The parametric equation for the tangent line to the curve is x = 1 - t, y = t, z = 1 - t.
For this question,
The curve is given as
x(t)=e^-t cos(t),
y(t) =e^-t sin(t),
z(t)=e^-t
The point is at (1,0,1)
The vector equation for the curve is
r(t) = { x(t), y(t), z(t) }
Differentiate r(t) with respect to t,
x'(t) = -e^-t cos(t) + e^-t (-sin(t))
⇒ x'(t) = -e^-t cos(t) - e^-t sin(t)
⇒ x'(t) = -e^-t (cos(t) + sin(t))
y'(t) = - e^-t sin(t) + e^-t cos(t)
⇒ y'(t) = e^-t ((cos(t) - sin(t))
z'(t) = -e^-t
Then, r'(t) = { -e^-t (cos(t) + sin(t)), e^-t ((cos(t) - sin(t)), -e^-t }
The parameter value corresponding to (1,0,1) is t = 0. Putting in t=0 into r'(t) to solve for r'(t), we get
⇒ r'(t) = { -e^-0 (cos(0) + sin(0)), e^-0 ((cos(0) - sin(0)), -e^-0 }
⇒ r'(t) = { -1(1+0), 1(1-0), -1 }
⇒ r'(t) = { -1, 1, -1 }
The parametric equation for line through the point (x₀, y₀, z₀) and parallel to the direction vector <a, b, c > are
x = x₀+at
y = y₀+bt
z = z₀+ct
Now substituting the (x₀, y₀, z₀) as (1,0,1) and <a, b, c > into x, y and z, respectively to solve for the parametric equation of the tangent line to the curve, we get
x = 1 + (-1)t
⇒ x = 1 - t
y = 0 + (1)t
⇒ y = t
z = 1 + (-1)t
⇒ z = 1 - t
Hence we can conclude that the parametric equation for the tangent line to the curve is x = 1 - t, y = t, z = 1 - t.
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can someone help me with the graph I don't get it thanks
Answer: Refer to the histogram diagram below
The frequency table is in another attachment, and may be optional.
======================================================
Explanation:
The first task is to sort the values from smallest to largest. Luckily the data set is already sorted for us.
The smallest number is 22. This is the min. Since we're doing intervals of 10 percent, this means the first bin spans from 20% to 29%, assuming we're only involving whole number percentages. Then the next bin is from 30% to 39%, and so on, until reaching the last bin of 90% to 99%
The bin labels are shown along the bottom of the histogram.
-----------
The height of each bar represents how many students are in that specific bin. The first bar is 1 unit tall because there's only one student in the 20% to 29% range (that 22% mentioned earlier).
The next bar is 2 units tall because of the scores of 30% and 37%. They are in the range from 30% to 39%
The next bar is 4 units tall because of the scores 45,46,46,49.
This process is done with the other values to get the histogram you see below.
The frequency table is shown in a separate attachment, but your teacher may only want the histogram part. I would ask him/her for clarification.
The third column of the frequency table is optional to show where all the scores fit (and to help confirm the frequencies). This will help confirm the answer. Most histograms commonly only show the first two columns.
------------
While it's a good thing to know how to do this by hand, it's recommend to use software to generate the histograms when it comes to real world applications. Eg: making a presentation to the boss.
The program I used was LibreOffice which is free and effectively identical to excel. There are other options you could pick from.
Side note: There are no gaps between the bars when it comes to histograms. In contrast, a bar chart will have gaps.
b)______ is the number of times a particular value occurs in a given data
Answer:
the frequency is the number of times a particular value occurs in a given data.
Find the slope of the line shown below
Answer:
[tex] \frac{3}{2} [/tex]
Step-by-step explanation:
From the attached photo, we can deduce:
(2,2) as (x1,y1)
(-2,-4) as (x2,y2)
Slope Formula =
[tex] \frac{y1 - y2}{x1 - x2} [/tex]
Now we can substitute these values into the formula to find the slope.
Slope of line =
[tex] \frac{2 - ( - 4)}{2 - ( - 2)} \\ = \frac{2 + 4}{2 + 2} \\ = \frac{6}{4} \\ = \frac{3}{2} (reduced \: to \: simplest \: form)[/tex]
Jonathan uploaded some original songs and also some pictures. alissa uploaded four times as many songs, but only uploaded twice as many pictures. jonathan uploaded 11 items while alissa uploaded 24. how many songs did jonathan upload? (1 point) 1 4 6 10
X is the number of songs and Y the number of pics
our 2 equations will be - X+y=11 & 4x+2y=24
consider x=11-y
Lets replace x in 2:
=4(11-y) +2y=24
=44-4y+2y=24
=44-24= 4y-2y
=20= 2y
=Y=20/2
=Y=10
Lets replace y in 1
=X+10=11
=X=11-10
=X=1
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Find the median of the data set. round your answer to the nearest tenth if necessary. 13,13,15,15,15,15,17,17,19,20
Answer:
Hey Desmariesin7587, the median of the data set that you gave is 15.9
Step-by-step explanation:
Since the amount of values in the data set was not even, I could not find a middle number. Instead, I added up all of the values and divided the total sum by the amount of values that there were; in this case, 10.
----------------------
Have a great day,
Nish
The area of the regular octagon is approximately 54 cm2.
A regular octagon has an apothem with length 4 centimeters and an area of 54 centimeters squared. Line segment A B is a side of the octagon.
What is the length of line segment AB, rounded to the nearest tenth?
3.4 cm
4.8 cm
24 cm
27 cm
The length of the line segment of octagon which is AB equal to 3.4 cm.
Given that the area of the regular octagon is 54 [tex]cm^{2}[/tex], apothem of regular octagon is 4 cm. Line segment AB is side of the octagon.
We are required to find the length of the line segment AB.
A regular octagon has eight equal sides.
Area of regular octagon=Perimeter*apothem/2-----------1
We have to put the value of area and apothem in equation 1.
54=perimeter*4/2
Perimeter=(54*2)/4
=27 cm
Perimeter=27 cm.
Now we know that the perimeter is equal to the sum of the lengths of all sides of figure. So,
Perimeter=8*side
27=8*AB
AB=27/8
=3.375 cm
After rounding off it will be equal to 3.4 cm.
Hence the length of line segment is equal to 3.4 cm.
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Answer:
A
Step-by-step explanation:
Michael has $15 and wants to buy a combination of school lunches to feed at least three classmates. A sandwich costs $2, and hot lunch costs $3.
This system of inequalities models the scenario:
2x + 3y ≤ 15
x + y ≥ 3
Part A: Describe the graph of the system of inequalities, including shading and the types of lines graphed. Provide a description of the solution set. (4 points)
Part B: Is the point (5, 1) included in the solution area for the system? Justify your answer mathematically. (3 points)
Part C: Choose a point in the solution set and interpret what it means in terms of the real-world context. (3 points)
The description of this graph tells us that the equation 2x + 3y ≤ 15 was shaded to the bottom while x + y ≥ 3 was shaded to the top as shown in the diagram.
How to create and explain the grapha. The first equation of the graph when it was drawn was shaded to the bottom, while the upper part of the shading has is that of the equation x + y ≥ 3.
b. (5,1) are the solutions for the equation 2x+3y<15. This is given that when x and y are substituted we would still have less than 15.
C The point in the graph would be 2,3. At this point, Michael would be able to buy 2 sandwiches and 3 hot lunches. This is still within his budget for 3 students.
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Answer:
Step-by-step explanation:
2,3 is the anwser
pleaseeee hurrryyyyyyyy
Answer:I THINK IT IS 1,-3
Step-by-step explanation:
Find two positive numbers satisfying the given requirements. the product is 242 and the sum is a minimum. (smaller value) (larger value)
The two positive numbers satisfying the given requirements are 15.56 and 15.56
For given question,
Let x and y be two positive numbers satisfying the given requirements.
⇒ xy = 242 .............(1)
The sum of given two positive numbers is a minimum.
Let the sum of given two positive numbers is S.
⇒ x + y = S ............(2)
From equation(1),
⇒ y = 242/x
Substitute above value of y in equation (2),
⇒ x + y = S
⇒ S = x + (242 / x)
Now, for above equation we find the derivative of x with respect to x.
⇒ 0 = 1 - [tex]\frac{242}{x^{2} }[/tex]
⇒ 242/x² = 1
⇒ x² = 242
⇒ x = ±15.56
Since the numbers are positive, x = 15.56
For x = 15.56
⇒ y = 15.56
Therefore, the two positive numbers satisfying the given requirements are 15.56 and 15.56
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If a polynomial function, f(x), with rational coefficients has roots 3 and startroot 7 endroot, what must also be a root of f(x)?
The conjugate which is 3 - √7 is also a root of f(x).
What is the root of a polynomial function?The root of a polynomial function f(x) is the value of x for which f(x) = 0.
Now if a polynomial function has a root x = a + √b then the conjugate of x which is x' = a - √b is also a root of the function, f(x).
What must also be a root of f(x)?
Given that the polynomial function, f(x), with rational coefficients has roots
3 + √7, then by the above, the conjugate which is 3 - √7 is also a root of f(x).
So, 3 - √7 is also a root of f(x).
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What are the solutions of the equation x4 6x2 5 = 0? use u substitution to solve.
The value of the x from the given equation is a +i and -i and +5i and -5i.
According to the statement
we have given that the equation [tex]x^4 + 6x^2 +5 =0[/tex]
and we have to find the value of x by the u substitution.
So,
The given equation is
[tex]x^4 + 6x^2 +5 =0[/tex]
Put [tex]x^2 = u[/tex]
By substitution method
Then the equation become
[tex]u^2 +6u +5 = 0[/tex]
And solve the equation then
[tex]u^2+ u +5u +5 = 0[/tex]
Then
u(u +1) + 5(u +1) = 0
(u+5) (u +1) = 0
Then
the value of u is -1 and -5 then
The value of x is
[tex]x = \sqrt{u}[/tex]
x = √-5 And √-1
Then the value of x is +i and -i and +5i and -5i.
So, The value of the x from the given equation is a +i and -i and +5i and -5i.
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The equation 2x2 − 12x 1 = 0 is being rewritten in vertex form. fill in the missing step. given 2x2 − 12x 1 = 0 step 1 2(x2 − 6x ___) 1 ___ = 0 step 2 2(x2 − 6x 9) 1 − 18 = 0 step 3 ✔ 2(x − 3)2 17 = 0 2(x − 3)2 − 17 = 0 2(x 3)2 − 17 = 0 2(x − 6)2 − 17 = 0
Given: [tex]2x^2 - 12x + 1 = 0[/tex]
Step 1: [tex]2(x^2- 6x + 3^2) + 1 - 2(3^2) = 0[/tex]
Step 2: [tex]2(x^2 - 6x + 9) + 1 - 18 = 0[/tex]
Second answer: [tex]2(x - 3)^2 - 17 = 0[/tex]
Explanation in detail:
The quadratic equation a[tex]x^{2} + bx + c = 0[/tex]'s vertex form is
where a[tex](x - h)^{2} + k[/tex] equals zero
A is the [tex]x^{2}[/tex] coefficient, and h is the vertex's x-coordinate on the equation's graph.
The vertex of the equation's graph, k, has the y-coordinate.
The completing square can be used to determine the vertex form.
[tex]2x^{2} - 12x + 1 = 0[/tex] is the equation.
Put[tex]2x^{2} -12x[/tex] in a bracket and subtract 2 from them as a common component to utilize the completing square.
∵ [tex]2(x^{2} - 6x) + 1 = 0[/tex]
To determine the product of the first and second terms in the binomial, divide the second term by two.
∵ [tex]6x / 2 = 3x[/tex]
∵[tex]3x = 3 * x[/tex]
∴ The binomial's first and second terms are x and 3, respectively.
The phrase in the bracket's middle is (-)
∴ Middle of the binomial's sign is (-)
∴ The binary value is [tex](x - 3)^2[/tex]
Square 3 is nine.
To keep the equation from altering, you must deduct the same number from the bracket after adding 9 there.
∴ [tex]2(x^2 - 6x + 9 - 9) + 1 = 0[/tex]
- Remove the brace and multiply -9 by 2.
∵ [tex]2 *-9 = -18[/tex]
∴ [tex]2(x^2- 6x + 9) + 1 - 18 = 0[/tex]
- Construct a square binomial from [tex](x^{2} - 6x + 9)[/tex]
∵ [tex]x^2 - 6x + 9 = (x - 3)^2[/tex]
∴ [tex]2(x - 3)^2 + 1 - 18 = 0[/tex]
Include related terms.
∴[tex]2(x - 3)^2 - 17 = 0[/tex]
∴ The equation's vertex form is[tex]2(x - 3)^2 - 17 = 0.[/tex]
Given that: [tex]2x^2 - 12x + 1 = 0[/tex]
Step 1: [tex]2(x^2- 6x + 3^2) + 1 - 2(3^2) = 0[/tex]
Step 2: [tex]2(x^2 - 6x + 9) + 1 - 18 = 0[/tex]
Step 3: [tex]2(x - 3)^2 - 17 = 0[/tex]
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the square root of 7956
Answer:
89.19
Step-by-step explanation:
8 9 . 1 9
79,56 . 00, 00
8 | 64
| 1556
169 | 1521
| 3500
1781 | 1781
1719
17829| 6200
Solve following equation:
[tex]4x+10=-2+4[/tex]
Step-by-step explanation:
4x + 10 = -2 + 4
4x = -2 + 4 - 10
4x = -8
x = -2
[tex]\huge\text{Hey there!}[/tex]
[tex]\huge\textbf{Equation:}[/tex]
[tex]\mathsf{4x + 10 = -2+ 4}[/tex]
[tex]\huge\textbf{Simplify it:}[/tex]
[tex]\mathsf{4x + 10 = 2}[/tex]
[tex]\huge\textbf{Subtract 10 to both sides:}[/tex]
[tex]\mathsf{4x + 10 - 10 = 2 - 10}[/tex]
[tex]\huge\textbf{Simplify it:}[/tex]
[tex]\mathsf{4x = 2 - 10}[/tex]
[tex]\mathsf{4x = -8}[/tex]
[tex]\huge\textbf{Divide 4 to both sides:}[/tex]
[tex]\mathsf{\dfrac{4x}{4} = \dfrac{-8}{4}}[/tex]
[tex]\huge\textbf{Simplify it:}[/tex]
[tex]\mathsf{x = \dfrac{-8}{4}}[/tex]
[tex]\mathsf{x = -2}[/tex]
[tex]\huge\textbf{Therefore, your answer should be:}[/tex]
~[tex]\frak{Amphitrite1040:)}[/tex]
A piece of PVC material weighs 0.063 pounds per cubic inch. What's the weight of a 36-inch piece of PVC pipe with an outside diameter of 0.82 inches and an inside diameter of 0.75 inches?
The weight of a 36-inch piece of PVC pipe is 0.20 pounds
How to determine the weight of a 36-inch piece of PVC pipe?The given parameters about the 36-inch piece of PVC pipe are
Height, h = 36 inches
Outside diameter, D = 0.82 inches
Inside diameter, d= 0.75 inches
The volume of the PVC pipe is then calculated as:
V = πh(D/2)^2 - πh(d/2)^2
Substitute the known parameters in the above equation
V = 3.14 * 36 * (0.82/2)^2 - 3.14 * 36 * (0.75/2)^2
Evaluate the quotients
V = 3.14 * 36 * 0.41^2 - 3.14 * 36 * 0.375^2
Evaluate the exponents
V = 3.14 * 36 * 0.1681 - 3.14 * 36 * 0.140625
Evaluate the products
V = 19.002024 - 15.89625
Evaluate the difference
V = 3.105774 cubic inches
From the question, we have:
Weight = 0.063 pounds per cubic inch
So, the total weight is
Total weight = 3.105774 cubic inches * 0.063 pounds per cubic inch
Evaluate the product
Total weight = 0.195663762 pounds
Approximate
Total weight = 0.20 pounds
Hence, the weight of a 36-inch piece of PVC pipe is 0.20 pounds
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A set of n = 25 pairs of scores (x and y values) has a pearson correlation of r = 0. 80. how much of the variance for the y scores is predicted by its relationship with x?
The amount of variance for y that is predicted by its relationship with x is 64%.
How much variance is predicted?
Variance measures the rate of dispersion of a data point around the dataset. It can be calculated by finding the square of the standard deviation of a dataset. Variance measures the variation of a data set.
Correlation is a statistical measure used to measure the linear relationship that exists between two variables. The greater the correlation coefficient is closer to one, the greater the linear relationship that exists between the two variables. A positive correlation occurs when the two variables move in the same direction.
Variance = (correlation coefficient²) x 100
(0.80²) x 100
0.64 x 100 = 64%
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The ratios of teachers:male students :female students is 2:17:18 the total number of students is 665 find the number of teachers
Answer: 38 teachers.
Step-by-step explanation:
Let the share ratio be x.
So, quantity of teachers = 2x, quantity of male students = 17x,
quantity of female students = 18x.
17x+18x=665
35x=665
Divide the left and right sides of the equation by 35:
х=19.
2x=2*19
2x=38.
What is the exact value of cos (67.5°)?
OA.
OB.
OC. -√2+√2
√2+√2
2
-
√2-√2
2
OD. √2+ √2
4
first off, make sure you have a Unit Circle, if you don't do get one, you'll need it, you can find many online.
let's double up 67.5°, that way we can use the half-angle identity for the cosine of it, so hmmm twice 67.5 is simply 135°, keeping in mind that 135° is really 90° + 45°, and that whilst 135° is on the 2nd Quadrant and its cosine is negative 67.5° is on the 1st Quadrant where cosine is positive, so
[tex]cos(\alpha + \beta)= cos(\alpha)cos(\beta)- sin(\alpha)sin(\beta) \\\\\\ cos\left(\cfrac{\theta}{2}\right)=\pm \sqrt{\cfrac{1+cos(\theta)}{2}} \\\\[-0.35em] ~\dotfill\\\\ cos(135^o)\implies cos(90^o+45^o)\implies cos(90^o)cos(45^o)~~ - ~~sin(90^o)sin(45^o) \\\\\\ \left( 0 \right)\left( \cfrac{\sqrt{2}}{2} \right)~~ - ~~\left( 1\right)\left( \cfrac{\sqrt{2}}{2} \right)\implies -\cfrac{\sqrt{2}}{2} \\\\[-0.35em] ~\dotfill[/tex]
[tex]cos(67.5^o)\implies cos\left( \frac{135^o}{2} \right)\implies \pm \sqrt{\cfrac{ ~~ 1-\frac{\sqrt{2} ~~ }{2}}{2}}\implies \stackrel{I~Quadrant}{+\sqrt{\cfrac{ ~~ 1-\frac{\sqrt{2} ~~ }{2}}{2}}} \\\\\\ \sqrt{\cfrac{ ~~ \frac{2-\sqrt{2}}{2} ~~ }{2}}\implies \sqrt{\cfrac{2-\sqrt{2}}{4}}\implies \cfrac{\sqrt{2-\sqrt{2}}}{\sqrt{4}}\implies \cfrac{\sqrt{2-\sqrt{2}}}{2}[/tex]
The function D(t)=18t2−130t gives the distance a car going a certain speed will skid in t seconds. Find the time it would take for the car to skid 150 feet.
The time it will take for the car to skid for 150 feet is 8.23 seconds.
How to find the time the car will skid?The function D(t) = 18t² − 130t gives the distance a car going a certain speed will skid in t seconds.
The time the it would take for the car to skid for 150 feet can be calculated as follows:
Therefore,
D(t) = 18t² - 130t
150 = 18t² - 130t
18t² - 130t - 150 = 0
divide by 3
9t² - 65t - 75 = 0
Hence,
t = 65 ± 5√277 / 18
t = 8.23425471586 or -1.01166666667
Hence,
t = 8.23 seconds.
Therefore, the time it will take for the car to skid for 150 feet is 8.23 seconds.
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3. Choose the two correct transformations for:
Ay
T
R'
RSTU to R'S'T'U'.
S'
A Rotation 180°clockwise
B Reflection across y = x
OC Reflection across y = -x
OD Rotation 180°counterclockwise
Find the value of x
a. 13 b. 14/5 c. 5 d. 8
Using the chord theorem, option C is the best solution since x = 5.
The four line segments that are created by two intersecting chords inside of a circle are explained by the chord theorem, sometimes referred to as the intersecting chords theorem, a statement in fundamental geometry. The products of the line segment lengths on each chord are equal, according to this assertion.
The value of x must be determined in order to answer the question.
We may determine the following using the chord property:
8*x = 4*10,
or, 8x = 40,
or, x = 40/8,
or x = 5.
Given that x = 5, option C is the best option according to the chord theorem.
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For what value of x does 4^x=(1/8)^x+5
A) -15
B)-3
C)3
D) 15
the value of x is -3. Option B
How to determine the valueWE have the expression to be;
4^x=(1/8)^x+5
With the knowledge of indices, we have that;
2 = 2 ^1
4 = 2^2
8 = 2^3
16 = 2^4
32 = 2^5
So from this given information,
we can express the expression as;
4^x = 2^2x
1/8^x = 2^-3x
Substitute into the formula
2^2x = 2^-3x - 15
Cancel out the like terms
2x = -3x - 15
collect like terms
2x + 3x = - 15
x = -15/ 5
x = -3
Thus, the value of x is -3. Option B
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Calculate the perimeter of 5y + 9 (3y+15) cm (7y+3)cm
Answer:
Step-by-step explanation:
assuming that the figure is a triangle with sides (5y + 9) and (3y + 15) and (7y + 3) we find the perimeter by adding everything
5y + 9 + 3y + 15 + 7y + 3 =
15y + 27
or
3(5y+9)
next time put all the data and figures, we are doing you a pleasure, at least put ALL the data.
Evaluate :11/3 + -3/7 + -4/9
Answer:
3.67+(-0.43)+(-0.44)
3.67-0.43-0.44
3.24-0.44
2.8 Ans
find PQ if possible
Answer:
11 units.
Step-by-step explanation:
The graph of y=√x is reflected in the x-axis and then translated
down 3 units. What is an equation that produces this transformation?
A. -√x-3
B. y= -3-(√x)
C. y= 3- (√x)
D. y= -(√x-3)
Given that………………………..
[tex]\sum^{\infty}_{n=1} (a/b)^n=5 \\ \\ =\frac{a/b}{1-\frac{a}{b}}=5 \\ \\ \frac{a}{b-a} =5 \\ \\ \frac{a}{b}=\frac{5}{6}[/tex]
So, we need to find
[tex]\sum^{\infty}_{n=1} n(5/6)^n
[/tex]
Let this sum be S.
Then,
[tex]S=(5/6)+2(5/6)^2 +3(5/6)^3+\cdots \\ \\ \frac{5}{6}S=(5/6)^2 + 2(5/6)^3+\cdots \\ \\ \implies \frac{1}{6}S=(5/6)+(5/6)^2+(5/6)^3+\cdots=5 \\ \\ \implies S=\boxed{30}[/tex]
Solve using the quadratic formula.
n2 + 2n + 1 = 0
Write your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.
n = ____
or n = ____
i wasnt ever smart lol please help quick
Answer:
-1
Step-by-step explanation:
n^2+2n = -1
n(n+2) = -1
4|a| + 8 -a, if a = -2
PLEASE HELP!!!!!!!!!!!!!! I'm stuck on this!!
4|a| + 8 - a, if a = -2
What we're doing here is plugging in the given value of a for every instance of a in the expression.
4|-2| + 8 - (-2)
---The absolute value of a number is its distance from 0. -2 is 2 units away from 0, so the absolute value of -2 is 2. And, remember two negatives make a positive!
4(2) + 8 + 2
8 + 8 + 2
16 + 2
18
Hope this helps!
what is 1016/86.89 include all your steps
Answer:
11.692945103
Step-by-step explanation:
take out a calculator divide 1016 by 86.89