PLEASE HELP ASAP: A particle is moving with velocity v(t) = t2 – 9t + 18 with distance, s measured in meters, left or right of zero, and t measured in seconds, with t between 0 and 8 seconds inclusive. The position at time t = 0 sec is 1 meter right of zero, that is, s(0) = 1.
The average velocity over the interval 0 to 8 seconds
The instantaneous velocity and speed at time 5 secs
The time interval(s) when the particle is moving right
The time interval(s) when the particle is
going faster
slowing down
Find the total distance the particle has traveled between 0 and 8 seconds
Answers
Answer:
1) Average velocity = 10/3 m/s
2) Instantaneous velocity = -2 m/s
Speed = 2 m/s to the left
3) (0, 3) ∪ (6, 8]
4) Going faster: (3, 4.5) ∪ (6, 8]
Slowing down: (0, 3) ∪ (4.5, 6)
5) Total distance = 35.67 m (nearest hundredth)
Step-by-step explanation:
The relationships between position (displacement), velocity and acceleration are:
[tex]\boxed{\boxed{\begin{array}{c}\textbf{POSITION (s)}\\\\\text{Differentiate} \downarrow\qquad\uparrow\text{Integrate}\\\\\textbf{VELOCITY (v)}\\\\\text{Differentiate}\downarrow\qquad\uparrow \text{Integrate}\\\\\textbf{ACCELERATION (a)}\end{array}}}[/tex]
Given a particle is moving with velocity v(t) = t² - 9t + 18, to find its position s(t) we can integrate v(t):
[tex]\begin{aligned}\displaystyle s(t)=\int v(t)\;\text{d}t&=\int(t^2-9t+18)\;\text{d}t\\\\&=\dfrac{t^{2+1}}{2+1}-\dfrac{9t^{1+1}}{1+1}+18t+C\\\\&=\dfrac{t^{3}}{3}-\dfrac{9t^{2}}{2}+18t+C\end{aligned}[/tex]
As s(0) = 1, then:
[tex]\begin{aligned}s(0)=\dfrac{(0)^{3}}{3}-\dfrac{9(0)^{2}}{2}+18(0)+C&=1\\0-0+0+C&=1\\C&=1\end{aligned}[/tex]
Therefore, the position function s(t) is:
[tex]\large\boxed{s(t)=\dfrac{t^3}{3}-\dfrac{9t^2}{2}+18t+1}[/tex]
Given a particle is moving with velocity v(t) = t² - 9t + 18, to find its acceleration a(t) we can differentiate v(t):
[tex]\begin{aligned}a(t)=\dfrac{\text{d}}{\text{d}t}[v(t)]&=2\cdot t^{2-1}-1\cdot9t^{1-1}+0\\&=2t-9\end{aligned}[/tex]
Therefore, the acceleration function a(t) is:
[tex]\large\boxed{a(t)=2t-9}[/tex]
[tex]\hrulefill[/tex]
Question 1To find the average velocity over the interval [0, 8], use the formula:
[tex]\textsf{Average Velocity}=\dfrac{s(t_2)-s(t_1)}{t_2-t_1}[/tex]
In this case:
t₁ = 0t₂ = 8Calculate the position at t₁ and t₂ by substituting t = 0 and t = 8 into s(t):
[tex]s(0)=\dfrac{(0)^3}{3}-\dfrac{9(0)^2}{2}+18(0)+1}=1[/tex]
[tex]s(8)=\dfrac{(8)^3}{3}-\dfrac{9(8)^2}{2}+18(8)+1}=\dfrac{83}{3}[/tex]
Therefore:
[tex]\textsf{Average Velocity}=\dfrac{s(8)-s(0)}{8-0}=\dfrac{\frac{83}{3}-1}{8}=\dfrac{10}{3}\; \sf m/s[/tex]
Therefore, the average velocity is 10/3 m/s.
[tex]\hrulefill[/tex]
Question 2To find the instantaneous velocity at t = 5 seconds, substitute t = 5 into v(t):
[tex]\begin{aligned}v(5)&=(5)^2-9(5)+18\\&=25-45+18\\&=-2\end{aligned}[/tex]
So, the instantaneous velocity at t = 5 seconds is -2 m/s.
Speed is a scalar quantity that measures how fast an object is moving regardless of its direction. Therefore, speed is the magnitude of velocity:
[tex]\textsf{Speed}=|v(5)|=|-2|=2\;\sf m/s[/tex]
Therefore, the speed at t = 5 is 2 m/s to the left.
[tex]\hrulefill[/tex]
Question 2The particle changes direction when v(t) = 0.
[tex]\begin{aligned}v(t)&=0\\\implies t^2-9t+18&=0\\t^2-6t-3t+18&=0\\t(t-6)-3(t-6)&=0\\(t-3)(t-6)&=0\\\\t-3&=0\implies t=3\\t-6&=0\implies t=6\end{aligned}[/tex]
Therefore, the particle changes direction at t = 3 and t = 6.
We know that the position of the particle at t = 0 is 1 meter right of zero. Therefore:
It is moving to the right in the interval (0, 3).It is moving to the left in the interval (3, 6).It is moving to the right in the interval (6, 8].Therefore, the time intervals between 0 ≤ t ≤ 8 when the particle is moving right is:
(0, 3) ∪ (6, 8][tex]\hrulefill[/tex]
Question 4When a(t) > 0:
[tex]\begin{aligned}a(t)& > 0\\2t-9& > 0\\2t& > 9\\t& > \dfrac{9}{2}\\t& > 4.5\; \sf s\end{aligned}[/tex]
When a(t) < 0:
[tex]\begin{aligned}a(t)& < 0\\2t-9& < 0\\2t& < 9\\t& < \dfrac{9}{2}\\t& < 4.5\; \sf s\end{aligned}[/tex]
Therefore:
Velocity is positive in the interval (0, 3) and (6, 8].Velocity is negative in the interval (3, 6).Acceleration is positive in the interval (4.5, 8].Acceleration is negative in the interval (0, 4.5).(Refer to the attachment).
If velocity and acceleration have the same sign, it means the object is speeding up.
If velocity and acceleration have opposite signs, it means the object is slowing down.
Therefore, the time intervals when the particle is going faster and slowing down are:
Going faster: (3, 4.5) ∪ (6, 8]Slowing down: (0, 3) ∪ (4.5, 6)[tex]\hrulefill[/tex]
Question 5To find the total distance the particle has traveled between 0 and 8 seconds, we need to consider the distance traveled between the intervals when it changes direction.
To do this, find the position of the particle at t = 0, t = 3, t = 6 and t = 8.
[tex]s(0)=\dfrac{(0)^3}{3}-\dfrac{9(0)^2}{2}+18(0)+1=1[/tex]
[tex]s(3)=\dfrac{(3)^3}{3}-\dfrac{9(3)^2}{2}+18(3)+1=23.5[/tex]
[tex]s(6)=\dfrac{(6)^3}{3}-\dfrac{9(6)^2}{2}+18(6)+1=19[/tex]
[tex]s(8)=\dfrac{(8)^3}{3}-\dfrac{9(8)^2}{2}+18(8)+1=\dfrac{83}{3}\approx27.67[/tex]
Therefore, in the interval 0 ≤ t < 3, the particle travels:
[tex]|s(3)-s(0)|=|23.5-1|=22.5\; \sf meters\;(to\;the\;right)[/tex]
In the interval 3 < t < 6, it travels:
[tex]|s(6)-s(3)|=|19-23.5|=4.5\; \sf meters\;(to\;the\;left)[/tex]
In the interval 6 < t ≤ 8, it travels:
[tex]|s(8)-s(6)|=|27.67-19|=8.67\; \sf meters\;(to\;the\;right)[/tex]
So the total distance the particle has traveled between 0 and 8 seconds is:
[tex]\textsf{Total distance}=22.5+4.5+8.67=35.67\; \sf meters[/tex]
Related Questions
Evaluate 3/2y - 3 + 5/3z when y = 6 and z = 3
Answers
3/2(6) -3 + 5/3(3)
18/2 - 3 + 15/3
Divide.
9 - 3 + 5
Subtract 9 - 3
6 + 5 = 11
Hope this helps!
To solve the expression, you should substitute the provided values for y and z then simplify the resulting expression. The solution is 11.
Explanation:To evaluate the expression 3/2y - 3 + 5/3z when y = 6 and z = 3, first, we should substitute the values of y and z into the expression.
So, the expression becomes 3/2*6 - 3 + 5/3*3. Multiply to reduce fractions: 9 - 3 + 5. This simplifies to 11.
So the value of the given expression when y = 6 and z = 3 is 11.
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A car averages 27 miles per gallon. If gas costs $4.04 per gallon, which of the following is closest to how much the gas would cost for this car to travel 2,727 typical miles? Individual Question $44.44 $109.08 $118.80 $408.04 $444.40
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hope it helped!
hope it helped!
A single fair die is rolled twice. determine the number of points in the sample space
Answers
There are a total of 36 outcomes if you rolled a fair single die twice.
The sample space for this event would be: (check in the attached file)
You are returning from a trip to England with €100 euro. The exchange rate is USD/EUR 1.3847 . How much in US dollars should you receive
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Using the exchange rate of USD/EUR 1.3847, you need to multiply your €100 by 1.3847 to determine the amount in US dollars, which is USD 138.47. Therefore, you will receive USD 138.47 for your €100.
You are returning from a trip to England with €100 euro, and the exchange rate is USD/EUR 1.3847. To calculate how much in US dollars you should receive, you use the exchange rate to convert euros to USD.
Start with the amount in euros: €100.Multiply by the exchange rate: 100 × 1.3847 = 138.47.Therefore, you should receive USD 138.47.
Mike is 11 years old. jim is twice as old as mike. when jim is 50 years old, how old will mike be? 28
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The average annual rainfall in California is 0.564 of a meter per year. What is the value of the digit 4 in that number
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Points J, K, L, and M are midpoints of the sides of the rectangle EFGH. Prove that quadrilateral JKLM is a rhombus by finding the lengths of the sides. The diagram is not drawn to scale.
Answers
Quadrilateral JKLM is a rhombus because its sides are all equal in length. By using the midpoints of the rectangle EFGH, we find that JK and LM are both 2a in length, and KL and MJ are both 2b in length, confirming that all four sides are equal.
To prove that quadrilateral JKLM is a rhombus, we find the lengths of its sides.
Since J, K, L, and M are midpoints of the sides of rectangle EFGH, the segments connecting them will be parallel to the sides of EFGH and half the length.
We can use the coordinates of F(-a,b), G(a,b), H(a,-b), and E(-a,-b) to find the lengths:
JK is parallel to EH and GH, so its length is half of EH and GH, and since E and G are on the same vertical line, the length of JK is a - (-a) = 2a.
KL is parallel to EG and FH, so its length is also half of EG and FH, and since E and F are on the same horizontal line, the length of KL is b - (-b) = 2b.
LM is parallel to EH and GH, hence its length is the same as JK which is 2a.
MJ is parallel to EG and FH, hence its length is the same as KL which is 2b.
Given that JK = LM and KL = MJ, and all are equal to 2a or 2b, quadrilateral JKLM has all sides of equal length. Therefore, it is a rhombus.
The probable question may be:
Points J, K, L, and M are midpoints of the sides of the rectangle EFGH. Prove that quadrilateral JKLM is a rhombus by finding the lengths of the sides. The diagram is not drawn to scale.
F(-a,b)
G(a,b)
H(a,-b)
E(-a,-b)
J(-a,0)
K(0,b)
L(a,0)
M(0,-b)
The radius of a large sphere is 8 times the radius of a small sphere. the surface area of the large sphere is how many times the surface area of the small sphere?
Answers
small sphere's radius = r, R = 8r
surface area of a sphere (SA) = 4×pi×radius^2
So what we need is the SA of the larger in terms of the smaller sphere, so if:
SA (of R) = 4×pi×R^2, then plug in "8r" for "R"...
SA = 4×pi×(8r)^2 = 4×3.14×64r^2
SA = 12.57×64r^2 = 804 r^2
Therefore the SA of the larger sphere is 804 times the SA of the smaller sphere.
I hope that makes sense!
The large sphere is 64 times the surface area of the small sphere.
What is a sphere?A sphere is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance r from a given point in three-dimensional space. That given point is the centre of the sphere, and r is the sphere's radius.
Given that, the radius of a large sphere is 8 times the radius of a small sphere.
Let the radius of a small sphere be x, then the radius of a large sphere is 8x.
We know that, the surface area of a sphere is 4πr².
Here, surface area of a small sphere = 4πx²
The surface area of a large sphere = 4π(8x)²
= 256πx²
Now, 256πx²/4πx² =64
Therefore, the large sphere is 64 times the surface area of the small sphere.
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Answers
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:)
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Answer:
D) 23.6 cm
Step-by-step explanation:
We first find the circumference of the circle. Circumference is given by the formula
C = 2πr, where π = 3.14 and r is the radius. The radius of our circle is 15 cm; this gives us
C = 2(3.14)(15) = 94.2.
Next we find the portion of the circumference associated with the given arc. The measure (in degrees) of the arc is 90°; this makes it 90/360 of the circle. 90/360 simplifies to 1/4; this means this arc is 1/4 of the circumference:
1/4(94.2) = 23.55
This rounds to 23.6.
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Answers
The measure of ∠DCE is 100° so option (B) will be correct.
What is an angle?An angle is a geometry in plane geometry that is created by 2 rays or lines that have an identical terminus.
The identical endpoint of the two rays—known as the vertex—is referenced as an angle's sides.
In another word, the angle is the measurement of the angular distance for example for linear motion we have a meter inch but for angular rotation, we don't have the measurement so the angle is useful to measure the angular rotation.
Given the angle,
∠DCG = ∠DCE + 20 + 25
145° = ∠DCE + 20 + 25
∠DCE = 100°
Hence "The measure of ∠DCE is 100°
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What is the value of x in the equation 1/5x – 2/3y = 30, when y = 15? 4 8 80 200
Answers
Then (1/5)x - 10 = 30, and (1/5)x = 40, and so x = 200 (answer)
1/5x - 2/3(15) = 30
1/5x - 30/3 = 30
1/5x - 10 = 30
1/5x = 30 + 10
1/5x = 40
x = 40 * 5
x = 200
hope this helped, God bless!
If it takes 5 machines 5 minutes to make 5 widgets, how long would it take 100 machines to make 100 widgets?
Answers
It takes 1 minute to make each widget.
WORTH 40 POINTS!
What is the value of the expression All of 2.6 multiplied by 10 to the power 9 over all of 1.3 multiplied by 10 to the power 2?
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2.0 × 1011
Which shows the expressions in the order they would appear on a number line from least to greatest2 to the power of 3, square root of 5, square root of 20, square root of 11, 11 over 9
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11 over 9, 2 to the power of 3, square root of 5, square root of 20, square root of 11
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1 over 9
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Answers
Answer:
4 to the power of 3, square root of 40, square root of 13, square root of 7, 14 over 9
Step-by-step explanation:
i took the test
it's the other way around.
Answer:
A
Step-by-step explanation:
Kathryn goes out lunch with Mia and Fran. each girl orders the $7 lunch speical. Kathryn agrees to pay the bill. How much will she have to psu?
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Planes Q and R are parallel. Lines a and b are shown on planes Q and R, respectively.
Which statement is true about lines a and b?
They are parallel lines.They are perpendicular lines.They are skew lines.They will intersect.Which inequality best represents that ice cream at −5°C is cooler than ice cream at 4°C?
−4°C > 5°C
4°C < −5°C
−5°C < 4°C
−5°C > 4°C
Answers
Answer:
IS D
Step-by-step explanation:
A stadium has 46,000 seats. Seats sell for $30 in Section A, $24 in Section B, and $18 in Section C. The number of seats in Section A equals the total number of seats in Sections B and C. Suppose the stadium takes in $1,193,400 from each sold-out event. How many seats does each section hold?
Answers
A=B=C
Total seat= A+B+C= A+A+A=3A
The total revenue for the stadium is $1,193,400. Then the total number of the seat would be:
revenue= total seats*price per seat
$1,193,400= A * $30 + B*24 + C*18
$1,193,400= A * $30 + A*24 + A*18
$1,193,400= A * $72
A= $1,193,400/ 72
A= 16575
Each section hold 16575 seats
The length of a rectangle is four times its width. if the area of the rectangle is 400 yd2 , find its perimeter.
Answers
L = 4W
A = LW
400 = (4W)(W)
400 = 4w^2
4W^2 = 400
W^2 = 100
W = 10
The width is 10 yd, and the length is 40 yd.
What is the missing term in the factorization?
18x2−32=2(3x+?)(3x−4)
Answers
2(9x^2 - 16) =
2(3x + 4)(3x - 4) <==
answer is 4
5. In “The Pigman & Me,” John announces that he is going to exact revenge on Paul. What does
he mean by exact?
(1 poin
in a precise way
sign into law
take by force
separate one object from another
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The angles with measures x and 4x are complementary angles. which is the value of x?
Answers
x+4x=90
5x=90
x=18
Answer:
x=18 degrees
Step-by-step explanation:
Hello
by definition, two angles are called complementary angles, if the sum of their degree measures is 90. Each angle is called complement of the other.
A = angle 1=x
B=angle 2 =4x
A and B are complementary angles,hence
A+B=90
replacing
[tex]x+4x=90\\\\5x =90\\x=\frac{90}{5} \\x=18\ degrees[/tex]
A=18 degrees
B=4*x
B=72 degrees
x=18 degrees
what system of linear inequalities is represented by the graph?
Answers
y=0.5x-1
second/dotted line:
y=2x-4
your allowed values are below the first line and above the second, so the system is
I: y<=0.5x-1
II: y>=2x-4
your dotted line might also mean > instead of >=, but this depends on the notation used in class
Answer:
y > 2x - 4 and y ≤ [tex]\frac{x}{2}-1[/tex]
Step-by-step explanation:
In this question we will find the equations of the lines first then decide the inequality sign.
Dotted line in the graph passes through two points (2, 0) and (0, -4)
Let the equation of the line is y = mx + c
Where m = slope and c = y-intercept
Slope (m) of the line will be = [tex]\frac{y-y'}{x-x'}[/tex]
m = [tex]\frac{0+4}{2-0}[/tex]
m = 2
y-intercept of the line = (-4)
So, equation of the line will be y = 2x - 4
Now we see the shaded area is above the dotted line so inequality will be
y > 2x - 4
Now for the bold line
Let the equation of this line is y = m'x + c'
This line passes through points (2, 0) and (0, -1)
then slope of the line m' = [tex]\frac{y-y'}{x-x'}[/tex]
m = [tex]\frac{0+1}{2-0}[/tex] = [tex]\frac{1}{2}[/tex]
y-intercept = c = -1
Then equation of the line will be
y = [tex]\frac{x}{2}-1[/tex]
Since shaded area is below the bold line therefore, inequality sign will be "less than equal to"
Inequality of the bold line will be y ≤ [tex]\frac{x}{2}-1[/tex]
System of the linear inequalities will be
y > 2x - 4 and y ≤ [tex]\frac{x}{2}-1[/tex]
Tim made a down payment of $1,000. He pays $450 for 36 months. What is the total amount he pays for the car?
Answers
1000 + 16,200 = 17,200.
He payed $17,200 for the car.
Hope this helps!
Answer:
$ 17,200
Step-by-step explanation:
Given,
The amount of down payment = $ 10000,
Also, monthly payment = $ 450,
Number of months = 36,
So, the total payment ( except down payment ) = number of months × monthly payment
= 36 × 450
= $ 16200,
Hence, the total amount paid for the car = down payment + total payment
= $ 1,000 + $ 16,200
= $ 17,200
Two angles form a complementary pair (sum is 90 degrees) one angle is 3x - 10 and the other angle is x + 8. Write an equation and solve for each angle.
Could somebody help? Thank you :)
Answers
4×-2=90
4x=92
X=23
One angle is 31. 23+8
The second one is 59. 3 (23)-10
Find the value of angle c?? Please help
Answers
180 = 5x + 30
Subtract 30 from both sides
150 = 5x
Divide both equations by 5
30 = x
Now since a and c are vertical angles a = c
So plug in 30 for x
2(30) + 10 = c
60 + 10 = c
70 = c
Sal currently has an account balance of $2,835.48. He opened the account five years ago with a deposit of $2310.72. If the interest compounds twice a year, what is the interest rate on the account?
Answers
A=p (1+r/k)^kt
A future value 2835.48
P present value 2310.72
R interest rate?
K compounded twice a year 2
T time 5 years
Solve the formula for R
R= [(A/p)^(1/kt)-1]×k
R=((2,835.48÷2,310.72)^(1÷10)−1)×2
R=0.0414×100
R=4.14%
Hope it helps!
Simplify (5ab4c)(-abc2). 4b3c -5a2b5c3 -5ab4c2
Answers
Your answer is :
-2a2b8c4−5a2b5c3−5ab4c2
Hoped I helped!
it will be simplified as [tex]\( -5a^2b^5c^3 \).[/tex]
Sure, let's simplify the expression step by step:
Given expression: [tex]\( (5ab^4c)(-abc^2) \)[/tex]
Step 1: Multiply the coefficients (numbers) together:[tex]\( 5 \times -1 = -5 \)[/tex]
Step 2: Multiply the variables with the same base together and add their exponents. Start with 'a': [tex]\( a \times a = a^{1+1} = a^2 \)[/tex]
Step 3: For 'b', multiply[tex]\( b^4 \) with \( b^1 \) which gives \( b^{4+1} = b^5 \).[/tex]
Step 4: For 'c', multiply[tex]\( c^1 \) with \( c^2 \) which gives \( c^{1+2} = c^3 \).[/tex]
Combine all these results together:[tex]\( -5a^2b^5c^3 \)[/tex]
So, the simplified expression is [tex]\( -5a^2b^5c^3 \).[/tex]
1. Start by multiplying the coefficients (numbers) together. In this case,[tex]\( 5 \times -1 = -5 \).[/tex]
2. Next, multiply the variables with the same base together and add their exponents. For 'a',[tex]\( a^1 \) times \( a^1 \) gives \( a^{1+1} = a^2 \). Similarly, for 'b', \( b^4 \) times \( b^1 \) gives \( b^{4+1} = b^5 \), and for 'c', \( c^1 \) times \( c^2 \) gives \( c^{1+2} = c^3 \).[/tex]
3. Combine all these results together to get [tex]\( -5a^2b^5c^3 \).[/tex]
This process ensures that each term in the expression is simplified according to the rules of exponents and multiplication of coefficients.
complete question
Simplify (5ab4c)(-abc2). 4b3c -5a2b5c3 -5ab4c2
2.1e-10 in standard form?
Answers
= 2.1 x 10^-10
= 0.00000000021
hope it helps
The value of 2.1e-10 = 0.00000000021
What is exponents?An exponent refers to the number of times a number is multiplied by itself.
Here e is used when the number is too large and small for the calculator.
We know 'e' represent exponents
2.1e-10, represents [tex]2.1* 10^{-10}[/tex]
= 2.1 x [tex]\frac{1}{10^{10}}[/tex]
= 2.1 x [tex]\frac{1}{10000000000}[/tex]
= 0.00000000021
Hence, 2.1e-10 = 0.00000000021
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What is the product of all positive divisors of $100$?
Answers
100*1=100
2*50=100
4*25=100
5*20=100
100*100*100*100*10= 1,000,000,000
Answer: 1,000,000,000
The product of all positive divisors of the number 100 is 1,000,000, using the property that the product of an integer's positive divisors is equal to the integer raised to the power of half its total number of divisors.
Explanation:The product of all positive divisors of 100 can be found using a special property. This property states that for any positive integer n, the product of all its positive divisors will be equal to n raised to the power of half its total number of divisors. Now, factors of 100 are 1, 2, 4, 5, 10, 20, 25, 50, and 100. So, there are 9 divisors, and the product of all positive divisors will be 100^((9/2)). The answer is 1,000,000.
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Using the given information, which lines, if any, can you conclude are parallel? Justify your conclusion with a theorem or postulate.
Answers
a and b are paralell