Which of the following are exterior angles? Check all that apply.
A.4
B.5
C.6
D.1
E.2
Understanding this proof for the proposition "For all integers a, gcd(9a+4, 2a+1) = 1.
Proof: gcd(9a+4, 2a+1) = gcd(2a+1, a) = gcd(a, 1). Since gcd(a, 1)=1, gcd(9a+4, 2a+1) =1.
First line because 4(2a+1)=8a+4 and 9a+4-(8a+4)= a
Second line because a times 2 =2a and 2a+1-2a=1
Although the second equality is more or less obvious since 2a+1 leaves a remainder of 1 when divided by a.
Find all points of extrema on the interval [0,2pi] if y=x-cosx ...?
The function y=x-cosx has an extremum on the interval [0,2pi] at x=3pi/2, determined by setting the first derivative equal to zero. The second derivative test is inconclusive at this point, but the point (3pi/2, 3pi/2) is a minimum due to the nature of the function.
To find all points of extrema on the interval [0,2pi] for the function y=x-cosx, we first need to find the derivative of the function and determine where it is equal to zero. The first derivative of y=x-cosx is y' = 1 + sinx. The critical points occur where y' = 0, which translates to 1 + sinx = 0 or sinx = -1. The only point in the interval [0,2pi] where sinx = -1 is at x = 3pi/2. To determine if this critical point is a maximum or minimum, we apply the second derivative test.
The second derivative of the function is y'' = cosx. At x = 3pi/2, the second derivative y'' = cos(3pi/2) = 0; since the second derivative is zero, the second derivative test is inconclusive. However, we can analyze the function around the critical point to conclude or use other methods like analyzing the first derivative's sign change.
For y=x-cosx, since the original function is a combination of a linear increasing function and cosine function, the critical point at x = 3pi/2 will be a minimum as the function increases both before and after this point.
Therefore, the point of extrema on the interval [0,2pi] for the function y=x-cosx are at (3pi/2, 3pi/2 - cos(3pi/2)), which simplifies to (3pi/2, 3pi/2).
What is the measure of angle TRV?
Measure of angle TRV is [tex]130^{0}[/tex].
What is an angle?An angle is formed when two straight lines or rays meet at a common endpoint. The common point of contact is called the vertex of an angle.
According to the question
< TRS + < TRV = [tex]180^{0}[/tex]
(x - 10) + (2x + 10) = [tex]180^{0}[/tex]
3x = [tex]180^{0}[/tex]
x = [tex]\frac{180}{3}[/tex]
x = [tex]60^{0}[/tex]
< TRV = 2x + 10
= 2([tex]60^{0}[/tex])+10
= 120 + 10
= [tex]130^{0}[/tex]
Hence, measure of angle TRV is [tex]130^{0}[/tex].
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m plus 4 equals minus 12
What is the fifth term in the binomial expansion of (x + 5)8?
175,000x3
43,750x4
3,125x5
7,000x5
The fifth term in the binomial expansion of [tex](x+5)^{8}[/tex] is [tex]\boxed{43,750\ x^{4}}[/tex].
Further explanation:
Given:
The binomial term is [tex](x+5)^{8}[/tex].
The expansion of [tex](x+5)^{8}[/tex] is as follows:
[tex]\boxed{{\left({a+b}\right)^n}=\sum\limits_{k=0}^n{{}^n{{\text{C}}_k}{a^{n - k}}{b^k}}}[/tex]
There are [tex]n+1[/tex] terms in the expansion of [tex](a+b)^{n}[/tex].
The sum of indices of [tex]a[/tex] and [tex]b[/tex] is equal to [tex]n[/tex] in every term of the expansion.
The general term [tex]T_{r+1}[/tex] of the binomial term [tex](a+b)^{n}[/tex] is as follows:
[tex]\boxed{{{\text{T}}_{r + 1}}={}^n{{\text{C}}_r}{a^{n - r}}{b^r}}[/tex]
For [tex]5^{th}[/tex] term the value of [tex]r[/tex] is calculated as follows:
[tex]\begin{aligned}r+1&=5\\r&=5-1\\r&=4\end{aligned}[/tex]
Now, the [tex]5^{th}[/tex] term of [tex](x+5)^{8}[/tex] is calculated as follows:
[tex]\begin{aligned}T_{5}&=T_{4+1}\\&=^8C_{4}\cdot x^{8-4}\cdot 5^{4}\\&=\dfrac{8\cdot 7\cdot 6\cdot 5}{4\cdot 3\cdot 2\cdot 1}\cdot x^{4}\cdot 625\\&=625\cdot 70x^{4}\\&=43,750x^{4}\end{aligned}[/tex]
Therefore, the fifth term of the binomial expansion [tex](x+5)^{8}[/tex] is [tex]\boxed{43,750\ x^{4}}[/tex].
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Answer details:
Grade: Senior school
Subject: Mathematics
Chapter: Binomial Theorem
Keywords: Binomial theorem, expansion, (x+5)^8, 175000x3, 43750x4, 3125x5, 7000x5, fifth term, binomial expansion, genral term, binomial, polynomial, indices.
The answer is:
[tex]\[\boxed{43750x^4}\][/tex]
To find the fifth term in the binomial expansion of [tex]\((x + 5)^8\)[/tex], we use the binomial theorem. The binomial theorem states that:
[tex]\[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\][/tex]
For the expansion of [tex]\((x + 5)^8\)[/tex], we identify:
[tex]\[a = x, \quad b = 5, \quad n = 8\][/tex]
The general term in the expansion is given by:
[tex]\[T_{k+1} = \binom{n}{k} a^{n-k} b^k\][/tex]
We need to find the fifth term, which corresponds to k = 4 (since k starts from 0):
[tex]\[T_5 = \binom{8}{4} x^{8-4} 5^4\][/tex]
First, calculate the binomial coefficient [tex]\(\binom{8}{4}\):[/tex]
[tex]\[\binom{8}{4} = \frac{8!}{4! \cdot 4!} = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = 70\][/tex]
Next, calculate [tex]\( x^{8-4} \)[/tex] and [tex]\( 5^4 \):[/tex]
[tex]\[x^{8-4} = x^4\]\[5^4 = 625\][/tex]
Now, combine these results:
[tex]\[T_5 = 70 \cdot x^4 \cdot 625\][/tex]
Finally, multiply the coefficients:
[tex]\[70 \times 625 = 43750\][/tex]
Thus, the fifth term is:
[tex]\[43750x^4\][/tex]
Therefore, the answer is:
[tex]\[\boxed{43750x^4}\][/tex]
What is eight times four?
n a division problem, the divisor is twenty times the quotient and five times the remainder. If remainder is 16, the number will be:
Option 1 : 3360 Option 2 : 336 Option 3 : 1616 Option 4 : 20516
Which type of triangle is RST?
a. equilateral
b. isosceles
c. scalene
d. right
9. Write an equation of the line that contains the median of traingle RST from S to RT .
a. x + 2 y = -10
b. x + 2 y = 6
c. x + 2 y = 8
d. x + 2 y = 10
10. Write the equation of the line that contains the altitude of triangle RST from T to RS .
a. 2x - 3y = -13
b. 3 x -2 y = -2
c. 2 x + 3 y = 29
d. 3 x + 2 y = 26
how many numbers are between 100 and 200?
Point G is the centroid of triangle abc use the information to find the value of x.
1. GC=3x +7 and CE= 6x
2. FG= x +8 and AF = 9x - 6
3. Bg=5x -1 and DG = 4x -5
The value of x is 7/6.
To find the value of x, we can utilize the given information about the lengths of the medians and segments within triangle ABC.
Using medians CE and GC:
Since G is the centroid of triangle ABC, it divides each median into two segments in a ratio of 2:1. Therefore, we can set up two equations based on the given lengths:
CE = 2/3 * GC
6x = 2/3 * (3x + 7)
Solving for x, we get:
6x = 2x + 14/3
4x = 14/3
x = 7/6
Using medians BG and FG:
Similarly, we can set up two equations based on the given lengths of medians BG and FG:
FG = 2/3 * BG
x + 8 = 2/3 * (5x - 1)
Solving for x, we get:
x + 8 = 10x/3 - 2/3
8/3 = 9x/3
x = 8/9
Using medians DG and AG:
Following the same approach, we can set up two equations based on the given lengths of medians DG and AG:
AG = 2/3 * DG
9x - 6 = 2/3 * (4x - 5)
Solving for x, we get:
9x - 6 = 8x/3 - 10/3
x - 6 = 8x/3 - 10/3
-5 = 5x/3
x = -3
Comparing the values obtained from each set of equations, we find that x = 7/6 is consistent across all three sets. Therefore, the value of x is 7/6.
One fifth of todays college students began using computers between the ages 5 and 8. if a college has 3,500 students, how many of the students began using computers between the ages 5 and 8?
1. Julia deposited $250 in a savings account that earns 2.7% simple interest. How much interest has Julia earned by the end of the first year?
$6.75
$92.59
$256.75
$675.00
2. Armand deposited $389.42 in a savings account that earns 3.2% simple interest. What is Armand’s account balance after seven years?
$87.23
$401.88
$476.65
$872.30
3. Andy deposited $1,567.12 in a savings account that earns 1.9% simple interest. What will Andy’s account balance be in nine months?
$1,567.12
$1,589.45
$1,596.90
$1,835.10
Answer:
Step-by-step explanation:
Simple interest formula is : [tex]p\times r\times t[/tex]
t is always in years here.
1.
p = 250
r = 2.7% or 0.027
t = 1
Simple interest earned = [tex]250\times0.027\times1[/tex] = $6.75
2.
p = 389.42
r = 3.2% or 0.032
t = 7
Simple interest earned = [tex]389.42\times0.032\times7[/tex] = $87.23
Amount after 7 years will be = [tex]389.42+87.23[/tex] = $476.65
3.
p = 1567.12
r = 1.9% or 0.019
t = [tex]9/12=0.75[/tex]
Simple interest earned = [tex]1567.12\times0.019\times0.75[/tex] = $22.33
Account balance after 9 months = [tex]1567.12+22.33[/tex] = $1589.45
Haley substituted the values of a, b, and c into the quadratic formula below.
If f(x) = {ln x for 0 < x less than or equal to 2
{x^2 ln2 for 2 < x less than or equal to 4, then the limit as x approaches 2 of f(x) is..
The function f(x) is a piecewise function. To find the limit as x approaches 2, you must consider it from the left and right. From the left, the limit is ln(2), from the right, it's 4ln(2). Since these are not equal, the limit does not exist.
Explanation:The function f(x) is a piecewise function where the first range covers 0 < x <= 2 and the second ranging from 2 < x <= 4. To find the limit of f(x) as x approaches 2, we need to find the limit from both directions, as x approaches 2 from the left (<) and the right (>).
For x approaches 2 from the left (<), f(x) boils down to ln(x), so the limit as x approaches 2 would be ln(2).
For x approaching 2 from the right (>), f(x) translates to x^2 * ln(2). Substituting x = 2 in this function gives us (2)^2 * ln(2) which is 4ln(2)
The limit does not exist since ln(2) not equal to 4ln(2).
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The limit of the function f(x) as x approaches 2 does not exist, because the values obtained when approaching from the left (ln2) and from the right (4ln2) do not match.
Explanation:The question is asking us to find the limit of the function f(x) as x approaches 2. This is a well-known concept in calculus, called the limit of a function. It is easy to solve using the rule that if f(x) and g(x) are two functions that agree at every point of a certain interval, except perhaps at one single point 'a', then their limits as x approaches 'a' are the same.
So let's find the limit as x approaches 2 by looking at each side independently. For x less than or equal to 2, the function is defined as ln(x). So when x approaches 2 from the left, we get ln(2). For x greater than or equal to 2, the function is defined as x^2ln(2). So when x approaches 2 from the right, we get 4ln(2).
Since the values obtained when approaching 2 from the left (ln2) and from the right (4ln2) do not match, we can conclude that the limit of f(x) as x approaches 2 does not exist.
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How do i write 7.6% as a fraction in simplest form and as a decimal?
Rectangle ABCD is reflected over the x-axis, followed by a reflection over the y-axis, and then rotated 180 degrees about the origin. What is the location of point A after the transformations are complete?
Rectangle ABCD is shown. A is at negative 5, 1. B is at negative 5, 3. C is at negative 1, 3. D is at negative 1, 1.
(−5, 1)
(5, −1)
(−5, −1)
(5, 1)
Answer:
First option is correct. The location of point A after the transformations is (-5,1).
Step-by-step explanation:
It is given that the coordinates of point A are (-5,1).
Rectangle ABCD is reflected over the x-axis. then x-coordinate remains the same but the sign of y-coordinate is changed.
[tex](x,y)\rightarrow (x,-y)[/tex]
Coordinates of point A are
[tex](-5,1)\rightarrow (-5,-1)[/tex]
After that ABCD is reflected over the y-axis, then y-coordinate remains the same but the sign of x-coordinate is changed.
[tex](x,y)\rightarrow (-x,y)[/tex]
Coordinates of point A are
[tex](-5,-1)\rightarrow (5,-1)[/tex]
After that ABCD rotated 180 degrees about the origin, then the sign of both coordinates are changed.
[tex](x,y)\rightarrow (-x,-y)[/tex]
[tex](5,-1)\rightarrow (-5,1)[/tex]
Therefore option 1 is correct.
Jordan keeps track of rainy days for 1 year . This year he counted 52 weeks in a year with at least I rainy day. How many weeks had no rainy days.
Answer:
A.
Step-by-step explanation:
What is 0.15% converted to a decimal?
Based on the figure, which pair of triangles is congruent by the Side Angle Side Postulate?
Answer
Triangle ABD and Triangle ECD
Triangle ABD and Triangle ADC
Triangle ADC and Triangle ABC
Triangle ABC and Triangle ECD
Answer: Triangle ABC and Triangle ECD
Step-by-step explanation:
In Triangle ABC and Triangle ECD
BD=CD and AD=ED [given in the figure]
∠BDA=∠EDC [Vertically opposite angles are equal]
⇒ΔABC ≅ ΔECD [By SAS postulate]
SAS postulate or Side Angle Side postulate tells that if two sides and their included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.
What are the possible rational zeros of f(x) = x4 + 6x3 – 3x2 + 17x – 15?
Help me please
...?
Answer with explanation:
The given fourth degree polynomial is:
[tex]f(x)=x^4+6 x^3-3 x^2+17 x-15[/tex]
By rational root theorem , the possible Zeroes of the polynomial are factors of 15 .
Factors of 15 are
[tex]=\pm 1, \pm 3, \pm 5, \pm 15[/tex]
There are 8 possible Zeroes of the given Polynomial expression.
Show that the equation represents a circle by rewriting it in standard form, and find the center and radius of the circle.
5x^2 + 5y^2 + 10x − y = 0
The equation 5x^2 + 5y^2 + 10x - y = 0 represents a circle. By completing the square, the equation is rewritten in the standard form as (x + 1)^2 + (y - 1/10)^2 = 101/100. The circle's center is at (-1, 1/10) with a radius of approximately 1.005.
Explanation:To show that the equation 5x^2 + 5y^2 + 10x - y = 0 represents a circle and to rewrite it in standard form, we can complete the square for both x and y terms. First, factor out the coefficients of the squared terms:
5(x^2 + 2x) + 5(y^2 - 1/5y) = 0Divide both sides by 5 to simplify the equation:
(x^2 + 2x) + (y^2 - 1/5y) = 0Now, complete the square by adding and subtracting the necessary constants inside each parenthesis:
(x^2 + 2x + 1) - 1 + (y^2 - 1/5y + 1/100) - 1/100 = 0By completing the square, the equation becomes:
(x + 1)^2 + (y - 1/10)^2 = 1 + 1/100This can be further simplified to:
(x + 1)^2 + (y - 1/10)^2 = 101/100The standard form of the equation for a circle is (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center and r is the radius of the circle. Therefore, our circle's center is at (-1, 1/10) and its radius is the square root of 101/100, which simplifies to 10.05/10 or approximately 1.005.
How many solutions are there to the following system of equations?
3x + 7y = -10
15x + 35y = -45
2
1
infinitely many
0
Which set of side lengths represents a triangle with 3 lines of reflectional symmetry?
Answer:
The answer is 5,5,5.
Step-by-step explanation:
Reflection mean same like a mirror, it says reflectional symmetry. So the sides are the same.
What is the next term of the following sequence? 20, 10, 5, 2 1/2, ... 1 1/2 1 1/4 3/4
The distance from the school to Brandi's house is 1,240 meters. Leaving the school, she rides her bicycle for 60 seconds at a speed of 5 meters per second. If Brandi continues cycling at this speed, how many more seconds will it take her to arrive at her house?
Brandi rides her bike at a speed of 5 m/s from school to her house. She will take an additional 188 seconds to cover the remaining distance of 940 meters to arrive at her house.
Explanation:Brandi's situation:
Distance to house: 1,240 metersSpeed leaving school: 5 m/sTime taken initially: 60 secondsTo find: Additional time needed to reach house.
Calculations:
Time taken initially: 60 secondsRemaining distance: 1,240 - (60*5) = 940 metersTime to cover the remaining distance: 940 / 5 = 188 secondsTotal time: 60 + 188 = 248 secondsA scale drawing for a restaurant is shown below.
In the drawing, 5cm represents 6m.
Assuming the dining hall is rectangular, find the area of the real dining hall.
What is the area?
The area of the real rectangular dining hall (assuming that 5cm on paper is 6m in real) is 72m².
A rectangle is a quadrilateral whose opposite edges are of equal length and the opposite angles are of equal measurements.
Given that
the length of the dining hall, l = 10cm
the width of the dining hall, b = 5cm
As evident from the figure, the dining hall is rectangular, and since it is known that the area of a rectangle is the product of its length and width, therefore,
Area of the hall (on paper) = l × b
= 10 × 5
= 50 square cm.
Also, as it is given in the question that 5cm on paper represents 6m in real, therefore
[tex]Area\ of\ the\ hall\ (in\ real)= (\dfrac{10}{5}\times6)\times(\dfrac{5}{5}\times6)[/tex]
[tex]\\= 2\times6\times6\\= 72\ m^2[/tex]
Hence, the area of the hall in real is 72m².
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What is 0.43 as a fraction in simplest form?
In a family with eight children, what is the probability that exactly six are boys? a. 7168 b. 28 c. 0.109375 d. 0.015625
The probability that exactly six children in the family are boys is 0.015625.
What is Combination?An arrangement of objects where the order in which the objects are selected does not matter.
The probability of having exactly six boys in a family with eight children can be calculated using the binomial distribution formula:
[tex]P(X = k) =^nC_{k} p^k(1-p)^(^n^-^k^)[/tex]
P(X = k) is the probability of having exactly k boys, n is the total number of children, k is the number of boys we want to find , p is the probability of having a boy and (1-p) is the probability of having a girl (also 1/2 in this case)
Substituting the values into the formula, we get:
P(X = 6) = ⁸C₆ (1/2)⁶ (1/2)⁸⁻⁶
= (8! / (6! ×2!)) (1/2)⁸
= (87/21) × (1/2)^8
= 0.015625
Therefore, the probability that exactly six children in the family are boys is 0.015625.
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How do you prove vertical angles are congruent?
Answer:
(<2 and <4 are vert angles) reason:given
(lines m and n intersect at p) reason: def of vertical angles
(<2 and <3 are a linear pair) reason: def of a linear pair
(<2 and <3 are a linear pair) reason: def of a linear pair
(m<2+m<3=180) reason:angle addition postulate
(m<3+m<4=180) reason:angle addition postulate
(m<2+m<3=m<3+m<4 reason:substitution property
(m<2 = m<4) reason:subtraction property
<2 ~<4 reason:definition of ~ angles
Step-by-step explanation:
edgenuity 2020