/tex], where:
P = $5000,
n = 10 years.
Plugging these values into the formula, we get:
V = $5000(1 + 0.11) [tex]x^{10}[/tex]
Performing the calculations gives us the future value of the stamp after 10 years.
You are picking up a friend and going to a concert. your friend lives 3 miles west of your house and the concert is 21 miles east of your friends house. when you are at the concert, how far away from home are you
An angle measuring 22 degrees is bisected what is the measure of the angle that are formed
What is the algebraic description for a transformation that translates the point (x,y) nine units to the right and five units down?
An algebraic description of the transformation that translates a point (x, y) nine units to the right and five units down is written as the new point (x + 9, y - 5).
The algebraic description of a transformation that translates a point (x, y) nine units to the right and five units down can be written as the following ordered pair: (x + 9, y - 5).
In this transformation, the x-coordinate of each point is increased by 9 units, reflecting a horizontal translation to the right, and the y-coordinate is decreased by 5 units, representing a vertical translation downwards. Thus, if you have a point (x, y), after the transformation, it becomes (x + 9, y - 5).
For example, if your initial point is (3, 4), after the transformation it would be at (3 + 9, 4 - 5) = (12, -1).
Kenya exchanges $150 for British pounds(£). Suppose the conversion rate is £1 = $1.60. How many pounds should she receive?
The composition is performed on . What are the coordinates of B'' and C''?
Solve for the variable: x – 2.8 = 1.6
3.4
-1.2
2.3
4.4
Simplify the expression.
55−4⋅5+12
Expression after the simplification is 62.5.
What is an expression?In mathematics, expression is the combining of variables with the application of operations and prescribed rules. It could take the shape of an equation, some numbers, etc.
Given, expression
55 - 4.5 +12
Simplifying,
first, add the numbers.
55 - 4.5 +12
= 67 - 4.5
= 62.5
Hence, the value of the expression is 62.5.
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Is each line parallel, perpendicular, or neither parallel nor perpendicular to the line x + 2y = 6?
Drag each choice into the boxes to correctly complete the table
y= -1/2x+5
-2x+y=-4
-x+2y=2
x+2y=-2
ANSWER
We need to rewrite [tex]x+2y=6[/tex], in slope intercept form.
[tex]\Rightarrow 2y=-x+6[/tex],
[tex]\Rightarrow y=-\frac{1}{2}x+3[/tex]
The given equation has a slope of [tex]-\frac{1}{2}[/tex].
ANSWER TO QUESTION 1
We now the write given options also in slope intercept form.
For the first one, we have
[tex]y=-\frac{1}{2}x+5[/tex]
This first equation also has a slope of [tex]-\frac{1}{2}[/tex].
Hence [tex]y=-\frac{1}{2}x+5[/tex] is parallel to [tex]x+2y=6[/tex].
ANSWER TO QUESTION 2:
The second equation is;
[tex]-2x+y=-4[/tex].
We need to write this one too in slope intercept form.
[tex]y=2x-4[/tex]
This second equation has a slope of 2
Since [tex]2\times -\frac{1}{2}=-1[/tex], the line [tex]-2x+y=-4[/tex]
is perpendicular to [tex]x+2y=6[/tex].
ANSWER TO QUESTION 3
We rewrite the equation [tex]-x+2y=2[/tex] in slope intercept form.
[tex]\Rightarrow 2y=x+2[/tex]
[tex]\Rightarrow y=\frac{1}{2}x+1[/tex]
This equation also has a slope of [tex]\frac{1}{2}[\tex]
since the slope of [tex]-x+2y=2[/tex] is not equal to the slope of [tex]x+2y=6[/tex], and the product of these two slopes [tex]\frac{1}{2} \times - \frac{1}{2} \ne -1[/tex], the two lines are neither parallel nor perpendicular.
QUESTION 4
The given line has equation [tex]x+2y=-2[/tex]. We rewrite this equation in the slope intercept form.
[tex]\Rightarrow 2y=-x-2[/tex]
[tex]\Rightarrow y=-\frac{1}{2}x-1[/tex]
This line has a slope of [tex]-\frac{1}{2}[/tex].
Since this slope is the same as the slope of the given line,
[tex]y=-\frac{1}{2}x-1[/tex] is parallel to [tex]x+2y=6[/tex].
See graph in attachment.
Manganese-56 is a beta emitter with a half-life of 2.6 h. what is the mass of manganese-56 in a 1.0-mg sample of the isotope at the end of 10.4 h?
