To find the centroid of the region bounded by the curves y = sin(x), y = 5cos(x), x = 0, and x = π/3, we need to first find the area of the region, which we can do by integrating the difference between the two curves over the interval [0, π/3]:
A = ∫[0,π/3] (5cos(x) - sin(x)) dx
Using the trigonometric identity cos(x) - sin(x) = sqrt(2)sin(x - π/4), we can rewrite the integrand as:
5cos(x) - sin(x) = 5cos(x) - sqrt(2)sin(x - π/4)sin(π/4)
We can then use the substitution u = x - π/4 to simplify the integral:
A = ∫[-π/4,π/12] (5cos(u + π/4) - sqrt(2)sin(u)sin(π/4)) du
= ∫[-π/4,π/12] (5sin(u) - sqrt(2)sin(u)sin(π/4)) du
= (5 - sqrt(2)sin(π/4)) ∫[-π/4,π/12] sin(u) du
= (5 - sqrt(2)sin(π/4)) [-cos(u)] [-π/4,π/12]
= (5 - sqrt(2)sin(π/4)) (cos(π/12) - cos(-π/4))
= (5 - sqrt(2)sin(π/4)) (sqrt(6)/4 + sqrt(2)/2)
= (5sqrt(6) + 5)/4 - (5sqrt(2) + sqrt(3))/4
Simplifying, we get:
A = (5sqrt(6) - sqrt(3))/2
To find the x-coordinate of the centroid, we need to find the first moment of area about the y-axis and divide by the area:
x¯ = (1/A) ∫[0,π/3] x(y2 - y1) dx
where y2 and y1 are the equations of the upper and lower curves, respectively. We have:
y2 = 5cos(x)
y1 = sin(x)
Substituting these into the formula, we get:
x¯ = (2/(5sqrt(6) - sqrt(3))) ∫[0,π/3] x(5cos(x) - sin(x)) dx
Using integration by parts with u = x and dv = (5cos(x) - sin(x)) dx, we get:
∫ x(5cos(x) - sin(x)) dx = x(5sin(x) + cos(x)) - ∫ (5sin(x) + cos(x)) dx
= x(5sin(x) + cos(x)) - 5cos(x) + sin(x)
Substituting this back into the formula for x¯ and simplifying, we get:
x¯ = (2/(5sqrt(6) - sqrt(3))) [(π/3)(5sin(π/3) + cos(π/3)) - ∫[0,π/3] (5cos(x) - sin(x)) dx]
= (2/(5sqrt(6) - sqrt(3))) [(5sqrt(3)/2 + 1/2)π/3 - (5sqrt(6) - sqrt(3))/2]
= (5sqrt(3) + 5)/(12sqrt(6)
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A taste test asks people from Texas and California which pasta they preferred brand a or brand b this table shows the results. A person is randomly selected for those tested. What is the probability that the person is from Texas given that the person prefers Brand b? Round your answer to two decimal places.
Answer: I just did it and the answer is 0.43
Step-by-step explanation:
Using it's concept, it is found that the probability that the person is from Texas given that the person prefers Brand b is given by:
A. 0.43.
What is a probability?A probability is given by the number of desired outcomes divided by the number of total outcomes.
In this problem, 105 people prefer brand B, and of those, 45 are from Texas, hence:
p = 45/105 = 0.43.
Which means that option A is correct.
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Can someone help me please? I'm struggling :(
Answer:
Ariel's plant is 1.6 inches taller than Dyani's.
It doesn't seem that you can multiply the measurement of Dyani's plant anymore than Ariel's. So I believe that would be 0 inches.
Step-by-step explanation: You can multiply Dyani's measurement with number like 1.0, 1.2, etc. You still wouldn't be able to multiply it anymore.
I hope this helps!
Help please : -5+6/3 + 3+6/-5 =
Step-by-step explanation:
please give me brainlest to me please
Answer:
look at top hahaha
Step-by-step explanation:
answer is on top
If you want to find the energy quantum of light, you multiply the frequency of the radiation (v) by "h". What is "h"?
Answer:
H is E
Step-by-step explanation:
In the formula of the energy of photons "h" signifies Planck's constant.
What is Planck's constant?The Planck constant, often known as Planck's constant, is a crucial physical constant in quantum physics. The mass-energy equivalency establishes the relationship between mass and frequency, and the constant establishes the relationship between a photon's energy and frequency.
In the equation energy E = h X v. The "h" there signifies Planck's constant
Planck's constant is a value, that shows the rate at which the energy of a photon increases/decreases, as the frequency of its electromagnetic wave changes.
Therefore, in the formula of the energy of photons "h" signifies Planck's constant.
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28 students travel on a field trip. They bring a van that can seat 12 students. Elena and Kiran's teacher asks parents to drive cars that seat 3 children each to transport the rest of the students.
Elena wonders if she should use the inequality
12+3>28
12
+
3
n
>
28
or
12+3≥28
12
+
3
n
≥
28
to figure out how many cars are needed. Kiran doesn't think it matters in this case. Solve the inequality to explain why Kiran is correct. (use / to represent fractional quantities and a space between whole numbers and the fraction. Example
112
1
1
2
type as 1 1/2)
n
Answer: Yes, you can use this inequality to find the numbers of cars required.
Step-by-step explanation:
12 + 3n > 28 where n = the number of cars required
3n > 28 -12
3n > 16
n > 5 1/3
Greater than 5 1/3 gives 6 cars.
Answer:
See below.
Step-by-step explanation:
12 + 3n ≥ 28
3n ≥ 16
n ≥ 5 1/3
The number of cars must be equal to or greater than 5 1/3.
Since there is no 1/3 of car, that means there must be 6 cars.
Whether the inequality is 12 + 3n > 28 with a solution of more than 5 1/5 or the inequality is 12 + 3n ≥ 28 with a solution of 5 1/3 or more, the answer is the same because the minimum number of cars that will be enough is 6.
–8, –4, 0, 4, 8, 12 What do these terms represent? an arithmetic series an arithmetic sequence a geometric series a geometric sequence
Answer:
Arithmetic sequence
Step-by-step explanation:
An list of numbers whereby the difference between each consecutive number is constant is called an arithmetic sequence. Given thelistvif numbers:
–8, –4, 0, 4, 8, 12 ;
We can see that each consecutive number in the list has a constant difference of 4. Hence x this is called an arithmetic sequence.
QUESTION 5 Find the least square polynomial approximation of degree two to the data. 1 -1 Let y = a + bx + cx². Find the following. a = b= c= X y least error = 0 -4 QUESTION 6 Solve the IVP using Taylor series( 3rd deg polynomial). dy/dx = 3x2y; y(1)=1 y'(1) = y"(1)= y"(1)= y(1.4)= True value at x=1.4 (2 decimal places) (2 decimal places) 2 4 3 11 4 20 25 points Save Answer 25 points Save Answer
We obtain the values a = -4, b = 5, and c = 1. These coefficients represent the best-fit quadratic function, which is y = -4 + 5x + x².
The least squares method is used to find the best-fitting polynomial curve to a given set of data points. In this case, we are trying to find a second-degree polynomial (a quadratic function) that approximates the data points (0, -4), (2, 4), and (3, 11). By minimizing the sum of the squared errors between the polynomial and the data points, we can determine the coefficients of the quadratic function.
To solve for the coefficients, we substitute the x-values of the data points into the polynomial equation and equate it to the corresponding y-values. This results in a system of equations that can be solved to find the values of a, b, and c.
After solving the system, we obtain the values a = -4, b = 5, and c = 1. These coefficients represent the best-fit quadratic function, which is y = -4 + 5x + x². This polynomial provides the least square approximation to the given data, minimizing the overall error between the data points and the curve.
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Find the point of diminishing retums (xy) for the function R(x), where R(x) represents revenue (in thousands of dollars) and x represents the amount spent on advertising (in thousands of dollars)
f(x)=11,000−x^3+36x^2+700x,05x≤20
The point of diminishing returns for the revenue function R(x) occurs when the amount spent on advertising is approximately $16.9 thousand.
To find the point of diminishing returns for the revenue function R(x) = 11,000 - x^3 + 36x^2 + 700x, we need to determine the value of x at which the marginal revenue, which is the derivative of R(x), equals zero. Let's find the derivative first.
R'(x) = d/dx (11,000 - x^3 + 36x^2 + 700x)
= -3x^2 + 72x + 700
Setting R'(x) equal to zero and solving for x, we get:
-3x^2 + 72x + 700 = 0
This is a quadratic equation, which can be solved using the quadratic formula. Applying the quadratic formula, we find two solutions: x ≈ -9.15 and x ≈ 26.15.
However, we are given the constraint 0 ≤ x ≤ 20, so the value of x cannot exceed 20. Therefore, we disregard the solution x ≈ 26.15.
Thus, the point of diminishing returns occurs when x is approximately 16.9 (rounded to one decimal place) thousand dollars. At this advertising expenditure, the rate of increase in revenue slows down, indicating diminishing returns.
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how do you turn fractions into decimals?
Answer:
You would need to divide the numerator by the denominator in order to convert a fraction into a decimal
Step-by-step explanation:
Answer:
You divide the numerator by the denominator.
Step-by-step explanation:
For example, you covert the fraction \(\frac{1}{2}\) into a decimal.
You just do: 1÷2=0.5, and there you have it, the decimal!
Have a nice day!
Select the correct answer.
3(* + 4) - 2 (1 - 1);
Which is the simplified form of the expression
OA.
39
-
- 1
11
2
67
B.
101 + 9
oc. Ir + 1
15 + 10
OD
1
Answer:
12Step-by-step explanation:
\(3(* + 4) - 2 (1 - 1);\\\\3(\times +4)-2(1-1)\\\\3\left(4\right)-2\left(1-1\right)\\\\\mathrm{Remove\:parentheses}:\quad \left(a\right)=a\\=3\times \:4-2\left(1-1\right)\\\\3\times \:4=12\\2\left(1-1\right)=0\\\\=12-0\\\\=12\)
The diameter of a circle is 13 cm. Find its area to the nearest tenth.
Area=\(\pi\) \(r^{2}\)
divide diameter with 2 to get radius 13/2= 6.5
area= pi 6.5x6.5
=\(\pi\)42.25
answer= 42.25\(\pi\)
Area of a circle is equal to \(\boldsymbol{132.8}\) square inches.
Area of a circleA circle is a collection of all the points that are at the fixed distance (radius) from a fixed point (center).
Diameter of a circle \(=\boldsymbol{13}\) centimeter
Radius of a circle is half of the diameter.
So,
Radius of a circle \(=\frac{13}{2}\) centimeter
Area of a circle \(=\boldsymbol{\pi r^2}\) where \(r\) denotes radius of a circle.
\(=\frac{22}{7}\times \left ( \frac{13}{2} \right )^{2}\\=132.7857\)
\(\boldsymbol{\approx 132.8}\) square inches
So, area of a circle is equal to \(\boldsymbol{132.8}\) square inches.
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How can parallel lines be used to compare figures E and F? Explain your reasoning.
Answer:
The similarities are
1) The length of E and F can be described by two parallel lines spaced 3 blocks apart where the parallel line on the left of both E and F is 2 blocks long, while the parallel line on the right is one block long
2) Both height of E and F can be described by two parallel lines spaced 2 blocks apart where the parallel line on the top of both E and F is 3 blocks long, while the parallel line at the bottom is one block long
3) The interior of E an F can be described by parallel lines 2 blocks long in the horizontal portion and one block long in the vertical portion
The differences are
1) The difference between E and F is that the parallel lines used for the construction of E are all one block higher than that of F
2) The other difference is that all the lines used for the construction of the figure E are 3 blocks to the left of those used for the construction of F
Figure F is essentially the translation of figure E three blocks to right and one block down
Step-by-step explanation:
course 3 chapter 5 triangles and the pythagorean theorem answer key
In the given right triangle, the perimeter is 12 in.
According to the Pythagorean theorem, in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the length of the two legs (a and b). Hence,
c^2 = a^2 + b^2
In the given triangle,
c = x in
a = 3 in
b = 4 in
Hence,
x^2 = 3^2 + 4^2
x^2 = 9 + 16
x^2 = 25
x = √25
x = 5
The perimeter of a geometric shape is the sum of all its side. Hence,
Perimeter = 3 + 4 + 5 = 12 in
Note: The question is incomplete. The complete question probability is: Using the Pythagorean theorem, find the perimeter of the triangle.
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The graph of the function f(x) = –(x + 3)(x – 1) is shown below.
On a coordinate plane, a parabola opens down. It goes through (negative 3, 0), has a vertex at (negative 1, 4), and goes through (1, 0).
Which statement about the function is true?
The function is positive for all real values of x where
x < –1.
The function is negative for all real values of x where
x < –3 and where x > 1.
The function is positive for all real values of x where
x > 0.
The function is negative for all real values of x where
x < –3 or x > –1.
The function is negative for all real values of x where x < –3 and where
x > 1, is the statement about the function is true.
Here, we have,
given that,
On a coordinate plane, a parabola opens down.
It goes through (negative 3, 0), has a vertex at (negative 1, 4), and goes through (1, 0).
It opens downward and crosses the x axis at (-3,0) and (1,0) this means for any x value less than -3 or greater than 1, the function is negative.
The answer would be:
The function is negative for all real values of x where
x < –3 and where x > 1.
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PLEASE HELP FAST!! HIGH POINTS!
What is the vertex form of the equation?
y = x^2 + 4x - 3
Oy = (x - 2)^2 - 7
Oy = (x + 2)^2-7
Oy = (x - 2)^2 + 7
Oy= (x + 2)^2 + 7
Answer:
he easiest way is: y=ax2+bx+c. axis on symmetry is: aos=−b2a. Vertex is: (aos,f(aos)). c = y-intercept. so your function: y=x2−4x.
Step-by-step explanation:
Find the value of x that makes m
n.
m
n
(180 - x)^
Answer:
x=180-x
2x=180
x=90
Step-by-step explanation:
On january 14, edamame industries purchased supplies of $700 on account. the entry to record the purchase will include?
The entry to record the purchase will include: a debit to Supplies and a credit to Accounts Payable
What are supplies?Supplies are the small items, used in the company's overall operation. Supplies used during the period will be reported as supplies expense and the unused supplies will be reported in the balance sheet as an asset.Now, given: Purchased supplies = $500 (on account).
To find: What does entry to record the purchase include?
Now,
The supply asset will grow, which will cause a debit to be applied to it. It is believed to have been purchased on account, which increased (credited) the account for payments due.Hence, the entry to record the purchase will include: a debit to Supplies and a credit to Accounts Payable.
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Prove the following statement using the given pieces of information.
Given: Line AB is parellel to Line DC and Line AD is parellel to Line BC.
Prove: Triangle ABC is congrent to Triangle CDA.
Triangle ABC is congruent to Triangle CDA using the SSS rule.
What is a triangle?It is a two-dimensional figure which has three sides and the sum of the three angles is equal to 180 degrees.
We have,
SSS rule states that,
If the three sides of one triangle are equal to the three sides of another triangle, then the triangles are congruent.
To prove:
ΔABC ≅ ΔCDA.
AB = DC ( given)
AD = BC (given)
AC = AC (common to both the triangles)
ΔABC ≅ ΔCDA ( By SSS rule )
Thus,
ΔABC ≅ ΔCDA by SSS rule.
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Help center of circle
Answer:
Center = (7, - 5)
Radius = 3
Step-by-step explanation:
\( {x}^{2} + {y}^{2} + 14x - 10y + 65 = 0 \\ {x}^{2} + 14x + {y}^{2} - 10 + 65 = 0 \\ ( {x}^{2} + 14x + {7}^{2}) - {7}^{2} + ({y}^{2} - 10y + {5}^{2}) \\- {5}^{2} + 65 = 0 \\ (x + 7)^{2} + (y - 5)^{2} - 49 - 25 + 65 = 0 \\ (x + 7)^{2} + (y - 5)^{2} - 49 + 40= 0 \\ (x + 7)^{2} + (y - 5)^{2} - 9= 0 \\ (x + 7)^{2} + (y - 5)^{2} = 9 \\ (x + 7)^{2} + (y - 5)^{2} = {3}^{2} \\ equating \: it \: with \: \\ (x - h)^{2} + (y - k)^{2} = {r}^{2} \\ - h = 7 \implies \: h = - 7 \\ - k = - 5 \implies \: k = 5 \\ center = ( - h, \: - k) = (7, \: \: - 5) \\ \\ radius \: r = 3\)
Use the graph to find the values below.
From the given graph, the numeric values of the expressions are given as follows:
f(-1) = 12.f(1) - g(1) = 8.g(x) = 4 when x = 2.f(x) = g(x) when x = 3.How to find the numeric value of a function or of an expression?To find the numeric value of a function, we replace each instance of the variable in the function by the desired value.
On the graph of the function, the equivalent to this is that we have to find where the x-axis has the value, and find the value of y for which the function passes through the y-axis.
From the graph, when x = -1, f(x) passes the point at y = 12, hence f(-1) = 12.
For the second item, applying the same logic, we have that:
f(1) = 10.g(1) = 2.Hence:
f(1) - g(1) = 10 - 2 = 8.
For the third item, we have to look at the graph of g(x), and find the value of x when y = 4, hence g(x) = 4 when x = 2.
For the fourth item, the functions intersect at point (3,8), hence f(x) = g(x) when x = 3.
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Look at the attachment for the question please hurry!!!
Answer:
Neither
Step-by-step explanation:
The two lines going through those points to not intersect at a 90-degree angle.
What are the 2 words typically associated with geometry
Step-by-step explanation:
In geometry, we have many formal definitions using defined words or terms. However, I can think of three words that are not formally defined. These words are point, line and plane. Even though these words are undefined we can describes them as follows:
1. Point. A point is an exact position or location in space.
It is important to understand that a point has no dimension (that is, actual size). So, it has no length, no width, and no height (thickness).
We usually named a point with a capital letter. In the coordinate plane, a point is named by an ordered pair, x,y .
Even though we represent a point with a dot, the dot can be very tiny or very large. Recall that a point has no size.
2. Line (Straight line)
A line has no thickness.
So, its length extends in one dimension.
The line extends in both directions without end (infinitely). A line has infinite length and has no width, no height.
We assume the line to be straight.
and can be drawn with arrowheads on both ends.
3. Plane
A plane is a flat surface with no thickness extending indefinitely in all directions and having two dimensions. So, it has infinite length, infinite width and has no height (thickness). Moreover, a plane is drawn as a four-sided figure resembling a parallelogram.
The representation of each concept is shown in the Figure below.
The ability to determine the age of some individuals can be difficult if there are not quality government records of birth.
Bone growth takes place at the growth plates at the end of long bones. Once all growth plates fuse, growth stops, and
an individual is considered a biological adult. The age at which growth plates fuse for males is approximately normally
distributed with a mean of 19.1 years and a standard deviation of 16.1 months. Complete parts (a) through (d).
The probability a male's growth plate will fuse before age 17 is 0.1112, for this we have to learn probability.
What is Probability?Probability of an event is a number that shows what is the possibility of an event occur.
we have to take entire area under the standardized curve from and including the left tail up to the z-score for 17.
\(= \frac{(17 - 18.6 years)}{1.308333 years}\)
\(= \frac{(-1.6) }{1.308333}\)
\(= -1.22293\)
\(= approx. 1.22.\)
The area from the mean to +ve or -ve 1.22 is .3888
We have to subtract .3888 from area from the tail to the mean (.5000) to get area from the tail to 1.22.
So, the growth plates will fuse before age 17 \(= .5000-.3888 = 0.1112\)
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Which values are solutions to the inequality –3x – 4 < 2? Check all of the boxes that apply. -4 -2 0 3
Answer:
0 and 3
Step-by-step explanation:
all you have to do is plug the numbers for x
so -3 x 0=0
-3 x 3=-9 then minus 4 equals -13
the others are bigger than 2
Answer:
0 and 3
Step-by-step explanation:
The height and weight of adults can be related by the equation y=48.3x−127 where x is height in feet and y is weight in pounds.
What does the slope of the line represent?
A. the average number of pounds per foot tall
B. the number of pounds heavier an adult one foot taller would weigh
C. the height of an adult weighing zero pounds
D. the number of adults in the sample
we conclude that the correct option is A, the slope is:
"the average number of pounds per foot tall"
What does the slope of the line represent?The linear equation:
y = 48.3*x - 127
Relates the height, x, and the weight, y of an adult.
Then the slope, which is 48.3, should have units of mass over height.
Now, if x is in feet and y is in pounds, then the slope has units of pounds/feet.
Then it represents the mean number of pounds per feet. So we conclude that the correct option is A:
"the average number of pounds per foot tall"
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Sabina bought a samsung mobile set for rs.15750 after getting the discount of 10%. Find the price of mobile set before discount.
Answer:
Step-by-step explanation:
90/100 * x = 15750
0.9 x = 15750
x = 15750/0.9
x = 17500
Now check this by taking 10% off. Do you get 15750?
Let's see
10/100 of 17500 = 1750
Subtract that from 17500
17500 - 1750 = 15750 which is exactly what you should get.
A bag of 20 counters 16 of the counters are green They are picked at random How likely is it for green to be picked
Answer:
4/5
Step-by-step explanation:
No: green = 16
Total counters = 20
Probabilty = 16/20 = 4/5
I hope im right!!
billy stacked 1 inch by 1 inch plastic blocks on his desk. he created a small wall. what is the volume of the wall of which he created?
Answer:
5
Step-by-step explanation:
conducting all dl dm dm dl k dl lwhat
5. Which triple integral in cylindrical coordinates gives the volume of the solid bounded below by the paraboloid z = x2 + y2 - 1 and above by the sphere x2 + y2+z2 = 7?
(a)
[
√3 √7-r2
r dz dr de
0
√3 Jr2-1
√2
√7-r2
(b)
(c)
(d)
(e)
0
-2π
2π √3
[ √
0
r dz dr de
-√2 Jr2-1
2π
√3 r2-1
r dz dr do
r dz dr dᎾ
r2-1
√7-2
r dz dr de
2-1
The correct triple integral in cylindrical coordinates that gives the volume of the solid bounded below by the paraboloid z = \(x^2 + y^2 - 1\)and above by the sphere \(x^2 + y^2 + z^2\)= 7 is (d) ∫∫∫ (r dz dr dθ).
Here are the limits of integration for each variable:
r: 0 to √(7 - \(z^2\))
θ: 0 to 2π
z: \(r^2\) - 1 to √3
The volume integral can be written as:
∫∫∫ (r dz dr dθ) from z = \(r^2\) - 1 to √3, θ = 0 to 2π, and r = 0 to √(7 - \(z^2\))
The limits of integration for r are determined by the equation of the sphere \(x^2 + y^2 + z^2\) = 7. Since we are in cylindrical coordinates, we have \(x^2 + y^2 = r^2\). Therefore, the expression inside the square root is 7 - \(z^2\),
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find the yy -component of vector a⃗ a→ = (5.0 m/s2m/s2 , −y−y -direction).
The yy-component of vector a a is -y-direction, which is equal to 0 m/s2.the product of the second term of the vector with unit vector j^j^, which is in the direction of the y-axis.
Given a vector, a⃗ a→ = (5.0 m/s2, −y−direction)Find the yy -component of the given vector a⃗ a→.Solution: The yy-component of the given vector a⃗ a→ = -y-directionTo find the yy-component of the given vector a⃗ a→, we have to take the product of the second term of the vector with unit vector j^j^, which is in the direction of the y-axis.Therefore, the yy-component of vector a⃗ a→ is :a_yy = -y-direction = (-1)(0) = 0 m/s² Answer: 0 m/s².
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The yy-component of vector a⃗ a→ is -y-direction.
Explanation:The yy-component of a vector represents the magnitude of the vector in the y-direction.
In this case, the vector a⃗ a→ is given as (5.0 m/s2, −y-direction).
Since the yy-component is the magnitude in the y-direction, the yy-component of the vector a⃗ a→ is -y-direction.
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