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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The sum of 4 times mike's age and 3 times susan's age is 79. mike is 1 year more than 3 times as old as susan. find each of their ages.
An article with a net weight of 10 lb (pounds) is packaged in a box that weighs ½ lb. If 20 of these boxed articles are put into one freight container that weighs 15 lb, what is the gross weight?
3y + 2z = 12 and y – z = 9.
Twice a number, increased by one, is between negative three and seven. find all the numbers.
To solve the inequality -3 < 2x + 1 < 7 for the number x, subtract 1 from all parts and then divide by 2. The solution is the range of numbers greater than -2 and less than 3.
Explanation:The problem statement here results in an inequality that can be expressed algebraically. If 'x' represents the unknown number, we can write the inequality as -3 < 2x + 1 < 7.
To solve for 'x', we will isolate the variable on one side by following the rules of inequalities similar to solving equations.
First, subtract 1 from all parts of the inequality to get the pure inequality without a constant.
-4 < 2x < 6
Now, divide all parts of the inequality by 2, which yields:
-2 < x < 3
All numbers greater than -2 and less than 3 satisfy the original problem statement. Therefore, the range of numbers that solve the inequality is from -2 to just below 3.
Suppose that the length of a certain rectangle is 5 meters less than four times its width. the perimeter of the rectangle is 40 meters. find the length and width of the rectangle.
given f(x)=x^3 and g(x)=1-5x^2, find (fog) (x) and its domain
Answer:
answer is b
Step-by-step explanation:
parallelogram ESTA has vertices E (10,0), S(14,3),T(6,9) and A (2,6). to calculate its area, Jamal will first determine the equation of the line through point E and perpendicular to ST. What is the equation of this line?
To determine the number of trout in a lake, a conservationist catches 180 trout, tags them and throws them back into the lake. later, 48 trout are caught; 12 of them are tagged. how many trout would the conservationist expect to be in the lake?
Six times a number decreased by ten is eight. Find the number.
A) 2
B) 3
C) 4
D) 5
What is the solution of the linear quadratic system or equations?
Y=x^2+5x-3
Y-x=2
The solution to the system of equations are (1, 3) and (-5, -3)
Given the following functions
Y=x^2+5x-3
Y-x=2; y = x + 2
Equating both outputs will give
x^2+5x-3 = x + 2
x^2+5x-3 - x - 2 = 0
x^2 + 4x - 5 = 0
Factorize the equation
x^2 + 5x - x - 5=0
x(x+5) - 1(x+5) = 0
(x-1)(x+5) = 0
x = 1 and -5
Recall that y = x + 2
y = 3 and -3
Hence the solution to the system of equations are (1, 3) and (-5, -3)
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The solution to a linear quadratic system of equations can be found by substitifying one equation into the other and solving. In this case, the solutions are (1,3) and (-5,-3).
Explanation:To find the solution of a linear quadratic system of equations, you need to substitute one equation into the other. In this case, for the equations [tex]Y = x^2 + 5x - 3[/tex] and [tex]Y - x = 2[/tex], you would first isolate Y in the linear equation, resulting in [tex]Y = x+2.[/tex]
Then, you substitute (x + 2) for Y in the quadratic equation, leading to [tex]x + 2 = x^2 + 5x - 3.[/tex]
Next, rearrange the equation to get it in the standard quadratic form: 0 = [tex]x^2 + 5x - x - 3 - 2.[/tex] Simplifying this results in the equation [tex]0 = x^2 + 4x - 5.[/tex]
Finally, solve this equation by factoring or using quadratic formula, which gives the solutions as x = 1, -5. Substituting both 1 and -5 into the equation Y = x + 2 gives the corresponding y-values, Y = 3 and -3. Therefore, the solutions to the system of equations are (1,3) and (-5,-3).
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Richard borrowed $950 from his parents to make some repairs on his car. He promised to repay the loan by giving his parents at least $85 from his paycheck each week. Assuming richard sticks to his promise, which of the following is not a possible scenario for the amount of debt Richard has remaining?
I think it's B but I'm not 100% sure
Answer:
B
Step-by-step explanation:
To solve this, you need to know first how many weeks will take Richard replacing the money.
We will call X the number of weeks so we say
$85 * X =$950
X= 11.17
This means it will take at least ≈12 weeks to pay his debt.
To know which scenario it's not possible, use the equation to choose the right option.
A: $85*5= $425
After 5 weeks, he paid $425, this mean he's $525 in debt.
B: $85*6= $510
After 6 weeks, he paid $510, this means he's $440 in debt.
C. $85*7= $595
After 7 week,s he paid $595, this means he's $355 in debt.
D. $85*8= $680
After 7 weeks, he paid $680, this means he's $270 in debt.
The not possible scenario is B because he will be paying at least $85 or more so he will debt $440 or less al week 6.
What type of transformation takes the graph of
f(x)=|x|
to the graph of
g(x)=â3+|x|
?
Â
vertical translation of 3 units up
horizontal translation of 3 units right
vertical translation of 3 units down
horizontal translation of 3 units left
In 2010, the national debt of the United States was about 14 trillion dollars. In 2003 it was about 7 x 10 to the 12 dollars. About how many times larger was the national debt in 2010 than 2003? THIS IS SCIENTIFIC NOTATION
Answer: About 2 times.
Step-by-step explanation:
Given : In 2010, the national debt of the United States was about 14 trillion dollars.
We know that 1 trillion [tex]=1,000,000,000,000= 10^{12}[/tex]
then, In 2010, the national debt of the United States was about [tex]14\times10^{12}[/tex]
dollars.
In 2003 it was about [tex]7\times10^{12}[/tex] dollars.
Now, the number of times the national debt was larger in 2010 than 2003 :-
[tex]\dfrac{ 14\times10^{12}}{7\times10^{12}}\\\\=\dfrac{14}{7}\times10^{12-12}\\\\=2\times10^0=2[/tex]
Hence, national debt was about 2 times larger in 2010 than 2003 .
find the digit that makes 2,72_ divisible by 6
To find the missing digit to make 2,72_ divisible by 6, add the digit 4.
To find the digit that makes 2,72_ divisible by 6, we need to focus on the divisibility rules for 6. In this case, a number is divisible by 6 if it is divisible by both 2 and 3. For 2, the last digit must be even, and for 3, the sum of the digits must be a multiple of 3.
Given the number 2,72_, the blank digit needs to be 4 to make the number divisible by 6. This is because when the digit 4 is added, the number becomes 2,724, which is divisible by both 2 and 3.
The height of a triangle is 4 cm more than twice the length of the base. the area of the triangle is 63 cm2. find the height of the triangle.
Answer:
THE ANSWER COULD BE 14
What are the steps for using a compass and straightedge to construct a square? Drag and drop the steps in order from start to finish. Use a straightedge to draw line a and label a point on the line as point P. Without changing the compass width, place the compass point on point S and draw an arc in the interior of ∠SPT . Use the straightedge to draw QS¯¯¯¯¯ and QT¯¯¯¯¯. Construct a line perpendicular to line a through point P. Label a point on this line as point R. With the compass open to the desired side length of the square, place the compass point on point P and draw an arc on line a and an arc on PR←→ . Label the points of intersection as points S and T. Keeping the same compass width, place the compass on point T and draw an arc in the interior of ∠SPT to intersect the previously drawn arc. Label the point of intersection as point Q.
Answer: The steps for using a compass and straightedge to construct a square are as follows :-
1. Use a straightedge to draw line a and label a point on the line as point P.
2. Construct a line perpendicular to line a through point P.
3. Label a point on this line as point R.
4. With the compass open to the desired side length of the square, place the compass point on point P and draw an arc on line a and an arc on PR←→.
5. Label the points of intersection as points S and T.
6. Keeping the same compass width, place the compass on point T and draw an arc in the interior of∠SPT to intersect the previously drawn arc.
7. Label the point of intersection as point Q.
8. Without changing the compass width, place the compass point on point S and draw an arc in the interior of∠SPT .
9. Use the straightedge to draw QS ,QT,SR and RT.
Here we have finished with the square RTQS.
What is the length of each side of a square room if the area is 400 ft²?
In a school election, 3/4 of the students vote. There are 1464 ballots. Write and solve and equation to find the number of students.
x = total number of students
equation is:
(3/4)x = 1464
to solve for x divide both sides by 3/4
x = 1464 / 3/4 = 1952
x = 1952
there are 1952 total students
Solve the equation
5x - 5 = 3x - 9
A. 1
B. -2
C. -3
D. -1
What is the greatest common factor of 60x4y7, 45x5y5, and 75x3y? 5xy 15x3y 45x3y5 75x5y7
Answer: 15x3y
Step-by-step explanation:
We rewrite the expressions by using factorisation method of their primes. This will give:
60x4y7= 2×2×3×5×x×x3×y×y6
45x5y5= 3×3×5×x×x4×y×y4
75x3y= 3×5×5×x×x2×y
Common factors= 3 × 5 × x3 × y
= 15x3y
Answer:
its B
Step-by-step explanation:
Hidemi is a waiter. He waits on a table of 4 whose bill comes to $69.98. If Hidemi receives a 20% tip, approximately how much will he receive?
$3.50
$14.00
$84.00
$7.00
The law of detachment can be used to draw a conclusion from the following statements: "It will be sunny on Tuesday." "I will always go outside if the weather is nice."
Final answer:
Using the law of detachment, if it is sunny on Tuesday and there is a habit of going outside in nice weather, one can conclude that they will go outside on Tuesday.
Explanation:
The law of detachment is a concept in logic used to form conclusions given a conditional statement and its hypothesis being true. For instance, in the scenario provided:
If it is sunny on Tuesday (hypothesis).
I will always go outside if the weather is nice (conditional statement).
By applying the law of detachment, we can conclude that:
I will go outside on Tuesday (conclusion).
This logical reasoning assumes that a 'nice' weather includes a sunny day. Hence, if it is indeed sunny on Tuesday, the condition for going outside is met.
Complete the expression so it forms a perfect-square trinomial. x² +___ x + 49 A. 14 B. 7 C. 7/2
The expression so it forms a perfect square trinomial is 14x.
We have given that,
The expression so it forms a perfect-square trinomial.
We have to determine the x² +___ x + 49.
What is the factor?
A divisor of an integer n also called a factor of n, is an integer m that may be multiplied by some integer to produce n.
In this case, one also says that n is a multiple of m.
(x + 7)(x + 7)
x^2 + 14x + 49
The expression so it forms a perfect square trinomial is 14x.
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Amy, who is 17 years old, is 5 years older than twice Moe's age.Write an equation that you could solve to find Moe's age. Solve the equation and describe your steps.
if TRAP is an isosceles trapezoid what is the value of x ?
the answer to this question is 19!
You and your family attend your brother’s championship baseball game. Between innings you decide to go to the snack stand. You go to the snack stand with $15 and find that sodas are $2.50 and that popcorn is $3.75. Write an inequality that models the number of sodas you can buy if you get a bag of popcorn too. What is the maximum number of sodas you can buy in this situation?
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Introduction to addition or subtraction of fractions with different denominator 3/5+1/3
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The probability of choosing either a girl or a boy is
Mrs. Howard is hosting a catered dinner party to celebrate her daughter's graduation. The catering company she plans to use charges $13.50 per plate of food and an additional fee of $540 to cover linens and tableware. The total bill for the catering services is $1,512. Which equation can be used to determine the total number of guests, g, who attend the dinner party? 13.50g + 540 = 1,512 13.50g - 1,512 = 540 5400g + 13.50 = 1,512 540g - 1,512 = 13.50
Answer:
The equation can be used to determine the total number of guests is 13.50g + 540 = 1512 and the total number of guest are 72 .
Step-by-step explanation:
As given
Mrs. Howard is hosting a catered dinner party to celebrate her daughter's graduation.
The catering company she plans to use charges $13.50 per plate of food and an additional fee of $540 to cover linens and tableware.
The total bill for the catering services is $1,512.
Total number of guests, g .
Than the equation becomes
1512 = 540 + 13.50 g
1512 - 540 = 13.50 g
972 = 13.50 g
[tex]g = \frac{972}{13.50}[/tex]
g = 72
Therefore the equation can be used to determine the total number of guests is 13.50g + 540 = 1512 and the total number of guest are 72 .