Answer: 0.0625
Step-by-step explanation:
Which graph represents y – 1 = 2(x – 2)?
Thank You! :)
Answer:
Option A
Step-by-step explanation:
Given : Equation [tex]y-1 = 2(x -2)[/tex]
To find : Which graph represents the given equation
Solution :
First thing we plot the graph of the equation.
We see that graph cuts at point (1.5,0) and (0,-3) to the coordinates.
From the given graph Option A graph cuts to these points.
Therefore, Option A is correct.
Refer the attached graph A.
What is the discriminant of the polynomial below 9x2 - 18x + 9?
Answer:
The discriminant of the given polynomial is 0.
Step-by-step explanation:
The discriminant of a quadratic polynomial [tex]p(x)=ax^2+bx+c[/tex] is
[tex]D=b^2-4ac[/tex]
The given expression is
[tex]9x^2-18x+9[/tex]
Here,
[tex]a=9,b=-18,c=9[/tex]
The discriminant of a given polynomial is
[tex]D=(-18)^2-4(9)(9)[/tex]
[tex]D=324-324[/tex]
[tex]D=0[/tex]
Therefore the discriminant of the given polynomial is 0.
70/b+5a-2b
Evaluate the expression when a=3 and b=10
which of the following are necessary when proving the opposite sides of a parallelogram are congruent? Check all that apply. A) Opposite sides are perpendicular B) Corresponding parts of similar triangles are similar C) Opposite sides are congruent D) Corresponding parts of congruent triangles are congruent
Answer:
C - Opposite sides are congruent
D - Corresponding parts of congruent triangles are congruent
Step-by-step explanation:
A parallelogram is a quadrilateral whose both pairs of opposite sides are parallel. In a parallelogram, the 2 sets of opposite sides are congruent. Also it has 2 sets of opposite angles congruent.
Therefore, the following are necessary when proving the opposite sides of a parallelogram are congruent.
C - Opposite sides are congruent
D - Corresponding parts of congruent triangles are congruent
Joey wants to buy a used car that costs $2500. He has already saved $1200 and makes $50 a week as a dog walker. Write and solve an equation to find the number of weeks, w, he must work until he has enough to purchase the car.
Joey must work 26 weeks until he has enough to purchase the car.
What is an Equation?An equation is a mathematical statement formed when two algebraic expressions are equated using an equal sign.
The equations are useful in determination of unknown parameters.
Joey wants to buy a used car,
The cost of the car is $2500.
The money he has saved is $1200
The money he makes by walking a dog is $50 per week.
Let w represent the number of weeks.
The equation for the condition can be written as:
The cost of the car is equated with the money he has and the money he earned in w weeks by dog walking.
2500 = 1200 + 50w
To determine the number of weeks he has to work to buy the car, the equation has to be solved for w,
50w = 2500 - 1200
50w = 1300
w = 1300 / 50
w = 26 weeks.
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10. Find the values of x, y, and z in the diagram below.
A man standing on a sidewalk looks up at the roof edge of a building. The building is 40 feet tall and the man stands 12 feet from the building.
What is the measure of the man's angle of inclination from where he stands?
Enter your answer, rounded to the nearest degree, in the box.
If its rounded it would be 73 degrees
You spend $154 to make wooden bookends and plan to sell them online for $8.50 each. You will charge an additional $3.25 per bookend to wrap and ship them. How many bookends must you sell in order for your profit to be at least $600?
To make a profit of at least $600, the number of bookends that need to be sold is calculated by adding the desired profit to the initial cost, and then dividing this sum by the net revenue per bookend. The result is approximately 144 bookends.
Explanation:The subject of this problem is Mathematics, specifically a topic within business math called 'breakeven analysis' or 'cost-volume-profit analysis'. It involves calculating the point at which revenue from selling a product equals the costs associated with producing and selling that product. In this case, you have spent $154 to make wooden bookends, and you plan on selling them online for $8.50 each. Additionally, there's a $3.25 fee per bookend for wrapping and shipping, which you charge to your customers. Your goal is to make a profit of at least $600.
To calculate the number of bookends that need to be sold, we first find the net revenue per bookend. This is the selling price minus the shipping cost, which comes out to $8.50 - $3.25 = $5.25 per bookend. The desired profit is added to the cost of making the bookends to find the total required revenue, which is $600 + $154 = $754.
Finally, total required revenue is divided by the net revenue per bookend to find the number of bookends that need to be sold: $754 ÷ $5.25 = approximately 144 bookends (rounded up to the nearest whole bookend since we can't sell a fraction of a bookend). So, in order for your profit to be at least $600, you must sell at least 144 bookends.
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Latex-free gloves can be purchased in a box of 100 for $44.95 or a box of 50 for $23.50. Which one would be the better deal
The length of a rectangle is 5 centimeters less than twice its width. The perimeter of the rectangle is 26 cm. What are the dimensions of the rectangle?
A. length = 7 cm; width = 6 cm
B. length = 4 cm; width = 9 cm
C. length = 6 cm; width = 7 cm
D. length = 5 cm; width = 5 cm
The correct option is [tex]\boxed{\text{\bf option C}}[/tex] i.e., the length is [tex]6\text{ cm}[/tex] and breadth is [tex]7\text{ cm}[/tex].
Further explanation:
The perimeter of arectangle is calculated as follows:
[tex]\boxed{\text{Perimeter}=2\cdot (l+b)}[/tex]
Here, [tex]l[/tex] is the length and [tex]b[/tex] is the breadth of the rectangle.
Given:
The length of the rectangle is [tex]5\text{ cm}[/tex] less than twice its breadth and the perimeter of the rectangle is [tex]26\text{ cm}[/tex].
Calculation:
Step 1:
First we write the dimensions in the form of variable [tex]x[/tex].
Consider [tex]x[/tex] as the breadth of the rectangle and [tex]y[/tex] as the length of the rectangle.
As per the question the length is [tex]5\text{ cm}[/tex] less than twice its breadth.
Therefore, the length can be expressed in the form of the equation as follows:
[tex]\boxed{y=2x-5}[/tex] …… (1)
Step 2:
Now, we have to calculate the dimensions of the rectangle.
So, substitute [tex]l=2x-5[/tex] and [tex]b=x[/tex] in the formula of the perimeter of the rectangle to obtain the value of [tex]x[/tex].
[tex]\begin{aligned}\text{Perimeter}&=2\cdot (l+b)\\26&=2\cdot (2x-5+x)\\26&=2\cdot (3x-5)\\26&=6x-10\\6x&=26+10\\6x&=36\\x&=6\end{aligned}[/tex]
Therefore, the breadth of the rectangle is [tex]6\text{ cm}[/tex].
Now, substitute [tex]x=5[/tex] in the equation (1) to obtain the value of the length.
[tex]\begin{aligned}l&=2x-5\\&=(2\cdot 6)-5\\&=12-5\\&=7\end{aligned}[/tex]
Therefore, the length of the rectangle is [tex]7\text{ cm}[/tex].
Thus, the correct option is [tex]\boxed{\text{\bf option C}}[/tex] i.e., the length is [tex]6\text{ cm}[/tex] and breadth is [tex]7\text{ cm}[/tex].
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Answer details:
Grade: Junior school
Subject: Mathematics
Topic: Area and Perimeter
Keywords: Length, breadth, perimeter, area, side, dimensions, rectangle, shapes, figure, triangle, square, equation, variable, constant, twice, area, mensuration.
9.12cm
10.36cm
11.34cm
12.61cm
Determine the sign of cos pi divided by seven without using a calculator.
Why might a person need an emergency fund?
Answer:
To prepare for the unexpected, it is important to have a reserve fund set aside. Having a reserve fund is important, but it should not always be your first priority. If you have credit card or other debt, pay that off before you start to put your money in a reserve. Credit card debt has a high interest rate, while the money that you will save in your reserve probably won’t gain a lot of interest. Why not? When you choose a place to save your financial reserve, low risk is key. Remember that your financial reserve fund is for emergencies only. If you have an emergency and need to use it, do. Once you are back on your feet, replenish it. Also, once you have saved the amount you need, stop. Your extra savings will serve you better in an investment that yields higher interest.
Step-by-step explanation:
Words taken from the online site 1 on edge for "Why might a person need an emergency fund?"
The term “emergency fund” refers to helping those times when a person is in need of money. An emergency fund allows you to save on rarer credit interest.
What is emergency fund?
The term “emergency fund” refers to the person's use of money at the time of need. Emergency money is saving money. They improve our financial security and safety. The emergency money is a small amount of money. For example, home repairs, paying medical bills, trip planning.
An emergency fund is money kept in a bank account to cover unexpected expenses like medical bills or car or home repairs. An emergency fund can also assist you in the event of a lack of income due to a job loss or prolonged illness.
Hence, the significance of the emergency fund is aforementioned.
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Alice's backyard is a rectangular piece of property that is twice as long as it is wide. The total area of her yard is 500 m^(2). What is the approximate width of her backyard?
Using only the values given in the table for the function, f(x), what is the interval of x-values over which the function is increasing?(–6, –3)(–3, –1)(–3, 0)(–6, –5)
The interval of x-values over which the function is increasing is (-3 , -1), the correct option is B.
What is a Function?A function is a law that relates two variables namely, a dependent and an independent variable.
A function always has a defined range and domain.
The x values and the corresponding function values are shown in the figure.
The x and y values are:
-6 -34
-5 3
-4 -10
-3 -11
-2 -6
-1 -1
0 -2
1 -15
The interval of x-values over which the function is increasing has to be determined.
The first option is (-6, -3) , the function is not increasing in this interval.
The second option is (-3 , -1) , the function is increasing in this interval.
The third option is(–3, 0), the function is not increasing in this interval.
The fourth option is (–6, –5), the function is not increasing in this interval.
Your question seems incomplete, the complete question is attached with the answer.
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When it is 11:30
a.m. in dublin, it is 2:30 p.m. in moscow. how far ahead of dublin is moscow?
Dublin is 3 hours ahead of Moscow.
What is Time difference?The difference between how time is measured in different ways; the length of a period of time. the time difference between two or more distinct time zones in terms of clock time.
Given Dublin time = 11:30 am
Moscow time = 2:30
we can military time and subtract.
11:30 a.m. = 11:30
2:30 p.m. = 2:30 + 12:00 = 14:30
14:30 - 11:30 = 3:00
Hence the time difference is 3 hours.
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In the diagram below bd is parallel to xy what is the value of y?
we know that
if the lines bd and xy are parallel. then
[tex]y+70\°=180\°[/tex] --------> by consecutive interior angles
so
Solve for y
Subtract [tex]70\°[/tex] both sides
[tex]y+70\°-70\°=180\°-70\°[/tex]
[tex]y=110\°[/tex]
therefore
the answer is
[tex]y=110\°[/tex]
Answer:
Option C. y = 110°
Step-by-step explanation:
In this question BD and XY are two parallel line and a transverse is intersecting both the lines at A and E.
Now ∠BAE = ∠AEY = 70° [ alternate interior angles ]
and ∠EAD + ∠BAE = 180° [ supplementary angles ]
or y + 70° = 180°
⇒ y + 70 - 70 = 180 - 70
⇒ y = 110°
Therefore Option C. measurement of y = 110° is the answer.
The table shows conversions for common units of capacity. Units of Capacity Customary System Units Metric System Units 1 gallon 3.79 liters 1 quart 0.95 liters 1 pint 0.473 liters 1 cup 0.237 liters What is the conversion factor to convert cups to liters?
The conversion factor to convert cups to liters is 0.237. When converting from cups to liters, multiply the number of cups by 0.237.
Explanation:According to the conversion table provided, the conversion factor to convert cups to liters is 0.237. That means for every 1 cup, it equals approximately 0.237 liters. To convert from cups to liters, you simply multiply the number of cups by this conversion factor, 0.237.
For example, to find out how many liters is 2 cups, you do the multiplication: 2 cups * 0.237 = 0.474 liters.
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The complete question is given below:
Given : A table showing conversions for common units of capacity.
Units of capacity
Customary System Units Metric System Units
1 gallon 3.79 liters
1 quart 0.95 liters
1 pint 0.473 liters
1 cup 0.237 liters
What is the conversion factor to convert cups to liters?
The conversion factor to convert cups to liters, according to the given table, is that 1 cup equals 0.237 liters. To convert any amount of cups to liters, multiply the number of cups by 0.237.
Explanation:The table in your question provides the conversion factors for common units of capacity, including the conversion of cups to liters in both the Customary and Metric systems.
According to the table, 1 cup is equivalent to 0.237 liters. This means that to convert any given number of cups to liters, you multiply the number of cups by 0.237.
For example, to convert 5 cups into liters, you would perform the following calculation: 5 cups × 0.237 = 1.185 liters. This conversion factor is useful in everyday scenarios such as cooking or reading a recipe that uses different measurement systems.
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