Find the point on the line 6x + y = 8 that is closest to the point
(−5, 2).
(x, y) =
Answers
9514 1404 393
Answer:
(x, y) = (31/37, 2 36/37) ≈ (0.8378, 2.9730)
Step-by-step explanation:
There are several ways we can go at this.
Method 1
Find the point of intersection with the perpendicular line through (-5, 2).
The perpendicular line will have the x- and y-coefficients swapped, one of them negated, and a new constant consistent with the given point.
x -6y = (-5) -6(2) = -17
So, we're looking for the solution of the system of equations ...
6x +y = 8x -6y = -17Using Cramer's Rule, we can find the solution to be ...
x = ((1)(-17) -(-6)(8))/((1)(1) -(-6)(6)) = 31/37 ≈ 0.837_837*
y = ((8)(1)-(-17)(6))/37 = 110/37 ≈ 2.972_972
The closest point is (x, y) = (31/37, 2 36/37).
__
Method 2
We can write the given line in parametric form, then find the point on it such that the direction along the line is perpendicular to the direction to the given point.
Solving the given equation for y, we have ...
y = 8 -6x
So, the parametric form of the given line can be written as ...
(x, y) = (0, 8) +t(1, -6) . . . . for some parameter t
The direction to the given point from (x, y) is then ...
(-5, 2) -(x, y) = (-5, 2) -((0, 8) +t(1, -6)) = (-5-0-t, 2-8+6t) = (-t-5, 6t-6)
The dot-product of this direction vector with the direction vector of the original line will be zero for the value of the parameter at the closest point.
(1, -6) · (-t-5, 6t-6) = 0 = (1)(-t -5) +(-6)(6t -6)
0 = -t -5 -36t +36 = -37t +31
31/37 = t
So, filling this value into the parametric equation gives the closest point as ...
(0, 8) +(31/37)(1, -6) = (31/37, 2 36/37) = (x, y)
__
Method 3
Find the point of tangency of the given line with a circle centered at (-5, 2).
The equation of a circle centered at (-5, 2) is ...
(x +5)^2 +(y -2)^2 = r^2
The given line will meet the circle at a point that satisfies this equation and the equation of the circle. Writing an expression for y using the given line, we can substitute into the circle equation to get ...
(x +5)^ +((8 -6x) -2)^2 = r^2
x^2 +10x +25 +36 -72x +36x^2 = r^2
37x^2 -62x + (some constant) = 0
This will have one solution when the constant is chosen to make the left-side expression a perfect square. That perfect square will be ...
37(x -31/37)^2 = 0
which is to say the tangent point of the line and the circle will have x-coordinate x = 31/37. The y-coordinate is 8 -6(31/37) = 2 36/37.
The closest point is (x, y) = (31/37, 2 36/37).
_____
* repeated decimal fractions are signified in typeset text with an overbar over the repeating digits. In plain text, we sometimes use a preceding underscore to identify the repeating digits.
Related Questions
What is the decay factor in the exponential decay function y=a(1−r)^t
Answers
Answer:
(1 - r)Step-by-step explanation:
Given function:
y = a (1−r)^tHere we have:
y - future value, a- present value, (1 - r) - decay factor, t- number of decay periodsQual a área total da pirâmide de Queóps ou Gizé?
Answers
Answer:
Step-by-step explanation:La Gran Pirámide de Giza (también conocida como la Pirámide de Keops o la Pirámide de Keops) es la más antigua y más grande de las pirámides del complejo piramidal de Giza que limita con la actual Giza en el Gran Cairo, Egipto. Es la más antigua de las Siete Maravillas del Mundo Antiguo y la única que permanece prácticamente intacta.
Los egiptólogos concluyen que la pirámide fue construida como una tumba para el faraón egipcio Jufu de la Cuarta Dinastía y estiman que fue construida en el siglo 26 aC durante un período de alrededor de 27 años. [3]
Inicialmente con 146,5 metros (481 pies), la Gran Pirámide fue la estructura más alta hecha por el hombre en el mundo durante más de 3.800 años. A lo largo de la historia, se eliminó la mayor parte de la cubierta de piedra caliza blanca lisa, lo que redujo la altura de la pirámide a los actuales 138,5 metros (454,4 pies). Lo que se ve hoy es la estructura central subyacente. Se midió que la base era de unos 230,3 metros cuadrados, lo que da un volumen de aproximadamente 2,6 millones de metros cúbicos (92 millones de pies cúbicos), que incluye un montículo interno. [4]
Las dimensiones de la pirámide, expresadas en unidades de medida del antiguo Egipto, eran de 280 codos de alto, una longitud de base de 440 codos, con un seked de 5+
1
/
2
palmeras (pendiente de 51 ° 50'40 ").
