please show all steps
4) [10 points) Find T(I), N(t), ay, and ay for the space curve r(t) =(21 - 1)i+rºj-4k.
Answers
To find the tangent vector T(t), the normal vector N(t), the binormal vector B(t), and the curvature κ(t) for the space curve r(t) = (21 - t)i + \(\sqrt{2t}\)j - 4k, we can use the formulas derived from the Frenet-Serret equations.
Given the space curve r(t) = (21 - t)i + \(\sqrt{2t}\)j - 4k, we can find the tangent vector T(t) by differentiating r(t) with respect to t and normalizing the resulting vector. Taking the derivative of r(t), we get dr/dt = ( \(\frac{-1}{\sqrt{2t} }\) + \(\frac{1}{\sqrt{2t} }\))j. Normalizing this vector, we obtain T(t) = (1, \(\frac{1}{\sqrt{2t} }\)), 0).
To find the normal vector N(t), we take the derivative of T(t) with respect to t and normalize the resulting vector. Differentiating T(t), we get dT/dt = (0, -\(\frac{1}{2t-\sqrt{2t} }\), 0). Normalizing this vector, we obtain N(t) = (\(\frac{1}{\sqrt{2t} }\), -1, 0).
The binormal vector B(t) can be found by taking the cross product of T(t) and N(t). The cross product of T(t) and N(t) is B(t) = (0, 0, -1).
To find the curvature κ(t), we use the formula κ(t) = ||dT/dt|| / ||dr/dt||, where ||...|| represents the magnitude. Calculating the magnitudes, we have ||dT/dt|| =\(\frac{1}{2t\sqrt{2t} }\) and ||dr/dt|| = \(\frac{1}{\sqrt{2t} }\). Thus, the curvature is κ(t) = \(\frac{1}{4t\sqrt{2t} }\).
Therefore, the tangent vector T(t) is (1, \(\frac{1}{\sqrt{2t} }\), 0), the normal vector N(t) is (\(\frac{1}{\sqrt{2t} }\), -1, 0), the binormal vector B(t) is (0, 0, -1), and the curvature κ(t) is \(\frac{1}{4t\sqrt{2t} }\) for the given space curve r(t) = (21 - t)i + \({\sqrt{2t} }\)j - 4k.
Learn more about vector here:
brainly.com/question/24256726
#SPJ11
Related Questions
PLEASE HELP ME!!!! THIS IS DUE TODAY!!!!!!!!
Answers
Answer:
sorry but can you put a picture or a question? thanks :)
Step-by-step explanation:
Simplify and then evaluate the equation
when x= 4 and y = 2.
5x + 2(9y – X) – y=[?]
I need help
Answers
Answer:
46
Step-by-step explanation:
(5)(4) + 2(9x2 - 4) -2
=20 + 2(18-4) -2
=20 + 2(14) -2
=20 + 28 -2
=48-2
=46
There are black and white counters in a bag in the ratio 20:17
There are 54
more black counters than white counters.
How many black counters are there?
Answers
There are 360 black counters and 306 white counter in 20:17 ratio.
Let's denote the number of black counters by B and the number of white counters by W. We know that the ratio of black to white counters is 20:17, which means that:
B/W = 20/17
We also know that there are 54 more black counters than white counters, which means that:
B = W + 54
We can use substitution to solve for B. Substituting the second equation into the first equation, we get:
(W + 54)/W = 20/17
Cross-multiplying, we get:
17(W + 54) = 20W
Expanding the left side, we get:
17W + 918 = 20W
Subtracting 17W from both sides, we get:
918 = 3W
Dividing both sides by 3, we get:
W = 306
Now we can use the second equation to find B:
B = W + 54 = 306 + 54 = 360
Therefore, there are 360 black counters in the bag.
To learn more about ratio click here
brainly.com/question/13419413
#SPJ4
Sally is making masks for her family. Her pattern calls for a rectangle that is 9 x 14. If one yard of fabric is 36 x 42 (after the salvage edge is removed), how many masks can she make from one yard of fabric? If her fabric was on sale for $3.00 per yard, what is the fabric cost of one mask? If her fabric is regularly priced at $9.00 per yard, what is the fabric cost of one mask? Answer: Number of masks: Cost using discount fabric: $ Cost using regular priced fabric:
Answers
Answer:
how many masks can she make from one yard of fabric?
the maximum she can obtain is 4 (9 x 4 = 36) x 3 (14 x 3 = 42) = 12 masks from one yard of fabric.If her fabric was on sale for $3.00 per yard, what is the fabric cost of one mask?
the fabric cost of 1 mask at a discount price = $3 / 12 = $0.25 per maskIf her fabric is regularly priced at $9.00 per yard, what is the fabric cost of one mask?
the fabric cost of 1 mask at regular price = $9 / 12 = $0.75 per maskGeometry 6.2
What is m∠N
Answers
Check the picture below.
Quinn has 6 bottles of water that each weigh 3/5 pound. What is the total weight of the water bottles?
Answers
Answer: Total weight of bottles = 18/5 pounds or 3.6 pounds.
Step-by-step explanation:
Weight of 1 water bottle is given as = 3/5 pound
Therefore weight of 6 water bottles will be
= 6 x 3/5 pounds=18/5 pounds or 3.6 pounds.
In a recent study on world happiness, participants were asked to evaluate their current lives on a scale from0 to 10, where 0 represents the worst possible life and 10 represents the best possible life. The meanresponse was 5.2 with a standard deviation of 2.4.(a) What response represents the 95th percentile?(b) What response represents the 60th percentile?(c) What response represents the first quartile?...
Answers
Solution
\(\begin{gathered} \text{Given} \\ \operatorname{mean},\text{ }\mu=5.2 \\ \text{standard deviation, }\sigma=2.4 \\ \\ \text{Recall the formula} \\ Z=\frac{x-\mu}{\sigma} \\ x-\mu=Z\sigma \\ x=Z\sigma+\mu \end{gathered}\)(a)
\(\begin{gathered} Z-\text{score for 95 percentile = 1.645} \\ x=Z\sigma+\mu \\ x=1.645(2.4)+5.2 \\ x=9.148 \\ x=9.15\text{ (2 decimal places)} \end{gathered}\)(b)
\(\begin{gathered} Z-\text{score for 60 percentile = }0.253 \\ x=Z\sigma+\mu \\ x=0.253(2.4)+5.2 \\ x=5.8072 \\ x=5.81\text{ (2 decimal places)} \end{gathered}\)(c)
\(\begin{gathered} Z-\text{score for first quartile (25\%) = }-0.674 \\ x=Z\sigma+\mu \\ x=-0.674(2.4)+5.2 \\ x=3.5824 \\ x=3.58\text{ (2 decimal places)} \end{gathered}\)When Shantel converted jmd$16200 to Us dollars she received usd$135. Which exchange rate did Shantel receive?
Answers
Answer: \(1\ \text{USD}\ \equiv\text{jmd}\$120\)
Step-by-step explanation:
Given
Shantel converted jmd$16,200 to USD $135
Using unitary method 1 USD is equivalent to
\(jmd\$16,200\equiv USD\$135\\\\USD\$135\equiv jmd\$16,200\\\\1\ \text{USD}\ \equiv \dfrac{16,200}{135}\\\\1\ \text{USD}\ \equiv\text{jmd}\$120\)
Answer:
Step-by-step explanation:
Write an equation in point slope form of the line that passes through the given point and with the given slope m
(-6,3); m=3
Answers
Answer: y-3=3(x+6)
Step-by-step explanation:
(-6,3) m=3
A line in the point-slope form is expressed as:
\(y-y_1=m(x-x_1)\\\\x_1=-6\ \ \ \ y_1=3\ \ \ \ m=3\)
Hence,
y-3=3(x-(-6))
y-3=3(x+6)
what is the estimated height of a building with six stories? does the least squares line give an accurate estimate of height? explain why or why not.
Answers
if the data is skewed or otherwise biased, the least squares line may not accurately represent the true relationship between the number of stories and building height.
The estimated height of a building with six stories would depend on the average height of each story, which can vary depending on the building's design and construction materials. However, a general rule of thumb is that each story in a building is typically between 10 and 14 feet in height, which would put a six-story building between 60 and 84 feet tall.
As for whether the least squares line gives an accurate estimate of height, that would depend on the data being used to generate the line. The least squares method is a statistical technique used to find the line of best fit for a set of data points, with the goal of minimizing the sum of the squared distances between each data point and the line.
If the data used to generate the line is representative of the population of buildings with six stories, then the least squares line should provide a reasonably accurate estimate of height. However, if the data is skewed or otherwise biased, the least squares line may not accurately represent the true relationship between the number of stories and building height. Therefore, it is important to carefully examine the data and consider potential biases before using the least squares method to make predictions.
To learn more about least squares visit: https://brainly.com/question/29834077
#SPJ11
Work out the value of 5 to the power of a if it equal 1/125
Answers
Answer:
a = -3
Step-by-step explanation:
Step 1: Since we're told that 5 to the power of a = 1/125, we can use the following equation to solve for a:
5^a = 1/125
Step 2: Take the log of both sides
log(5^a) = log(1/125)
Step 3: According to the power rule of logs, we can bring a down and multiply it by log(5):
a * log(5) = log (1/125)
Step 4: Divide both sides by log(5) to solve for a:
(a * log(5) = log(1/125)) / log(5)
a = -3
Optional 5: We can check our answer by plugging in -3 for a and seeing if we get 1/125 when completing the operation:
5^-3 = 1/125
1/(5^3) = 1/125 (rule of exponents states that a negative exponent creates a fraction with 1 as the numerator and the base (5) and exponent (-3 becoming 3) as the denominator
1/125 = 1/125
State the equation of the graphed function.
Answers
The equation of the graphed function is given as follows:
f(x) = x³ + 2x² - 5x - 6.
How to obtain the equation of the function?
The equation of the function is obtained considering the Factor Theorem, as a product of the linear factors of the function.
From the graph, the zeros of the function are:
x = -3.x = -1.x = 2.Hence the function is:
f(x) = a(x + 3)(x + 1)(x - 2).
In which a is the leading coefficient.
Expanding the product, we have that:
f(x) = a(x² + 4x + 3)(x - 2)
f(x) = a(x³ + 2x² - 5x - 6).
When x = 0, y = -6, hence the leading coefficient a is obtained as follows:
-6a = -6
a = 1.
Hence the function is:
f(x) = x³ + 2x² - 5x - 6.
More can be learned about functions at https://brainly.com/question/24808124
#SPJ1
what is the slope of (7,-8) and (9,9).....
Answers
Answer:
(7,8) = 0,0) (−6,6)
(5,4) (−9,7)
(0,1) (1,0)
(9,9) = (0,9) (8,6)
(−5,9) (3,0)
(1,−2) (3,6)
Investment increase exponentially by about 26% every 3 years. if you start with $2,000 investment how much money would you have after 45 years
Answers
Using an exponential function, it is found that you would have an amount of $64,060 in 45 years.
What is an exponential function?An increasing exponential function is modeled by:
\(A(t) = A(0)(1 + r)^t\)
In which:
A(0) is the initial value.r is the growth rate, as a decimal.In this problem, considering the initial value of A(0) = 2000, with a growth rate of r = 0.26 each 3 years, the equation is given by:
\(A(t) = 2000(1.26)^{\frac{t}{3}}\)
In 45 years, the amount will be given by:
\(A(45) = 2000(1.26)^{\frac{45}{3}} = 64060\)
More can be learned about exponential functions at https://brainly.com/question/25537936
#SPJ1
Please answer fast! (100 points)
Answers
Answer:
x = 2.25
Step-by-step explanation:
X equals to 2.25. This is because the scale factor between 8 and 6 is 1 1/3.
Next, we take 3 and divide it with 1 1/3, ending up with 2.25.
A lock requires a 3 number combination using the numbers 0 through 9, none of which may be repeated. How many outcomes are possible?
48
720
12010
Answers
Answer:
720
Step-by-step explanation:
The lock requires that repetition is not allowed.
The possibles numbers are 0 - 9, that is, 10 digits.
The first number in the combination can be picked from the 10 digits.
This means that there are 10 possible choices.
The second number in the combination can now only be picked from 9 digits.
This means that there are 9 possible choices.
The third number in the combination can now only be picked from 8 digits.
This means that there are 8 possible choices.
Therefore, the number of possible outcomes in the combination is:
10 * 9 * 8 = 720 outcomes
Help ASAP
22.41 + 8.9
Answers
Answer:
22.41 + 8.9= 31.31
hope this helps <3
What is a half of 3/4
Answers
Answer:
1.5/2 or 75%
Step-by-step explanation:
3/2=1.5
4/2=2
Lily walks 2 kilometers during each trip to school. Write an equation that shows the
relationship between the number of trips x and the total distance walked y.
Answers
Answer:
y = 2x
Step-by-step explanation:
Total distance walked = y
Number of trips = x
Lily walks 2 kilometers during each trip to school
Equation that shows the
relationship between the number of trips x and the total distance walked y:
y = 2x
For example, if the number of trips (x) is 3
Total distance walked,y = 2(3)
y = 6 kilometers
Again PLEASE ANSWER
You have designed a concrete patio which is the shape of a parallelogram and a triangle combined. The parallelogram has a length of 12 m with a height of 4 m. The triangle has a base of 12 m with a height of 5 m. Calculate both the area and perimeter of this patio.
Answers
Answer:
perimeter is 35.6 area is 108
Step-by-step explanation:
it helps to draw a picture. first the perimeter of the rectangle is 4x2 +12=20 we wont be needing one of the twelves because it is connected to the triangle the area or the rectangle is 12x4=48. now for the triangle. we will need the Pythagorean theorem to find the sides first split the base in half 12/2=6 now we use the height and the base \(6^{2} +5x^{2} =\)61 the square root of 61 is about 7.8 the perimeter is 7.8+7.8 +12 but we wont need the base because it is attached to the rectangle =15.6 the area of the triangle is like half a square Base times height though 12x5=60
the whole perimeter if 15.6+20=35.6 the area is 60+48=108
Answer:
50 m squared
Step-by-step explanation:
You have to solve for the areas and then add
Which angles are alternate exterior angles with angle 11?
Answers
Answer:
<6 and <16 are alternate exterior angles.
This table shows the number of girls enrolled in school by class. If a student is chosen, which is the probability that a senior will be chosen?
freshman: 165
sophomore: 145
junior: 114
senior: 102
Answers
we need to multiply this number by 100 to get the as a percentage:
0.1806 * 100 = 18.06%
The probability of choosing a senior can be calculated using the formula:
P(senior) = (102/562) * 100
P(senior) = 18.06%
This means that there is an 18.06% chance of choosing a senior if a student is randomly selected from the school.
First, we need to calculate the total number of students in the school by adding together the number of students in each class:
Total = 165 + 145 + 114 + 102 = 562
Next, we need to calculate the probability of choosing a senior by taking the number of seniors enrolled in the school (102) and dividing it by the total number of students in the school (562).
102/562 = 0.1806
Finally, we need to multiply this number by 100 to get the probability as a percentage:
0.1806 * 100 = 18.06%
Learn more about probability here
https://brainly.com/question/11234923
#SPJ4
Kite ABCD is drawn with diagonals. To find the area of ABCD, Alex imagined the kite divided into two triangles. What is the area of ABCD?
Answers
The area of kite ABCD is equal to half the product of the sum of its diagonals and the height between them.
How to find the area of ABCD?Since kite ABCD is divided into two triangles by its diagonals, we can find the area of the kite by finding the sum of the areas of these two triangles.
Let AC and BD be the diagonals of the kite, intersecting at point E. Then triangle ABE and triangle CDE are the two triangles formed by the diagonals.
The area of a triangle can be found using the formula: Area = (base x height) / 2
For triangle ABE, the base is AB and the height is the distance from E to the line containing AB. Similarly, for triangle CDE, the base is CD and the height is the distance from E to the line containing CD.
Since the diagonals of a kite are perpendicular and bisect each other, the distance from E to the line containing AB is the same as the distance from E to the line containing CD.
Therefore, the heights of the two triangles are equal.
So, we can find the area of ABCD as follows:
Area of ABCD = Area of triangle ABE + Area of triangle CDE
= (AB x height)/2 + (CD x height)/2
= (AB + CD) x height / 2
Therefore, the area of kite ABCD is equal to half the product of the sum of its diagonals and the height between them.
Learn more about area of the kite
brainly.com/question/15640807
#SPJ11
A five-digit integer will be chosen at random from all possible positive five-digit integers. What is the probability that the number's units digit will be less than 5
Answers
The probability that the number's units digit will be less than 5 is 0.5 that it is equal to 1/2.
Probability is a branch of mathematics concerned with numerical descriptions of how likely an event is to occur or how likely a statement is to be true. The probability of an event is a number between 0 and 1, where, roughly speaking, 0 indicates the impossibility of the event and 1 indicates certainty.
There are 10 possible digits, 0, 1, 2, ... 9 that make up the 5-digit number made up of integral values.
There are 5 digits that are less than 5 which are 0, 1, 2, 3, and 4 so the probability will be found by taking only those nubers which have unit's place less than 5,
By probability formula , Probability= no. of favourable outcomes/total no. of outcomes
So probability will be found that will be equal to 5/10, or 1/2
For more information about probability, visit https://brainly.com/question/24756209
#SPJ4
What is 69.37 minus 1.93
Answers
Answer:
the answer is 67.44 hope it helps smile
A large helium balloon is tethered to the ground by two taut lines. One line is 100 ft. Long and makes an 80 degree angle with the ground. The second line makes a 40 degree angle with the ground. How long is the second line?
Answers
The second line is 153ft long.
We are making a figure out of the given statment. Where we have the Triangle ABC and other two Triangle ADB and ADC inside Triangle ABC. Here Angle ADC = 80 degress and Angle ACD = 40 egress.
According to given statement and figure,
In triangle ABO, AD= 100 ft
Hence, BD/AD = Cos80
or, BD = (100 * Cos80)
or, BD = 17.36 ft
Now, AB/AD = Sin80
or, AB = (100* Sin 80)
or, AB = 98.48
Now, In case of triangle ABC
AB/AC = Sin40
or, AC = AB/ Sin 40
or, AC = 98.48/ Sin40
or, AC = 153.2 ft (nearly equal to 153 ft)
The second line is 153ft long.
To know more about Trigonometric solution here:
https://brainly.com/question/29088508#
#SPJ4
There is a 70% chance of getting stuck in traffic when leaving the city. On two separate days, what is the probability that you get stuck in traffic both days
Answers
The probability of getting stuck in traffic on any given day when leaving the city is 70%. When considering two separate days, we can use the multiplication rule of probability to find the probability of getting stuck in traffic on both days.
The multiplication rule of probability states that the probability of two independent events occurring together is the product of their individual probabilities. In this case, the events of getting stuck in traffic on two separate days are independent, meaning that the occurrence of one does not affect the probability of the other.
To find the probability of getting stuck in traffic on both days, we can multiply the probability of getting stuck on the first day (0.7) by the probability of getting stuck on the second day (also 0.7):
P(getting stuck on both days) = P(getting stuck on day 1) x P(getting stuck on day 2)
P(getting stuck on both days) = 0.7 x 0.7
P(getting stuck on both days) = 0.49 or 49%
Therefore, the probability of getting stuck in traffic on both days is 49%. This means that there is a less than 50% chance of getting stuck in traffic on both days, despite the 70% chance of getting stuck on each individual day.
Learn more about probability here: brainly.com/question/30034780
#SPJ11
A backpack that normally sells for $39 is on sale for $25. Find the
percent of change.
Answers
Answer: To find the discount, simply multiply the original selling price by the %discount:
ie: 39 x 33/100= $12.87
So, the discount is $12.87.
Step-by-step explanation: To find the sale price, simply minus the discount from the original selling price:
ie: 39- 12. 87= 26.13
So, the sale price is $26.13
Which two numbers below will you set the inside of the absolute value to in the beginning of your work?
|3x-8|=4
A. 4 and -4
B. 2 and -2
C. 3 and -4
D. 8 and -8
Answers
Answer:
I think the answer is A. I hope this is helpful
A grocery store surveyed 3,000 of its customers and asked what brand of butter they purchased the last time that they bought butter. 500 respondents said they purchased the store brand. The grocery store says that there is a 16.67% probability that a customer who buys butter will buy the store brand. What type of probability is this
Answers
Probability is the possibility that something will happen or is the degree of likelihood of an event. It's a number that reflects the likelihood of an occurrence.
Probability is expressed as a number ranging from 0 to 1, with 0 indicating that an event is impossible and 1 indicating that it is certain to happen. A probability of 0.5, for example, represents a 50% chance that the event will occur. Types of Probability1. Empirical Probability: Empirical probability is a measure of how frequently an event occurs based on past experiences or experiments.
Theoretical probability is the probability of an event occurring based on mathematical calculations or formulas.3. Subjective Probability: Subjective probability is a probability based on personal judgment, experience, and intuition. It is not based on data or calculations but on what the person believes or feels to be true. The probability mentioned in the problem is a subjective probability.
To know more about Probability visit:-
https://brainly.com/question/31828911
#SPJ11