(1 pt) find the coordinates of the point (x, y, z) on the plane z = 3 x + 4 y + 4 which is closest to the origin.
Answers
The solution to the problem involves setting up and solving a system of equations derived from the distance function in 3D. The minimum point of the distance function corresponds to the point in the plane that's closest to the origin.
Explanation:
To find the coordinates of the point (x, y, z) on the plane z = 3x + 4y + 4 which is closest to the origin, we need to find the minimum distance from origin (0,0,0) to any point on the plane. The distance equation is given by √[(x₂−x₁)²+(y₂−y₁)²+(z₂−z₁)²]. This is simply Euclidean distance in 3D.
In this case, we can define the distance function d = √(x² + y²+ (3x + 4y + 4)²). To find the minimum distance, we take the derivative of this distance function with respect to both x and y, set them to 0, and solve the resulting equations. The solutions to these equations yield x and y. We substitute x and y into z = 3x + 4y + 4 to get z.
Finally, solving these equations we obtain the coordinates of the point (x, y, z) on the plane that is closest to the origin.
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Answer:
(-6/13, -8/13, 2/13)
Step-by-step explanation:
You want the point on the plane z = 3x +4y +4 that is closest to the origin.
Normal vectorWhen the plane equation is written in standard form, the variable coefficients comprise a vector that is normal to the plane. The line normal to the plane and passing through the origin can be written in terms of this vector. Solving for the variable in that equation lets us find the point on the plane where the line intersects.
The standard form equation for the plane is ...
3x +4y -z = -4 . . . . . . . subtract 4+z from the given equation
Then the normal to the plane is ...
N = (3, 4, -1)
and the vector equation of the line through the origin is ...
L = (0, 0, 0) +tN = (3t, 4t, -t)
Point of intersectionThe value of t that makes a point on the line satisfy the plane equation is ...
3(3t) +4(4t) -(-t) = -4
26t = -4
t = -4/26 = -2/13
Then the point tN is ...
(-2/13)(3, 4, -1) = (-6/13, -8/13, 2/13)
The point on the plane closest to the origin is (-6/13, -8/13, 2/13).
Related Questions
which of the following values of x makes the rational expression below equal to zero? 2x+16/x+5
a. 5
b. -5
c. 8
d. -8
Answers
For this case what you should know is that an indefinite expression are those in which mathematically no numerical value can be obtained.
We have then that a division between zero causes an expression to be undefined.
For x = -5 we have:
(2 (-5) +16) / ((- 5) +5)
(-10 + 16) / (- 5 + 5) = 6/0 (undefined)
Answer:
x = -5
b. -5
Answer:
d. -8
Step-by-step explanation:
Remember that you cannot possibly divide anything by 0, so if the denominator gives as a result 0, you are dealing with an impossibility, in order for a franction to give as a result 0, the numerator has to be 0, so we just need to equalize the numerator to 0 and solve for x to know how can we make the rational expression 0:
2x+16=0
2x=-16
x=-16/2
x=-8
So in order to make the expression equal to 0 x needs to be equal to -8,
Which trinomial is the product of the expression below ? (2R-3) (R+12)
Answers
2R^2 + 21R - 36
Explanation:
To answer this question, we will follow these steps:
1- expand the brackets
2- gather like terms
Steps are as follows:
(2R-3)(R+12) = 2R(R) + 2R(12) - 3(R) - 3(12)
= 2R^2 + 24R - 3R - 36
= 2R^2 + 21R - 36
Hope this helps :)
(2R-3) (R+12)---------- > 2R*R+12*2R-3*R-3*12---------- > 2R²+24R-3R-36
2R²+24R-3R-36---------------> 2R²+21R-36
the answer is 2R²+21R-36
miss Roberts simplified this ratio 2:6 explain how you know she is incorrect can you tell where she might have made a mistake
Answers
Do you go to k12 btw ?
Question 7 plz help me saner this question
Answers
[tex] \frac{8}{4000} * 100 [/tex]
1/5 %
= 0.2%
I hope this helps!
In a probability experiment, Craig rolled a six-sided die 57 times. The die landed on a number greater than three 36 times. What is the ratio of rolls greater than three to rolls less than or equal to three?
Answers
36:21 →12:7
The ratio of rolls greater than three to rolls less than or equal to three is 36:21, which simplifies to 12:7 after dividing both terms by the greatest common divisor of 3.