La Gran Pirámide se construyó extrayendo aproximadamente 2,3 millones de bloques grandes con un peso total de 6 millones de toneladas. La mayoría de las piedras no son uniformes en tamaño o forma y solo están vestidas de manera tosca. [5] Las capas exteriores se unieron con mortero. Se utilizó principalmente piedra caliza local de la meseta de Giza. Otros bloques fueron importados en barco por el Nilo: piedra caliza blanca de Tura para la carcasa y bloques de granito de Asuán, con un peso de hasta 80 toneladas, para la estructura de la Cámara del Rey. esto podría ayudar lo siento si no
Layla has 280 ounces of juice to fill 8-ounce cups. Which statement is true?
( I will love and comment if answered correctly and reasonable )
A - She can fill at least 35 cups.
B - She can fill fewer than 35 cups.
C - She can fill more than 35 cups.
D - She can fill no more than 35 cups.
Answers
Answer:
D
Step-by-step explanation:
she has 280 and no more so 280 divided by 8 is 35
Answer:
D is the correct answer
Step-by-step explanation:
280/8=35
WILL GIVE BRAINLIEST!! do not answer if you dont know what the answer is.
3(n + 2) = 9(6 - n)
Answers
With the equation use the distributive property to multiply 3 into (n+2). And 9 into (6-n).
Then you should have 3n+6=54-9n
You add 9n to both sides to get 12n+6=54.
After you should subtract six from both sides to get 12n=48
The last step is to divide 12 from both sides to get n=4 :)
Sam built a circular fenced-in section for some of his animals.The section has a circumstance of 55 meters.What is the Approximate area, in square meters, of the section? Use 22/7 for for pi
A) 240.625 m squared
B) 926.5 m squared
C) 2,376.79 m squared
D) 240.8 m squared
Answers
Answer:
i think option C is correct hope it helps
Answer:
C) 2,376.79 m squared
Step-by-step explanation:
A boat traveled 17 miles in 2 hours. At this rate, how many
miles will the boat travel in 72 hour
Answers
Answer:
1 hour=8.5miles
Step-by-step explanation:
if the boat traveled 17 miles in 2 hours then in 1 hour the boat traveled, because 17/2=8.5
idek if this answers your question but 8.5*72=612
PLZ ANSWER I WILL MARK BRAINLIEST
Answers
Step-by-step explanation:
There is a difference between the numbers that is 6
So...
y=6x+5
Answer:
6x+5
Step-by-step explanation:
1 x 6=6+5=11
Dan bought 20 bunches of seedless green grapes for $48. How many bunches can Kali buy if she has $12?
5
29
4
6
Answers
Answer:
Quantity = 5 grapes
Step-by-step explanation:
Given the following data;
Quantity of grapes = 20
Cost = 48
To find the cost of each grape;
Cost of each grape = total cost/quantity
Cost of each grape = 48/20
Cost of each grape = $2.4
Now, to find how many grapes $12 can buy;
Quantity = total cost/cost of each grapes
Quantity = 12/2.4
Quantity = 5 grapes
Therefore, $12 can buy 5 grapes.
Solve using the tangent formula(real answers please)
Answers
Answer:
21.17
Step-by-step explanation:
tan= opposite/ adjacent
tan(36)= x/29
0.73= x/29
x= 21.17
(50 pts will mark brainliest) The perimeter of a rectangle is 48 inches. The width of the rectangle is 4 inches more than 3 times the length of the rectangle. What are the dimensions of the rectangle?
Answers
Answer:
The rectangle is 5 inches long by 19 inches wide.
Step-by-step explanation:
Let's start by listing what we know:
The perimeter of a rectangle is 48 inches.The width of the rectangle is 4 inches more than 3 times the length of the rectangle.Let's represent the width with "w" and length with "l", and make a system of equations based on what we know.
[tex]\left \{ {{2w + 2l = 48} \atop {w = 3l + 4}} \right.[/tex]
Now, let's solve the system of equations using substitution:
[tex]2w + 2l = 48[/tex]
[tex]2 (3l + 4) + 2l = 48[/tex]
[tex]6l + 8 + 2l = 48[/tex]
[tex]8l + 8 = 48[/tex]
[tex]8l = 40[/tex]
[tex]l = 5[/tex]
The length of the rectangle is 5 inches.
