Find the distance of the point (4,4,−4) from the line r(t)=⟨−1+2t,1+2t,3−3t⟩.
Answers
Translate the given point and line together so that you get a new point and a new line that passes through the origin. This turns the problem into finding the distance between the new point,
p = (4, 4, -4) - (-1, 1, 3) = (5, 3, -7)
and the new line,
r*(t) = r(t) - ⟨-1, 1, 3⟩ = ⟨2t, 2t, -3t⟩
Let p = ⟨5, 3, -7⟩, the vector starting at the origin and pointing to p. Then the quantity ||p - r*(t)|| is the distance from the point p to the line r*(t).
Let u be such that ||p - r*(t)|| is minimized. At the value t = u, the vector p - r*(t) is orthogonal to the line r*(t), so that
(p - r*(u) ) • r*(u) = 0
I've attached a sketch with all these elements in case this description is confusing. (The red dashed line is meant to be perpendicular to r*(t).)
Solve this equation for u :
p • r*(u) - r*(u) • r*(u) = 0
p • r*(u) = r*(u) • r*(u)
and x • x = ||x||² for any vector x, so
p • r*(u) = ||r*(u)||²
⟨5, 3, -7⟩ • ⟨2u, 2u, -3u⟩ = (2u)² + (2u)² + (-3u)²
10u + 6u + 21u = 4u ² + 4u ² + 9u ²
17u ² - 37u = 0
u (17u - 37) = 0
==> u = 0 or u = 37/17
We ignore u = 0, since the dot product of any vector with the zero vector is 0.
Then the minimum distance distance between the given point and line is
||p - r*(u)|| = ||⟨5, 3, -7⟩ - 37/17 ⟨2, 2, -3⟩|| = √(42/17)
Related Questions
What is log10^6, considering log10^2=a and log10^3=b?
Answers
The answer is simply just a+b.
Solution:
log10^6=log10^2+log10^3
Since log10^2=a and log10^3=b,
The answer is a+b.
I think that the answer is a+b.
what is the inverse of y=3x+3?
Answers
Answer:
The inverse is x/3 -1
Step-by-step explanation:
y = 3x+3
To find the inverse, exchange x and y
x = 3y+3
Solve for y
x-3 =3y+3-3
x-3 = 3y
Divide by 3
x/3 - 3/3 = 3y/3
x/3 -1 = y
The inverse is x/3 -1
I need help with this question
Answers
Differentiate 4x^2+y^2=36 with respect to x. Hence find the turning points of the curve.
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If y = y(x), then the derivative with respect to x is dy/dx. Differentiating both sides of the given equation gives
d/dx [4x ² + y ²] = d/dx [36]
8x + 2y dy/dx = 0
2y dy/dx = -8x
dy/dx = -4x/y
The turning points of the curve, taken as a function of x, are those points where the derivative vanishes.
-4x/y = 0 ===> x = 0
This value of x corresponds to two points on the curve,
4×0² + y ² = 36 ===> y ² = 36 ===> y = ±6
So there are two turning points, (0, -6) and (0, 6).
Using the formula D = s:t where D equals distance traveled, r equals the average rate of
speed, and t equals the time traveled, choose the expression or equation that correctly
represents this information.
Mary drove 150 miles in three hours. What was her average rate of speed?
=
150 = 3
r = 3 = 150
O p + 150 · 3
Answers
Answer: r = 50 miles/h
Step-by-step explanation:
Let r be the rate of average speed.
Then
r = D/t
r = 150/3
r = 50 miles/h
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Answers
The last part answers the first part for you, just look at the y-values.
In other words:
A' (-8, 2)
B' (-4, 3)
C' (-2, 8)
D' (-10, 6)
Explanation:
When you reflect any point over the x-axis, the y-value of the ordered pair is going to change.
This makes sense especially considering that the x-axis is horizontal, so the only way you could cross is to move up or down. If you were to move left or right, you'd only be able to cross the y-axis, since it's vertical.
Now for the last part, as I mentioned above, if you are reflecting across the y-axis, the x-values of the ordered pair is going to change.
A'' (8, 2)
B'' (4, 3)
C'' (2, 8)
D'' (10, 6)
Take note that the only thing that changes for the respective value is its sign, while the number itself stays the same.
If car eyelashes sold for $13.99. If you bud double that, how much would you have paid for them? (Hint if needed: if they had been exactly $14, how different would your answer be?)
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Answer:
13.99 x 2 = 27.98 dollars
now if they were 14 dollars exactly and you doubled that it would be 28 dollars so the difference would be 0.02 cents
Step-by-step explanation:
0.18 divided by 0.04
Answers
evaluate -7x^2 -5 + y^2- forx = -2, y=4
Answers
Answer:
The (y^2 ) or (y^–2) It is not clear, I solved it in both ways (y^2) and (y^-2) .^_^
[tex] - 7 {x }^{2} - 5 + {y}^{2} \\ - 7( { - 2}^{2} ) - 5 + {4}^{2} \\ - 7( - 4) - 5 + 16 \\ 28 - 5 + 16 \\ = 39[/tex]
_______o_____o_______
[tex] - 7 {x }^{2} - 5 + {y}^{ - 2} \\ - 7( { - 2}^{2} ) - 5 + {4}^{ - 2} \\ - 7( - 4) - 5 + \frac{1}{16} \\ 28 - 5 + \frac{1}{16} \\ = 23.06[/tex]
I hope I helped you^_^
find coordinate of a point on x axis which is at a distance of 5 units from (5,4)
Answers
Step-by-step explanation:
Hi there!
Please see attached picture for your answer.
Hope it helps!
Business/multivariable calc question
help needed asap!!!!
Answers
Answer:
There is a min value of 376 located at (x,y) = (9,7)
============================================================
Explanation:
Solve the second equation for y
x+y = 16
y = 16-x
Then plug it into the first equation
f(x,y) = 3x^2+4y^2 - xy
g(x) = 3x^2+4(16-x)^2 - x(16-x)
g(x) = 3x^2+4(256 - 32x + x^2) - 16x + x^2
g(x) = 3x^2+1024 - 128x + 4x^2 - 16x + x^2
g(x) = 8x^2-144x+1024
The positive leading coefficient 8 tells us we have a parabola that opens upward, and produces a minimum value (aka lowest point) at the vertex.
Let's compute the derivative and set it equal to zero to solve for x.
g(x) = 8x^2-144x+1024
g ' (x) = 16x-144
16x-144 = 0
16x = 144
x = 144/16
x = 9
The min value occurs when x = 9. Let's find its paired y value.
y = 16-x
y = 16-9
y = 7
The min value occurs at (x,y) = (9,7)
Lastly, let's find the actual min value of f(x,y).
f(x,y) = 3x^2+4y^2 - xy
f(9,7) = 3(9)^2+4(7)^2 - 9*7
f(9,7) = 376
The smallest f(x,y) value is 376.
Put the shapes in the venn diagram. Is this right? if wrong tell me please.
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Answer:
I'm not sure if the kite outside has at least a pair of equal sides. The rest I think are right though? Correct me if I'm wrong.
I need to find the distance B in the special counter sink shown
Answers
Answer:
Step-by-step explanation:
87°32' = 86°92'
(86°92')/2 = 43°46'
B = 13/(16cos(43°46')) = 1.125
Answer:
Step-by-step explanation:
I will mark as brainliest:)
Answers
Answer:
Cube C
Step-by-step explanation:
[tex]{ \sf{because \: it \: is \: identical}}[/tex]
Armando's carpet has an area of 220 square feet. He hires the Carpet Pro carpet cleaning service, which is able to clean 9 square feet of his carpet every minute. Therefore, in the first minute, CarpetPro is able to clean 9 square feet of Armando's carpet (leaving 211 square feet remaining to be cleaned). In the second minute, the cleaner is again able to clean 9 square feet of carpet (leaving 202 square feet to be cleaned), etc. Which of the following functions expresses the number of square feet that still remain to be cleaned after t minutes from the time that the carpet cleaner began cleaning this carpet.
a. A(t) = 220 - 9
b. A(t) = 220 - 0.1
c. A(t) = 220(0.1)
d. A(t) = 220(0.96)
e. A(t) 220 - 0.96
Answers
Answer:
The answer is A
Step-by-step explanation:
The answer needs to be in slope-intercept form or y=mx+b because the amount of carpet being cleaned every minute is a constant decrease of 9 square feet.
So our b value (or how much carpet cleaned at 0 minutes) is 220
And our m value (or slope or how much carpet is being cleaned every minute) is 9
So if we plug the variables with the tangible numbers we get A(t) = -9x+220 or A(t) = 220-9x.
provided by gauth math
May I get some help with this question?
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hence
XS=ZS
third choice
Least you greatest (help please but don’t give wrong answer pls)
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Answer:
84.16, 84.31, 675/8, 423/5
Step-by-step explanation:
Simplify i^38 ????????
Answers
Answer:
i is defined as the square root of -1.
i^2 = -1
i^3 = -i
i^4 = 1
Following the pattern, we see that i^40 = 1, so i^38 is two above, or equal to -1.
So, i^38 = -1.
Let me know if this helps!
The probability that a graduate of the Faculty of Finance will defend the diploma “excellent” is 0.6. The probability that he will defend his diploma “perfectly” and receive an invitation to work at the bank is 0.4. Suppose a student defends a diploma. Find the probability that he will receive an invitation to work in a bank?
Answers
The probability that he receives an invitation to work in a bank given that he defends his diploma excellently is [tex]\mathbf{0. \overline 6}[/tex]
The reason for the above probability value is as follows;
The known parameters are;
The probability that a graduate of the Faculty of Finance will defend the diploma excellently, P(A) = 0.6
The probability that he will defend perfectly and receive an invitation to work at a bank, P(A∩B) = 0.4
The unknown parameter is;
The probability that the student will receive an invitation from the bank given that the student defend the diploma excellent, [tex]\mathbf {P(B \ | \ A)}[/tex]
The process;
[tex]\mathbf{ P(B \ | \ A)}[/tex] is found using the conditional probability formula as follows;
[tex]\mathbf {P(B \ | \ A) = \dfrac{P(A \cap B) }{P(A)}}[/tex]
Plugging in the values, we get;
[tex]P(B \ | \ A) = \dfrac{0.4 }{0.6} = \dfrac{2}{3} = 0. \overline 6[/tex]
The probability that the student will receive an invitation from the bank given that the student defend the diploma excellent, [tex]P(B \ | \ A)[/tex] = [tex]\mathbf {0. \overline 6}[/tex]
Learn more about conditional probability here;
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Suppose that the manufacturer of a gas clothes dryer has found that, when the unit price is p dollars, the revenue R (in dollars) is R(P) = - 8p^2 + 24,000p. What unit
price should be established for the dryer to maximize revenue? What is the maximum revenue?
The unit price that should be established to maximize revenue is $|
(Simplify your answer.)
Answers
Here we have a problem of maximization and quadratic equations.
The unit prize that maximizes the revenue is $1,500, and the maximum revenue is $18,000.
We know that the revenue equation is:
R(P) = - 8p^2 + 24,000p
Where the variable p is the price.
Now we want to find the value of p that maximizes the revenue.
To do it, we can see that the revenue equation is a quadratic equation with a negative leading coefficient.
This means that the arms of the graph will go downwards, then the maximum point of the graph will be at the vertex.
Remember that for an equation like:
y = a*x^2 + b*x+ c
The x-value of the vertex is at:
x = -b/(2*a)
Then for the equation:
R(P) = - 8p^2 + 24,000p
The vertex is at:
p = -(24,000)/(2*-8) = 1,500
The value of p that maximizes the revenue is p = $1,500
To get the maximum revenue, we need to evaluate the revenue equation in that p value.
R(1,500) = - 8*(1,500)^2 + 24,000*1,500 = 18,000
And the revenue equation is in dollars, then the maximum revenue is 18,000 dollars.
If you want to learn more, you can read:
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p = 1500 $ the unit price
R(p) = 18000000 $ maximum revenue
We will use two different procedures to calculate the maximum revenue.
That is equivalent to solve the problem and after that to test the solution
The first one is:
R(p) = - 8*p² + 24000*p
we realize that R(p) is a quadratic function ( a parabola) of the form:
y = a*x² + b*x + c ( c = 0 in this case)
We also know that as the coefficient of p² is negative the parabola opens downwards then the vertex is a maximum value for R(p), and the x coordinate of p is:
x = p = - b/2*a then by substitution
p = - ( 24000)/ 2 ( - 8)
p = 1500 $ and for that value of p
R(p) = - 8 ( 1500)² + 24000* (1500) = - 18000000 + 36000000
R(p) = 18000000 $
The second procedure is solving with the help of derivatives.
R(p) = - 8*p² + 24000*p
Tacking derivatives on both sides of the equation we get:
R´(p) = -16p + 24000
If R´(p) = 0 then -16p + 24000 = 0
p = 24000/ 16 p = 1500
if we check for the second derivative
R´´(p) = -16 -16 < 0 therefore there is a maximum value for R(p) when p = 1500, and that value is:
By substitution in R(p)
R(p) = -8 *(1500)² + 24000* 1500
R(p) = - 18000000 + 36000000
R(p) = 18000000 $
Convert 4.206 m into mm
Answers
Answer:
4206 is the answer of this question
Answer:
I think it will help you a lot.
plzzz help me i need help picture below
A local bakery buys their flour in bulk. The relationship between the cost of the flour and its weight is shown in the graph below.
Is the relationship shown a direct variation? Explain your reasoning using complete sentences.
The point (1, 0.50) is on this graph. What does this ordered pair represent?
Approximately how much does the bakery pay for 40 pounds of flour?
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Answer:
It is a direct variation...
The price of flour does not increase or decrease the
more that you purchase (no discount for large purchases)
(1,0.5) means that a pound of flour costs 50
40 pounds would cost $20
Step-by-step explanation:
The probability of winning a raffle is 2/5. What is the probability of not winning the raffle?
0
3/5
2/5
Answers
Answer:
3/5
Step-by-step explanation:
um 3/5+2/5 = 1
help me plzzzzzzzzzzzzzzzzzzzzzzzzzz
Answers
[2＋(－6)]÷2＝－2
The midpoint is (3, －2)
Hope my answer helped u :)
Janet invests a sum of EUR in an account that offers 3.5% simple interest. After ten years her investment is worth 7425 EUR. How much did she invest?
Answers
We need to find the amount of money Janet invested in 10 years to yield 7425 EUR
She invested EUR 21,214.29
Simple interest = P × R × T
Where,
P = principal = ?
R = interest rate = 3.5% = 0.035
T = Time = 10 years
Simple interest = 7425 EUR
Simple interest = P × R × T
7425 = p × 0.035 × 10
7425 = p × 0.35
7425 = 0.35p
Divide both sides by 0.35
P = 7425 / 0.35
= 21,214.285714285
Approximately,
P = EUR 21,214.29
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construct a rectangle PQRS, In which AB= 8cm and diagonal AC= 10cm
Answers
A rectangle can be constructed using a straightedge, and a setsquare or compass
Please find attached the drawing of rectangle ABCD
The steps to construct the rectangle ABCD are as follows:
Question:
The missing part of the question is the name of the rectangle = ABCD
The given parameters are;
The length of the side AB = 8 cm
The length of the diagonal of the rectangle ABCD = AC = 10 cm
The steps to construct a rectangle are;
Draw the segment [tex]\overline{AB}[/tex] = 8 cm on a planeDraw perpendicular lines at points A and B with length h given by Pythagoras's theorem as followsh² = [tex]\overline{AC}[/tex]² - [tex]\overline{AB}[/tex]²
∴ h² = 10² - 8² = 36
h = √36 = 6
h = 6 cm = The length of the sides [tex]\mathbf{\overline{AD}}[/tex] and [tex]\mathbf{\overline{CB}}[/tex]
Draw arcs with radius 6 cm from points A and B to intersect the perpendicular lines drawn from points A and B on the same side of the line [tex]\overline{AB}[/tex] at points D and CJoint point C to D with a straight line which is segment [tex]\overline{CD}[/tex] and which completes the rectangle ABCDLearn more about the construction of basic shapes here;
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Find f(-1) given f(x) = –2x^3 + 3x^2 – 22
Answers
[tex]\\ \sf\longmapsto f(-1)[/tex]
[tex]\\ \sf\longmapsto -2x^3+3x^2-22[/tex]
[tex]\\ \sf\longmapsto -2(-1)^3+3(-1)^2-22[/tex]
[tex]\\ \sf\longmapsto -2(-1)+3(1)-22[/tex]
[tex]\\ \sf\longmapsto 2+3-22[/tex]
[tex]\\ \sf\longmapsto 5-22[/tex]
[tex]\\ \sf\longmapsto -17[/tex]
Dividing integers
7. (-154) ➗ (-14) =
11. (-40) ➗10=
15. 90 ➗ (-15)=
16. 108 ➗ (-9)=
17. (-48) ➗ (-6)=
18. (-105) ➗ 7=
Answers
first we shall learn the rules.when numbers with same sign are divided it gives pisitive sign but, when numbers of different signs are divided it gives negetive sign.
here,
7. (-154) ➗ (-14) =11
11. (-40) ➗10=-4
15. 90 ➗ (-15)=-6
16. 108 ➗ (-9)=-12
17. (-48) ➗ (-6)=8
18. (-105) ➗ 7=-15
hope it helps you..........
Taylor wants to find the perimeter of a rectangular playground. The lenght of the playground measures (3x-20) metres. The width of the playground measures (2x+4) metres. What is the perimeter of the playground?
Answers
Answer:
Step-by-step explanation:
P = 2(3x-20) + 2(2x+4) = (6x-40) + (4x+8) = 10x-32
The required perimeter of the playground is 10x-32.
The length of the playground measures (3x-20) metres.
The width of the playground measures (2x+4) metres.
What is the perimeter?
Perimeter, is the measure of the figure on its circumference.
The Required perimeter is for the playground is given by
= 2(3x-20) + 2(2x+4)
= 10x-32
Thus the required perimeter of the playground is 10x-32.
learn more about perimeter here:
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The number of bacteria in a certain culture grows exponentially at a rate of 1% per hour. Assuming that 5,000 bacteria are present initially, find the time required for the bacteria population to reach 45,000. (Round your answer to the nearest hour.)
Answers
9514 1404 393
Answer:
221 hours
Step-by-step explanation:
The population is given by the exponential equation ...
population = (initial value) × (1 +growth rate)^t
where the units of t are the same as the units of growth rate.
This lets us write ...
p(t) = 5000×1.01^t
We want this to be 45000, so ...
45000 = 5000×1.01^t
9 = 1.01^t . . . . . . . . . . . . divide by 5000
log(9) = t×log(1.01) . . . . take logs
t = log(9)/log(1.01) ≈ 220.8
It will take about 221 hours for the population to reach 45000.
