Find the minimum distance from the parabola x-y^2=0
to the point (0,3).
Answers
Answer:
√5
Step-by-step explanation:
The distance from the given point to the curve can be found using the distance formula. The resulting expression can be minimized to find the minimum distance.
d = √((x2 -x1)² +(y2 -y1)²)
__
Point on the ParabolaA point on the parabola can be described by ...
x -y² = 0
x = y² . . . . . add y²
Then a point on this curve is ...
(y², y)
Distance to Given PointThe distance to it from (0, 3) is ...
d = √((0 -y²)² +(3 -y)²) = √(y⁴ +y² -6y +9)
Find the MinimumThe distance will be minimized when the derivative of the radical expression is zero:
d(y⁴ +y² -6y +9)/dy = 0 = 4y³ +2y -6
0 = 2y³ +y -3 . . . . . . . . remove a factor of 2
This will have a rational root in the set {±1, ±3}.
Trial and error shows that y=1 is the only real solution to this equation.
__
Then the minimum distance is ...
d = √(1⁴ +1² -6·1 +9) = √5
_____
Additional comment
The attached graph shows that a circle with radius √5 centered at (0, 3) will intersect the parabola at exactly one point. That confirms our solution.
Related Questions
A triangle is inside of a rectangle. The triangle has a height of h and a base of b. Section 1 is to the left of the triangle, and section 2 is to the right.
Look at the drawing of a triangle within a rectangle. The triangle and the rectangle have the same base, b, and height, h.
If the area of the triangle is 48 square units, what is the total area of sections 1 and 2?
square units
Answers
Answer:
I think its 40
Step-by-step explanation:
Answer:
48
Step-by-step explanation:
Got it right on the assignment.
Hope this helps!!
The mean of the ages of 5 brothers is 13 years.
12, 16, O, 14, 8
Answers
Answer:
O = 15
Step-by-step explanation:
.mean ' is the average
(12 + 16 + O + 14 + 8) / 5 = 13
50 + O = 65
O = 15
f(X)= _3 x²_7x_9
f(_5) +f(2)=
Answers
Answer:
Step-by-step explanation:
[tex]-3x^2-7x-9=F(x)[/tex]
[tex]F(-5)=-75+40-9=-49[/tex]
[tex]F(2)=-12-14-9=-35[/tex]
[tex]f(-5)+f(2)=-49-35=-84[/tex]
What is the name given to a polynomial with 10 terms
Answers
Answer:
It should be called A decic polynomial
Step-by-step explanation:
Find the chance of having a face card if a card is drawn randomly from a deck of cards.
Answers
Answer:
3/13
Step-by-step explanation:
P(Choosing a Face card) =
12/52= 3/13
pls mark as brainliest
The best way to convince employers of your skills and abilities is to _____. a. provide a written list, such as in your resume b. offer proof by example in answering interview questions c. have the employer contact your references d. none of the above
Answers
Answer:
A.
Step-by-step explanation:
B and C seem to stray away from what the question originally asked.
Answer:
the answer is B ( offer proof by example in answering interview questions)
Step-by-step explanation:
hope this helps:)
ed 2023
Which of the equations below is a line parallel to the line y = 5x - 2?
Answers
Answer:
An unlimited number of lines are parallel to the line described by y = 5x + 2. The requirement is that it is in the form y = 5x + b, b ≠ 2.
Step-by-step explanation:
Given the functions f(x) = –4x + 5 and g(x) = x3 + x2 – 4x + 5, what type of functions are f(x) and g(x)? Justify your answer. What key feature(s) do f(x) and g(x) have in common? (Consider domain, range, x-intercepts, and y-intercepts.)
Answers
The f(x) has a degree of 1, hence it is linear in nature while g(x) is a polynomial function because of the third degree.
The key features that they have in common are the y-intercept, domain, and range
Linear and quadratic equationsLinear equations are equations that has a degree of 1 while polynomial equations are equations that has a degree of 2 and above.
Given the following functions
f(x) = –4x + 5 and;
g(x) = x^3 + x^2 – 4x + 5
Since the f(x) has a degree of 1, hence it is linear in nature while g(x) is a polynomial function because of the third degree.
The key features that they have in common are the y-intercept, domain and range
Learn more on functions here: https://brainly.com/question/25638609
When K is negative the parabola _____
Answers
Answer:
Is shifted downwards
Step-by-step explanation:
K determines where the parabola is on the y axis basically. If the k value was positive it would be shifted upwards and if it was negative it would be shifted downwards.
[tex] \displaystyle\red{ \sf\lim_{x \to {0}^{ + } }\frac{1}{ {sin}^{2}x } \int _{ \frac{x}{2} }^{x} {sin}^{ - 1} t \: dt}[/tex]
Answers
Use L'Hopital's rule twice:
[tex]\displaystyle \lim_{x\to0^+} \frac1{\sin^2(x)} \int_{\frac x2}^x \sin^{-1}(t) \, dt = \lim_{x\to0^+} \frac1{\sin^2(x)} \left(\int_0^x \sin^{-1}(t) \, dt - \int_0^{\frac x2} \sin^{-1}(t) \, dt\right) \\\\ = \lim_{x\to0^+} \frac{\sin^{-1}(x) - \frac12 \sin^{-1}\left(\frac x2\right)}{2\sin(x)\cos(x)} ~~~~~~~~~~ (LHR) \\\\ = \lim_{x\to0^+} \frac{2\sin^{-1}(x) - \sin^{-1}\left(\frac x2\right)}{2\sin(2x)} \\\\ = \lim_{x\to0^+} \frac{\frac2{\sqrt{1-x^2}} - \frac1{2\sqrt{1-\frac{x^2}4}}}{4\cos(2x)} ~~~~~~~~~~ (LHR) \\\\ = \lim_{x\to0^+} \frac{\frac2{\sqrt{1-x^2}} - \frac1{\sqrt{4-x^2}}}{4\cos(2x)}[/tex]
The remaining limand is continuous at x = 0, and the limit is
[tex]\displaystyle \lim_{x\to0^+} \frac1{\sin^2(x)} \int_{\frac x2}^x \sin^{-1}(t) \, dt = \frac{2-\frac12}4 = \boxed{\frac38}[/tex]
If Alex did 3/6 and then 1/4 of his homework, how much more does he need in order to finish ALL of his homework?
Answers
I SUCK AT MATH PLS HELP
WORTH 1 POINT
GIVING 10
Answers
Answer:
34 degrees
Step-by-step explanation:
Assuming that ADJ is a right angle, we can add 4x and 3x-8 and set equal to 90 degrees to simply to get get x.
7x-8=90
Add 8
7x = 98
x= 14
Now that we know x, we can plug in the value to find angle CDJ.
3x-8
3(14)-8
34 degrees
Suppose f(x) = x². Find the graph of f(x-4)
4
Click on the correct answer.
1
2 3
Click on each graph to enlarge it.
graph 1
graph 2
graph 3
graph 4
What’s the answer ?
Answers
Answer:
1
Step-by-step explanation:
Translation of a function h units to the right is accomplished by replacing x in the function definition with (x-h).
Here, x has been replaced by x-4, so we expect to see the graph look like the original function graph shifted 4 units to the right.
That matches Graph 1.
*50 POINTS* Pls HELP. ANSWER THE QUESTION IN THE PIC!
Answers
Answer:
D
Step-by-step explanation:
Here's why I think it's D.
y = x + 44 is the equation the car wash owner uses
x represents the number of days
y represents the number of cars (washed)
so if we plug in one of the given numbers (from the chart)
example:
Day 2 Numbers of Cars Washed 44
2 will represent the "number of days" also known as x
44 will represent the "number of cars" also known as y
so if we plug in those values to the equation the car wash owner uses
y = x + 44, it doesn't match
__________________________________
(44) = 2 + (44)
44 [tex]\neq[/tex] 46
Therefore, I believe the answer is D
Answer:
D
Step-by-step explanation:
It is not a good fit for the dataThere should equal number of points below and above the x-axis in the residual plotThe given equation : y = x + 44 does not match the given dataWhich expression is a factored form of
3n² - 4n - 4
Answer choices.............................
................................................
ANSWER
for reference of other who need it. cuz I needed it and it wasn't on here
A: (n+2) (3n +1)
Answers
Answer:
The answer is (n – 2) (3n + 2)
Step-by-step explanation:
3n² – 4n – 4
3n² – 6n + 2n – 4
3n(n – 2) + 2(n – 2)
(n – 2) (3n + 2)
Thus, (n – 2) (3n + 2) is a factored of
3n² - 4n - 4
40% if what number is 46?
Answers
Answer:
40 percent of 46 is 18.4.
A ladder is leaning against a building. The distance from the bottom of the ladder to the building is 12 ft shorter than the length of the ladder. How high up the side of the building is the top of the ladder if that distance is 6 ft less than the length of the ladder? C The top of the ladder is ___ ft up the side of the building?
Answers
Answer:
The answer for this question is 2 ft
Step-by-step explanation:
12ft÷6ft=2
please show me how you also got the answer as well, thanks a bunch!!
Answers
Answer:
Step-by-step explanation:
how do i multiply 30 by -20
Answers
Multiplication of a positive number and negative number
⇒ is like any other multiplication
30 * -20 = 30 * ((-1) * 20) = 30 * -1 * 20 = (-1) * 600 = -600
Hope that helped!
The percentage error in measuring a clothe is 40% if the error committed is 10cm what is the original Value of the measurement??
Answers
cross multiply:
40% - 10
100% - x
so 40x = 1000
x=25
Express with fractional exponents instead of radicals.
6√x² y³
Answers
Answer:
using of fractional exponents give us the value 6(x^2y^3)^1/2 or 6xy^3/2.
5 questions! Math, answer by saying "question __ is ____ " thank you! picture included! 50 points!
Answers
Answer:
1.c
2.a
3.d
4.b
5.b
Step-by-step explanation:
I found the answers on a web site.
Answer:
1.A
2.D
3.C
4.B
5.A
Step-by-step explanation:
Is this statement true or false in the ordered pair (4,3) the y coordinate is 4
Answers
false
For this question, I solve Q.1, but I cannot solve Q2.
Q1. Suppose that the balance of a person’s bank account in Superior, WI is normally distributed with mean $580 and standard deviation $125. Find the amount of money which would guarantee a person has more money in their account than 80% of Superior, WI residents.
Q2. For the same setup as in Problem 2, find the probability a random person from
Superior has less than $400 or more than $1000 in their bank account.
Answers
The amount of money which would guarantee a person has more money in their account than 80% of Superior, WI residents is $680 approx. The probability a random person from Superior has less than $400 or more than $1000 in their bank account is 0.9247
How to get the z scores?If we've got a normal distribution, then we can convert it to standard normal distribution and its values will give us the z score.
If we have
[tex]X \sim N(\mu, \sigma)[/tex]
(X is following normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex])
then it can be converted to standard normal distribution as
[tex]Z = \dfrac{X - \mu}{\sigma}, \\\\Z \sim N(0,1)[/tex]
(Know the fact that in continuous distribution, probability of a single point is 0, so we can write
[tex]P(Z \leq z) = P(Z < z) )[/tex]
Also, know that if we look for Z = z in z tables, the p-value we get is
[tex]P(Z \leq z) = \rm p \: value[/tex]
Let we take:
X = The balance of a person’s bank account in Superior, WI
Then, by the given data, we have:
[tex]X \sim N(\mu = 580, \sigma = 125)[/tex]
Let the amount of money which would guarantee a person has more money in their account than 80% of Superior, WI residents be X = x dollars. This is the money, below which lie 80% of Superior, WI residents, and so as their income, or values of X.
Symbolically, we have:
[tex]P(X < x) = 80\% = 0.8[/tex]
Converting X to standard normal variate Z, we have:
[tex]Z = \dfrac{X - \mu}{\sigma} = \dfrac{X - 580}{125}\\[/tex]
Thus, the probability statement can be rewritten as:
[tex]P(X < x) =0.8\\P\left( Z < z = \dfrac{x -580}{125} \right) = 0.8[/tex]
From the z-tables, the value of Z for which the p-value comes out as between 0.84 and 0.85, let it be their average 0.845.
Thus, we get:
[tex]P(Z < z = 0.845) \approx 0.8[/tex]
Thus, we get:
[tex]z = \dfrac{x - 580}{125} \approx 0.8\\\\x \approx 0.8 \times 125 + 580 = 680[/tex]
Thus, the amount of money which would guarantee a person has more money in their account than 80% of Superior, WI residents is $680 approx.
Now, for second question, we need:
The probability a random person from Superior has less than $400 or more than $1000 in their bank account.
So, income of a random person from Superior is a random value of X.
So, this probability is written as:
[tex]P(400 < X < 1000)[/tex]
Rewritting it, we get:
[tex]P(400 < X < 1000) = P(X < 1000) - P(X \leq 400)[/tex]
Converting X to Z (standard normal variate), we get:
[tex]P(400 < X < 1000) = P(X < 1000) - P(X \leq 400)\\\\P(400 < X < 1000) = P\left(Z < \dfrac{1000-580}{125} \right) - P\left(Z \leq \dfrac{400-580}{125} \right)\\\\P(400 < X < 1000) = P( Z \leq 3.36) - P(Z \leq -1.44)[/tex]
The p-value for Z = 3.36 is 0.9996
The p-value for Z = -1.44 is 0.0749
Thus, we get:
[tex]P(400 < X < 1000) = P( Z \leq 3.36) - P(Z \leq -1.44)\\\\P(400 < X < 1000) = 0.9996 - 0.0749 = 0.9247[/tex]
Thus, the probability a random person from Superior has less than $400 or more than $1000 in their bank account is 0.9247
Thus, the amount of money which would guarantee a person has more money in their account than 80% of Superior, WI residents is $680 approx. The probability a random person from Superior has less than $400 or more than $1000 in their bank account is 0.9247
Learn more about z-score here:
https://brainly.com/question/21262765
#SPJ1
If a chain is on 24 tooth front gear and 8 tooth rear gear determine whether this combination will result in high speed than a 2 to 1 gear ratio, assuming the speed you are pedalling is the same
Answers
24/8 = 3
the 24 to 8 gear = 3:1 ratio which is greater than 2:1
WHOEVER ANSWERS CORRECT AND FIRST WILL GET BRAINLIEST! How long does it take an elevator to go from the ground floor to the observation deck at a height of 869 feet?
Answers
so for a height of 869 feet the time will be 21.725s
[tex]1s = 40ft \\ x = 869ft \\ x= \frac{869}{40} \\ x = 21.725ft[/tex]
Solve the linear equations below by using:
2x + y = -3
3x - 2y = -8
a. Addition
b. Substitution
c. Graphing
Answers
X=2
4x+2y=-6
+(3x-2y=-8
The mass of 1 tablespoon of syrup is as heavy as 5 teaspoons of cooking oil. The mass of 2 teaspoons of cooking oil is 12 g lighter than 1 tablespoon of syrup.find the mass of 1 teaspoon of cooking oil
Answers
Answer:
1 tsp of oil = 4 g
Step-by-step explanation:
1 Tbsp of syrup = 5 tsp of oil
2 tsp of oil = 1 Tbsp of syrup - 12 g
2 tsp of oil = 5 tsp of oil - 12 g
-3 tsp of oil = -12 g
1 tsp of oil = 4 g
What is the mean of this data set? 12, 14, 16, 17, 18, 25
Answers
Find the value by using logarithms
25.753
0.5341
Answers
Step-by-step explanation:
589292 is the correct answer:-)