Now, let's use this to solve for the width.
[tex]w = 3l + 4[/tex]
[tex]w = 3(5) + 4[/tex]
[tex]w = 15 + 4[/tex]
[tex]w = 19[/tex]
The width of the rectangle is 19 inches.
Let's check our work:
[tex]2w + 2l = 48[/tex]
[tex]2(19) + 2(5) = 48[/tex]
[tex]38 + 10 = 48[/tex]
[tex]48 = 48[/tex]
The results match, so our answer is correct.
The rectangle is 5 inches long by 19 inches wide.
What is the length of PQ to the nearest hundreth place.(use 3.14 for *)
P
135° Q
A
12 in.
a
169.6 in.
56.5 in.
Ob
Ос
84.8 in.
Od
28.3 in.
Answers
Given:
The radius of the circle = 12 in.
The central angle of the intercepted arc PQ = 135 degrees.
To find:
The measure of arc PQ.
Solution:
The length of arc is:
[tex]s=2\pi r\times \dfrac{\theta}{360^\circ}[/tex]
Where, r is the radius and [tex]\theta [/tex] is the central angle.
Substituting [tex]r=12,\pi=3.14,\theta=135^\circ[/tex] in the above formula, we get
[tex]s=2(3.14)(12)\times \dfrac{135^\circ}{360^\circ}[/tex]
[tex]s=75.36\times 0.375[/tex]
[tex]s=28.26[/tex]
[tex]s\approx 28.3[/tex]
The measure of arc PQ is 28.3 in. Therefore, the correct option is (d).
Please help!
If the measure of O equals 69°, then what is the measure of N?
A.
81°
B.
21°
C.
111°
D.
69°
Answers
Answer:
D. 69°
Step-by-step explanation:
Opposite angles are congruent.
Answer:
N is also 69 so the an is D) 69
Step-by-step explanation:
please help meeeeeeeeeeee
Answers
Answer:
for example
Step-by-step explanation:
(1,3)
the one would go on the bottom.
and the 3 would go on the vertical side!
hope that helped! :)
Write down the first 4 terms of the sequence given by f(n) = 2n + 3
Answers
Explanation:
a1= 2×1+3= 5
a2= 2x2+3= 7
a3=2x3+3= 9
a4= 2x4+3= 11
SOLVE
FIND THE AMOUNT OF DISCOUNT WHEN THE ORIGINAL PRICE IS $44.00 AND THE DISCOUNT RATE IS 40%
WILL GUVE BRIANLIST
XXX
X
X
X
X
X
X
X
X
X
X
Answers
Answer:
$26.40
Step-by-step explanation:
40% of 44 is 17.6.
44-17.6 = 26.40
plzzz help for brianly
Answers
Answer:
Add up the area of all the sides
Step-by-step explanation:
Surface area is the shell of a 3d shape, not the volume so you add the outside of the shape.
Hope this helps! :)
3. The graph of an equation intersects the y-axis at some point. What do the coordinates of the intersection indicate?
O A the input when the output is zero
O B. the output when the input is zero
O C. the input when the output is 1
O D. the output when the input is 1
4. The equation for this graph is Output = 2 .
Answers
Answer:
Area: ½ × base × height
Perimeter: sum of side lengths of the triangle
of vertices: 3
Number of edges: 3
Internal angle: 60° (for equilateral)
Sum of interior angles: 180°
Step-by-step explanation:
The price of a stock went from $100 to $99.80. What was the percentage of decrease?
Answers
7 1/2 x 3/4 = 45/8 =
Answers
Answer:
90/6 is the answer
Step-by-step explanation:
Answer:
15/71
Step-by-step explanation:
(
71
2
x)(3)
4
=
45
8
(
71
2
x)(3)*(8)=(45)*(4)
852x=180
Step 2: Divide both sides by 852.
852x
852
=
180
852
What is the surface area of the cylinder with height 5 m and radius 7 m? Round your answer to the nearest thousandth.
Answers
Answer:
It SHOULD be 527.79(m2) If I'm not mistaken.
Step-by-step explanation:
Answer:
7000
Step-by-step explanation:
7×1000=7000
5×5=25÷5=5-5=0
What is the image point of (5,-4)(5,−4) after a translation left 4 units and down 3 units?
Answers
Answer:
(1,-7)
Step-by-step explanation:
1) (5,-4). 4 units to the left will be (1,-4)
2) (1,-4). 3 units down will be (1,-7)
Is the answer
X is left/ right.
