Answers
Answer:
See below for answers and explanations
Step-by-step explanation:
Problem 1
Recall that the projection of a vector [tex]u[/tex] onto [tex]v[/tex] is [tex]\displaystyle proj_vu=\biggr(\frac{u\cdot v}{||v||^2}\biggr)v[/tex].
Identify the vectors:
[tex]u=\langle-10,-7\rangle[/tex]
[tex]v=\langle-8,4\rangle[/tex]
Compute the dot product:
[tex]u\cdot v=(-10*-8)+(-7*4)=80+(-28)=52[/tex]
Find the square of the magnitude of vector v:
[tex]||v||^2=\sqrt{(-8)^2+(4)^2}^2=64+16=80[/tex]
Find the projection of vector u onto v:
[tex]\displaystyle proj_vu=\biggr(\frac{u\cdot v}{||v||^2}\biggr)v\\\\proj_vu=\biggr(\frac{52}{80}\biggr)\langle-8,4\rangle\\\\proj_vu=\biggr\langle\frac{-416}{80} ,\frac{208}{80}\biggr\rangle\\\\proj_vu=\biggr\langle\frac{-26}{5} ,\frac{13}{5}\biggr\rangle\\\\proj_vu=\langle-5.2,2.6\rangle[/tex]
Thus, B is the correct answer
Problem 2
Treat the football and wind as vectors:
Football: [tex]u=\langle42\cos172^\circ,42\sin172^\circ\rangle[/tex]
Wind: [tex]v=\langle13\cos345^\circ,13\sin345^\circ\rangle[/tex]
Add the vectors: [tex]u+v=\langle42\cos172^\circ+13\cos345^\circ,42\sin172^\circ+13\sin345^\circ\rangle\approx\langle-29.034,2.481\rangle[/tex]
Find the magnitude of the resultant vector:
[tex]||u+v||=\sqrt{(-29.034)^2+(2.481)^2}\approx29.14[/tex]
Find the direction of the resultant vector:
[tex]\displaystyle \theta=tan^{-1}\biggr(\frac{2.841}{-29.034}\biggr)\approx -5^\circ[/tex]
Because our resultant vector is in Quadrant II, the true direction angle is 6° clockwise from the negative axis. This means that our true direction angle is [tex]180^\circ-5^\circ=175^\circ[/tex]
Thus, C is the correct answer
Problem 3
We identify the initial point to be [tex]R(-2,12)[/tex] and the terminal point to be [tex]S(-7,6)[/tex]. The vector in component form can be found by subtracting the initial point from the terminal point:
[tex]v=\langle-7-(-2),6-12\rangle=\langle-7+2,-6\rangle=\langle-5,-6\rangle[/tex]
Next, we find the magnitude of the vector:
[tex]||v||=\sqrt{(-5)^2+(-6)^2}=\sqrt{25+36}=\sqrt{61}\approx7.81[/tex]
And finally, we find the direction of the vector:
[tex]\displaystyle \theta=tan^{-1}\biggr(\frac{6}{5}\biggr)\approx50.194^\circ[/tex]
Keep in mind that since our vector is in Quadrant III, our direction angle also needs to be in Quadrant III, so the true direction angle is [tex]180^\circ+50.194^\circ=230.194^\circ[/tex].
Thus, A is the correct answer
Problem 4
Add the vectors:
[tex]v_1+v_2=\langle-60,3\rangle+\langle4,14\rangle=\langle-60+4,3+14\rangle=\langle-56,17\rangle[/tex]
Determine the magnitude of the vector:
[tex]||v_1+v_2||=\sqrt{(-56)^2+(17)^2}=\sqrt{3136+289}=\sqrt{3425}\approx58.524[/tex]
Find the direction of the vector:
[tex]\displaystyle\theta=tan^{-1}\biggr(\frac{17}{-56} \biggr)\approx-17^\circ[/tex]
Because our vector is in Quadrant II, then the direction angle we found is a reference angle, telling us the true direction angle is 17° clockwise from the negative x-axis, so the true direction angle is [tex]180^\circ-17^\circ=163^\circ[/tex]
Thus, A is the correct answer
Problem 5
A vector in trigonometric form is represented as [tex]w=||w||(\cos\theta i+\sin\theta i)[/tex] where [tex]||w||[/tex] is the magnitude of vector [tex]w[/tex] and [tex]\theta[/tex] is the direction of vector [tex]w[/tex].
Magnitude: [tex]||w||=\sqrt{(-16)^2+(-63)^2}=\sqrt{256+3969}=\sqrt{4225}=65[/tex]
Direction: [tex]\displaystyle \theta=tan^{-1}\biggr(\frac{-63}{-16}\biggr)\approx75.75^\circ[/tex]
As our vector is in Quadrant III, our true direction angle will be 75.75° counterclockwise from the negative x-axis, so our true direction angle will be [tex]180^\circ+75.75^\circ=255.75^\circ[/tex].
This means that our vector in trigonometric form is [tex]w=65(\cos255.75^\circ i+\sin255.75^\circ j)[/tex]
Thus, C is the correct answer
Problem 6
Write the vectors in trigonometric form:
[tex]u=\langle40\cos30^\circ,40\sin30^\circ\rangle\\v=\langle50\cos140^\circ,50\sin140^\circ\rangle[/tex]
Add the vectors:
[tex]u+v=\langle40\cos30^\circ+50\cos140^\circ,40\sin30^\circ+50\sin140^\circ\rangle\approx\langle-3.661,52.139\rangle[/tex]
Find the magnitude of the resultant vector:
[tex]||u+v||=\sqrt{3.661^2+52.139^2}\approx52.268[/tex]
Find the direction of the resultant vector:
[tex]\displaystyle\theta=tan^{-1}\biggr(\frac{52.139}{-3.661} \biggr)\approx-86^\circ[/tex]
Because our resultant vector is in Quadrant II, then our true direction angle will be 86° clockwise from the negative x-axis. So, our true direction angle is [tex]180^\circ-86^\circ=94^\circ[/tex].
Thus, B is the correct answer
Related Questions
Stacy’s goal is to run at an average speed of 1 mile per 10 minutes. What is the number of laps that she must run in 1/2 hour to reach her goal?
Answers
Answer:
3 miles (3 laps)
Step-by-step explanation:
Assuming 1 lap is a mile, and 1/2 hour has 30 minutes, 1 mile per 10 minutes gives us 3 miles.
2root 3 + root 3 is equal to?
Answers
Answer:
i believe the answer is C
Step-by-step explanation:
Answer:
(C) 3√3
Step-by-step explanation:
Think of √3 as x.
Then 2x + x = 3x, so
2√3 + √3 = 3√3
PLS ANSWER IT PLS PLS
Answers
Answer:
The one above the orange answer in the picture
Step-by-step explanation:
85% of 20 is 17 every thing else is not 85%
The correct answer would be the one on top left
Emily is 4 years Jacob's junior. Eight years ago. she was half Jacob's age. How old are they?
Answers
Answer:
Emily is now 12.
Jacob is now 16.
Step-by-step explanation:
since Emily is 4yrs younger than Jacob, let Jacob be x and Emily be (x - 4) yrs.
eight yrs ago: Emily is ( x-4-8) = ( x - 12)
Jacob is ( x - 8)
Emily was half Jacob's age: so, we have to balance the ages by either multiplying Emily's age by 2 or dividing Jacob's age by 2. ( I multiplied by 2)
2( x - 12) = x - 8
2x - 24 = x - 8............ Bring the like terms together and solve for x
x = 16, which is Jacob's current age
Emily ( x - 4) = 12 yrs
Use the distribution property to explain how 30 flowers in a bunch can be counted in groups of 2s and 3s.
Answers
The way 30 flowers in a bunch can be counted in groups of 2s and 3s is 2(3(3) + 3(2))
Prime factorization of numbersFrom the question, the interpretation is that we are to write the prime factorization of 30
30 = 2 * 15
30 = 2{3 (3 + 2)}
Expand using the distribution property
30 = 2(3(3) + 3(2))
30 = 2(9 + 6)
Hence the way 30 flowers in a bunch can be counted in groups of 2s and 3s is 2(3(3) + 3(2))
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Find the absolute value of x: (x+8)=-2
Answers
Given that the absolute value of a function is its positive value and the value of x is -10, hence the absolute value of x is 10
How to find the absolute value of a function?The absolute value is the positive value of any given value.
Given the equation
x + 8 = -2
Determine the value of x
x = -2 - 8
x = -10
Since x is -10, hence the absolute value of x is 10
Learn more on absolute value here: https://brainly.com/question/24368848
find the length of the missing side. if your answer is not an integer, leave it in simplest radical form.
Answers
Answer:15h
Step-by-step explanation: because 8m+7m = 15h and the side the H is on is way bigger than the other sides so thats how you know your supposed to add.
Use the order of operations to find the value of the expression.
2
+
3
×
16
−
2
×
21
−
3
A.
5
B.
14
C.
35
D.
44
Answers
Using the order of operations to find the value of the expression, we have; A. 5
How to carry out order of operations?We want to use order of operations to solve the problem;
2 + 3 × 16 − 2 × 21 − 3
Now, the order is called BODMAS which means;
Bracket, Of, Division, Multiplication, Addition, Subtraction.
From the order above and looking at our question, we will first carry out multiplication to get;
2 + 48 - 42 - 3
Next is addition to get;
50 - 42 - 3 = 5
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Laurence earned scores of 87, 80, 75, and 85 on four biology tests. What score must he earn on his next test to have an 85 average?
A. 95
B. 98
C. 80
D. 85
Answer: 98
Answers
To get an 85 average, Laurence needs a score of 25 on his upcoming biology exam.
What is math average?
The middle number, which is determined by dividing the sum of all the numbers by the total number of numbers, is the average value in mathematics.
When calculating the average for a set of data, we first sum up all the values and divide the result by the total number of values.
Average = (sum of values) / (number of values)
The total of Laurence's marks on the first four biology exams can be calculated using the following formula
Sum of scores = 87 + 80 + 75 + 85
= 327
We need the total of all five scores to be 85 x 5 = 425 in order to obtain an average of 85 for five tests.
Therefore, We need the total of all five scores to be 85 x 5 = 425 in order to obtain an average of 85 for five tests.
Next score = (sum of all five scores) - (sum of first four scores)
= (85 x 5) - 327
= 25
Therefore, To get an 85 average, Laurence needs a score of 25 on his upcoming biology exam.
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8 + X = 14 what does X equal to?
Answers
Answer:
x=6
Step-by-step explanation:
Rearrange terms:
8+x=14
x+8=14
Subtract 8 from both sides and simplify.
Given :-
8 + X = 14To find :-
The value of XSolution :-
8+x = 14
Transferring 8 to Right sidex = 14-8 (Signs change when transferred to opposite side)
x = 6
Factor x2 + 3x + 2 A) (x + 3)(x + 1) B) (x + 2)(x + 1) C) (x + 2)(x - 1) D) (x - 2)(x + 1) E) (x - 2)(x - 1)
Answers
Answer:
(x + 1) (x + 2)
Step-by-step explanation:
x² + 3x + 2
We can Split the Middle Term of this expression to factorise it
In this technique, if we have to factorise an expression like ax² + bx + c,
we need to think of 2 numbers such that:
N₁ · N₂ = a · c = 1 · 2 = 2
And,
N₁ + N₂ = b = 3
After trying out a few numbers we get N₁ = 2 and N₂ = 1
2 · 1 = 2 and 2 + 1 = 3
x² + 3x + 2 = x² + 2x + x + 2
x (x + 2) + 1 (x + 2)
(x + 1) (x + 2) , is the factorized form.
Answer:
(x+2)(x+1)
Step-by-step explanation:
[tex]x^{2} +3x+2[/tex]
→Multiply the first and last term's coefficient =1×2=2
Now,
→Factors of 2=2,1
→Using 2 and 1 we must add up to 3
Now we get,
[tex]\hookrightarrow x^{2} +3x+2\\\\\hookrightarrow x^{2} +2x+x+2[/tex]
Here when we add 2x and x we get 3x.
Now,
Take common x from x²+2x as well as 1 from x+2
Then we get,
[tex]\hookrightarrow x^{2} +2x+x+2\\\\\hookrightarrow x(x+2)+1(x+2)\\\\[/tex]
Here, x+2 is common. So we can write both of them as one.
[tex]\hookrightarrow (x+2)(x+1)[/tex]
And it is our final answer.
Hence, the final factors are (x+2)(x+1)
HELP ASAP!!!
This circle is centered at the origin, and the length of its radius is 3. What is
the circle's equation?
A. x + y3 = 27
B. x² + 7 = 9
c. x² + 7 = 3
D. x+y = 3
Answers
Which of the following is the correct factorization of the polynomial below?
p^3 - 125q^3
A. (p - 5q)(p^2 + 5pq + 25q^2)
B. (p - 25q)(p^2 + 25pq + 25q^2)
C. (p^2 + 10q)(p^3 + 25pq + 5q^2)
D. The polynomial is irreducible.
Answers
The correct factored form of the expression p³ - 125q³ is,
(p - 5q)(p² + 5pq + 25q²).
What is a polynomial?A polynomial is an algebraic expression.
A polynomial of degree n in variable x can be written as,
a₀xⁿ + a₁xⁿ⁻¹ + a₂xⁿ⁻² +...+ aₙ.
What is a factor of a polynomial?We know that if x = a is one of the roots of a given polynomial x - a = 0 is a factor of the given polynomial.
To confirm if x - a = 0 is a factor of a polynomial we replace f(x) with f(a) and if the remainder is zero then it is confirmed that x - a = 0 is a factor.
We know, a³ - b³ = (a - b)(a² + ab + b²).
Given, An expression p³ - 125q³.
= p³ - (5q)³.
= (p - 5q)(p² + p.5q + (5q)²).
= (p - 5q)(p² + 5pq + 25q²).
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A tree of 14m height is broken by the wind so that its top touches the ground and makes an angle of 60° with the ground. Find the length of the broken part of the tree.
Please help me,
I'll will mark as BRAINLIEST...
Answers
Answer:
7.50 m
Step-by-step explanation:
The side lengths of the "special triangle" with angles 30°-60°-90° have the ratios 1 : √3 : 2. These ratios can be used to solve the problem.
__
The parts of the tree of interest correspond to the long side and the hypotenuse of a 30-60-90 triangle.
In terms of the above ratio units, the fraction of tree height that is broken off is ...
[tex]\dfrac{\text{hypotenuse}}{\text{hypotenuse $+$ long side}}=\dfrac{2}{2+\sqrt{3}}\approx0.535898[/tex]
So, the length of the broken part of the tree is ...
0.535898 × 14 m ≈ 7.50 m
Please help me yall… my math is due tommorrow!
Answers
Answer:
Step-by-step explanation:
Remark
I think the easiest way to do this is to calculate the probability of those that are 12 to 24. Then subtract the decimal you get from 1. That saves you adding up 3 numbers to get the same answer.
Givens
Total number of movie goers = 3880
Total number that are between 12 and 24 = 1180
Solution
Probability of a random person being drawn from 1180
people between 12 and 24 = Those between 12 and 24 = 1180
Total Number = 3880
P(12 - 24) = 1180 / 3880
P(12 -24) = 0.3041
Remember, that's not what you want. You want everyone else
P(not(12-24)) = 1 - 0.3041
Answer 0.6959
Brendan is building a dog house, and the
dimensions of the roof are shown below. What is
the lateral surface area of the roof?
2.7f
5 ft
3 f+
A. 24.84 f+2
B. 46 f+2
C. 54.1 f+2
D. 43.2 f+2
Answers
Answer:
B. 46 ft²
Step-by-step explanation:
Two sides are rectangles 3.1 ft by 5 ft.
One side is a rectangle 5 ft by 3 ft.
area = 2(3.1 ft × 5 ft) + (5 ft × 3 ft)
area = 46 ft²
PLEASE HELP ILL GIVE FIVE STARS AND GIVE BRAINLIEST
Ray is x years old. His brother Ron is y years older than Ray
What is the sum of their ages now?
Answers
Answer: The sum will be x+y because their ages are variables.
Step-by-step explanation:
35\40 in simplest form
Answers
Answer:
7/3
Step-by-step explanation:
pls mark me brainliest :)
Answer:
7/8
Step-by-step explanation:
Divide both the numerator and denominator by the GCD
35 ÷ 5
40 ÷ 5
A student wants to interview other students about their favorite sport. The student interviews every fifth student who enters the school in the morning. Is this a random sample or a biased sample?
A
Random sample
B
Biased sample
Answers
Answer:
Biased
Step-by-step explanation:
Ice is placed around a bowl of water to lower the temperature. The equation D=−75t+22 shows the time, t, measured in minutes and temperature, D, measured in degrees Celsius.
Answers
Answer:
Before cooling begins, the temperature of the water is 22°C.
Step-by-step explanation:
need steps??
have a good day
rate my answer
Solve for X
Pls give your working out as well x
Answers
[tex]3^{x-2} = \left(\dfrac 13 \right)^{\sqrt x}\\ \\\implies 3^{x -2} = \left(3^{-1} \right)^{\sqrt x} \\\\\implies 3^{x-2} = 3^{-\sqrt x}\\\\\implies x -2 = -\sqrt x\\\\\implies (x-2)^2 = \left(-\sqrt x \right)^2\\\\\implies x^2 -4x +4 = x\\\\\implies x^2 -5x +4 =0\\\\\implies x^2 -4x-x+4=0\\\\\implies x(x-4) -(x-4)=0\\\\\implies (x-1)(x-4)=0\\\\\implies x = 1~~~~~~;[x = 4 ~ \text{Does not satisfy the original equation}]\\\\\text{Hence, x =1 is the solution.}[/tex]
Q10. Consider the following algorithm:
g_1 = 9
g_2 = 5
Git
For k starting at 0 and ending with 8:
gk = (k-1)*gk-1 + gk-2
What is the last term, 8, of the recursive sequence generated as a result of
executing this algorithm?
Answers
Answer:
61,940
Step-by-step explanation:
For a recursive sequence of reasonable length, it is convenient to use a suitable calculator for figuring the terms of it. Since each term not only depends on previous terms, but also depends on the term number, it works well to use a spreadsheet for doing the calculations. The formula is easily entered and replicated for as many terms as may be required.
__
The result of executing the given algorithm is shown in the attachment. (We have assumed that g_1 means g[-1], and that g_2 means g[-2]. These are the starting values required to compute g[0] when k=0.
That calculation looks like ...
g[0] = (0 -1)×g[-1] +g[-2} = (-1)(9) +5 = -4
The attachment shows the last term (for k=8) is 61,940.
Rewrite the parametric equations in Cartesian form: X(t) = -3sin t, y(t) = 3 cos t
Answers
Answer:
[tex]( \frac{x}{3} ) {}^{2} + ( \frac{y}{3}) {}^{2} = 1[/tex]
or x² +y²= 9
Step-by-step explanation:
In Cartesian form, the equation is expressed only in terms of y and x.
x= -3sin(t) -----(1)
y= 3cos(t) -----(2)
I've written x instead of x(t) as in the later part of the working, we will be having an equation of only x and y, thus x will no longer be a function of t. This applies to equation 2, where I have replaced y(t) with y.
Relating sine to cosine:
sin²(t) +cos²(t)= 1
[sin(t)]² +[cos(t)]²= 1 -----(3)
From (1):
[tex] \frac{x}{ - 3} = \sin(t) [/tex]
[tex] \sin(t) = - \frac{x}{3} [/tex] -----(3)
From (2):
[tex] \frac{y}{3} = \cos(t) [/tex]
[tex] \cos(t) = \frac{y}{3} [/tex] -----(4)
Substitute (4) &(5) into (3):
[tex]( - \frac{x}{3} )^{2} + ( \frac{y}{3}) {}^{2} = 1[/tex]
[tex]( \frac{x}{3} ) {}^{2} + ( \frac{y}{3} ) {}^{2} = 1[/tex]
The steps below are optional as the above is already considered to be the Cartesian form.
[tex] \frac{ {x}^{2} }{9} + \frac{y {}^{2} }{9} = 1[/tex]
Multiplying both sides by 9:
x² +y²= 9
Dylan has 58912 cookies he eats 34512 how much does he have now?
Answers
Answer:
24,400
Step-by-step explanation:
58912 minus 34512 equals 24400.
Hope this Helps.
BTW it's my birthday
Answer:
24400
Step-by-step explanation:
No explanations; 58912-34512=24400
Yes I wanted to know what is the answer
Answers
Answer:
14 flats
5 plants
Step-by-step explanation:
First you would do 89/6 but since that doesn’t divide nicely you can do 90/6 to get 15 then subtract 1 since the original number was 89 and you would get 14
The 14 times 6 would get you 84
Then subtract 84 from 89 and you get 5 plants left over ^^
Factor the equation 4x^2-20x+25
Answers
Answer:
[tex](2x - 5)^2[/tex]
Step-by-step explanation:
[tex]y = 4x^2 -20x + 25\\\rightarrow (2x - 5)(2x - 5)\\\rightarrow (2x - 5)^2\\\\\text{checking...}\\(2x - 5)^2 = (2x - 5)(2x - 5) = 4x^2 - 10x - 10x + 25 = 4x^2 - 20x + 25[/tex]
Q9.
Solve the simultaneous equations
7x – 2y = 34
3x + 5y = -3
Show clear algebraic working.
Answers
After solving simultaneously we found the values for x and y are
2.76 and 5.65 respectively
Simultaneous EquationsWe have the following two expressions
7x – 2y = 34 --------1
3x + 5y = -3 ---------2
Multiply 1 by eqn 5 and eqn 2 by 2
35x - 10y = 170-------3
6x + 10y = -6 ----------4
subtracting 4 from 3 we have
29 x= 164
divide both sides by 29
x = 164/29
x = 5.65
Put x = 5.65 in eqn 1 to find y
7*5.65 – 2y = 34
39.55 - 2y = 34
39.55-34 = 2y
5.55 = 2y
divide both sides by 2
x = 5.55/2
x = 2.76
Learn more about simultaneous equations here:
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make y the subject...
Answers
Answer:
y = qx/p -qr
Step-by-step explanation:
To solve for a variable, you must undo the operations done to the variable. Here, the variable is divided by a negative number (-q), and the result is added to another term (x/p).
We undo the addition by adding the opposite of the term that is added:
[tex]\dfrac{x}{p}-\dfrac{x}{p}-\dfrac{y}{q}=r-\dfrac{x}{p}\qquad\text{add $-\dfrac{x}{p}$}\\\\-\dfrac{y}{q}=r-\dfrac{x}{p}\qquad\text{simplify}[/tex]
We undo the division by multiplying by the reciprocal of the coefficeint, -q.
[tex]\left(-\dfrac{y}{q}\right)(-q)=\left(r-\dfrac{x}{p}\right)(-q)\qquad\text{multiply by $-q$}\\\\\boxed{y=-qr+\dfrac{qx}{p}}\qquad\text{simplify}[/tex]
I need help with this please
Answers
Answer:
4 and 7
2 and 5
Step-by-step explanation:
4 and 7 are co interior angles
2 and 5 are co interior angles
sum of co interior angles are 180° hence they are Supplementary
Write an equation of the line with a slope of -3 and y -intercept of 0 .
y=
Answers
Answer:
y=-3x
Step-by-step explanation:
