a.
[tex]f(x,y)=e^{-(x-a)^2-(y-b)^2}\implies\begin{cases}f_x=-2(x-a)e^{-(x-a)^2-(y-b)^2}\\f_y=-2(y-b)e^{-(x-a)^2-(y-b)^2}\end{cases}[/tex]
Critical points occur where [tex]f_x=f_y=0[/tex]. The exponential factor is always positive, so we have
[tex]\begin{cases}-2(x-a)=0\\-2(y-b)=0\end{cases}\implies(x,y)=\boxed{(a,b)}[/tex]
b. As the previous answer established, the critical point occurs at (-3, 8) if [tex]\boxed{a=-3}[/tex] and [tex]\boxed{b=8}[/tex].
c. Check the determinant of the Hessian matrix of [tex]f(x,y)[/tex]:
[tex]\mathbf H(x,y)=\begin{bmatrix}f_{xx}&f_{xy}\\f_{yx}&f_{yy}\end{bmatrix}[/tex]
The second-order derivatives are
[tex]f_{xx}=(-2+4(x-a)^2)e^{-(x-a)^2-(y-b)^2}[/tex]
[tex]f_{xy}=4(x-a)(y-b)e^{-(x-a)^2-(y-b)^2}[/tex]
[tex]f_{yx}=4(x-a)(y-b)e^{-(x-a)^2-(y-b)^2}[/tex]
[tex]f_{yy}=(-2+4(y-b)^2)e^{-(x-a)^2-(y-b)^2}[/tex]
so that the determinant of the Hessian is
[tex]\det\mathbf H(x,y)=f_{xx}f_{yy}-{f_{xy}}^2=\left((4(x-a)^2-2)(4(y-b)^2-2)-16(x-a)^2(y-b)^2\right)e^{-2(x-a)^2-2(y-b)^2}[/tex]
[tex]\det\mathbf H(x,y)=(16(x-a)^2(y-b)^2-8(x-a)^2-8(y-b)^2)+4)e^{-2(x-a)^2-2(y-b)^2}[/tex]
The sign of the determinant is unchanged by the exponential term so we can ignore it. For [tex]a=x=-3[/tex] and [tex]b=y=8[/tex], the remaining factor in the determinant has a value of 4, which is positive. At this point we also have
[tex]f_{xx}(-3,8;a=-3,b=8)=-2[/tex]
which is negative, and this indicates that (-3, 8) is a local maximum.
The critical point of the given function is (a, b). For it to be at (-3, 8), a and b should be -3 and 8 respectively. This point (-3, 8) is a local maximum for the function.
Explanation:The critical point of the given equation f(x, y) = e−(x − a)² − (y − b)² are the coordinates (a, b).
To find the values of a and b such that the critical point is at (−3, 8), we simply set (a, b) = (−3, 8). This implies that a = -3 and b = 8.In the context of the function f(x, y), the given point (-3,8) represents a local maximum. This is because the function f(x, y) achieves its maximum value when its exponent is at a minimum, which occurs at x=a and y=b. Hence we can say that (-3, 8) is a local maximum for the function f(x, y) with a = -3 and b = 8.Learn more about Critical points here:https://brainly.com/question/32077588
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Find the twenty-fifth term of an arithmetic sequence if the first term is-1 and the common difference is 5. Write the first three terms of an arithmetic sequence in which the twenty-first term is 17 and the fiftieth term is 75. 10. 11
Final answer:
The twenty-fifth term of the arithmetic sequence is 119. The first three terms of another arithmetic sequence are 17, 19, 21.
Explanation:
To find the twenty-fifth term of an arithmetic sequence, we can use the formula:
nth term = first term + (n-1) * common difference
Substituting the given values:
nth term = -1 + (25-1) * 5 = -1 + 24 * 5 = -1 + 120 = 119
Therefore, the twenty-fifth term of the arithmetic sequence is 119.
For the second part of the question, to find the common difference, we can use the formula:
common difference = (fiftieth term - twenty-first term) / (50 - 21)
Substituting the given values:
common difference = (75 - 17) / (50 - 21) = 58 / 29 = 2
Using the first term of 17 and the common difference of 2, we can write the first three terms of the arithmetic sequence:
17, 19, 21
Random variables X Poisson~ ( a) ,Y Poisson ~ ( a) . X and Y are independent. If 2 1, 2 1. U =2X+ Y-1, V=2X- Y +1. Find: ) Cov (U ,V ).
By definition of covariance,
[tex]\mathrm{Cov}(U,V)=E[(U-E[U])(V-E[V])]=E[UV-E[U]V-UE[V]+E[U]E[V]]=E[UV]-E[U]E[V][/tex]
Since [tex]U=2X+Y-1[/tex] and [tex]V=2X-Y+1[/tex], we have
[tex]E[U]=2E[X]+E[Y]-1[/tex]
[tex]E[V]=2E[X]-E[Y]+1[/tex]
[tex]\implies E[U]E[V]=(2E[X]+E[Y]-1)(2E[X]-(E[Y]-1))=4E[X]^2-(E[Y]-1)^2=4E[X]^2-E[Y]^2+2E[Y]-1[/tex]
and
[tex]UV=(2X+Y-1)(2X-(Y-1))=4X^2-(Y-1)^2=4X^2-Y^2+2Y-1[/tex]
[tex]\implies E[UV]=4E[X^2]-E[Y^2]+2E[Y]-1[/tex]
Putting everything together, we have
[tex]\mathrm{Cov}(U,V)=(4E[X^2]-E[Y^2]+2E[Y]-1)-(4E[X]^2-E[Y]^2+2E[Y]-1)[/tex]
[tex]\mathrm{Cov}(U,V)=4(E[X^2]-E[X]^2)-(E[Y^2]-E[Y]^2)[/tex]
[tex]\mathrm{Cov}(U,V)=4V[X]-V[Y]=4a-a=\boxed{3a}[/tex]
PLEASE HELP PRECALC MARK BRAINLIEST
let's recall that a rational whose numerator and denominator are of the same degree, has a horizontal asymptote at the fraction provided by the leading term's coefficients.
so we can simply pick any two polynomials, make them the same degree and give their leading term 2 and 9 respectively.
hmmmm say for the numerator x⁴ - 3x³.... and the denominator hmm say x⁴ + 7x, so then let's give them 2 and 9 respective... so
[tex]\bf \cfrac{\stackrel{\stackrel{\textit{leading term}}{\downarrow }}{2x^4}-3x^3}{\underset{\underset{\textit{leading term}}{\uparrow }}{9x^4}+7x}\implies \stackrel{\textit{horizontal asymptote}}{y=\cfrac{2}{9}}[/tex]
12x + 1 = 25
Solve the following equation. Then place the correct number in the box provided.
For this case we have the following equation:
[tex]12x + 1 = 25[/tex]
We must find the solution!
Subtracting 1 from both sides of the equation we have:
[tex]12x = 25-1\\12x = 24[/tex]
Dividing between 12 on both sides of the equation we have:
[tex]x = \frac {24} {12}\\x = 2[/tex]
Thus, the solution is given by[tex]x = 2[/tex]
Answer:
[tex]x = 2[/tex]
Solve for x
a.
b.
Will mark brainliest!
Answer:
a) x = 7
b) x = 2
Step-by-step explanation:
* Lets revise some facts in the circle
- If two secant segments are drawn to a circle from a point outside the
circle, the product of the length of one secant segment and its
external part is equal to the product of the length of the other secant
segment and its external part.
# Example:
- If AC is a secant intersects the circle at points A and and B
- If DC is another secant intersects the circle at points D and E
- The two secants intersect each other out the circle at point C
∴ AC × CB = DC × CE , where AC is the secant and CB is its external
part and DC is the secant and CE is its external part
* Lets solve the problem
a) There are two secants intersect each other at point outside the circle
∵ The first secant is x + 5
∵ Its external part is 5
∵ the second secant is 4 + 6 = 10
∵ Its external part is 6
∴ (x + 5) × 5 = 10 × 6 ⇒ simplify
∴ 5x + 25 = 60 ⇒ subtract 25 from both sides
∴ 5x = 35 ⇒ divide both sides by 5
∴ x = 7
* x = 7
b) There are two secants intersect each other at point outside the circle
∵ The first secant is 5 + 3 = 8
∵ Its external part is 3
∵ the second secant is x + 4
∵ Its external part is 4
∴ 8 × 3 = (x + 4) × 4 ⇒ simplify
∴ 24 = 4x + 16 ⇒ subtract 16 from both sides
∴ 8 = 4x ⇒ divide both sides by 4
∴ 2 = x
* x = 2
what is the solution to the equation 9^(x+1) =27
ANSWER
[tex]x = \frac{1}{2} [/tex]
EXPLANATION
The given exponential equation is
[tex] {9}^{x + 1} = 27[/tex]
The greatest common factor of 9 and 27 is 3.
We rewrite the each side of the equation to base 3.
[tex]{3}^{2(x + 1)} = {3}^{3} [/tex]
Since the bases are equal, we can equate the exponents.
[tex]2(x + 1) = 3[/tex]
Expand the parenthesis to get:
[tex]2x + 2 = 3[/tex]
Group similar terms
[tex]2x = 3 - 2[/tex]
[tex]2x = 1[/tex]
[tex]x = \frac{1}{2} [/tex]
For this case we must solve the following equation:
[tex]9 ^ {x + 1} = 27[/tex]
We rewrite:
[tex]9 = 3 * 3 = 3 ^ 2\\27 = 3 * 3 * 3 = 3 ^ 3[/tex]
Then the expression is:
[tex]3^ {2 (x + 1)} = 3 ^ 3[/tex]
Since the bases are the same, the two expressions are only equal if the exponents are also equal. So, we have:
[tex]2 (x + 1) = 3[/tex]
We apply distributive property to the terms within parentheses:
[tex]2x + 2 = 3[/tex]
Subtracting 2 on both sides of the equation:
[tex]2x = 3-2\\2x = 1[/tex]
Dividing between 2 on both sides of the equation:
[tex]x = \frac {1} {2}[/tex]
Answer:
[tex]x = \frac {1} {2}[/tex]
solve 8 + 5^x = 1008. Round to the nearest ten-thousandth.
Answer: 4.29203
Explanation:
5^x=1008-8
5^x=1000
take the log of both sides
x=3 log 5 (10)
x=3+3log5(2)
or 4.29203
For this case we must solve the following equation:
[tex]8 + 5 ^ x = 1008[/tex]
Subtracting 8 on both sides of the equation:
[tex]5 ^ x = 1008-8\\5 ^ x = 1000[/tex]
We apply Neperian logarithm to both sides of the equation:
[tex]ln (5 ^ x) = ln (1000)[/tex]
We use the rules of the logarithms to draw x
of the exponent.
[tex]xln (5) = ln (1000)[/tex]
We divide both sides of the equation between[tex]ln (5)[/tex]:
[tex]x = \frac {ln (1000)} {ln (5)}\\x = 4.29202967[/tex]
Rounding:
[tex]x = 4.2920[/tex]
Answer:
[tex]x = 4.2920[/tex]
What is the value of x?
Answer:
It's B
Step-by-step explanation:
180 - 105 = 75 +55 = 130; the recangle has a sum of 180, hence 180-130 =50
Please help!!!!!>>>>>>
Answer:
-3.3
Step-by-step explanation:
-2.3-(4.5-3 1/2)=-2.3-(4.5-3.5)=-2.3-(1)=-3.3
A bag contains 3 red marbles, 2 blue marbles, and 2 green marbles. One marble is picked, then another marble. Assume that the selections are made with replacement. A) Find the probability of picking two red marbles with replacement. B) find the probability of picking a red marble and a blue marble. Assume the selections are made without replacement. C) Find the probability of picking two red marbles without replacement. D) find the probability of picking a red marble and a blue marble without replacement.
Answer:
A.) 3/7
B.) 5/7
C.) 2/7
D.) 2/7
what is the logarithmic function modeled by the following table? x f(x) 9 2 27 3 81 4
Answer:
The logarithmic function modeled by the given table:
f(x) = log₃x
Step-by-step explanation:
Given Table:
x f(x)
9 2
27 3
81 4
We can see that x increases as powers of 3. And f(x) is the power.
We assume that f(x) = log₃x
Checking using the table:
for x = 9
f(x) = log₃9 = 2
for x = 27
f(x) = log₃27 = 3
for x = 81
f(x) = log₃81 = 4
Hence proved.
Use your knowledge of the process of "Writing an equation given two points" to solve the following problem: A vendor has learned that, by pricing his deep fried bananas on a stick at $1.00, sales will reach 100 per day. Raising the price to $2.00 will cause the sales to fall to 52 per day. Let y be the number of the vendor sells at x dollars each. Write a linear equation that models the number of sold per day when the price is x dollars each.
Answer: Our required linear equation would be [tex]x+48y=148[/tex]
Step-by-step explanation:
Since we have given that
Cost of deep fried bananas on a stick = $1.00
Number of sales reached = 100 per day
Cost of deep fried bananas on a stick becomes = $2.00
Number of sales reached = 52 per day.
Let x is the number of dollars each.
Let y be the number of vendors sale.
So, we need to form the linear equation:
As we know the formula for two point slope form:
[tex]y-y_1=\dfrac{y_2-y_1}{x_2-x_1}(x-x_1)\\\\y-1=\dfrac{2-1}{52-100}(x-100)\\\\y-1=\dfrac{1}{-48}(x-100)\\\\-48(y-1)=(x-100)\\\\-48y+48=x-100\\\\-48y=x-100-48\\\\-48y=x-148\\\\x+48y=148[/tex]
Hence, our required linear equation would be [tex]x+48y=148[/tex]
Solve the linear Equation . -10x -3= -10 -3x
-10x - 3 = -10 - 3x
Bring -10x to the other side by adding it to both sides
(-10x + 10x) - 3 = -10 + (-3x + 10x)
0 - 3 = -10 + 7x
-3 = -10 + 7x
Bring -10 to the oposite side by adding 10 to both sides
-3 + 10 = (-10 + 10) + 7x
7 = 0 + 7x
7 = 7x
Isolate x by dividing 7 to both sides
7/7 = 7x/7
x = 1
Hope this helped!
~Just a girl in love with Shawn Mendes
-10x-3= -10-3x
-10x+10x-3= -10x+10x-3x
-3=-3x
divide by -4 for -3 and -3x
-3/-3= -3x/-3
1=x
x= 1
check answer by using substitution method
-10x-3= -10-3x
-10(1)-3=-10-3(1)
-13=- 13
Answer is x= 1
The shoe store has 720 pairs of shoes for sale. Each pair of shoes is being sold for $25.00. If all of the shoes are sold, how much will the shoe store make if their profit is 25% of the selling price?
$4,500.00
$4,200.00
$4,750.00
$4,250.00
Answer:4,500
Step-by-step explanation:
720 multiple by 25 gives you 18,000 then you find 25% of 18,000 by multiplying 18,000 times 25/100 which gives you 4,500. Or you can find 25%of 25 then multiple the answer by 720
Answer: 4,500.00
Step-by-step explanation:
Experimental versus theoretical probability. SOMEONE HELP ME
Theoretical probability is calculated based on the number of expected outcomes, while experimental probability is based on observed outcomes. They are used in different situations for predictions and estimates, respectively.
Explanation:Theoretical probability is calculated by dividing the number of times an event is expected to occur by the number of possible outcomes.
For example, if you flip a fair coin, there is one way to obtain heads and two possible outcomes, so the theoretical probability of heads is 1/2 or 0.5.
Experimental probability, on the other hand, is based on observations from an experiment.
If you flip a coin 10 times and get 6 heads, the experimental probability of heads is 6/10 or 0.6.
Both theoretical and experimental probability have their uses in different situations, but theoretical probability is often used to make predictions based on known probabilities, while experimental probability provides a more accurate estimate based on actual observations.
Help with this math question
Rationalize the denominator or is it rationalized already ? -13/√x
Answer:
[tex]-\frac{13\sqrt{x} }{x}[/tex]
Step-by-step explanation:
We have been given the following expression;
-13/√x
In order to rationalize the denominator, we multiply the numerator and the denominator by √x;
[tex]-\frac{13}{\sqrt{x}}=-\frac{13\sqrt{x} }{\sqrt{x}\sqrt{x}}\\ \\-\frac{13\sqrt{x} }{\sqrt{x^{2} } }\\\\-\frac{13\sqrt{x} }{x}[/tex]
Final answer:
To rationalize the denominator, multiply both the numerator and denominator by the conjugate of the denominator. In this case, the conjugate of √x is -√x. Multiply -13/√x by -√x/-√x to get -13√x / x.
Explanation:
To rationalize the denominator, we need to eliminate the square root from the denominator. We can do this by multiplying the numerator and denominator by the conjugate of the denominator. The conjugate of √x is -√x. So, multiplying the numerator and denominator by -√x gives us:
-13/√x * (-√x)/(-√x) = 13√x/(-x) = -13√x / x
Therefore, the rationalized form of -13/√x is -13√x / x.
A random sample of 133 people was taken from a very large population. Sixty-five of the people in the sample were females.The standard error of the proportion of females is (Round your answer to 4 decimal places.)
Answer: 0.0433
Step-by-step explanation:
Given: Sample size : n= 133
The number of females in sample = 65
Then the proportion of females : [tex]P=\dfrac{65}{133}[/tex]
The formula to calculate the standard error of the proportion is given by :-
[tex]S.E.=\sqrt{\dfrac{P(1-P)}{n}}[/tex]
[tex]\Rightarrow S.E.=\sqrt{\dfrac{\dfrac{65}{133}(1-\dfrac{65}{133})}{133}}\\\\\Rightarrow\ \Rightarrow S.E.=0.0433444676341\approx0.0433[/tex]
Hence, the standard error of the proportion of females is 0.0433.
The standard error of the proportion of females in the given sample is calculated using the formula SE = sqrt[p * (1-p) / n]. In this case, the proportion (p) is 0.4887 and the sample size (n) is 133, giving a standard error of 0.0432.
Explanation:The question is asking for the standard error of the proportion of females in the said sample. The standard error (SE) of a proportion is a measure of uncertainty around a proportion estimate. It is calculated using the formula SE = sqrt[p * (1-p) / n], where p is the proportion and n is the sample size.
So, we have a sample size, n = 133, and the proportion of females, p = 65/133 = 0.4887.
Substitute these values into the formula, we get: SE = sqrt[0.4887 * (1 - 0.4887) / 133] = 0.0432 (rounded to four decimal places).
Therefore, the standard error of the proportion of females in this sample is 0.0432.
Learn more about Standard Error here:https://brainly.com/question/13179711
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Which of the following pairs of numbers contains like fractions? A. 5⁄6 and 10⁄12 B. 3⁄2 and 2⁄3 C. 3 1⁄2 and 4 4⁄4 D. 6⁄7 and 1 5⁄7
The answers are:
A.
[tex]\frac{5}{6}[/tex] and [tex]\frac{10}{12}[/tex]
D.
[tex]\frac{6}{7}[/tex] and [tex]1\frac{5}{7}[/tex]
Why?To find which of the following pairs of numbers contains like fractions, we must remember that like fractions are the fractions that share the same denominator.
We are given two fractions that are like fractions. Those fractions are:
Option A.
[tex]\frac{5}{6}[/tex] and [tex]\frac{10}{12}[/tex]
We have that:
[tex]\frac{10}{12}=\frac{5}{6}[/tex]
So, we have that the pairs of numbers
[tex]\frac{5}{6}[/tex]
and
[tex]\frac{5}{6}[/tex]
Share the same denominator, which is equal to 6, so, the pairs of numbers contains like fractions.
Option D.
[tex]\frac{6}{7}[/tex] and [tex]1\frac{5}{7}[/tex]
We have that:
[tex]1\frac{5}{7}=1+\frac{5}{7}=\frac{7+5}{7}=\frac{12}{7}[/tex]
So, we have that the pair of numbers
[tex]\frac{6}{7}[/tex]
and
[tex]\frac{12}{7}[/tex]
Share the same denominator, which is equal to 7, so, the pairs of numbers constains like fractions.
Also, we have that the other given options are not like fractions since both pairs of numbers do not share the same denominator.
The other options are:
[tex]\frac{3}{2},\frac{2}{3}[/tex]
and
[tex]3\frac{1}{2},4\frac{4}{4}[/tex]
We can see that both pairs of numbers do not share the same denominator so, they do not contain like fractions.
Hence, the answers are:
A.
[tex]\frac{5}{6}[/tex] and [tex]\frac{10}{12}[/tex]
D.
[tex]\frac{6}{7}[/tex] and [tex]1\frac{5}{7}[/tex]
Have a nice day!
Radar detectors are either powered by their own battery or plug into the cigarette lighter socket. All radar detectors come in two models: no-frills and fancy. In addition, detectors powered by their own batteries detect either radar or laser, or both, whereas the plug-in types come in models that detect either radar or laser, but not both. How many different radar detectors can you buy?
Final answer:
There are 10 different radar detectors one can buy, considering the power source, detection capabilities (radar, laser, or both), and model type (no-frills or fancy).
Explanation:
To find out how many different radar detectors one can buy, we need to consider the options presented and calculate the total number of combinations. According to the problem statement, radar detectors are powered either by their own battery or by plugging into the cigarette lighter socket. They come in two models: no-frills and fancy. Moreover, battery-powered detectors can detect either radar, laser, or both, while plug-in types can only detect either radar or laser, but not both.
Battery-powered detectors:No-frills model (detects radar) - 1 optionFancy model (detects radar) - 1 optionNo-frills model (detects laser) - 1 optionFancy model (detects laser) - 1 optionNo-frills model (detects both) - 1 optionFancy model (detects both) - 1 optionPlug-in detectors:No-frills model (detects radar) - 1 optionFancy model (detects radar) - 1 optionNo-frills model (detects laser) - 1 optionFancy model (detects laser) - 1 optionAdding these up, we get a total of 6 options for battery-powered and 4 options for plug-in detectors, making a grand total of 10 different radar detectors one can buy.
Suppose the population of a town was 40,000 on January 1, 2010 and was 50,000 on January 1, 2015. Let P(t) be the population of the town in thousands of people t years after January 1, 2010.
6 (a) Build an exponential model (in the form P(t) = a bt ) that relates P(t) and t. Round the value of b to 5 significant figures.
a = ?
b = ?
Answer:
Given,
The initial population ( on 2010 ) = 40,000,
Let r be the rate of increasing population per year,
Thus, the function that shows the population after t years,
[tex]P(x)=40000(1+r)^t[/tex]
And, the population after 5 years ( on 2015 ) is,
[tex]P(5)=40000(1+r)^{5}[/tex]
According to the question,
P(5) = 50,000,
[tex]\implies 40000(1+r)^5=50000[/tex]
[tex](1+r)^5=\frac{50000}{40000}=1.25[/tex]
[tex]r + 1= 1.04563955259[/tex]
[tex]\implies r = 0.04653955259\approx 0.04654[/tex]
So, the population is increasing the with rate of 0.04654,
And, the population after t years would be,
[tex]P(t)=40000(1+0.04654)^t[/tex]
[tex]\implies 40000(1.04654)^t[/tex]
Since, the exponential function is,
[tex]f(x) = ab^x[/tex]
Hence, by comparing,
a = 40000,
b = 1.04654
Answer:
maryland. (c)
wyoming. (b)
reduced burning of fossil fuels. (b)
Step-by-step explanation:
if you're looking for the answer the question im looking for then those are the answers
Graph f (x) = 1.6x-2 +1
describe the domain and range
describe x- and y- intercepts
Answer:
We need to find the domain, range, x-intercept and y-intercept of the following function:
[tex]f(x) = 1.6x^{-2} + 1[/tex] ⇒ [tex]f(x)=\frac{1.6}{x^{2} }+1[/tex]
To find the y-intercept, we have to make 'x=0'
[tex]f(x) = \frac{1.6}{x^{2} } + 1[/tex] ⇒ [tex]f(x) = \frac{1.6}{0} + 1[/tex]. Given that divisions by zero are not possible, we conclude that there's no y-intercept. In other words, the function does not cross the y-axis,
To find the x-intercept, we have to make 'y=0'
[tex]f(x) = \frac{1.6}{x^{2} } + 1[/tex] ⇒ [tex]\frac{1.6}{x^{2} } + 1 = 0[/tex]
⇒ [tex]x^{2} = -1.6[/tex]
Given that we cannot take the square rooth of a negative number, we can conclude that there's no x-intercept. In other words, the function does not cross the x-axis.
The domain is all the possible values that the independent variable 'x' can take. Given that we can not divide by zero, the domain is all real numbers except zero. In set notation: ℝ - {0}.
The Range is all the possible values that the dependent variable 'y' can take. Solving the expression for 'x' we have:
[tex]\frac{1.6}{x^{2} } + 1 = y[/tex] ⇒ [tex]\frac{1.6}{x^{2} }= y-1[/tex]
⇒ [tex]\sqrt{(\frac{1.6}{y-1 })}= x[/tex]
Given that square roots can not be negative, and the denominator can't be equal to zero, the range is y>1. In set notation: Range: (1, +∞)
Determine whether the pair of triangles is congruent. If yes, include the theorem or postulate that applies.
Question 6 options:
yes; The triangles are congruent by hypotenuse-angle congruence.
yes; The triangles are congruent by leg-leg congruence.
no; The triangles are not congruent.
There is not enough information to determine congruency.
---------------------------------------------------------------------
Determine whether the pair of triangles is congruent. If yes, include the theorem or postulate that applies.
Question 9 options:
yes; The triangles are congruent by hypotenuse-angle congruence.
yes; The triangles are congruent by hypotenuse-leg congruence.
no; The triangles are not congruent.
there is not enough information to determine congruency.
ANSWER IN 5 MINUTES TO RECEIVE BRAINLIEST.
Answer:
Question 6
There is not enough information to determine congruency
Question 9
yes; The triangles are congruent by hypotenuse-leg congruence
Step-by-step explanation:
* Lets revise the cases of congruence
- SSS ⇒ 3 sides in the 1st Δ ≅ 3 sides in the 2nd Δ
- SAS ⇒ 2 sides and including angle in the 1st Δ ≅ 2 sides and
including angle in the 2nd Δ
- ASA ⇒ 2 angles and the side whose joining them in the 1st Δ
≅ 2 angles and the side whose joining them in the 2nd Δ
- AAS ⇒ 2 angles and one side in the first triangle ≅ 2 angles
and one side in the 2ndΔ
- HL ⇒ hypotenuse - leg of the first right angle triangle ≅ hypotenuse
- leg of the 2nd right angle Δ
- HA ⇒ hypotenuse - angle of the first right angle triangle ≅ hypotenuse
- angle of the 2nd right angle Δ
- LL ⇒ leg - leg of the first right angle triangle ≅ leg - leg of the
2nd right angle Δ
* Now lets solve the problem
# Question 6
- There are two right angle triangles
- They have two different legs
- There is no mention abut legs equal each other
- There is no mention about hypotenuses equal each other
- There is no mention about acute angels equal each other
∴ There is not enough information to determine congruency
# Question 9
- There are two right angle triangles
- Their hypotenuse is common
- There are two corresponding legs are equal
∵ The hypotenuse is a common in the two right triangles
∵ Two corresponding legs are equal
- By using hypotenuse-leg congruence
∴ yes; The triangles are congruent by hypotenuse-leg congruence
Find the geometric means in the following sequence. –6, ? , ? , ? , ? , –1,458?
Answer:
The sequence is,
-6, -18, -54, -162, - 486, -1458, -4374
Step-by-step explanation:
It is given a geometric sequence,
–6, ? , ? , ? , ? , –1,458?
From the given sequence we get first term a₁ = -6 and 6th term a₆ = -1458
To find the common ratio 'r'
6th term can be written as
a₆ = ar⁽⁶ ⁻ ¹⁾
-1458 = 6 * r⁽⁶⁻¹⁾
r⁵ = -1458/-6 = 243
r = ⁵√243 = 3
To find the sequence
We have a = -6, r = 3
a₂ = -6 * 3 = -18
a₃ = a₂*3 = -18* 3 = -54
a₄ = a₃*3 = -54 * 3 = -162
a₅ = a₄*3 = -162* 3 = -486
a₆ = - 1458
a₇ = a₂*3 =-1458 * 3= -4374
The sequence is,
-6, -18, -54, -162, - 486, -1458, -4374
Compute the value of the following improper integral if it converges. If it diverges, enter INF if it diverges to infinity, MINF if it diverges to minus infinity, or DIV otherwise (hint: integrate by parts). ∫∞18ln(x)x2dx Determine whether ∑n=1∞(8ln(n)n2) is a convergent series. Enter C if the series is convergent, or D if it is divergent.
Answer:
INF for first while D for second
Step-by-step explanation:
Ok I think I read that integral with lower limit 1 and upper limit infinity
where the integrand is ln(x)*x^2
integrate(ln(x)*x^2)
=x^3/3 *ln(x)- integrate(x^3/3 *1/x)
Let's simplify
=x^3/3 *ln(x)-integrate(x^2/3)
=x^3/3*ln(x)-1/3*x^3/3
=x^3/3* ln(x)-x^3/9+C
Now apply the limits of integration where z goes to infinity
[z^3/3*ln(z)-z^3/9]-[1^3/3*ln(1)-1^3/9]
[z^3/3*ln(z)-z^3/9]- (1/9)
focuse on the part involving z... for now
z^3/9[ 3ln(z)-1]
Both parts are getting positive large for positive large values of z
So the integral diverges to infinity (INF)
By the integral test... the sum also diverges (D)
To compute the value of the improper integral, we can integrate by parts. Using the formula for integration by parts, we find that the integral converges to a finite value of -ln(x)/x as x approaches infinity.
Explanation:To compute the value of the improper integral ∫∞18ln(x)/x2dx, we can integrate by parts. Let u = ln(x) and dv = 1/x2dx. Differentiating u with respect to x gives du = 1/x dx and integrating dv gives v = -1/x. Applying the formula for integration by parts, we get:
∫∞18ln(x)/x2dx = -ln(x)/x + ∫∞181/x2dx.
Simplifying the integral, we have:
∫∞181/x2dx = -1/x
As x approaches infinity, 1/x approaches 0. Therefore, the improper integral converges to a finite value of -ln(x)/x.
A collegiate video-game competition team has a 0.70 probability of winning a match. Over the course of a season, 8 matches are played. Individual matches are independent of any other matches. Calculate the probability that the team will win exactly 7 matches over the course of one season.
The probability that the team will win exactly 7 matches over the course of one season is:
0.1977
Step-by-step explanation:We know that the probability of k successes out of n successes is given by the binomial distribution as:
[tex]P(X=k)=n_C_kp^k(1-p)^{n-k}[/tex]
where p is the probability of success .
Here we are asked to find the probability that the team will win exactly 7 matches over the course of one season.
Since, there are 8 matches over the course of season.
This means n=8
and k=7
and p=0.70
(Since, 0.70 probability of winning a match )
Hence, we get:
[tex]P(X=7)=8_C_7\times (0.70)^7\times (1-0.70)^{8-7}\\\\i.e.\\\\P(X=7)=8\times (0.70)^7\times 0.30\\\\i.e.\\\\P(X=7)=0.1977[/tex]
Hence, the answer is:
0.1977
Final answer:
The probability that a collegiate video-game competition team with a 0.70 chance of winning will win exactly 7 out of 8 matches is approximately 25.41%.
Explanation:
The question asks to calculate the probability that a collegiate video-game competition team, which has a 0.70 probability of winning a match, wins exactly 7 out of 8 matches in a season. This scenario can be modeled using the binomial distribution formula, which is given by P(X = k) = (n C k) * p^k * (1 - p)^(n - k), where 'n' is the total number of trials (matches), 'k' is the number of successful outcomes (wins), and 'p' is the probability of a single success.
To find the probability of winning exactly 7 matches, we set n = 8, k = 7, and p = 0.70. Thus, the calculation becomes P(X = 7) = (8 C 7) * (0.70)⁷ * (0.30)¹. Calculating further, we have P(X = 7) = 8 * (0.70)⁷ * (0.30) = 0.254121. Therefore, the probability that the team will win exactly 7 matches over the course of one season is approximately 25.41%.
The equation of the tangent plane to the ellipsoid x2/a2 + y2/b2 + z2/c2 = 1 at the point (x0, y0, z0) can be written as xx0 a2 + yy0 b2 + zz0 c2 = 1 Find the equation of the tangent plane to the hyperboloid x2/a2 + y2/b2 − z2/c2 = 1 at (x0, y0, z0) and express it in a form similar to the one for the ellipsoid.
The equation of the tangent plane to the ellipsoid at the given point is [tex]\frac{xx^0}{a^2} + \frac{yy^0}{b^2} - \frac{zz^0}{c^2} = 1[/tex]
The equation of the tangent plane to the ellipsoid is given as:
[tex]\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1[/tex]
The point on the ellipsoid is given as:
(x0, y0, z0)
The equation of the tangent plane to the ellipsoid at the given point can be written as:
[tex]\frac{xx^0}{a^2} + \frac{yy^0}{b^2} + \frac{zz^0}{c^2} = 1[/tex]
Given that the equation of the tangent plane to the hyperboloid is
[tex]\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1[/tex]
The equations at the tangents of the ellipsoid and the hyperboloid take the same form.
So, the equation of the tangent plane to the ellipsoid at the given point is [tex]\frac{xx^0}{a^2} + \frac{yy^0}{b^2} - \frac{zz^0}{c^2} = 1[/tex]
Read more about tangent planes at:
https://brainly.com/question/15465847
3x+6y=18 solve for x
3x+6y=18
then
3x+6y+−6y=18+−6y
next
3x=−6y+18
then
3x /3 =( −6y+18) /3
answer
x=−2y+6
First subtract 6y from both sides of the equation.
[tex]3x+6y-6y=18-6y\Longrightarrow 3x=18-6y[/tex]
Then divide both sides of the equation with 3.
[tex]3x=18-6y\Longrightarrow x=\dfrac{18-6y}{3}[/tex]
Which further simplifies to.
[tex]x=\dfrac{18}{3}-\dfrac{6y}{3}\Longrightarrow\boxed{6-2y}[/tex]
Hope this helps. I tried to made the steps very clear and easy.
r3t40
f(x)= x-2/ x-4
Graph this equation and identify the points of discontinuity, holes, vertical asymptotes, x-intercepts, and horizontal asymptote.
Answer:
The discontinuity is x = 4
There no holes
The equation of the vertical asymptote is x = 4
The x intercept is 2
The equation of the horizontal asymptote is y = 1
Step-by-step explanation:
* Lets explain the problem
∵ [tex]f(x)=\frac{x-2}{x-4}[/tex]
- To find the point of discontinuity put the denominator = 0 and find
the value of x
∵ The denominator is x - 4
∵ x - 4 = 0 ⇒ add 4 to both sides
∴ x = 4
* The discontinuity is x = 4
- A hole occurs when a number is both a zero of the numerator
and denominator
∵ The numerator is x - 2
∵ x - 2 = 0 ⇒ add 2 to both sides
∴ x = 2
∵ The denominator is x - 4
∵ x - 4 = 0 ⇒ add 4 to both sides
∴ x = 4
∵ There is no common number makes the numerator and denominator
equal to 0
∴ There no holes
- Vertical asymptotes are vertical lines which correspond to the zeroes
of the denominator of the function
∵ The zero of the denominator is x = 4
∴ The equation of the vertical asymptote is x = 4
- x- intercept is the values of x which make f(x) = 0, means the
intersection points between the graph and the x-axis
∵ f(x) = 0
∴ [tex]\frac{x-2}{x-4}=0[/tex] ⇒ by using cross multiplication
∴ x - 2 = 0 ⇒ add 2 to both sides
∴ x = 2
* The x intercept is 2
- If the highest power of the numerator = the highest power of the
denominator, then the equation of the horizontal asymptote is
y = The leading coeff. of numerator/leading coeff. of denominator
∵ The numerator is x - 2
∵ The denominator is x - 4
∵ The leading coefficient of the numerator is 1
∵ The leading coefficient of the denominator is 1
∴ y = 1/1 = 1
* The equation of the horizontal asymptote is y = 1
Brandon has a jar of quarters and dimes with a total value of $7.05. The number of quarters is 3 less than twice the number of dimes. How many quarters and how many dimes does Brandon have?
Number of quarters =
Number of dimes =
Answer:
quarters: 23dimes: 13Step-by-step explanation:
Let d represent the number of dimes. Then the number of quarters is 2d-3 and the total value of the coins is ...
0.10d + 0.25(2d-3) = 7.05
0.60d -0.75 = 7.05 . . . . . . . simplify
d = (7.05 +0.75)/0.60 = 13 . . . . add 0.75, divide by 0.60
2d-3 = 2·13 -3 = 23
Brandon has 23 quarters and 13 dimes.
A chef plans to mix 100% vinegar with Italian dressing. The Italian dressing contains 12%
vinegar. The chef wants to make 160 milliliters of a mixture that contains 23% vinegar. How much vinegar and how much Italian dressing should she use?
Answer:
Amount of vinegar. 100% : 20 milliliters
Amount of Italian dressing: 140 milliliters
Step-by-step explanation:
Let's call A the amount of vinegar. 100%
Let's call B the amount of Italian dressing . 12% vinegar
The resulting mixture should have 23% vinegar, and 160 milliliters.
Then we know that the total amount of mixture will be:
[tex]A + B = 160[/tex]
Then the total amount of pure antifreeze in the mixture will be:
[tex]A + 0.12B = 0.23 * 160[/tex]
[tex]A + 0.12B = 36.8[/tex]
Then we have two equations and two unknowns so we solve the system of equations. Multiply the first equation by -1 and add it to the second equation:
[tex]-A -B = -160[/tex]
[tex]-A -B = -160[/tex]
+
[tex]A + 0.12B = 36.8[/tex]
--------------------------------------
[tex]-0.88B = -123.2[/tex]
[tex]B = \frac{-123.2}{-0.88}[/tex]
[tex]B = 140\ milliliters[/tex]
We substitute the value of B into one of the two equations and solve for A.
[tex]A + 140 = 160[/tex]
[tex]A = 20\ milliliters[/tex]