Since Craig rolled the die 57 times, and it landed on a number greater than three 36 times, then it landed on a number less than or equal to three the remaining number of times. Subtracting the number of greater rolls from the total rolls:
Total rolls: 57
Rolls greater than three: 36
Rolls less than or equal to three = 57 - 36 = 21
Now, we can establish the ratio of rolls greater than three to rolls less or equal to three:
Ratio = Rolls greater than three : Rolls less than or equal to three = 36 : 21
And if needed, we can simplify this ratio by dividing both sides by their greatest common divisor, which is 3:
Simplified Ratio = 12 : 7
The glass container is filled to a height of 2.25 inches. What percent of the container is filled with sand?
Answers
Part A) How much sand is currently in the container?
[sand currently in the container]=(5)*(4 1/2)*(2.25)-----> (5)*(4.5)*(2.25)
[sand currently in the container]=50.625 in³
the answer Part a) is 50.625 in³
Part B) How much more sand could the container hold before?
[sand could the container hold before]=[5*4.5*3]-[50.625]
[sand could the container hold before]=[67.5]-[50.625]------> 16.875 in³
the answer Part B) is 16.875 in³
Part C) What percent of the container is filled with sand?
the volume of container is [5*4.5*3]=67.5 in³
the volume filled with sand=50.625 in³
therefore
if 100%----------------> 67.5 in³
X---------------------> 50.625 in³
X=(50.625*100)/67.5=75 %
the answer Part C) is 75%
Assume that when adults with smartphones are randomly selected, 54% use them in meetings or classes (based on data from an lg smartphone survey). if 12 adult smartphone users are randomly selected, find the probability that fewer than 3 of them use their smartphones in meetings or classes.
Answers
Using the binomial probability formula, sum up the probabilities for 0, 1, and 2 smartphone users using their phones in meetings, which will give you the total probability for fewer than 3.
Explanation:The subject of this question appears to be probability theory, a branch of mathematics that studies the likelihood of different outcomes. The problem presented involves calculating the probability of a certain number of smartphone users behaving in a specified way. In this case, it's about finding the likelihood that fewer than 3 out of 12 selected individuals use their smartphones in meetings or classes.
To solve this, we'll use a binomial probability formula: P(X=r) = (nCr) * p^r * (1-p)^(n-r), where:
n is the number of trials (in this case, 12 smartphone users),r is the desired number of successful outcomes,p is the probability of success on a single trial (54%), andnCr is the combination of n items taken r at a time.
You will need to sum up the probabilities for 0, 1, and 2 smartphone users using their phones in meetings or classes to get the total probability for fewer than 3.
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add or subtract.
[tex] \binom{7 \sqrt{2} }{2} - \binom{ \sqrt{8} }{2} [/tex]
Answers
This is the fraction of 5*sqrt(2) all over 2
"sqrt" is shorthand for "square root"
--------------------------------------------
Work Shown:
[tex]\frac{7\sqrt{2}}{2}-\frac{\sqrt{8}}{2}[/tex]
[tex]\frac{7\sqrt{2}}{2}-\frac{\sqrt{4*2}}{2}[/tex]
[tex]\frac{7\sqrt{2}}{2}-\frac{\sqrt{4}*\sqrt{2}}{2}[/tex]
[tex]\frac{7\sqrt{2}}{2}-\frac{2*\sqrt{2}}{2}[/tex]
[tex]\frac{7\sqrt{2}-2*\sqrt{2}}{2}[/tex]
[tex]\frac{5\sqrt{2}}{2}[/tex]
Which describes the difference between the graph of f(x) = 3x2 + 2 and g(x) = 9x2?
B) The graph of f(x) is obtained by shifting g(x) up 2 units and compressing vertically by a factor of 3.
C) The graph of g(x) is obtained by shifting f(x) right 2 units and stretching vertically by a factor of 3.
D) The graph of f(x) is obtained by shifting g(x) right 2 units and stretching vertically by a factor of 3.
Answers
The solution is, the graph of f(x) is vertically stretched by a factor of 3 and shifted 7 units down, as f(x) = x changes to h(x) = 3x² - 7.
What is transformation of graph?Graph transformation is the process by which an existing graph, or graphed equation, is modified to produce a variation of the proceeding graph.
here, we have,
Given that,
here is a graph of f(x) = x which is transformed into h(x) = 3x² - 7.
The graph of a the equation -
f(x) = x
is a straight line.
now, we get, When it is squared and multiplied by 3, we get -
h(x) = 3x²
which is a parabola stretched by a factor of 3 in vertical direction along
+ y axis.
When 7 is subtracted, it will shift the graph down by 7 units.
h(x) = 3x² - 7
Therefore, the graph of f(x) is vertically stretched by a factor of 3 and shifted 7 units down, as f(x) = x changes to h(x) = 3x² - 7.
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Complete question:
What is the effect on the graph of f(x) = x when it is transformed to
h(x) = 3x2 - 7?
A. The graph of f(x) is vertically stretched by a factor of 3 and shifted
7 units down.
B. The graph of f(x) is horizontally compressed by a factor of 3 and
shifted 7 units to the right.
C. The graph of f(x) is horizontally stretched by a factor of 3 and
shifted 7 units down.
D. The graph of f(x) is vertically stretched by a factor of 3 and shifted
7 units to the right
Shirley is drawing triangles that have the same area. The base of each triangle varies inversely with the height. What are the possible base and height of a second triangle if the first triangle's base is 16 and its height is 6 ?
Answers
base 8, height 12
base 4, height 24
base 2, height 48
base 1, height 96
base 32, height 3
base 48, height 2
base 96, height 1
Which of these is equivalent to 92 + 42? (9 + 4)2 (9 × 4)2 (9 × 9) + (4 × 4) (9 + 9) + (4 + 4)
Answers
9 ^ 2 + 4 ^ 2.
We have then that the equivalent expression for this case is:
(9 × 9) + (4 × 4)
Both expressions give the same result 97
Answer:
(9 × 9) + (4 × 4)
Can anyone help me on this question?:) tysm ily
Answers
V=25*40*40+25*35*40=40*1875=75000 cm³
answer: 75000 cm³.
Va doctor prescribes 10 mg of cough syrup. available is 5 mg/5 ml. how many teaspoons would you give?
Answers
ans: 2 teaspoons
Which equation could be the equation of this parabola?
A. y = -2x^2
B. x = -2y^2
C. x = 2y^2
D. y = 2x^2
Answers
Parabolas in the form of [tex]\sf x=y^2[/tex] open to the left or to the right.
Since this parabola opens upward, we know it must be in the form of [tex]\sf y=x^2[/tex]. Also, if the coefficient of 'x' is positive, it opens upward, but if it is negative, it opens downward. This parabola opens upward, so we know that the coefficient of 'x' must be positive. Therefore, our answer must be D, [tex]\sf y=2x^2[/tex]
Given the lengths of two sides of a triangle, find the range for the length of the third side (between what two numbers should the length of the third side be). Write the inequalities for each case. 8 and 13
Answers
5 < third side < 21
Final answer:
The length of the third side of a triangle with side lengths of 8 and 13 must be greater than 5 units and less than 21 units according to the triangle inequality theorem.
Explanation:
To find the range for the length of the third side of a triangle given two sides with lengths of 8 and 13, we can use the triangle inequality theorem.
This theorem states that the length of any side of a triangle must be less than the sum of the lengths of the other two sides and greater than the absolute value of their difference.
The inequalities for the third side, let's call it c, are:
c < 8 + 13c > |13 - 8|Therefore, we can say:
c < 21c > 5So the length of the third side must be greater than 5 units and less than 21 units.
Simplify this equation.
(equation and choices in image)
Answers
b.)x³-7x²+3x+36
(x-4)(x²-3x-9)
Distribute x and -4 to x²-3x-9
=x²(x)-3x(x)-9(x)+x²(-4)-3x(-4)-9(-4)
=x³-3x²-9x-4x²+12x+36
Combine all liked terms
=x³-7x²+3x+36
How do you turn 43% into a fraction
Answers
To write a percent as a fraction in lowest terms, first remember that a percent is a ratio that compares a number to 100. 43% can be written as the ratio 43 to 100 or 43/100.
Therefore, 43% can be written as the fraction 43/100.
The heights in centimeters of 5 students are: 165, 175, 176, 159, 172 find the sample median, sample mean and sample variance.
Answers
The mean is: 169.4 cm
The sample variance steps to solve are:
"Work out the Mean (the simple average of the numbers)Then for each number: subtract the Mean and square the result (the squared difference).Then work out the average of those squared differences."
The value of the sample median is 172.
The value of the sample mean is 169.4.
The value of the sample variance is 52.3
What is a median?It is the middle value of the given set of numbers after arranging the given set of numbers in order.
We have,
The heights in centimeters of 5 students are: 165, 175, 176, 159, 172
Now,
Arrange the heights in order.
159, 165, 172, 175, 176
Median = 172
Mean = (159 + 165 + 172 + 175 + 176) / 5 = 847 / 5 = 169.4
Sample variance.
= ∑ (x - Mean)² / (n - 1)
= 209.2 / 4
= 52.3
∑ (x - Mean)²
= (159 - 169.4)² + (165 - 169.4)² + (172 - 169.4)² + (175 - 169.4)² + (176 - 169.4)²
= 108.16 + 19.36 + 6.76 + 31.36 + 43.56
= 209.2
Thus,
The sample mean = 169.4
The sample median = 172
The sample variance = 52.3
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Find three consecutive odd integers such that the sum of the smallest and 7 times the largest is 68
Answers
x = smallest
y = middle
z = largest
x + 7z = 68 ← This equation represents the given information
We can assume that "z" is less than 11 because 7*11 = 77, which is greater than 68.
Our possible consecutive odd integers could be:
a) 1, 3, 5
b) 3, 5, 7
c) 5,7, 9
Lets to plug in the "x" and "z" values
a) 1 + 7(5) = 1+ 35 = 36 This is incorrect, because 36 ≠ 68
b) 3 + 7(7) = 3 + 49 = 52 This is incorrect as well, because 52 ≠ 68
That leaves our final choice. Let's check to be certain
c) 5 + 7(9) = 5 + 63 = 68
5, 7, 9 is the 3 consecutive integers such that the sum of the smallest and 7 times the largest is 68.
The three consecutive odd integers such that the sum of the smallest and 7 times the largest is 68 are: Integers 5, 7 and 9.
Let;
x = smallest odd integer(x+2) = middle odd integer(x +4) = largest integerThe question however suggests that;
The sum of the smallest number and 7 times the largest number is 68.
Therefore;
x + 7(x+4) = 688x + 28 = 688x = 40x = 5.In essence,
the smallest number is x = 5.the middle number is x+2 = 7.the largest number is x+4 = 9Read more:
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One-third the sum of 6 and 3 (u get 10 points)
Hurry
Answers
Final answer:
The sum of 6 and 3 is 9, and one-third of this sum is 3 when you divide 9 by 3.
Explanation:
In mathematics, a sum is the result of adding two or more numbers or quantities together. It represents the total or aggregate of those quantities. The process of finding this total is called addition.
Thus,
To calculate one-third the sum of the numerals 6 and 3, we first find the sum of 6 and 3, which is 9.
Then, we need to take one-third of 9.
To find one-third of a number, you divide the number by 3.
So one-third of 9 is 9 divided by 3, which equals 3.
Therefore, one-third the sum of 6 and 3 is 3.
Please help me with question 5
Answers
By similar triangles ACD is similar to BCD
on the left AD/CD is one ratio
On the right CD/DB is the other. These two ratios are equal because they are corresponding parts.
AD / CD = CD / DB
AD = 3
DB = 12
CD = ??
Substitute.
3/CD = CD/12 Cross multiply
36 = CD^2
sqrt(36) = sqrt(CD^2)
CD = 6 <<<<==== Answer.
A <<<<====Answer.
How would you write the following phrase as a number expression eight more than triple a number
Answers
In a survey, 60 baseball fans were asked whether they thought the designated hitter rule should be changed. Forty-one fans thought the rule should be changed. WHat percent of the fans surveyed said that the designated hitter rule should be changed.
Answers
HELP MEEE!!! PLEASE!!!
Answers
In 2015, the price of a business math text rose to $150. This is 8% more than the 2014 price. What was the old selling price?
Answers
150=1.08x
x=$138.88
Answer: old selling priceof businesss math text rose = $138.89
Explanation:
In year 2015, Price of business math text rose = $150
Suppose privious price (year 2014)= x dollars
Previous price is 8% less than the price in 2015 because given that $150 is 8% more than previous year price.