Y is up/down.
5-4 (going left decreases the x) = 1
-4-3 (going down decreases the y) = -7
Remember that two negatives when adding/subtracting stays negative.
What are the prime factors of 24 and 32?
Answers
Step-by-step explanation:
Prime Factors for 24: 2, 2, 2, and 3.
Prime Factors for 32: 2, 2, 2, 2, and 2.
Answer:
Prime Factors for 24: 2, 2, 2, and 3.
Prime Factors for 32: 2, 2, 2, 2, and 2.
Step-by-step explanation:
Divide £42 in the ratio 4:3
Answers
Answer:
Should be 24:18
Step-by-step explanation:
E
The base of the parallelogram is 8 units and its height is half of the base.
Find the area of the parallelogram.
A. 32 sq. units
B. 12 sq. units
C. 24 sq. units
D. 4 sq. units
Answers
Answer:
A. 32 sq. units
Step-by-step explanation:
To find the area of a parallelogram, multiply the base by the height.
In this case, the base is 8.
However, the height is unknown.
According to the word problem, the height is HALF the base.
To find half of the base, we divide it by 2.
8 / 2 = 4
4 is the height.
Lastly, multiply as usual.
Base ⋅ height:
8 x 4 = 32.
Therefore, the area of the parallelogram is 32 sq. units.
NEED ANWSERS FAST
33-38
Find the value of x
Answers
Answer:
33. 18
34. 50
35. 50
36. 5
37. 1
38. 24
Step-by-step explanation:
Explanation found on photo! Hope it helped and my apologies for my messy work
You get a single payment loan of $4,400 at an interest rate of 12% the term of the loan is 172 days how much exact interest will you pay
Answers
9514 1404 393
Answer:
$248.81
Step-by-step explanation:
The interest formula is ...
I = Prt
where principal P earns interest at annual rate r for t years. Here, t is a fraction of a year: 172/365. The interest is ...
I = $4400×0.12×172/365 ≈ $248.81
You will pay interest of $248.81.
Answer:
$ 248. 81 is the answer.Step-by-step explanation:
[tex] \carryonlearning[/tex]In triangle MNO the measure of angle O =90 degrees the measure of angle M =66 degrees and NO = 60 feet Find the length of MN to the nearest tenth of a foot
Answers
Answer:
Length of MN = 65.68 feet (Approx.)
Step-by-step explanation:
Given:
Height of NO = 60 feet
Angle ∠M = 66°
Find:
The length of MN
Computation:
Given triangle is a right angled triangle
So,
NO is a perpendicular
MN is hypotenuse
Using trigonometry functions
Sin θ = Perpendicular / Hypotenuse
Sin 66 = NO / MN
0.9135 = 60 / Length of MN
Length of MN = 60 / 0.9135
Length of MN = 65.68 feet (Approx.)
Answer: approximately 65.6782 = 65.7
Step-by-step explanation:
Someone know this please help geometry if you are good at it
Answers
Answer:
Option A) outside
[tex]----------[/tex]
hope it helps..
have a great day!!!
Add.write the answer as a mixed number in the simplest form.
Answers
Answer:
5[tex]\frac{11}{12}[/tex]
Step-by-step explanation:
GCF of 4 and 6 is 12
1/6 = 2/12
3/4 = 9/12
4 + 1 = 5
2/12 + 9/12 = 11/12
5 11/12
Answer:
71/12Step-by-step explanation:
First you want to make the denominators in the fractions the same number. To do so, we need to multiply both the numerators and the denominators. Do this until the denominators are the same.
[tex]\frac{1}{6} *\frac{4}{4} = \frac{4}{24}[/tex]
[tex]\frac{3}{4} *\frac{6}{6} = \frac{18}{24}[/tex]
Now add
[tex]4\frac{4}{24} + 1\frac{18}{24} = 5\frac{22}{24}[/tex]
After adding, you need to multiply the whole numbers by the denominator and add that number to the numerator.
[tex]5*24=120[/tex]
[tex]\frac{120}{24} +\frac{22}{24} = \frac{142}{24}[/tex]
Finally simplify
[tex]\frac{142}{24} / \frac{2}{2} = \frac{71}{12}[/tex]
Let a line l be given in the coordinate plane. What linear equation is the graph of line l?
6
5
1
3-
3 2
2
1
0
&
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
-1
Answers
Answer:
y=2/7x+2.5
Step-by-step explanation: