Answer:
[tex]y = 5[/tex].
Or, equivalently:
[tex]\begin{aligned}\begin{bmatrix}0 \\ -3 \\ 0\end{bmatrix} \left(\begin{bmatrix}x \\ y \\ z\end{bmatrix} - \begin{bmatrix}1 \\ 5 \\ -3\end{bmatrix}\right) = 0\end{aligned}[/tex].
Step-by-step explanation:
Every plane in [tex]\mathbb{R}^{3}[/tex] could be represented with a vector equation of the form [tex]\vec{n}\, (\vec{r} - \vec{r}_{0}) = 0[/tex], where:
[tex]\vec{n}[/tex] is a vector normal to the plane (a normal vector), and [tex]\vec{r}_{0}[/tex] is the position vector of a point in the plane.Notice that in this question, the coordinates (and hence the position vectors) of the points in this plane are already given. For example, the position vector of the point [tex](1,\, 5,\, -3)[/tex] is the vector:
[tex]\begin{aligned}\vec{r}_{0} = \begin{bmatrix}1 \\ 5 \\ -3\end{bmatrix}\end{aligned}[/tex].
Specifically in [tex]\mathbb{R}^{3}[/tex], normal vectors of a plane could be found by:
finding two distinct directions parallel to that plane, and taking the cross product between the two directions.Subtracting position vectors of points in this plane from each other would give directions that are parallel to this plane:
[tex]\begin{aligned}\begin{bmatrix}2 \\ 5 \\ -3\end{bmatrix} - \begin{bmatrix}1 \\ 5 \\ -3\end{bmatrix} = \begin{bmatrix}1 \\ 0 \\ 0\end{bmatrix}\end{aligned}[/tex].
[tex]\begin{aligned}\begin{bmatrix}3 \\ 5 \\ 2\end{bmatrix} - \begin{bmatrix}1 \\ 5 \\ -3\end{bmatrix} = \begin{bmatrix}2 \\ 0 \\ 5\end{bmatrix}\end{aligned}[/tex].
The cross product between these two vectors in [tex]\mathbb{R}^{3}[/tex] would be:
[tex]\begin{aligned} \vec{n} = \begin{bmatrix}1 \\ 0 \\ 0\end{bmatrix} \times \begin{bmatrix}2 \\ 0 \\ 5\end{bmatrix} = & \begin{bmatrix}0 \times 5 - 0 \times 0 \\ 0 \times 2 - 1 \times 5\\ 1 \times 0 - 0 \times 2\end{bmatrix} = \begin{bmatrix}0 \\ -3 \\ 0\end{bmatrix}\end{aligned}[/tex].
(Note that the cross product between other directions parallel to the plane might give other normal vectors that are parallel to the one in this example.)
Using the position vector of the point [tex](1,\, 5,\, -3)[/tex] as [tex]\vec{r}_{0}[/tex], one possible vector equation for this plane would be:
[tex]\begin{aligned}\begin{bmatrix}0 \\ -3 \\ 0\end{bmatrix} \left(\begin{bmatrix}x \\ y \\ z\end{bmatrix} - \begin{bmatrix}1 \\ 5 \\ -3\end{bmatrix}\right) = 0\end{aligned}[/tex].
Expand the dot product and simplify to obtain a scalar equation for this plane:
[tex]\begin{aligned}\begin{bmatrix}0 \\ -3 \\ 0\end{bmatrix} \begin{bmatrix}x - 1 \\ y - 5 \\ z - (-3)\end{bmatrix} = 0\end{aligned}[/tex].
[tex]0\, (x - 1) + (-3)\, (y - 5) + 0\, (z - (-3)) = 0[/tex].
[tex](-3)\, (y - 5) = 0[/tex].
[tex]y = 5[/tex].
Contagious diseases can spread quickly. Suppose 3 people are ill during the first week of an epidemic, and each person who is ill spreads the disease to five people by the end of the next week. By the end of the 8th week of the epidemic, how many people have become ill?
Answer:
1875
Step-by-step explanation:
3x5 = 15
15x 5 = 75
75x 5 = 375
375 x 5 = 1875
1875 because at the end of the next week, which basically turns 8 weeks into 4, and the number of contaminants multiply by 5 each other week.
Select the correct answer.
What is the range of this absolute value function?
y
-10
-10.
-8
9
N-
N
A
-6
-8
-10
10
Solve this fraction, thanks!
[tex] \frac{10}{5} + \frac{ - 12}{5} [/tex]
Answer:
-2/5
Step-by-step explanation:
I actually didn't see the negative sign sorry and thanks for correcting my mistake!!!
Answer:
-2/5
Step-by-step explanation:
+ × - = -
10/5-12/5
apply the fraction rule a/c-b/c =a-b/c
10-12/5
substract 10-12 =-2
-2/5
Help please?????????
If you reflect point B(- 2, 5) over the y- axis, what is the reflected point ?
Answer:
(2,5)
Step-by-step explanation:
Since it is reflected across the y-axis only the x point will change and vise-versa
Which one geos first to last 3/4 or 1/3 or 5/8
Find the vertex of the parabola y=x^2 + 5x + 10
The coordinates of the vertex of the given parabola [tex]y=x^2+5x+10[/tex] will be [tex](\dfrac{5}{2},\dfrac{15}{4})[/tex].
What is the vertex of a parabola?Vertex of the parabola is the point of intersection of symmetry axis of the parabola and the parabola itself.
The given equation of parabola is [tex]y=x^2 + 5x + 10[/tex].
To get the coordinates of vertex of this parabola, we have to write the equation of parabola in the standard form.
So, write the equation of parabola in its standard form as,
[tex]y=x^2 + 5x + 10\\y=x^2+2\times \dfrac{5}{2}x+\dfrac{25}{4}-\dfrac{25}{4}+10\\y=(x-\dfrac{5}{2})^2+\dfrac{15}{4}\\(y-\dfrac{15}{4})=(x-\dfrac{5}{2})^2[/tex]
The standard form of parabola is,
[tex]y-b=(x-a)^2[/tex] where (a, b) is the coordinate of the vertex.
So, by comparing the equations, the vertex of the parabola will be [tex](\dfrac{5}{2},\dfrac{15}{4})[/tex].
Therefore, the coordinates of the vertex of the given parabola [tex]y=x^2+5x+10[/tex] will be [tex](\dfrac{5}{2},\dfrac{15}{4})[/tex].
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Need help on this one please
A path in the park is 15 kilometers in. A trash bin is placed every 1/3 kilometers how many trash cans are along the path ?
Determine the exact value of sin 30° tan 45° + tan 30° sin 60⁰
Answer:
1.
Step-by-step explanation:
1) if sin30=0.5; tan45=1; tan30=1/√3; sin60=√3/2, then
2) 0.5+0.5=1.
A lawn sprinkler sprays water 12 feet in every direction as it rotates. What is the area of the sprinkled lawn?
If the diameter of the circle is 14 units, calculate its area: then calculate its circumference
Answer:
Area: 153.86
Circumference: 43.96
Which polynomial function has a root of 1 with
multiplicity 2 and a root of 6 with multiplicity 1?
Of(x) = (x - 1)(x-6)
Of(x) = 2(x - 1)(x - 6)
Of(x) = (x - 1)(x - 1)(x-6)
O f(x) = (x - 1)(x-6)(x-6)
Answer:
C) f(x) = (x - 1)(x - 1)(x-6)
Explanation:
Multiplicity means number of times a factor appears in a particular equation.
For example, a root of 3 has multiplicity of 3 then = (x - 3)(x - 3)(x - 3)
Breakdown for this question:
If a polynomial where root of 1 has multiplicity of 2, then it means (x - 1)²
Where polynomial of root of 6 has multiplicity of 1, then it means (x - 6)
Joining them together:
(x - 1)²(x - 6) = (x -1)(x - 1)(x - 6) Hence option C is correct.
Martin has a piece of purple yarn that is 3 1/2 inches long he has a piece of yellow yarn that is 3 5/6 inches Long how much longer is the yellow yarn?
Answer:
2 1/3 longer
Step-by-step explanation:
yw ;)
Answer:
⅓
Step-by-step explanation:
Expression : 3 ⅚ - 3 ½
3 ½ would become 3 3/6.
3 ⅚ - 3 3/6 = 2/6 = 1/3
Enter the number that belongs in the green box.
Answer:
The answer is x = 10.
Step-by-step explanation:
Two options:
1. Looking at the image it looks like a parallelogram. So, most side lengths are equal to each other or are the same.
2. We can use the Pythagorean Theorem to solve for x.
This table represents a proportional relationship between x and y.
x 1.3 1.6 1.9 2.2
y 1.82 2.24 2.66 3.08
What is the constant of proportionality?
Enter your answer as a decimal in the box.
ill give brainliest this is for k12
The table is an illustration of a linear table, and the constant of proportionality is 1.4
How to determine the constant of proportionality?The table is given as:
x 1.3 1.6 1.9 2.2
y 1.82 2.24 2.66 3.08
The constant of proportionality (k) is calculated using:
k = y/x
Using the values on the table, we have:
k = 1.82/1.3
Divide
k = 1.4
Hence, the constant of proportionality is 1.4
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4/8 plus 2/4 find the common denomnator
Answer:
The common denominator is 4.
Step-by-step explanation:
4/8 simplified is 2/4.
Therefore, we can now add them together.
2/4 + 2/4 = 1.
Hope this helps:) Goodluck!
I can’t solve this problem
Answer:
f(-2) = -8f(-1) = -5f(0) = 0f(1) = 7Step-by-step explanation:
f(w) = [tex]w^{2}[/tex] + 6w
w = -2f(w) = f(-2) = [tex](-2)^{2}[/tex] + 6*(-2) = 4 - 12 = -8
w = -1f(w) = f(-1) = [tex](-1)^{2}[/tex] + 6*(-1) = 1 - 6 = -5
w = 0f(w) = f(0) = [tex]0^{2}[/tex] + 6*0 = 0 + 0 = 0
w = 1f(w) = f(1) = [tex]1^{2}[/tex] + 6*1 = 1 + 6 = 7
Find the product with the exponent in simplest
form. Then, identify the values of x and y.
X
1
3
6³.6
64 6
=
X ⇒7, y = 12/7
X ⇒12
x= 7/12
COMPLETE
7 , 12 are the answers
Answer:
x = 7
y = 12
Step-by-step explanation:
The key to solving this problem is understanding the properties of exponents:
If you are multiplying two powers with the same base (in this case the base is 6), the result is the base raised to the power of the two exponents added together.
6^1/3+1/4 = 6^x/y
6^7/12 = 6^x/y
So the product would be 6^7/12
Therefore, the answers are:
x = 7
y = 12
Fred figured out the volume of the glass is 141.37cm. Show work
To solve this problem, we have to calculate the volume of a hemisphere and the volume of a cylinder and add them together. The volume of the glass can be only be calculated when we find the volume of the hemisphere and the volume of the cylinder and then add then up together.
Volume of a HemisphereThe volume of a hemisphere is given as
[tex]v = \frac{2}{3} * \pi * r^3[/tex]
Data
r = 3cmLet's substitute the values into the equation and solve.
[tex]v = \frac{2}{3} * \pi * r^3\\v = 56.556cm^3\\[/tex]
Volume of a CylinderThe formula of volume of a cylinder is
[tex]v = \pi r^2 h[/tex]
data given
h = 3cmr = 3cmLet's substitute the values into the equation and solve
[tex]v = \pi r^2 h\\v = 3.142 * 3^2 * 3\\v = 84.834cm^3[/tex]
Let's add both volumes together so we have the volume of the glass cup
[tex]56.556 + 84.834 = 141.39cm^3[/tex]
The volume of the glass can be only be calculated when we find the volume of the hemisphere and the volume of the cylinder and then add then up together.
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Bob walks dogs after school. He charges $43. 50 for 6 hours and $65.25.75 for 9 hours.
Enter an equation to represent the relationship. Let x represent the number of hours Bob works
and y represent the amount he charges.
The equation of the amount charged by Bob will be y = 7.25x.
What is an equation?
The equation in mathematics is the relationship between the variables and the number and establishes the relationship between the two or more variables.
Given that:-
Bob walks dogs after school. He charges $43. 50 for 6 hours and $65.25.75 for 9 hours.The equation will be generated as follow:-
6 hours = $ 43.50
1 hours = $ 7.25
9 hours = $ 65.2575
1 hours = $ 7.25
Let x represent the number of hours Bob works and y represent the amount he charges. The equation will be:-
y = 7.25x dollars.
Therefore the equation of the amount charged by Bob will be y = 7.25x.
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(x+3)^2+4=46
what is the solution of this equation
Answer:
x = -3 + √42
Step-by-step explanation:
1) Expand the square
2) Distribute
3) Distribute
4) Distribute
5) Combine like terms
6) Add the numbers
7) Move terms to the left side
8) Subtract the numbers
9) Use the quadratic formula
10) Simplify
11) Separate the equations
12) Solve
Therefore, the solution of this equation is x = -3 + √42
Step-by-step explanation:
(x + 3)² + 4 = 46
(x + 3)² = 42
x + 3 = ±sqrt(42)
x = ±sqrt(42) - 3
x1 = sqrt(42) - 3 = 3.480740698...
x2 = -sqrt(42) - 3 = -9.480740698...
What comes next in this pattern? 7/2, 6/3, 5/4, ... 4/9 4/7 4/5
Answer:4/3
4/3
Step-by-step explanation:
√343k please simplify this
Answer:
the answer of√343k is 18k
amusement on a particularly boring transatlanfic fight one of the authors am used himseft by counting the heads of the people in the seats in front of him. He noticed that all 37 of them either had black hair or had a whole row to themselves or both. Of this total 33 had black hair and 6 were fortunate enough to have a whole row of seats to themselves. How many black haired people had whole rows to themselves?
1. What is the volume of this composite figure?
i6m
16 m
10 m
4 m
5 m
Answer:
560
Step-by-step explanation:
16 times 5 times 4=320
6 times 16 divide 2 then multiple 5
A pedestrian walks 7.4 kilometers west
and then 9.2 kilometers south.
What is the magnitude of the
pedestrian's resultant vector?
Hint: Draw a vector diagram.
[ ? ] kilometers
Round your answer to the nearest tenth.
Answer: 11.8 km
Step-by-step explanation:
The resulting figure is a right angled triangle shown below
[tex]$$Apply the Pythagorean theorem to find out the magnitude of the resultant vector $A B$ in the triangle $A B C$ as follows -$$\begin{aligned}&A B^{2}=B C^{2}+C A^{2} \text { (Hypotenuse }{ }^{2}=\text { Base }^{2}+\text { Adjacent }^{2} \text { ) } \\&=7.4^{2}+9.2^{2} \\&=139.4 \\&A B=11.806777 \\&\quad A B \approx 11.8 \mathrm{~km} \text { (Rounded off to the nearest tenth) }\end{aligned}$$[/tex]
Choose ALL the numbers that round down when rounding to the nearest hundred! Is it!: 47,035, 3,140, 76,654, 1,298, 607,882, 8,560?
Answer:
The answer is:
the first one
second one
:D
Step-by-step explanation:
in this case, any number in the tens place less than five will make the number round down
7th grade math perimeter!!! Help
Answer:30.28 cm^2
Step-by-step explanation: area of the rectangle = 6x4 = 24
area of semicircle -> (πr^2)/2
radius of semcircle = 4/2 = 2
area of this question semicricle = (π2^2)/2
= 6.28 (3 s.f.)
total area = 24+6.28 =30.28 cm^2
remember put units ^^
a line passes through the point (-9,1) and has a slope of 4/3. Write an equation in slope-intercept form for this line.
Answer: y=4/3x+1
Step-by-step explanation:
The formula is Y=mx+b
b is the y-intercept and MX is slope
