The dot product associated to sides AB and BC is 0.
What is the dot product generated by two sides of a triangle?
In this question we must find the dot product of two vectors associated with two surrounding sides of a triangle. First, we need to find the vectors associated to sides AB and BC:
Side BA
BA = (5, 3) - (0, 0)
BA = (5, 3)
Side BC
BC = (- 3, 5) - (0, 0)
BC = (- 3, 5)
By definition of dot product, the result associated to the two sides of the triangle is:
BA • BC = (5, 3) • (- 3, 5)
BA • BC = (5) · (- 3) + (3) · (5)
BA • BC = - 15 + 15
BA • BC = 0
The dot product associated to sides AB and BC is 0.
Remark
The statement is poorly formatted, the correct form is shown below:
If A(x, y) = (5, 3), B(x, y) = (0, 0) and C(x, y) = (- 3, 5) are the vertices of a triangle then the value of the dot product between vectors AB and BC is:
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S is a geometric sequence.
a) (√x + 1), 1 and (√x-1) are the first three terms of S.
Find the value of x.
You must show all your working.
Evaluate the integral.
√
S-
5
√x (4+5√x)²
2
dx
Substitute [tex]y=4+5\sqrt x[/tex] and [tex]dy=\frac5{2\sqrt x}\,dx[/tex]. Then the integral is
[tex]\displaystyle \int \frac5{\sqrt x (4+5\sqrt x)^2} \, dx = 2 \int \frac{1}{(4+5\sqrtx)^2} \frac{5}{2\sqrt x} \, dx = 2 \int y^{-2} \, dy[/tex]
By the power rule,
[tex]\displaystyle \int y^{-2} \, dy = -y^{-1} + C[/tex]
so that
[tex]\displaystyle \int \frac5{\sqrt x (4+5\sqrt x)^2} \, dx = \boxed{-\frac2{4+5\sqrt x} + C}[/tex]
Use the distributive property to simplify the expression.
-6(2²+3)-2(1²-2)
A. 4² +22
B. 4:² +14
C. -8²-22
D. -8:²-14
[tex] \huge\mathbb{ \underline{SOLUTION :}}[/tex]
Given:[tex]\bold{-6(2^2+3)-2(1^2-2)}[/tex][tex]\\[/tex]
The [tex]\mathrm{distributive \: property}[/tex] states that an expression that is given in the form of [tex]\small\sf{ A(B + C)}[/tex] can be solved as [tex]\small\sf{A \times (B + C) = AB + AC}[/tex] . So:
[tex]\small\longrightarrow\sf{-24-18-2+4}[/tex]
[tex]\small\longrightarrow\sf{-42+2}[/tex]
▪ [tex]\large\tt{All \: \: options \: \: are \: \: wrong}[/tex]
[tex]\\[/tex]
[tex]\huge \mathbb{ \underline{ANSWER:}}[/tex]
[tex]\small\longrightarrow\sf{−6 (2^2+3) − 2 (1^2 - 2) = \underline{-6(4+3)}}[/tex]
Find the limit. use l'hospital's rule if appropriate. if there is a more elementary method, consider using it. lim x→0 e3x − 1 − 3x x2
It looks like the limit is
[tex]\displaystyle \lim_{x\to0} \frac{e^{3x} - 1 - 3x}{x^2}[/tex]
L'Hôpital's rule works in this case; applying it twice gives
[tex]\displaystyle \lim_{x\to0} \frac{e^{3x} - 1 - 3x}{x^2} = \lim_{x\to0} \frac{3e^{3x} - 3}{2x} = \lim_{x\to0} \frac{9e^{3x}}{2} = \boxed{\frac92}[/tex]
How can x² = x² + 2x + 9 be set up as a system of equations? (1 point)
1. y= x²-9
y = x² + 2x + 9
2. y=x²
y = x² + 2x +9
3. y = x² + 2x
y = x² +9
4. y=x²
y = 2x + 9
Answer: 2
Step-by-step explanation:
Each of the two sides of the equation is set equal to y
1,615×10 to the 2 power
simplifying 1,615×10 to the 2 power would give 161500
Simplifying the index form
Index notation is known as a way of representing numbers (constants) and variables (e.g. x and y) that have been multiplied by themselves a number of times.
Index notations, or indices are use to simplify expressions or solve equations involving powers.
For instance;
8 × 8 × 8 × 8
8 is multiplied by itself 4 times
In index form , it is written as 8 ^4, that is, 8 to the 4 power
From the information given, we have to simply 1,615×10 to the 2 power
It can be written as;
= 1, 615 × 10 × 10
= 1, 615 × 100
= 161500
Thus, simplifying 1,615×10 to the 2 power would give 161500
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Please please please help
In a lottery, the probability of the jackpot being won in any draw is
a What is the probability that the jackpot prize will be won in each of four consecutive draws? 1/60^4
b How many consecutive draws need to be made for there to be a greater than 98% chance that at least
one jackpot prize will have been won?
The probability that the jackpot prize will be won in each of four consecutive draws is (1/60)⁴.
The number of consecutive draws needed will be, n = 233
What is probability?Probability is the likelihood or chance of an event happening or not.
Probability = number of expected outcomes/number of possible outcomes.From the given question, the probability of the jackpot being won in any draw is 1/60.
The probability that the jackpot prize will be won in each of four consecutive draws will be:
1/60 * 1/60 * 1/60 * 1/60 = (1/60)⁴
b. The number of consecutive draws that needs to be made for there to be a greater than 98% chance that at least one jackpot prize will have been won is calculated as follows:
There is a 100% - 98% chance that that none has been won = 2% that none has been won.
Also, the probability of the jackpot not being won in a draw is = 1 1/60 = 59/60
The number of consecutive draws needed will be (59/60)ⁿ ≤ 0.02
Solving for n by taking logarithms of both sides:
n = 233
In conclusion, probability measures chances of an event occurring or not.
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Pls help me answer this question3 x 2 2/5
Answer:
7 1/5
Step-by-step explanation:
Test the claim that the proportion of people who own cats is larger than 60 t the 0. 10 significance level?
The null hypothesis to test the claim that the proportion of people who owns cats is larger than 60% of the significance level is [tex]H_{0}[/tex]:μ<0.06.
Given that the significance level is 0.10.
We are required to form the null hypothesis to test the claim that the proportion of people who owns cats is larger than 60% the significance level.
Hypothesis is a statement which is tested for its validity. Null hypothesis is the statement which is accepted or not by z test,t test,f test ,chi-square test or any other test.
We have to take opposite of the statement to form a null hypothesis. Since we have to check whether the proportion of people who owns cats is larger than 60% of the significance level, we have to assume that it is smaller than 60% of the significance level.
60% of the significance level=0.60*0.10=0.06.
Null hypothesis is [tex]H_{0}[/tex]:μ<0.06
Hence the null hypothesis to test the claim that the proportion of people who owns cats is larger than 60% of the significance level is [tex]H_{0}[/tex]:μ<0.06.
Question is incomplete.The question should include the following:
Find the null hypothesis for the testing.
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The difference between the record high and low temperaturs in Charlotte, North Carolina, is 109°F. The record low temperature was -5°F. Write and solve an equation to find the record high temperature.
Answer: 104 degrees farenheit
Step-by-step explanation: H = record high temperature. -5 + 109 = H. -5+109 = 109 + (-5) = 109-5 = 104. H = 104.
Answer:
104
Step-by-step explanation:
Let x = record high and y = record low temperature in Charlotte. The difference between the records high and low, x and y, is 109 degrees Fahrenheit, so x - y = 109. Record low is -5, so x - (-5) = 109.
x + 5 = 109
x = 104
Find the area of a circle with a diameter of 16.
Either enter an exact answer in terms T or use 3.14 for TT and enter your answer as a decimal
Answer:
Step-by-step explanation:
Area of circle:
area = π · r · r
Radius= [tex]\frac{16}{2}[/tex]= 8
[tex]3.14\times { 8 }^{ 2 }[/tex] = 200.96 [tex]cm^2\\[/tex]
Find the dicontinuities of the function. f(x) = x2 12x 27 x2 4x 3 . there is a removable discontinuity at ( , ).
The removable discontinuity of the given function is (-3, -3).
What are the discontinuities of the function?
Discontinuous functions are functions that are not a continuous curve - there is a hole or jump in the graph. It is an area where the graph cannot continue without being transported somewhere else.The given function is [tex]f(x) = \frac{x^{2}+ 12x + 27 }{x^{2} + 4x +3}[/tex]
f(x)=(x²+12x+27)/(x²+4x+3)=(x²+9x+3x+27)/(x²+3x+x+3+
=(x+9)(x+3)/(x+3)(x+1)=(x+9)/(x+1)
The holes in the graph by factoring and cancelling are (-3, -3).
Therefore, the removable discontinuity of the given function is (-3, -3).
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Answer: first answer is -3 for both and second is x=-1
Step-by-step explanation:
Find the absolute maximum and minimum values of the function, subject to the given constraints. g(x,y)=9x2 6y2; −1≤x≤1 and −1≤y≤7
For function g(x, y) = 9x² + 6y²,
the absolute minimum is 15 and the absolute maximum is 303
For given question,
We have been given a function g(x, y) = 9x² + 6y² subject to the constraint −1≤x≤1 and −1≤y≤7
We need to find the absolute maximum and minimum values of the function.
First we find the partial derivative of function g(x, y) with respect to x.
⇒ [tex]g_x=18x[/tex]
Now, we find the partial derivative of function g(x, y) with respect to x.
⇒ [tex]g_y=12y[/tex]
To find the critical point:
consider [tex]g_x=0[/tex] and [tex]g_y=0[/tex]
⇒ 18x = 0 and 12y = 0
⇒ x = 0 and y = 0
This means, the critical point of function is (0, 0)
We have been given constraints −1 ≤ x ≤ 1 and −1 ≤ y ≤7
Consider g(-1, -1)
⇒ g(-1, -1) = 9(-1)² + 6(-1)²
⇒ g(-1, -1) = 9 + 6
⇒ g(-1, -1) = 15
And g(1, 7)
⇒ g(1, 7) = 9(1)² + 6(7)²
⇒ g(1, 7) = 9 + 294
⇒ g(1, 7) = 303
Therefore, for function g(x, y) = 9x² + 6y²,
the absolute minimum is 15 and the absolute maximum is 303
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A newspaper started an online version of its paper 14 years ago. In a recent presentation to stockholders, the lead marketing executive states that the revenues for online ads have more than doubled that of the revenues for printed ads since starting the online version of the paper. Use the graph below to justify the lead executive’s statement and to determine the approximate year that the two ad revenues were equal.
It is to be noted that at the seven and half year, is when the revenue of both ads became equal. This is a graph problem. See the explanation below.
What is the explanation for the above answer?Step 1:
Note that the amount of money earned by routine company activities is known as revenue, which is calculated by dividing the average sales price by the number of units sold.
Step 2:
Note that the graph is to be used to justify the statement by the lead executive.
Step 3
From the graph, we know that the revenue in the 10th year for printed ads was $ 2,000,000 and $ 3,000,000 for online ads. Represented on as coordinates, that would be (0,3); (10, 2).
Thus, we can create an equation that states:
(y-2) = [(3-2)/(0-10)] * (x - 10)
⇒ y - 2
= [-1/10] * [x - 10]
Hence,
10y - 20 = - x + 10
10 y + x = 30 ........Lets call this equation A
We can also state that:
Online revenue coordinates on the graph are (0,0,) (10, 3)
Thus,
(y-0) =
[(3-0)/(10-0)] (x -0)
⇒ y = [3x/10] [10y - 3x] = 0.........Lets make this equation B
For printed Ad Revenue:
Year 12= x
10y + 12 = 30
Y = 18/10
y = 1.8
For online Ad revenue
Year = 12 = x
10y = 36
Y = 36/10
y = 3.6
From the above, it is clear that in year 10, the online ad revenue got doubled as same as that of revenue from printed ads.
In order to get the year in which the revenue were equal, we solve both equations simultaneously:
+ 10y + x = 30
± 10y ≠ 3x = 3
4x = 30
x thus, = 30/4
= 7.5
Thus, it is correct to state that both revenue's became equal by the mid of the 7th year going to the eight year.
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Full Question:
Missing graph is attached.
What is the scale factor of the dilation shown ?
Work Shown:
k = scale factor
k = (A'B')/(AB)
k = 8/12
k = (4*2)/(4*3)
k = 2/3
Triangle A'B'C' (image) has side lengths that are 2/3 as long compared to the side lengths of triangle ABC (preimage).
A parabola opens up and passes through (-4, 2) and (6, -3). How do you know that (-4, 2) is not the vertex
Answer:
Step-by-step explanation:
The minimum is at the vertex of this parabola because it opens up.
Now if (-4, 2) is the minimum then all the y values on the parabola must be > 2,
But we are given that y = -3 is on the graph ( the point (6,-3) - that is y < 2 here,
Therefore (-4, 2) cannot be the vertex .
What is the sum of this infinite geometric series?
[tex]\qquad \qquad \textit{sum of an infinite geometric sequence} \\\\ \displaystyle S=\sum\limits_{i=0}^{\infty}\ a_1\cdot r^i\implies S=\cfrac{a_1}{1-r}\quad \begin{cases} a_1=\stackrel{\textit{first term}}{\frac{1}{8}}\\ r=\stackrel{\textit{common ratio}}{\frac{2}{3}}\\ \qquad -1 < r < 1 \end{cases}[/tex]
[tex]\displaystyle\sum_{k=0}^{\infty} ~~ \underset{a_1}{\frac{1}{8}}\underset{r}{\left( \frac{2}{3} \right)}^k\implies S=\cfrac{ ~~ \frac{1}{8} ~~ }{1-\frac{2}{3}}\implies S=\cfrac{ ~~ \frac{1}{8} ~~ }{\frac{1}{3}}\implies S=\cfrac{3}{8}[/tex]
What type of construction is illustrated in the figure?
A
The bisection of ∠D
B
A line segment congruent to segment AB
C
An angle congruent to ∠D
D
The bisection of segment BD
Option A is correct. The type of construction that we have here is the bisection of the <D.
What is the bisection of an angle?The bisection of angle can be defined to be the construction of a ray that would help to divide a particular angle into two equal halves.
In this diagram we can see that the angle here is at D. Hence the construction is aimed at dividing this particular angle into 2. Therefore the answer to the question is The bisection of ∠D.
The bisector does the job of creating an equal measure. The given bisector is known to have the midpoint of the segment. It cuts through this angle and creates two different angles that are of the same size.
If we draw the line in that shape, we will be having the division of that angle. From the explanation here, we can see the answer is the first option
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Complete the equation.
2 x 4 =
X 2
Answer:
4 is correct answer.
Step-by-step explanation:
That because it contains property of communicative. That is a×b=b×a.
the point in the graph of the equation 2x+5y=20, where x coordinate is 5/2, is
Answer: (5/2, 3)
Step-by-step explanation:
Substituting in x=5/2,
[tex]2(5/2)+5y=20\\\\5+5y=20\\\\5y=15\\\\y=3[/tex]
So, the point is (5/2, 3)
you spent $14.95 for a new shirt. you now have $12.48. write and solve an equation to find how much money you had before you bought the shirt
Answer:
x - 14.95 = 12.48
x = 27.43
Step-by-step explanation:
a homeowner has budgeted $10,000 for some home remodeling. a contractor has told him the labor and the cost of materials will be about the same amount. the homeowner wants to have about $3,000 left over for furnishings. how much will the homeowner be able to spend on labor and on materials?
Answer:
$3,500 labor and $3,500 materials
Step-by-step explanation:
furnishings + labor + materials = 10,000
furnishings = 3000
3000 + labor + materials = 10,000
labor = materials
3000 + labor + labor = 10,000
2(labor) = 7,000
labor = 7,000/2
labor = 3,500
labor = materials = 3,500
At the movie theatre, child admission is $6.80 and adult admission is $9.90. On Thursday, twice as many adult tickets as child tickets were sold, for a total sales
of $984.20. How many child tickets were sold that day?
Number of child tickets:0
Answer:
37 child tickets / 74 adult tickets
Step-by-step explanation:
I randomly picked a number and increased or decreased whether the solution was too high or low (guess and check)
The number of child tickets sold that day is 37.
We have,
Let's assume the number of child tickets sold is "C" and the number of adult tickets sold is "A."
The cost of a child ticket: $6.80
The cost of an adult ticket: $9.90
The total sales for the day: $984.20
The number of adult tickets sold is twice the number of child tickets sold:
A = 2C
To find the number of child tickets sold, set up an equation based on the total sales:
6.80C + 9.90A = 984.20
Substituting the value of A from equation 4:
6.80C + 9.90(2C) = 984.20
Simplifying the equation:
6.80C + 19.80C = 984.20
26.60C = 984.20
C = 984.20 / 26.60
C ≈ 37
Therefore,
37 child tickets were sold that day.
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What is the domain of the function y = X+ 6 -7?
x>-7
x>-6
x>6
x>7
The domain of the function y = √(x + 6) - 7 is x > -6
How to determine the domain of the function?The equation of the function is given as
y = √(x + 6) - 7
Set the radical greater than 0
x + 6 > 0
Subtract 6 from both sides of the equation
x > -6
Hence, the domain of the function y = √(x + 6) - 7 is x > -6
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The domain of the function in discuss described as; y = √x+6 -7 is; x >= -6.
What is the domain of the function described as in the task content above?According to the task content, it follows that the domain of.the function can be evaluated by means of the characteristics associated with the square root.
The function given is; y = √x+6 -7
Since, the square root of a negative number renders a complex number as it's results, it follows that; x+6 >= 0.
Hence, x >= -6.
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Select the correct answer from each drop-down menu.
The endpoints of the longest chord on a circle are (4, 5.5) and (4, 10.5).
The center of the circle is at the point and its radius is
units.
The required answers are:
1) The center of the circle = (4, 8)
2) The radius of the circle = 2.5 units
3) The equation of the circle = (x - 4)² + (y - 8)² = 6.25
What is the equation of a circle?The equation of the circle which has a center at (h, k) and a radius of 'r' units is (x - h)² + (y - k)² = r²
To calculate radius 'r', we have r = sqrt( (x1 - h)² + (y1 - k)²)
Where (x1, y1) is the point that lies on the circle.
Calculation:Given that,
The endpoints of the longest chord on a circle are (4, 5.5) and (4, 10.5)
We know that the longest chord on a circle is nothing but the diameter of the circle.
So, the center is the midpoint of the diameter. I.e.,
(h, k) = ([tex]\frac{4+4}{2}[/tex], [tex]\frac{5.5+10.5}{2}[/tex])
⇒ (h, k) = (4, 8)
Therefore, the center of the circle is (4, 8)
Then, the radius is calculated by
r = sqrt( (x1 - h)² + (y1 - k)²)
⇒ r = [tex]\sqrt{(4-4)^2+(5.5-8)^2}[/tex]
⇒ r = 2.5 units
Thus, the radius of the circle is 2.5 units.
So, the equation of the circle with center (4, 8) and radius of 2.5 units is,
(x - h)² + (y - k)² = r²
⇒ (x - 4)² + (y - 8)² = 2.5²
⇒ (x - 4)² + (y - 8)² = 6.25
Thus, the equation of the circle is x - 4)² + (y - 8)² = 6.25.
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In the diagram below, if < ACD = 48 °, find the measure of < ABD.
Answer:
d
Step-by-step explanation:
the opposite angles of a cyclic quadrilateral sum to 180° , that is
∠ ABD + ∠ ACD = 180°
∠ ABD + 48° = 180° ( subtract 48° from both sides )
∠ ABD = 132°
Given the points (–3,k) and (2,0), for which values of k would the distance between the points be 34‾‾‾√ ?
The distance between the points (–3,k) and (2, 0) exists k = ± 3.
How to estimate the distance between points (–3, k) and (2, 0)?
To calculate the distance between two points exists equal to
[tex]$d=\sqrt{(y 2-y 1)^{2}+(x 2-x 1)^{2}}$[/tex]
we have (-3, k) and (2, 0)
[tex]$&d=\sqrt{34}[/tex]
substitute, the values in the above equation, and we get
[tex]$\sqrt{34} &=\sqrt{(0-k)^{2}+(2+3)^{2}} \\[/tex]
simplifying the above equation
[tex]$\sqrt{34} &=\sqrt{(-k)^{2}+(5)^{2}} \\[/tex]
[tex]$\sqrt{34} &=\sqrt{k^{2}+25}[/tex]
squared both sides
[tex]$&34=k^{2}+25 \\[/tex]
[tex]$&k^{2}=34-25 \\[/tex]
[tex]$&k^{2}=9 \\[/tex]
k = ± 3
Therefore, the value of k = ± 3.
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How do you determine the
solution to a system of equations
when graphing? Is it possible to
have more than 1 solution when
graphing? Is it possible to have no
solutions? How?
When graphing, the intersections of the graphs represent the solutions of the system.
How to determine the solutions of a system by graphing?
When graphing a system of equations, you just need to graph both equations in the same coordinate axis.
The solutions of the system are all the points where the graphs of the two equations intersect.
This means that if there is only one intersection, there is one solution.
But we can have more than one intersection, like in the case where at least one of the equations is a polynomial of degree 2 or more.
And there is also the case that the graphs never intersect, like in parallel lines, in these cases we have no solutions.
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ANSWER ASAP!
what is the value of a₁₇ if a₁₃=26 and the recursive form of a geometric sequence is aₙ=1/2aₙ₋₁
please answer with atleast some detail
The 17th term of the geometric sequence given in the problem is:
[tex]a_{17} = \frac{13}{8}[/tex]
What is a geometric sequence?A geometric sequence is a sequence in which the result of the division of consecutive terms is always the same, called common ratio q.
The nth term of a geometric sequence is given by:
[tex]a_n = a_1q^{n-1}[/tex]
In which [tex]a_1[/tex] is the first term.
As a function of the mth term, the nth term can also be given as follows:
[tex]a_n = a_mq^{n - m}[/tex]
In this problem, we have that:
[tex]a_{13} = 26, q = \frac{1}{2}[/tex]
Hence the 17th term is:
[tex]a_{17} = a_{13}q^{4}[/tex]
[tex]a_{17} = 26 \times \frac{1}{16}[/tex]
[tex]a_{17} = \frac{13}{8}[/tex]
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Which function is positive for the entire interval [-3, -2]?
Answer:
A function that is positive in the entire interval [-3, -2] is -x2 - 5x - 5.
Answer:
The second function (second graph and choice)
Step-by-step explanation:
If you look at the second function you will see that within the closed interval [-3,-2] the graph y values are positive
First choice is incorrect since at x = -2 the y value is negative
Third choice incorrect since at x = -2, y value is negative
Fourth choice incorrect since y value is negative for x = -2