Now write the linear equation for the given information:
x + 8% of x = 150
Solve for x
x + [tex] \frac{8}{100} [/tex]x = 150
x + 0.08x = 150
1.08x = 150
divide 1.08 both the side
x = [tex] \frac{150}{1.08} [/tex]
x = $138.89
Old selling price = $138.89
1. Solve w=x+y/z for y.
2. Solve the equation above for z.
Answers
Answer:
y = z(w -x)z = y/(w -x)Step-by-step explanation:
As in any "solve for ..." problem, you reverse (undo) the operations done to the variable of interest. Generally, you do that in reverse order.
1. y is divided by z then the quotient added to x. To solve for y, we subtract x, then multiply by z.
... w - x = y/z . . . . . . . subtract x from the given equation
... z(w -x) = y . . . . . . . multiply by z
___
2. To solve for z, it is more convenient to start with the equation for y that is the result of the first question. We divide that by the coefficient of z.
... z = y/(w -x)
From w=x+y/z, the value of y = z(w - x)
w=x+y/z
This is changing the subject formula. The first thing to do is to subtract x from both sides. This will be:
w - x = x + y/z - x
w - x = y/z
Then, we cross multiply
z(w - x) = y
Therefore, y = z(w - x)
Since y = z(w - x)
Then, to get the value of z goes thus:
z = y/(w - x)
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Jane has $10000 to invest in AAA and AA bonds. The AAA bond yields a 5% and requieres a minimum investment of $1000. The AA bond yields an average of 8% and requires a minimum investment of $2000. Jane requires that at least three times as much money should be invested in the AAA bond as in the AA bond
Answers
$10,000 to invest
AAA bonds - minimum of $1,000 yields 5%
AA bonds - minimum of $2,000 yields 8%
Jane requires that at least three times as much money should be invested in the AAA bond as in the AA bond
Let x represent the investment in AA bond.
x + 3x = 10,000
4x = 10,000
x = 10,000 / 4
x = 2,500
Jane should invest 2,500 to AA bond.
Jane should invest 7,500 to AAA bond. This amount is 3 times the amount invested in AA bond.
2,500 * 8% = 200
2,500 + 200 = 2,700
7,500 * 5% = 375
7,500 + 375 = 7,875
Jane's $10,000 investment will earn $575 interest for a total of $10,575.
Help!
Could it be...??
a. /_ A
b. /_ BCA
c. /_ b
d. /_ CBA
Answers
You are wearing a pair of cargo pants with six pockets. You've put $10 in one of the pockets, but you cannot remember which one. After checking two pockets without success, what is the probability that the money will be in the next pocket you check? NextReset
Answers
The total number of ways left over is 4 for your next try
The number of pockets that will be successful is 1
Since there are 6 pockets all together. And you checked two of them and neither of them had the $10. so you cannot forget those 2 when figuring this out, and now there are 4 pockets left unchecked
The probability that the money will be in next pocket is 0.25.
We have a pair of cargo pants with six pockets and $10 in one of the pockets.
We have to find out the probability that the money will be in the next pocket you check after checking two pockets without success.
What is the formula to calculate the probability of an event ?The formula to calculate the probability of an event is -
[tex]P(A) = \frac{Total\;number\;of\;favorable\;outcomes}{Total\;number\;of\;outcomes\;in \;sample\; space} =\frac{n(A)}{n(S)}[/tex]In the question it is given -
Total money = $10
Total no. of pockets = 6 = n(S)
After two unsuccessful check in two pockets, new sample space =
n(S') = n(S) - 2 = 4
Hence, the probability that the money will be in the next pocket you check is -
P(A) = [tex]\frac{1}{4} = 0.25[/tex]
Hence, the probability that the money will be in next pocket will be 0.25.
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y - 5 = 0.5x2 + 6x - 3 How many x-intercepts does the graph of this quadratic have?
Answers
Y - 5 = 0.5x2 + 6x - 3--------------- > y=0.5x2 + 6x -3+5=0.5x2 + 6x+2
y=0.5x2 + 6x +2
using a graphic tool
see the attached figure
the graph have two x-intercepts----------> (-11.657,0) and (0.343,0)
Answer:
its (A)
Step-by-step explanation: