Suppose triangle ABC has vertices at A(1, 0), B(10, 0), and C(2, 6). After a 60° counterclockwise rotation about the origin, vertex B' has coordinates (5, ?).
Answers
Let OB be the radius of circle with center O.
Let B' be the image of B after the described rotation
OB and OB' are sides of the equilateral triangle OBB'.
The x coordinate of B' is the midpoint of OB, that is 5.
In the right triangle B', point (5, 0) and B:
Distance point (5, 0) to B is 5
|B'B|=|OB|=10
so by the pythagorean theorem:
[tex]a= \sqrt{ 10^{2} - 5^{2} } = \sqrt{ 2^{2} *5^{2} - 5^{2} }= \sqrt{5^{2}(4-1)}=5 \sqrt{3} [/tex] units
Answer: [tex]5 \sqrt{3}[/tex]
Related Questions
Assume a plane is flying directly north at 200 mph, but there is a wind blowing west at 23 mph. Part I: Express both the velocity of the plane and the velocity of the wind as vectors, using proper notation to represent each direction of motion. Part II: What is the velocity vector of the plane? Part III: What is the ground speed of the plane?
Answers
Define j as a unit vector in the northern direction.
Part I
Because the wind is blowing west, its velocity vector is
-23i mph or as (-23, 0) mph
Because the plane is traveling north, its velocity vector is
200j mph or as (0, 200) mph
Part II
The actual velocity of the plane is the vector sum of the plane and wind velocities.
That is,
200j - 23i or (-23, 200) mph
Part III
The ground speed of the plane is the magnitude of its vector.
The ground speed is
√[200² + (-23)²] = 201.32 mph
The ground speed of the plane is 201.3 mph (nearest tenth)
Not:
The direction of the plane is
tan⁻¹ 23/200 = 6.56° west of north.
The velocity of the plane is 200 mph due north and the velocity of the wind is 23 mph due west. The velocity vector of the plane is 200 mph due north minus 23 mph due west. The ground speed of the plane can be found using the Pythagorean theorem.
Explanation:Part I: The velocity of the plane can be represented as 200 mph due north, and the velocity of the wind can be represented as 23 mph due west.
Part II: To find the velocity vector of the plane, we subtract the velocity of the wind from the velocity of the plane. The resultant velocity vector of the plane is 200 mph due north minus 23 mph due west.
Part III: The ground speed of the plane is the magnitude of the resultant velocity vector of the plane. We can calculate it using the Pythagorean theorem: ground speed = square root of (200^2 + 23^2).
The circumference of a circle is 19πinches. find the radius.c=2πr
Answers
c=2πr
19 = 2 π r
r = 19/ 2 π
r = 19/ (2×3.14) =19/6.28
r = 3.03 nches
A polynomial function, f(x), with rational coefficients has roots of –2 and square root of 3. The irrational conjugates theorem states that which of the following must also be a root of the function?
Answers
Answer:
Step-by-step explanation:
option b
Given a polynomial with rational coefficients, if √3 is a root, its irrational conjugate -√3 must also be a root due to the Irrational Conjugates Theorem, ensuring that the polynomial maintains rational coefficients.
Explanation:The subject of this question is a polynomial function with rational coefficients, which is encountered in mathematics, particularly in algebra. When a polynomial has rational coefficients and one of its roots is an irrational number, such as the square root of 3 (√3), the Irrational Conjugates Theorem states that its conjugate, in this case, the negative square root of 3 (-√3), must also be a root of the polynomial. Therefore, the polynomial function in question must include both √3 and -√3 as roots to have rational coefficients.
Given that the function already has -2 as a root, we know that (x + 2) is a factor of the polynomial. Additionally, because √3 is a root, the factors (x - √3) and (x + √3) will also be part of the polynomial. To find the roots of a quadratic equation or higher-order polynomials, one can set the function equal to zero and solve for x, either by factoring, applying the quadratic formula, or other algebraic methods.
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Law of sines:
Triangle ABC has measures a = 2, b = 2, and m∠A = 30°. What is the measure of angle B?
15°
30°
45°
60°
Answers
since a and b are equal to number 2, they are the same value. Because A is 30 degrees, B is 30 degrees too.
Answer: Second option is correct.
Step-by-step explanation:
Since we have given that
ΔABC has measures a=2, b=2, m∠A=30⁰
As we know the "Law of sines " i.e.
[tex]\frac{a}{\sin A}=\frac{b}{\sin B}\\[/tex]
so, we put the given values in above formula:
[tex]\frac{2}{\sin 30\textdegree}=\frac{2}{\sin B}\\\\\implies \sin 30\textdegree=\sin B\\\\\implies B=30\textdegreee[/tex]
Hence, Second option is correct.
What's 9080 each number in expanded notation
Answers
Alex has been serving 2/3 cup of lemonade to each student. If he has 1 1/3 cups of lemonade left, how many students can still get lemonade?
Question 2 options:
1
2
3
0
Answers
In this case, every student is served 2/3 cup of lemonade and Alex has 1 1/3 cups of lemonade left. Then you just need to divide the lemonade left with lemonade served per student. The calculation would be:
Number of students served= lemonade left / lemonade served per student
Number of students served= 1 1/3 cup / (2/3 student/cup) = 2 student
One bundle contains 500 $20 bills. what would be the total value of 44 bundles
Answers
let y = total amount
x = number of bundles
y = $20 * 500x
y = $10,000 per bundle * x
So, using this equation, we can determine the total amount of money by simply substituting the number of bundles that we have.
To solve our problem, let x = 44 bundles
y = $10,000 per bundle * 44 bundles
y = $440, 000
Therefore, the total amount is $440,000.
What is the area of the region completely bounded by the curve y=-x^2+x+6?
Answers
y = -1 (x² - x - 6)
y = -1 (x - 3)(x + 2)
From this, we can tell the x-axis intercepts are -2 and 3.
(To do this, simply equate any of the expressions involving x to 0 and rearrange to give x).
Now, integrate between the limits -2 and 3:
R = area of region bounded
R = ³∫₋₂ -x² + x + 6 .dx
R = ³[-1/3x³ + 1/2x² + 6x]₋₂
R = (-1/3(3)³ + 1/2(3)² + 6(3)) - (-1/3(-2)³ + 1/2(-2)² +6(-2))
R = (-9 + 9/2 + 18) - (8/3 + 2 - 12)
R = (27/2) - (-22/3)
R = 27/2 + 22/3 = 125/6 units²
To find the area of the region completely bounded by the curve y=-x^2+x+6, you can integrate the equation with respect to x and evaluate it between the x-values where the curve intersects the x-axis. By solving the quadratic equation -x^2+x+6=0, you can determine the x-values. Then, evaluate the definite integral between these x-values to find the area.
Explanation:The area of the region completely bounded by the curve y=-x^2+x+6 can be found by integrating the equation with respect to x and evaluating it between the appropriate bounds. The integral of the given equation is ∫(-x^2+x+6) dx. To find the area, we need to find the definite integral between the x-values where the curve intersects the x-axis. First, set the equation equal to zero and solve for x:
-x^2+x+6=0
This quadratic equation can be factored as: (x-2)(x+3). Therefore, the curve intersects the x-axis at x=2 and x=-3.
By evaluating the definite integral between x=-3 and x=2, we can find the area of the region:
Area = ∫-32 (-x^2+x+6) dx
Integrating this equation will give you the area of the region bounded by the curve y=-x^2+x+6.
Two dice are rolled. Are the events, “rolling doubles” and “rolling an even sum”, mutually exclusive? Justify your response.
There are 8 students lined up at the classroom door. What is the probability that Laura and Kimiko will end up next to each other if the students arrange themselves blindfolded?
Answers
A specific example is if we rolled a pair of '3's an we get 3+3 = 6 as a result.
So in conclusion, the events "rolling doubles" and "getting an even sum" are not mutually exclusive. If the first event happens, then the second event definitely happens. The same can't be said the other way around.
-----------------------------------------------------
Problem 2)
We have 8 students. Of those 8, we have two students Laura and Kimiko that we want to keep together. In other words, we want them to be adjacent to each other. If we think of Laura and Kimiko as one student, then we have 8-2+1 = 7 "students", 6 of which are individuals who can move on their own and 2 (Laura and Kimiko) who are tied together as one "student".
There are 7! = 7*6*5*4*3*2*1 = 5040 ways to arrange these 7 "students". For any given permutation, we can have Laura on the left side or Kimiko on the left side. So there are 2 ways to arrange Laura and Kimiko.
In total there are 2*5040 = 10080 different ways to arrange all of the students where Laura and Kimiko will be next to each other.
This is out of 8! = 8*7*6*5*4*3*2*1 = 40320 ways to arrange the students where we don't worry about Laura and Kimiko sticking together.
Divide the two values: 10080/40320 = 1/4
Answer:
The answer as a fraction is 1/4
The answer as a decimal value is 0.25 (since 1/4 = 0.25)
In percent form, the answer is 25% (0.25 turns into 25% after moving the decimal 2 spots to the right)
On a floor plan drawn at a scale of 1:100, the area of a rectangular room is 30
c. what is the actual area of the room?
Answers
such that a*b=30.
Since the scale is 1:100, a represents 100a, and b represents 100b.
So the actual dimensions are 100a by 100b.
The area is 100a*100b=a*b*10,000=30*10,000=300,000 (square units)
Answer: 300,000 (square units)
using the formula C=2TTr, find the circumference of a circle with a diameter of 28 in. round your answer to the nearest inch
A. 56in
B. 28 in
C. 44in
D.88in
Answers
radius = diameter/2
so 28/2=14
using 3.14 for PI
2*3.14*14 = 87.92 = 88 inches
so radius = 28/2 = 14
Circumference = 2 x pi x 14 = 28 pi
or from calculator = 87.96 inches ....correct to 2 decimal places
A jar of 57 coins contains only dimes and quarters. The value of all the coins in the jar is $10.05. How many dimes are in the jar?
Answers
The dimes are 28 dimes
what is Algebra?Algebra is the part of mathematics that helps represent problems or situations in the form of mathematical expressions.
Given:
total coins = 57
So,
D + Q = 57
D= 57 - Q....(1)
0.10D + 0.25 Q = 10.05
Using (1), we get
0.10(57 - Q) + 0.25Q= 10.05
5.7 - 0.10Q + 0.25Q = 10.05
0.15Q = 4.35
Q= 29
So, dimes = 57- 29= 28 dimes
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2x-5y=-6; 2x-7y=-14
Answers
2x-7y = -14
-2x+5y = 6
2x - 7y = -14
-2y= - 8
y = 4
What is the arc length of a circle that has a 6-inch radius and a central angle that is 65 degrees? Use 3. 14 for pi and round your answer to the nearest hundredth
Answers
C=2π6
C=37.6991118431
That's the arc length of the whole circle, i.e. the arc length of 360°.
65° is 18.055555555555555555555555555556% of 360°, (65/360*100)
so 18.055555555555555555555555555556% of 37.6991118431 is
6.8067840827819444444444444444444. Rounded to the nearest hundredth, the answer is 6.81 inches.
That's real pi, lets see if it makes a difference to use "stupid pi".
C=2πr
C=2π6
C=37.68
That's the arc length of the whole circle, i.e. the arc length of 360°.
65° is 18.055555555555555555555555555556% of 360°, (65/360*100)
so 18.055555555555555555555555555556% of 37.68 is
6.803333333333333333333333. Rounded to the nearest hundredth, the answer is 6.80 inches.
Yep makes a difference. That's why you don't use stupid pi. 3.14159 is what we always used in engineering, or just the pi button and using a ton of digits.
Answer: 6.80 inches.
At a store, 2 gallons of milk cost $6. Which is the value of the ratio of dollars to gallons of milk?
Answers
Answer:
3:1 or 3/1
Step-by-step explanation:
We have the next information:
Gallons of Milk Cost
2 ⇒ $6
dividing both quantities by two, we get the price for a single gallon of milk:
Gallons of Milk Cost
1 ⇒ $3
We are asked for the ratio of dollars to gallons of milk (the order is important since dollars go first than gallons) so we will have the following:
dollars:gallons or dollars/gallons
and since $3 dollars pay for 1 gallon of milk, the ratio is:
3:1 or 3/1
Find the equation of the quadratic function with zeros 10 and 14 and vertex at (12, -8).
Answers
our vertex is at (12, -8) thus h = 12, k = -8
and our zeros are 10, 0 and 14,0, so.. .we'll use say hmm (10,0) to get the "a" coefficient.
[tex]\bf \qquad \textit{parabola vertex form}\\\\ \begin{array}{llll} \boxed{y=a(x-{{ h}})^2+{{ k}}}\\\\ x=a(y-{{ k}})^2+{{ h}} \end{array} \qquad\qquad vertex\ ({{ h}},{{ k}})\\\\ -------------------------------\\\\ vertex\ (12,-8)\ \begin{cases} h=12\\ k=-8 \end{cases}\implies y=a(x-12)^2-8 \\\\\\ \textit{we also know that } \begin{cases} y=0\\ x=10 \end{cases}\implies 0=a(10-12)^2-8 \\\\\\ 8=a(-2)^2\implies 8=4a\implies \boxed{2=a} \\\\\\ thus\qquad \boxed{y=2(x-12)^2-8}[/tex]
Carol spends 17 hours in a 2-week period practicing her culinary skills. How many hours does she practice in 5 weeks?
Answers
Final answer:
Carol practices for 8.5 hours per week, so in a 5-week period, she would practice for a total of 42.5 hours.
Explanation:
The student asked how many hours Carol practices her culinary skills in a 5-week period, if she practices for 17 hours in a 2-week period. To find the answer, we calculate how many hours Carol practices per week by dividing the total hours she practices in two weeks by two. Then we multiply the weekly hours by the number of weeks in question, which is five.
The calculation is as follows: Carol practices for 17 hours / 2 weeks = 8.5 hours per week. Then, 8.5 hours/week x 5 weeks = 42.5 hours in total for a 5-week period.
The variable is Z is inversely proportional to X. When X is 6, Z has the value 0.5. What is the value of Z when X = 10
Answers
zα1/x
hence;
z=k/x
where k is the constant of proportionality;
this implies that:
k=xz
when x=6,z=0.5
thus;
k=0.5*6
k=3
hence;
z=3/x
when x=10,
z=3/10
the answer is 3/10
Graph the ellipse with equation x squared divided by twenty five plus y squared divided by four = 1.
Answers
thus, check the picture below.
Which phrases can be used to represent the inequality mr024-1.jpg? Check all that apply. The product of 6.5 and the sum of a number and 1.5 is no more than 21. The sum of 1.5 and the product of 6.5 and a number is no greater than 21. The product of 6.5 and a number, when increased by 1.5, is below 21. The product of 6.5 and the sum of a number and 1.5 is at minimum 21. The sum of 1.5 and the product of 6.5 and a number is at least 21. The product of 6.5 and a number, when increased by 1.5, is at most 21.
Answers
The product of 6.5 and a number, when increased by 1.5, is below 21.
6.5x + 1.5 < 21
The sum of 1.5 and the product of 6.5 and a number is at least 21.
6.5x + 1.5 ≥ 21
The product of 6.5 and a number, when increased by 1.5, is at most 21.6.5x + 1.5 ≤ 21
Answer:
b and c
Step-by-step explanation:
2020 edge assignment
The expression 4 square root of 81^3 can be rewritten as_____.
A. 81^3/4
B.81^4/3
C. 81^12
D. 81^1/12
Answers
Answer:
81^1/12
HAVE A GREAT DAY
The expression ''4 square root of 81³'' can be rewritten as,
⇒ [tex]81^{\frac{3}{4} }[/tex]
What is an expression?Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.
Given that;
The expression to write is,
⇒ 4 square root of 81³
Now, It can be written as;
⇒ 4 square root of 81³
⇒ [tex]\sqrt[4]{81^{3} }[/tex]
By rule of exponent we get;
⇒ [tex]81^{\frac{3}{4} }[/tex]
Thus, The expression ''4 square root of 81³'' can be rewritten as,
⇒ [tex]81^{\frac{3}{4} }[/tex]
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What equation is solved by the graphed systems of equations? Two linear equations that intersect at the point negative 1, negative 4.
Answers
To solve this problem, we have to manually solve for the value of x for each choices or equations. The correct equation will give a value of -1 since the linear equations intersects at point (-1, -4).
1st: 7x + 3 = x + 3
7x – x = 3 – 3
6x = 0
x = 0 (FALSE)
2nd: 7x − 3 = x – 3
7x – x = 3 – 3
6x = 0
x = 0 (FALSE)
3rd: 7x + 3 = x − 3
7x – x = - 3 – 3
6x = -6
x = -1 (TRUE)
4th: 7x − 3 = x + 3
7x – x = 3 + 3
6x = 6
x = 1 (FALSE)
Therefore the answer is:
7x + 3 = x − 3
In this exercise, we are going to solve using our knowledge of systems and in this way we will find that the equation that satisfies the points.
As we know that the equation that will satisfy will have to have the values of X=-1, we will solve each one of the alternatives as:
First equation is:[tex]7x + 3 = x + 3\\6x = 0\\x = 0[/tex]
We realize that the value of x is not what we want so it doesn't satisfy us.
second equation is:[tex]7x - 3 = x - 3\\7x- x = 3- 3\\6x = 0\\x = 0[/tex]
We realize that the value of x is not what we want so it doesn't satisfy us.
third equation is:
[tex]7x + 3 = x − 3\\6x = -6\\x = -1[/tex]
fourth equation is:[tex]7x − 3 = x + 3\\7x – x = 3 + 3\\6x = 6\\x = 1[/tex]
We realize that the value of x is not what we want so it doesn't satisfy us.
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Linda is putting money into a savings account. She starts with $450 in the savings account, and each week she adds $70 .
Let S represent the total amount of money in the savings account (in dollars), and let W represent the number of weeks Linda has been adding money. Write an equation relating S to W . Then use this equation to find the total amount of money in the savings account after 19 weeks.
Answers
S=450+70W would be the equation
then afet 19 weeks
W=19
S=450+70(19)
S=450+1330
S=1780
equation is
S=70W+450
after 19 weeks, has 1780
S (w)=450+70w
The total amount of money in the savings account after 19 weeks is
S (19)=450+70×19
S (19)=1,780
Is it possible for a line segment to have more than one bisector?
Answers
Yes, it is possible to have more than one bisector in a line segment.
Bisector is a line that divides a line or an angle in to two equivalent parts. There are two types of Bisectors based on what geometrical shape it bisects.
Bisector of a Line Angle BisectorIn general 'to bisect' something means to cut it into two equal parts. The bisector is the one that doing the cutting process.
With a line bisector, we cut a line segment into two equal parts with another line - the bisector. Just imagine the line PQ is being cut into two equal lengths (PF and FQ) by the bisector line AB.
Whenever AB intersects at a right angle, it is called the "perpendicular bisector" of PQ. If it crosses at any other angle it is simply called a bisector. Drag the points A or B and see both types.
For obvious reasons, the point F is called the midpoint of the line PQ,
Expand (2x-3y)^4 using Pascal's Triangle. Show work
Answers
1) let 2x = a and 3y = b
(a+b)⁴ = a⁴ + a³b + a²b² + ab³ + b⁴
Now let's find the coefficient of each factor using Pascal Triangle
0 | 1
1 | 1 1
2 | 1 2 1
3 | 1 3 3 1
4 | 1 4 6 4 1
0,1,2,3,4,.. represent the exponents of binomials
Since our binomial has a 4th exponents, the coefficients are respectively:
(1)a⁴ + (4)a³b + (6)a²b² + (4)ab³ + (1)b⁴
Now replace a and b by their real values in (1):
2⁴x⁴ +(4)8x³(3y) + (6)(2²x²)(3²y²) + (4)(2x)(3³y³) + (1)(3⁴)(y⁴)
16x⁴ + 96x³y + 216x²y² + 216xy³ + 81y⁴
Answer:
16x^4 - 96x^3y + 216x^2y^2 - 216xy^3 + 81y^4
Step-by-step explanation:
(2x - 3y)^4
Fifth line on a Pascal Triangle
1, 4, 6 4, 1
(1) 2x^4
2^4 = 16
2x^4 = 16x^4
16x^4
(4) 2x^3 (-3y)^1
2^3 = 8
-3^1 = -3
8 times -3 times 4 = -96
-96x^3y
(6) 2x^2 (-3y)^2
2^2 = 4
-3^2 = 9
4 times 9 times 6 = 216
216x^2y^2
(4) 2x^1 (-3y)^3
2^1 = 2
-3^3 = -27
2 times - 27 times 4 = -216
-216xy^3
(1) (-3y)^4
-3^4 = 81
81y^4
16x^4 - 96x^3y + 216x^2y^2 - 216xy^3 + 81y^4
Help? @texaschic101
Parabola and its vertex
Answers
f (x) = a(x-4)^2 - 1
15 = a(0-4)^2 - 1
15 = a(-4)^2 - 1
15 = a(16) - 1
15 +1 = 16a
16 = 16a
16/16 = 16a/16
1 = a
A must be equal to 1
y = (x-4)^2 - 1
y = x^2 - 8x + 16 - 1
y = x^2 - 8x + 15
Factor (find two number that summed give -8 and multiplied +15)
(x-3)(x-5) = 0
(x-3) = 0 -> x = 3
(x-5) = 0 -> x = 5
So answer is: (3;0) and (5;0)
Solve by factoring and list only the positive solution: 2x2 - 5x = 88
Answers
The general form of a quadratic equation is,
Ax² + Bx + C = 0
where A and B are numerical coefficients and C is the constant. If we are to express the given equation in this form,
2x² - 5x - 88 = 0
The sum of the roots, x₁ and x₂ is -B/A and the product is equal to C/A.
Sum: x₁ + x₂ = -(-5/2) = 5/2
Product: x₁x₂ = -88/2 = -44
The values of x₁ and x₂ are 8 and -11/2.
The factors are (x - 8) and (x + 11/2)
ANSWERS: 8 and -11/2.
What is the area of the composite figure?
(6π + 4) cm2
(6π + 16) cm2
(12π + 4) cm2
(12π + 16) cm2
Answers
in this figure there are 3 semi circle and one square, one semi circle radius is 2 , square all sides 2.2=4
one semi circles area =pi.r²/2
one semi circle area = pi.2²/2=2.pi
three semi circle area =3.2.pi =6.pi
square area = 4² =16
area of composite figure = 6.pi+16
The area of the composite figure is 6π + 16 cm²
Composite Figure:Composite figures are composed of different dimensional figures. The area of a composite figure is the sum of the whole 2 dimensional figures that forms the composite figure.
Therefore, the figure above has 3 semi circle and 1 square.
Therefore, the area can be calculated as follows;
area = sum of the area of the 3 semi circle + area of the squarearea = 1 / 2 πr² + 1 / 2 πr² + 1 / 2 πr² + L²
area = 3 / 2 (πr²) + L²
where
r = 2 cm
L = 4 cm
Therefore,
area of the composite figure = 3 / 2(π × 4) + 4²
area of the composite figure = 3 / 2(4π) + 16
area of the composite figure = 6π + 16 cm²
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What is a rule for the total cost of the tickets ? Give the rule in words and as a algebraic expression
Answers
ur equation would be : y = 12x + 150....where x is the number of students and y is the total cost...but that is an equation, and u want an expression...so I guess the expression would be 12x + 150
the cost of each ticket (12) multiplied by the number of students (x), added to 150 will give u the total cost
so theh rule is 12s+150 where s is the number of students
What is the equation of the line that is parallel to y=-2/3x+4 and that passes through (–2,–2)?
Answers
-
-
-
y=-2/3x-4/3
y=-2/3x-10/3
y=-2/3x-2/3
y=-2/3x-17/4
Answers
y=-2x/3+b, using the point (-2,-2) we can solve for "b", the y-intercept.
-2=-2(-2)/3+b
-2=4/3+b
-6/3-4/3=b
-10/3=b so our line is:
y=-2x/3-10/3 (or more neatly in my opinion :P)
y=(-2x-10)/3
It's the second one down from the top :)
The equation of the line parallel to y=-2/3x+4 and passing through (-2, -2) is y = -2/3x - 10/3.
Explanation:To find the equation of a line parallel to y = -2/3x + 4 and passing through the point (-2, -2), we need to use the fact that parallel lines have the same slope. The given equation has a slope of -2/3, so the parallel line will also have a slope of -2/3. Using the point-slope form of a line, we can write the equation as:
y - y1 = m(x - x1)
where (x1, y1) is the given point and m is the slope. Plugging in the values, we have:
y - (-2) = -2/3(x - (-2))
Simplifying the equation, we get:
y - (-2) = -2/3(x + 2)
y + 2 = -2/3x - 4/3
y = -2/3x - 4/3 - 2
y = -2/3x - 4/3 - 6/3
y = -2/3x - 10/3
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