Answer:
a. i. (i + tj + 2tk)/√(1 + 5t²)
ii. (-5ti + j + 2k)/√[25t² + 5]
b. √5/[√(1 + 5t²)]³
Step-by-step explanation:
a. The unit tangent
The unit tangent T(t) = r'(t)/|r'(t)| where |r'(t)| = magnitude of r'(t)
r(t) = (t, t²/2, t²)
r'(t) = dr(t)/dt = d(t, t²/2, t²)/dt = (1, t, 2t)
|r'(t)| = √[1² + t² + (2t)²] = √[1² + t² + 4t²] = √(1 + 5t²)
So, T(t) = r'(t)/|r'(t)| = (1, t, 2t)/√(1 + 5t²) = (i + tj + 2tk)/√(1 + 5t²)
ii. The unit normal
The unit normal N(t) = T'(t)/|T'(t)|
T'(t) = dT(t)/dt = d[ (i + tj + 2tk)/√(1 + 5t²)]/dt
= -5ti/√(1 + 5t²)⁻³ + [-5t²j/√(1 + 5t²)⁻³] + [-10tk/√(1 + 5t²)⁻³]
= -5ti/√(1 + 5t²)⁻³ + [-5t²j/√(1 + 5t²)⁻³] + j/√(1 + 5t²)+ [-10t²k/√(1 + 5t²)⁻³] + 2k/√(1 + 5t²)
= -5ti/√(1 + 5t²)⁻³ - 5t²j/[√(1 + 5t²)]⁻³ + j/√(1 + 5t²) - 10t²k/[√(1 + 5t²)]⁻³ + 2k/√(1 + 5t²)
= -5ti/√(1 + 5t²)⁻³ - 5t²j/[√(1 + 5t²)]⁻³ - 10t²k/[√(1 + 5t²)]⁻³ + j/√(1 + 5t²) + 2k/√(1 + 5t²)
= -(i + tj + 2tk)5t/[√(1 + 5t²)]⁻³ + (j + 2k)/√(1 + 5t²)
We multiply by the L.C.M [√(1 + 5t²)]³ to simplify it further
= [√(1 + 5t²)]³ × -(i + tj + 2tk)5t/[√(1 + 5t²)]⁻³ + [√(1 + 5t²)]³ × (j + 2k)/√(1 + 5t²)
= -(i + tj + 2tk)5t + (j + 2k)(1 + 5t²)
= -5ti - 5²tj - 10t²k + j + 5t²j + 2k + 10t²k
= -5ti + j + 2k
So, the magnitude of T'(t) = |T'(t)| = √[(-5t)² + 1² + 2²] = √[25t² + 1 + 4] = √[25t² + 5]
So, the normal vector N(t) = T'(t)/|T'(t)| = (-5ti + j + 2k)/√[25t² + 5]
(b) Use Formula 9 to find the curvature.
The curvature κ = |r'(t) × r"(t)|/|r'(t)|³
since r'(t) = (1, t, 2t), r"(t) = dr'/dt = d(1, t, 2t)/dt = (0, 1, 2)
r'(t) = i + tj + 2tk and r"(t) = j + 2k
r'(t) × r"(t) = (i + tj + 2tk) × (j + 2k)
= i × j + i × 2k + tj × j + tj × 2k + 2tk × j + 2tk × k
= k - 2j + 0 + 2ti - 2ti + 0
= -2j + k
So magnitude r'(t) × r"(t) = |r'(t) × r"(t)| = √[(-2)² + 1²] = √(4 + 1) = √5
magnitude of r'(t) = |r'(t)| = √(1 + 5t²)
|r'(t)|³ = [√(1 + 5t²)]³
κ = |r'(t) × r"(t)|/|r'(t)|³ = √5/[√(1 + 5t²)]³
Please help me with this question
A medical device company knows that the percentage of patients experiencing injection-site reactions with the current needle is 11%.
What is the probability that an injection-site reaction occurs for the first time on the 6th patient of the day?
0.0001
0.0614
0.3685
0.4970
Answer:
0.4970
Step-by-step explanation:
I might be wrong
The probability that an injection-site reaction occurs for the first time on the 6th patient of the day will be 0.4970. Then the correct option is D.
What is probability?Its fundamental concept is that someone will nearly surely occur. The proportion of positive events in comparison to the total of occurrences.
Then the probability is given as,
P = (Favorable event) / (Total event)
A medical device company knows that the percentage of patients experiencing injection-site reactions with the current needle is 11%.
The probability that an injection-site reaction occurs for the first time on the 6th patient of the day is given as,
P = (1 - 0.11)⁶
P = (0.89)⁶
P = 0.4970
The probability that an injection-site reaction occurs for the first time on the 6th patient of the day will be 0.4970. Then the correct option is D.
More about the probability link is given below.
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Sketch the region enclosed by the given curves and calculate its area.
y=4-x^2 ,y=0
The answer is 32/3. But how do I get to that answer?
Answer:
Step-by-step explanation:
1.) we need to find the bounds of integration which is just the points of intersection
here is it (-2,0) and (2,0)
which means we will integrate from -2 to 2
next, we take the upper equation and subtract that from the lower one
kind of confusing but it would look like (sketch it out if you're not sure)
(4-x²)-0= 4-x²
then we can integrate
[tex]\int\limits^2_{-2} {4-x^2} \, dx =4x-\frac{x^3}{3}|_{-2}^{2}=(4*(2)-\frac{2^3}{3})-(4(-2)-\frac{-2^3}{3})=5.333333-(-5.3333333)= 10.666666667=\frac{32}{3}[/tex]
Four students want to have their picture taken together. They will stand side-by-side for the picture. In how many different ways can the four students be arranged to take a picture?
Show your work, please :')
Answer:
24 waysStep-by-step explanation:
This is the permutation of 4:
4P4 = 4! = 1*2*3*4 = 24 ways[tex]\huge\qquad \mathbb{\fcolorbox{red}{lavenderblush}{✰Answer}}[/tex]
✶⊶⊷⊶⊷❍❁❥❀❥❁❍⊶⊷⊶⊷✶
Four students want to have their picture taken together. They will stand side-by-side for the picture. In how many different ways can the four students be arranged to take a picture
so we have to find the permutation of 4
4×3×2×124.°. In 24 different ways can the four students be arranged to take a picture
Find the area of the rectangle if the perimeter is 52 cm
Answer:
No solution is possible from the information provided
Step-by-step explanation:
help pleaseeee it’s timed!!!
Answer:
C
Step-by-step explanation:
The solution triangle is attached below :
Tonobtinnthe Angle formed, θ; we apply trigonometry ;
Using ;
Cos θ = Adjacent / hypotenus
Cos θ = 4 / 7
θ = Cos^-1(4/7)
θ = 55.15°
θ = 55°
20 POINTS MATH PROBLEM
Answer:
D. x=36
Step-by-step explanation:
3x-5=103
3x=103+5
3x=108
x=36
Can anyone help me with this question
Answer:
50%
Step-by-step explanation:
If angle f is 50 degrees and cuts it to 75 degrees
Find the difference between the two numbers
75-50 = 25
Divide by the correct value
25/50 = 1/2
Change to percent form
50%
Suppose that 48% of high school students would admit to lying at least once to a teacher during the past year and that 25% of students are male and would admit to lying at least once to a teacher during the past year.20 Assume that 50% of the students are male. What is the probability that a randomly selected student is either male or would admit to lying to a teacher, during the past year? Be sure to show your work and indicate all the rules that you use to find your answer.
Answer:
0.73 = 73% probability that a randomly selected student is either male or would admit to lying to a teacher, during the past year.
Step-by-step explanation:
I am going to treat these events as Venn probabilities, considering that:
Event A: Lying to the teacher.
Event B: Male
48% of high school students would admit to lying at least once to a teacher during the past year and that 25% of students are male and would admit to lying at least once to a teacher during the past year
This means that [tex]P(A) = 0.48, P(A \cap B) = 0.25[/tex]
Assume that 50% of the students are male.
This means that [tex]P(B) = 0.5[/tex]
What is the probability that a randomly selected student is either male or would admit to lying to a teacher, during the past year?
This is:
[tex]P(A \cup B) = P(A) + P(B) - P(A \cap B)[/tex]
Considering the values we were given:
[tex]P(A \cup B) = 0.48 + 0.5 - 0.25 = 0.73[/tex]
0.73 = 73% probability that a randomly selected student is either male or would admit to lying to a teacher, during the past year.
Một đề thi trắc nghiệm có 10 câu, mỗi câu có 4 phương án trả lời và chỉ có một đáp án đúng. Một sinh viên trả lời một cách ngẫu nhiên, xác suất để sinh viên được 5 điểm là
Câu trả lời:
0,0584
Giải thích từng bước:
Câu hỏi đưa ra đáp ứng điều kiện cần thiết cho phân phối xác suất nhị thức:
Số câu hỏi, số lần thử, n = 10
Xác suất, p = 1 / số lựa chọn = 1/4 = 0,25
q = 1 - p = 1 - 0,25 = 0,75
Tính xác suất để sinh viên đó được 5 điểm, x = 5;
Gợi lại:
P (x = x) = nCx * p ^ x * q ^ (n-x)
P (x = 5) = 10C5 * 0,25 ^ 5 * 0,75 ^ 5
P (x = 5) = 252 * 0,25 ^ 5 * 0,75 ^ 5
P (x = 5) = 0,0584
Suppose that g(x)= f(x)+ 6. Which statement best compares the graph of g(x) with the graph of f(x)?
A. The graph of g(x) is the graph of f(x) shifted 6 units down.
B. The graph of g(x) is the graph of f(x) shifted to the right.
C. The graph of g(x) is the graph of f(x) shifted 6 units to the left.
D. The graph of g(x) is the graph of f(x) shifted 6 units up.
Answer:
D
Step-by-step explanation:
The + 6 moves it up 6 units.
The correct answer is (D) "The graph of g(x) is the graph of f(x) shifted 6 units up."
What is the function?A relationship between a group of inputs and one output is referred to as a function. In plain English, a function is an association between inputs in which each input is connected to precisely one output. A domain, codomain, or range exists for every function. Typically, f(x), where x is the input, is used to represent a function.
When we add a constant to a function, such as in the case of g(x) = f(x) + 6, it will shift the graph of f(x) upward by 6 units.
This is because, for any value of x, the value of f(x) will be added to 6, resulting in a vertical shift of the entire graph.
Option (A) is incorrect because adding 6 to f(x) would shift the graph up, not down.
Option (B) is incorrect because adding a constant to a function does not cause it to shift horizontally.
Option (C) is incorrect because adding 6 to f(x) would shift the graph right, not left.
D. The graph of g(x) is the graph of f(x) shifted 6 units up. Adding a constant term to a function will shift the graph of the function vertically. In this case, adding 6 to f(x) will shift the graph of f(x) upward by 6 units, resulting in the graph of g(x).
Learn more about function here:
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What is the value of z2 if 3z1 = 6 + 18i and z1 + z2 = 5 + 9i?
Answer:
z2=-3-3i
Step-by-step explanation:
3z1+0z2=6+18i
z1+z2=5+9i (×3)
3z1+3z2=15+27i
3z1+0z2=6+18i (-) =
-3z2=9+9i
z2=-3÷(9+9i)
z2=-3-3i
A cyclist travels at an average speed of 18 mph for 2 hours.
How far did she travel in miles?
Answer:
36
Step-by-step explanation:
18 miles per hour so for 2 hours which is double, she should have traveled 36 miles.
Good Luck!
Jack is 2 years older than his sister jill. The sum of their ages is 24 years. If jack is x years, find x
Answer:
x = 13
Jack is 13 years old and Jill is 11 years old.
====================================================
Explanation:
x = Jack's age
x-2 = Jill's age, since she is 2 years younger than Jack
Add up the ages and set the sum equal to 24 to solve for x
x+(x-2) = 24
2x-2 = 24
2x = 24+2
2x = 26
x = 26/2
x = 13
Jack is 13 years old and his sister is x-2 = 13-2 = 11 years old
Check: 13+11 = 24, so the answer is confirmed.
Find the value of n?
Answer:
[tex]n^{2} -3=39.2=78\\n^2=78+3=81\\n=\sqrt{81} \\n=9[/tex]
Step-by-step explanation:
If f(x) = x -2 and g(x) = 2x – 6, then g(4)/f(3) =
Answer:
Step-by-step explanation:
(2×4-6)/(3-2)=2
Answer:
[tex]{ \tt{f(x) = x - 2}} \\ { \bf{f(3) = 3 - 2 = 1}} \\ \\ { \tt{g(x) = 2x - 6}} \\ { \bf{g(4) = 2(4) - 6 = 2}} \\ \\ { \boxed{ \tt{ \frac{g(4)}{f(3)} = \frac{2}{1} = 2}}}[/tex]
Look at the illustration.
What is WX?
Answer:
O 0.5 units
Step-by-step explanation:
so the first thing we have to do is to calculate for the dilation factor. Taking point G as the reference point, we can see that the distance of point G from rectangle W'X'Y'Z is 1.5 while the distance from rectangle WXYZ is (1.5 + 7.5) = 1.5 / 9 = 1/6
Since WX has an initial measure of 3 units, therefore the measure of W'X' is:
W'X' = 3 units *(1/6) = 0.5 units
35 of 39
Complete the item by determining if the equation is true or false,
5x 3 1/2 = 3 x 5 1/2
0 17.5 = 16.5 (false)
O 17.5 = 17.5 (true)
O 18,5 = 18,5 (true)
O 18.5 = 19.5 (false)
Step-by-step explanation:
5x 3 1/2 = 3 x 5 1/2 is false
Add the following fractions. See the image below
Answer:
[tex]\dfrac{19}{840}[/tex]
Step-by-step explanation:
The given fraction is:
[tex]\dfrac{1}{168}+\dfrac{3}{180}[/tex]
We need to solve it.
The LCM of 168 and 180 is 2520.
So,
[tex]\dfrac{1}{168}+\dfrac{3}{180}=\dfrac{15+3\cdot14}{2520}\\\\=\dfrac{19}{840}[/tex]
So, the required answer is equal to [tex]\dfrac{19}{840}[/tex].
Multiple choice math problem
Answer:
ok so its a triangle with one side being
11
on side being
2
and one side is x
so we just use the formula
11^2+2^2=c^2
11^2+2^2=125
125 squared is 25
so
3 times the square root of 13 is 10.8166538264
so no
5 times the square root of 5 is 11.1803398875
so the answer i guess might be a?
Hope This Helps!!!
C
What is the equation of the line that is parallel to the line y - 1 = 4(x + 3) and passes through the point (4, 32)?
SER
O y=-*x+33
y=-*x+36
O y =
O y = 4x - 16
O y = 4x + 16
Mark this and retum
Save and Exit
Nex
Submit
9514 1404 393
Answer:
y = 4x + 16
Step-by-step explanation:
You can replace the point coordinates in the given point-slope form to get ...
y - 32 = 4(x -4)
Then rearrange to put this into slope-intercept form.
y = 4x -16 +32 . . . . eliminate parentheses, add 32
y = 4x +16 . . . . . . . . simplify
What is the slope intercept form of the equation of the line shown below
Answer:
[tex]y=\frac{4}{3}x-4[/tex]
Step-by-step explanation:
----------------------------------------
The slope-intercept form formula is: [tex]y=mx+b[/tex]
The [tex]m[/tex] stands for the slope and the [tex]b[/tex] stands for the y-intercept.
By looking at the graph, I can figure out that the y-intercept is -4 because y-intercept is where the lines cross the y-axis and in this graph, the line crosses the y-axis at (0,-4).
The slope is [tex]\frac{4}{3}[/tex] because to get to the ordered pair (3,0), from (0,-4), you would have to go up 4 and over 3 to the right so it's [tex]\frac{4}{3}[/tex]
So now, if we insert the values in the formula, it would be [tex]y=\frac{4}{3}x-4[/tex]
----------------------------------------
Hope this is helpful.
Answer:
y = 4/3x -4
Step-by-step explanation:
First find the slope
m = ( y2-y1)/(x2-x1)
= ( 0 - -4)/( 3 - 0)
= (0+4)/( 3-0)
= 4/3
The y intercept is -4
The slope intercept form of the equation is
y = mx+b where m is the slope and b is the y intercept
y = 4/3x -4
The football and soccer teams at Juan's middle school are selling movie tickets. He wants to determine the average
number of tickets sold at the school. First, he surveyed eight random players on the football team. Then, he surveyed
every third athlete on the football and soccer team. Which sampling technique should produce a more representative
sample?
The first sampling method, surveying eight random football players, is the most representative.
O Neither sample will be representative.
Both samples should be exactly the same.
The second sampling method, survey.ing the both teams, is the most representative.
Step-by-step explanation:
both samples should be exactly the same
After a new product is launched the cumulative sales S(t) (in $1000) t weeks after launch is given by:
S(t) = 72/1 + 9e^-0.36t
Required:
a. Determine the cumulative amount in sales 3 weeks after launch.
b. Determine the amount of time required for the cumulative sales to reach $70,000.
c. What is the limiting value in sales?
Answer:
$17.750 ; 15.979 ; 72
Step-by-step explanation:
Given that :
Cummulative sales, S(t) is represented by the equation :
S(t) = 72/(1 + 9e^-0.36t)
Cummulative sales after 3 weeks :
Put t = 3 in the equation, as t = time after launch
S(3) = 72/(1 + 9e^-0.36(3))
S(3) = 72 / (1 + 9e^-1.08)
S(3) = 72 / (1 +3.0563597)
S(3) = 72 / 4.0563597
S(3) = 17.749905 = $17.750 thousands
Amount of time required for sales to reach 70000
S(t) = 72/(1 + 9e^-0.36t)
S(t) = 70
70 = 72/(1 + 9e^-0.36t)
70 * (1 + 9e^-0.36t) = 72
(1 + 9e^-0.36t) = 72 / 70
1 + 9e^-0.36t = 1.0285714
9e^-0.36t = 1.0285714 - 1
9e^-0.36t = 0.0285714
e^-0.36t = 0.0285714 / 9
e^-0.36t = 0.0031746
Take the In of both sides ;
In(e^-0.36t) = In(0.0031746)
-0.36t = - 5.752573
t = - 5.752573 / - 0.36
t = 15.979
About 16 weeks
The limiting value in sales :
Take the limit as t - - > ∞
S(t - - > ∞) = 72/(1 + 9e^-0.36t)
Put t = 0
S(0) - - > 72 / (1 + 0)
72 / 1
= 72
The length of a rectangle is four times its width.
If the area of the rectangle is 100 yd”, find its perimeter.
Answer: 50yd
Step-by-step explanation:
We know that the area of any rectangle is length times width. The perimeter is the sum of twice the length and twice the width.
Let width = x
Let length = 4x
Area = 100m2
Next, we can write an equation using these variables and formula for area.
4x2 = 100
x2 = 25
x = -5 and x = 5
Since the dimensions cannot be negative, we accept the positive value:
x = 5
Next, we can substitute this value of x into the variables.
width = 5 m
length = 20 m
Finally, we can find the perimeter by plugging in these dimensions into the perimeter formula,
Perimeter = 2(5 m) + 2(20 m)
= 10 m + 40 m
= 50 m
15 points What’s the area‼️‼️‼️‼️‼️ please help me attach work too if you can
Answer:
66
Step-by-step explanation:
5x2= 10
7x8= 56
10+56= 66
Answer:
106
Step-by-step explanation:
split the shape, so you have 2 rectangles.
Multiply 8 and 7 = 56
add 2 and 8 =10, then multiply 10 by 5. =50
add 50 and 56
therefore your answer is 106.
If 21% of kindergarten children are afraid of monsters, how many out of
each 100 are afraid?
Answer:
The appropriate answer is "21".
Step-by-step explanation:
Given:
Afraid percentage,
p = 21%
or,
= 0.21
Sample size,
n = 100
As we know,
⇒ [tex]X=np[/tex]
By putting the values, we get
[tex]=0.21\times 100[/tex]
[tex]=21[/tex]
The upwards acceleration of a small rocket at time t s is given by a = 16 − 1.5t. The rocket is subject to this
acceleration for 3 seconds. Given that it starts from rest at t = 0, calculate the height reached by the rocket in
this time
Answer:
11.5
Step-by-step explanation:
plug 3 in for x
then solve
Determine which value best approximates the length of the arc represented by the integral ∫_0^1 √1 + [d/dx(4/x+1)]² dx.
(Make your selection on the basis of a sketch of the arc and not
by performing any calculations.)
(a) 10
(b) -5
(c) 2
(d) 4
(e) 1
Answer:
Option C
Step-by-step explanation:
From the question we are told that:
Length of arc integral
[tex]l=\int_0^1 \sqrt{1 + [\frac{d}{dx}(\frac{4}{x+1})]^2 dx}[/tex]
The Sketch is attached below
From the Graph
Approximation gives length of arc as
[tex]l=\sqrt{5}[/tex]
[tex]l=2[/tex]
Option C
I'm not good with ordering fractions from smallest to largest. Can help anyone help with this problem?