The equation of oscilating plane at the point on the curve corresponding to the indicated value of t, r(t)=i + 2tj +t^2k; t=1 is
x - 1 = 0.
Unit tangent vector, T(t) at t = 1 is
T(t) = (√2/2 ) j +(√2/2 ) k
Principle Normal vector at t = 1 is N(1) = - (√2/2 ) j + (√2/2 ) k
We have , x,y,z for the osculating plane at the point on the curve corresponding to the indicated value of t.
r(t) = i + 2t j + t² k
So, r'(t) = 0 i + 2 j + 2t k
Unit tangent vector , say T(t) is
T(t) = (0 i + 2 j + 2t k )/√0+(2)²+ (2t)²
T(t) =( 2 j + 2t k)/√4+ 4t² = 2(j+k)/2√1+t²
T(t) = (j + t k) / √1+t²
At t = 1 , T(t) = (j + k )/√1+1 = (j+k)/√2
T(t) = (√2/2 ) j + (√2/2 ) k
Now, tangent vector and normal vector prependicular to each other.
So, principle vector is N(1) = - (√2/2 ) j + (√2/2 ) k
Now, binormal vector is B = T×N
= | i j k |
| 0 √2/2 √2/2 |
| 0 -√2/2 √2/2 |
= <1,0,0>
Equation of oscilating plane is
1( x- 1) + 0(y - 2) + 0( z - 1) = 0
=> x = 1
Hence, equation of oscilating plane is x-1 = 0
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James bought a 3 pound sack of oranges at the grocery store. There are about 0.454 kilograms in 1 pound. Which measurement is closest to the weight of the sack of oranges in kilograms?
Answer:
1.362 kilograms
Step-by-step explanation:
3 x .454 = 1.362 Kilograms
Factor completely and find all the zero(s)
Answer:
Factors are (x-1) (4x²-1)
Zero(s) are x = 1, x = 1/2, x = -1/2
Step-by-step explanation:
Given function is
4x³ - 4x² - x + 1
= 4x³ - x - 4x² + 1
= x(4x² - 1) - 1(4x² - 1)
= (x - 1) (4x² - 1) (Factors of the equation)
To find the zero(s),
Either, Or,
x - 1 = 0 4x² - 1 = 0
or, x = 1 or, x² = 1/4
or, x = √(1/4)
or, x = +- 1/2
How many 1/2 inch cubes fit along it’s 2 1/2 length?
Five 1/2 inch cubes fit along it’s 2 1/2 length.
What is meant by length?An object's longest or largest dimension is a measured distance. Dimensions: 10 ft. see Weights and Measures, Metric System Table
An object's length, which is a measurement of its longest side, is its most stretched dimension. Your dog, for instance, would measure its length from the point of its nose to the end of its tail. The space between its top and bottom feet is greater than that of the creature's head.
In addition to utilizing handspan, foot span, and other methods, one can measure length in many units, such as centimeters, inches, meters, and so on.
Let x be the number of 1/2 cubes to fit 2 1/2 length
(1/2)x= 2 1/2
(1/2)x=5/2
x=5
Therefore, five 1/2 inch cubes fit along it’s 2 1/2 length.
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is the following a qualitative or quantitative variable?: the type of flowers you plant in your garden
Qualitative variables include all observable qualities or characteristics of a group or population that can not be measured numerically.
What is Qualitative?
The goal of qualitative research is to collect and examine non-numerical data in order to comprehend people's attitudes, beliefs, and motivations in relation to their social reality.
Quantitative Variable
Quantitative variables, as the name implies, are those that can be expressed by a numerical value.
1. Quantitative (Discrete)
2. Quantitative (Continuous)
3. Quantitative (Discrete)
4. Qualitative (nominal)
5. Qualitative (nominal)
6. Qualitative (nominal)
7. Qualitative (nominal)
8. 9. Quantitative (Continuous)
10. Qualitative (nominal)
11. Quantitative (Continuous)
12. Qualitative (ordinal)
13. Qualitative (nominal)
14. Quantitative (Discrete)
15. Qualitative (nominal)
Complete question: Which is the following a qualitative or quantitative variable?
1)ordinal
2)Continuous
3)nominal
4) All of these
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The measures of the angles of a triangle are shown in the figure below. Solve
for x.
Answer:
x = 5
Step-by-step explanation:
To solve this problem, we need to utilize the triangle sum theorem.
⭐What is the triangle sum theorem?
the sum of all angles in a triangle equals to 180°1. Set up the equation as per the triangle sum theorem
(5x+6) + 43 + 106 = 180
2. Add the constants
5x + 155 = 180
3. Subtract 155 from the LHS and RHS
5x = 25
4, Divide the LHS and RHS by 5 to isolate x
∴ x = 5
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Given: ABCD is a parallelogram with AE = 9x−5, AC = 14x + 34. Find AC
The value of AC according to given equation of Parallelogram is 188 units.
What is parallelogram?
In elementary geometry, a parallelogram may be a quadrilateral with 2 pairs of parallel sides. the alternative or facing sides of a quadrangle square {measure} of equal length
Main body:
according to question:
AE = 9X-5
AC = 14X+34
as E is midpoint of AC so , we can say
2AE = AC
2(9x-5)= 14x+34
18x-10= 14x+34
4x = 44
x = 11
Now we need to find AC = 14x+34
= 14*11+34
= 188 units
Hence the value of AC according to given equation of Parallelogram is 188 units.
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The value of y varies directly with x. When the value of y is , the value of x is 6.
What is the value of y when the value of x is ?
A. 4
B. 2 1/8
C. 12 3/4
D. 34
As x and y are directly proportional the value of y is 9 when the value of x is 15.
What are ratio and proportion?A ratio is a comparison between two similar quantities in simplest form.
Proportions are of two types one is the direct proportion in which if one quantity is increased by a constant k the other quantity will also be increased by the same constant k and vice versa.
In the case of inverse proportion if one quantity is increased by a constant k the quantity will decrease by the same constant k and vice versa.
Given, The value of y varies directly with x.
So, y ∝ x, Or y = kx, where k is the constant of proportionality.
When the value of y is 6 the value of x is 10.
Therefore, 6 = 10k.
k = 6/10 ⇒ 3/5.
Now the value of x is 15 hence the value of y will be,
y = (3/5).15.
y = 3×3 = 9.
The question can be formed as follows :
Q. x and y are directly proportional, when x is 10, y is 6. what is the value
y when x is 15?
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The following coordinates are each a result of a single transformation. What type of transformation occurred for each image?
D (3,7)---> D' (-7,3) Select One
Answer:
This is a 90° counterclockwise rotation
Step-by-step explanation:
When you rotate a point 90 °counterclockwise the x x and y coordinate values interchange and the new x coordinate has a negative sign while the y-coordinate retains the original sign
Find the distance traveled by a particle with position (x, y) as t varies in the given time interval. Compare with the length of the curve.
x = sin2t, y = cos2t, 0≤t≤3π
The distance traveled by a particle with position (x, y) as t varies in the time interval 0 ≤ t ≤ 3π be, 6π units.
Given, a particle with position (x, y) move across a distance as t varies in the given time interval 0 ≤ t ≤ 3π
where, x = sin2t, y = cos2t
On using the length of the curve concept, we get
d = ∫√((dx/dt)² + (dy/dt)²) dt
as, x = sin2t and y = cos2t
dx/dt = 2cos2t
dy/dt = -2sin2t
On substituting the values, we get
d = ∫√(4cos²2t + 4sin²2t) dt
d = ∫2 dt
d = 2∫dt
d = 2×(3π - 0)
d = 6π
So, the distance traveled by the particle be, 6π
Hence, the distance traveled by the particle be, 6π
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a professor at a certain school polled 12 colleagues about the number of meetings they attended in the last five years (x) and the number of papers they submitted to peer reviewed journals (y) during the same period. the summary data are as follows: n
The number of meetings they attended in the last five years (x) and the number of papers they submitted to peer reviewed journals (y) during the same period is [tex]-8.6+3.15[/tex].
What is formula for slope and intercept is?
[tex]$$\begin{aligned}& b=\frac{n \sum x y-\left(\sum x\right)\left(\sum y\right)}{n \sum x^2-\left(\sum x\right)^2} \\& a=\bar{y}-b \bar{x} \\& \hat{y}=a+b x\end{aligned}$$[/tex]
The slope is
[tex]$$\begin{aligned}b & =\frac{n \sum x y-\left(\sum x\right)\left(\sum y\right)}{n \sum x^2-\left(\sum x\right)^2} \\& =\frac{12 \times 318-(12 \times 4)(12 \times 4)}{12 \times 232-(12 \times 4)^2} \\& =3.15\end{aligned}$$[/tex]
The intercept is
[tex]$$\begin{aligned}a & =\bar{y}-b \bar{x} \\& =4-3.13 \times 4 \\& =-8.6\end{aligned}$$[/tex]
The regression equation is
[tex]$$\begin{aligned}\hat{y} & =a+b x \\& =-8.6+3.15 x\end{aligned}$$[/tex]
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q.16 find the number of ways to put eight different books in five boxes, of no box is allowed to be empty. (5 points)
The number of ways to put eight different books in five boxes, of no box is allowed to be empty will be 403200.
As, it is given that,
No box is allowed to be empty.
An arrangement of items in a specific order is referred to as a permutation. Here, the components of sets are organized in a linear or sequential manner. For instance, the set A=1,2 has a permutation of 2, which is 1,2, or 2,1. There is no alternative way to organize the components of set A,
So, we can put one book in each box in [tex]8 P_5[/tex] mays.
Now remaining 3 books we can put in by [tex]${ }_5 P_3$[/tex] ways
Total number of ways will be [tex]$=8 P_5 \cdot 5 P_3$[/tex]
Now calculating the value,
[tex]=\frac{8 !}{(8-5) !} * \frac{5 !}{(5-3) !}\\\\=\frac{8 !}{3 !} \times \frac{5 !}{2 !}\\\\=403200[/tex]
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A stamp collection consists of 3c 12c and 15c stamps. The number of 3c stamps is five times the number of 12c stamps. The number of 15c stamps is four less than the number of 12c stamps. The total value of the stamps in the collection is $3.18. What is the number of 15c stamps in the collection?
For a data set of weights (pounds) and highway fuel consumption amounts (mpg) of ten types of automobile, the linear correlation coefficient is found and the P-value is 0.013. Write a statement that interprets the P-value and includes a conclusion about linear correlation
The P-value indicates that the probability of a linear correlation coefficient that is at least as extreme is 1.3% which is low so there is sufficient evidence to conclude that there is a linear correlation between weight and highway fuel consumption in automobiles.
Given that,
The complete statement is,
The p-value is given as:
[tex]p=0.013[/tex]
Express as percentage
[tex]p=1.3[/tex] %
This means that the probability that the linear correlation coefficient is at least as extreme is 1.3%.
From the z-table, we have:
[tex]z = 1.81[/tex], when the p-value is 0.013
This value of z-score is low, compared to the percentage of the scores have a z-score of between -1 and 1.
So, the p-value implies that it is likely that there is a linear correlation between the variables.
Hence, the texts that complete the three blanks are: "1.3%", "low" and "is"
Therefore,
The P-value indicates that the probability of a linear correlation coefficient that is at least as extreme is 1.3% which is low so there is sufficient evidence to conclude that there is a linear correlation between weight and highway fuel consumption in automobiles.
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Johanntakes a random sample of 101010 of his coworkers and asks them each howmany hours they worked in the past week. Assume that their hours workedare independent, and let HHH represent the average number of hours worked. Is HHH a binomial variable
The average number of hours worked is not a binomial variable as there is no success or failure in the scenario.
What is binomial variable?In probability theory and statistics, the discrete probability distribution of the number of successes in a series of n independent experiments, each asking a yes-or-no question and each with its own Boolean-valued outcome: success or failure, is known as the binomial distribution with parameters n and p.
A variable with two possible forms is said to have a binomial distribution. It could result in success or failure, heads or tail, and yes or no. If a coin is tossed, for instance, it could land on its head or its tail. It is a binomial variable as a result.
Here, there may be more than two values for the typical hours worked. It is not a binomial variable as a result.
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Help Please!
Aaden took a taxi from his house to the airport. The taxi company charged a pick-up fee of $3.10 plus $2.75 per mile. The total fare was $27.85, not including the tip. Write and solve an equation which can be used to determine x, the number of miles in the taxi ride.
The number of miles the taxi rode are 9 miles.
What is an equation?In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign =.
Given that, the taxi company charged a pick-up fee of $3.10 plus $2.75 per mile. The total fare was $27.85.
Let the number of miles be x.
Now, 3.10+2.75x=27.85
2.75x=27.85-3.10
2.75x=24.75
x=24.75/2.75
x=9 miles
Therefore, the number of miles the taxi rode are 9 miles.
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What is the fourth root of 625
The required answer is the fourth root of 625 is 5. In summary, the fourth root of 625 is 5.
To find the fourth root of 625, we need to determine the number that, when raised to the power of 4, equals 625. Here's the step-by-step process:
Step 1: Start with the number 625.
Step 2: Take the fourth root of 625.
Since we're looking for the fourth root, we need to find a number that, when raised to the power of 4, equals 625.
Step 3: Determine the number that, when raised to the power of 4, equals 625.
To find this number, we can use the concept of exponentiation. Let's denote the unknown number as "x". Mathematically, we have [tex]x^4 = 625[/tex]..
Step 4: Solve the equation [tex]x^4 = 625[/tex].
To solve this equation, we need to find the value of x. Taking the fourth root of both sides of the equation, we have [tex]\sqrt[4]{(x^4)} =\sqrt[4] {625}.[/tex]
The fourth root of 625 is (625)^(1/4).
Step 5: Simplify the expression.
We can simplify [tex]\sqrt[4]{ 625}[/tex] as follows:
[tex]\sqrt[4]{ 625} = \sqrt[4]{ 5^4}= 5^{4/4} = 5^1 = 5.[/tex]
Therefore, the fourth root of 625 is 5. In summary, the fourth root of 625 is 5.
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which polynomial is the product of (4x-3)(x^2+2x-6)
Answer:
4x³ + 5x² - 30x + 18
Step-by-step explanation:
(4x - 3)(x² + 2x - 6)
each term in the second factor is multiplied by each term in the first factor.
4x(x² + 2x - 6) - 3(x² + 2x - 6) ← distribute parenthesis
= 4x³ + 8x² - 24x - 3x² - 6x + 18 ← collect like terms
= 4x³ + 5x² - 30x + 18
In a conceptual replication, the _________ are the same, but the _________ is/are different from the original study.a. independent variables; dependent variablesb. variables; operationalizationc. methods; participantsd. researchers; outcomes
In a conceptual replication, the variables are the same, but the operationalization is/are different from the original study.
The researcher seeks to recreate the basic idea of the research in a conceptual replication. An exact replication involves the researcher reproducing, or attempting to duplicate, the precise circumstances of the original study.
A replication attempt in which the primary effect of interest is the same but tested in a different sample and captured in a different way than that reported originally .
A conceptual replication is sometimes used to investigate what constraints limit the extent to which an effect may be observed and generalised (e.g., only within certain contexts, with certain samples, and using certain measurement methodologies) in order to evaluate and advance theory.
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Multiply.
(3x-w) (3x-7w)
Simplify your answer.
Answer:
7w^2+3x^2-24wx
Step-by-step explanation:
we distribute
(3x-w)(3x-7w)
3x(3x)+3x(-7w)-w(3x)-w(-7w)
3x^2-21wx-3wx+7w^2
combine like terms
3x^2-24wx+7w^2
order correctly
7w^2+3x^2-24wx
hopes this helps
Answer:
[tex]9x^2-24xw+7w^2[/tex]
Step-by-step explanation:
Given expression:
[tex](3x-w)(3x-7w)[/tex]
[tex]\boxed{\begin{minipage}{5.5 cm}\underline{FOIL Method}\\\\$(a+b)(c+d)=ac+ad+bc+bd$\\ \end{minipage}}[/tex]
Apply the FOIL method to expand the brackets:
[tex]\implies 3x \cdot 3x +3x \cdot(-7w)+(-w) \cdot 3x+(-w) \cdot (-7w)[/tex]
[tex]\implies 9x^2-21xw-3xw+7w^2[/tex]
Combine like terms:
[tex]\implies 9x^2-24xw+7w^2[/tex]
What percent of 120 is to 48?
a. 40%
b. 57.6%
c. 0.4%
d. 2.5%
Answer: a. 40%
Step-by-step explanation: 48% of 120 is 40
Answer: a. 40%
Step-by-step explanation:
48/120 = x/100
butterfly method
120x/120 = 4800/120
simplify.
x = 40%
If 2w - 1 = 13, what is the value of 2w?
6
14
7
12
Answer: 7
Step-by-step explanation: 13+1=14
2*?=14 2*7=14
What is 4 + 4? ....................................
The value of 4 plus 4 will be 8.
How to illustrate the expression?Expression simply refers to the mathematical statements which have at least two terms which are related by an operator and contain either numbers, variables, or both.
It should be noted that addition, subtraction, multiplication, and division are all possible mathematical operations. As an illustration, the expression x + y has the terms x and y with an addition operator between them.
In this case:
4 + 4
= 8
The value is 8.
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State the Domain, Range, in interval notation.
Domain:
Range:
The range and domain of the graph shown is
domain -4 ≤ x < ∞
range ∞ ≤ x < 3
How to find the domain and the rangeAll of a function's x-values, or inputs, make up the domain, and all of a function's y-values, or outputs, make up the range.
The domain of a graph is every value in the graph, from left to right.
Examining the curve the extreme left has x coordinate value of -4 and the right end have an arrow which means the value continues. this is written as
-4 ≤ x < ∞
The graph's entire range, from lower to higher numbers, in the vertical line represents the range.
Examining the curve the lowest point the curve reached in y-coordinate have an arrow which means the value continues and the top end have the value of 3. this is written as
∞ ≤ x < 3
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Due to the tide, the water level rises and falls daily in Buzzard's Bay. D gives the depth of the water, in meters, t hours after midnight on a certain day. What is the best interpretation for the following statement? The value of the derivative of D at t = 8 is equal to -0.5. Choose 1 answer a. At 8 a.m. the water level decreased at a rate of 0.5 meters per hour. b. At 8 a.m. the water level decreased at a rate of 0.5 meters. c. At 8 a.m. the water level was 0.5 meters below sea level. d. Until 8 a.m. the water level decreased at a rate of 0.5 meters per hour.
The correct interpretation for the statement D'(8) = -0.5 is (c) At 8 a.m. the water level was 0.5 meters below sea level.
How to interpret the function notation D'(n)From the question, we have the following parameters that can be used in our computation:
The function D gives the depth of the water, in meters, t hours after midnight on a certain day
This means that
D = depth in meters
t = time in hours
From the question, we have
Notation = D(n)
Also, from the question, we have
D'(8) = -0.5
This is the inverse of the function D(t)
So, the values of the parameters are
D(t) = -0.5
t = 8
This means that the number of hours is 8 and the depth is -0.5 meters
Hence, the interpretation of D'(8) = -0.5 is (c) At 8 a.m. the water level was 0.5 meters below sea level.
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Max found a car he wants to buy that costs $16,000. He can
afford to pay $250 a month for the car. His bank offers him a car
loan of 7.3%. How will the length of his loan be for this payment
amount?
Answer: To calculate the length of a loan, you need to know the total cost of the loan, the monthly payment amount, and the interest rate. In this case, the total cost of the car is $16,000, the monthly payment amount is $250, and the interest rate is 7.3%.
First, you need to calculate the total interest that will be paid on the loan. This can be done by multiplying the total cost of the loan by the interest rate, and then dividing by 100 to convert the result to a percentage. In this case, the total interest paid would be $16,000 * 7.3 / 100 = $1,168.
Next, you need to subtract the total interest from the total cost of the loan to find the total amount of the loan that will be used to pay for the car. In this case, the total amount of the loan used to pay for the car would be $16,000 - $1,168 = $14,832.
Finally, you can use this total amount and the monthly payment amount to calculate the length of the loan by dividing the total amount by the monthly payment amount. In this case, the length of the loan would be $14,832 / $250 = 59.32 months, or just over four years. The answer is 59.32 months.
Step-by-step explanation:
Using the graph below, find f(-2)
Answer:
f(-2) =3
Step-by-step explanation:
f(-2) = value of the function when x = -2
Using the graph find the value of the function when x = -2
The value is 3
f(-2) =3
Question 7
Provide an appropriate response.
Suppose x is a uniform random varlable with a = 40 and b = 80. Find the probability that a randomly selected observation is between 43 and 75.
Mary weighed 7 1/2pounds when she was born. What number makes the statement true? 7 1/2 = ?/2
Complete the square for the expression. Also, identify the resulting expression as a binomial squared.
x2 + 15x + ____
The complete square for the binomial expression is x² + 15x + [ 15²/4 ].
Square of a binomial:→ The squaring of a binomial will always give a trinomial.
For example,
(a + b)(a + b) = (a + b)² = a² + 2ab + b²
(ax - b) (ax - b) = (ax - b)² = ax² - 2axb + b²
→ If a trinomial (ax² + bx + c ) is perfect square then
The formula for constant term c = [ b/2√a ]²
Here we have
x² + 15x + __ is a square of the expression
As we know constant term c = [ b/2√a ]²
=> Constant term of (x² + 15x + __) = [ 15/2√1 ]² = [ 15²/4 ]
Therefore the complete expression is x² + 15x + [ 15²/4 ]
Here divide and multiply by 2 with 15x
=> [tex]x^{2} + 2 (\frac{15x}{2}) (1) + (\frac{15}{2})^{2}[/tex]
The above expression is in the form of (a + b)² = a² + 2ab + b²
=> [tex]x^{2} + 2 (\frac{15x}{2}) (1) + (\frac{15}{2})^{2}[/tex] = [tex][x + \frac{15}{2} ]^{2}[/tex]
Therefore,
The complete square for the binomial expression is x² + 15x + [ 15²/4 ].
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Mr. Sharp spent the day at the races. His horses were placed in 8 out of 20 races. In what % of the
races did his horses place?
Answer:
40 %
Step-by-step explanation:
To find the percentage, take the number won and divide by the total
8/20 = .4
Then multiply by 100%
.4 * 100%
40 %
