Divide both sides by [tex]x^2-1[/tex] to get a linear ODE,
[tex]\dfrac{\mathrm dy}{\mathrm dx}+\dfrac2{x^2-1}y=\dfrac{x+1}{x-1}[/tex]
In order for this operation to be valid in the first place, we require that [tex]x\neq\pm1[/tex] (since that would make [tex]\dfrac1{x^2-1}[/tex] undefined, which we don't want to happen). Then we are forcing any solution to the ODE to exist on any of the three intervals, [tex](-\infty,-1)[/tex], [tex](-1, 1)[/tex], or [tex](1,\infty)[/tex], and either the first or third of these can be chosen as the largest interval.
In case you also need to solve the ODE: Multiply both sides by [tex]\dfrac{1-x}{1+x}[/tex], so that
[tex]\dfrac{1-x}{1+x}\dfrac{\mathrm dy}{\mathrm dx}-\dfrac2{(1+x)^2}y=-1[/tex]
Then the left side can be condensed as the derivative of a product, since
[tex]\dfrac{\mathrm d}{\mathrm dx}\left[\dfrac{1-x}{1+x}\right]=-\dfrac2{(1+x)^2}[/tex]
and we have
[tex]\dfrac{\mathrm d}{\mathrm dx}\left[\dfrac{1-x}{1+x}y\right]=-1[/tex]
Integrate both sides:
[tex]\displaystyle\int\frac{\mathrm d}{\mathrm dx}\left[\frac{1-x}{1+x}y\right]\,\mathrm dx=-\int\mathrm dx[/tex]
[tex]\dfrac{1-x}{1+x}y=-x+C[/tex]
[tex]\implies\boxed{y=\dfrac{(-x+C)(1+x)}{1-x}}[/tex]
The largest interval over which the general solution is defined for the given differential equation is [-1, ∞).
Here's how:
Rewrite the differential equation in proper form.Analyze the coefficients to determine the interval of definition.In this case, the interval is determined by the denominator of the coefficient of dy/dx.The annual snowfall in a town has a mean of 35 inches and a standard deviation of 11 inches. Last year there were 60 inches of snow. How many standard deviations from the mean is that
Answer:
z=2.27
Step-by-step explanation:
[tex]z=\frac{x-\mu}{\sigma}[/tex]
where z is the deviation from mean.
mean (μ) = 35 inches
standard deviation (σ) = 11 inches
last year snow fall (x) = 60 inches
[tex]z=\frac{x-\mu}{\sigma}[/tex]
[tex]z=\frac{60-35}{11}[/tex]
z=2.27
now, the standard deviation for the 60 inches snow from the mean is calculated to be 2.27
The functions q and r are defined as follows.
q(x) = -2x +1
r(x) = 2x^2 - 1
Find the value of .
q(r(4))
Answer:
q(r(4)) = -61
Step-by-step explanation:
q(x) = -2x +1
r(x) = 2x^2 - 1
q(r(4))
First find r(4)
f(4) = 2 (4)^2 -1
= 2 *16 -1
= 32-1
= 31
Then put this value in for x in q(x)
q(r(4)) = q(31) = -2(31)+1
= -62+1
= -61
Answer:
The value of q( r(4) ) = -61
Step-by-step explanation:
It is given that,
q(x) = - 2x +1
r(x) = 2x^2 - 1
To find the value of q(r(4))
r(x) = 2x^2 - 1
r(4) = 2( 4^2) - 1 [Substitute 4 instead of x]
= 2(16) - 1
= 32 - 1 = 31
q( x ) = -2x +1
q( r(4) ) = q(31) [Substitute 31 instead of x)
= (-2*31) +1
= -62 + 1 = -61
Therefore the value of q(r(4)) = -61
Please help me with this
Answer:
The correct answer is first option
24
Step-by-step explanation:
From the figure we get, mAXM = 72° and m<AMR = 38°
Also it is given that, all triangles are isosceles triangles and
m<FXA = 96°
To find the measure of <FXM
From the figure we get,
m<FXA = m<AXM + m<FXM
m<FXM = m<FXA - m<AXM
= 96 - 72
= 24
Therefore the correct answer is first option
24
Problem 4. Let m and n be two integers. Show that m^3- n^3 is even if and only if m n is even.
Answer:
The expression [tex]m^3-n^3[/tex] is even if both variables (m and n) are even or both are odd
Step-by-step explanation:
Let's remember the logical operations with even and odd numbers
odd*odd=odd
even*even=even
odd*even=even
odd-odd=even
even-even=even
even-odd=odd
Now, the original expression is:
[tex]m^3-n^3[/tex] which can be expressed as:
[tex](m*(m*m))-(n*(n*n))[/tex]
If m and n are both odd, then:
[tex](m*(m*m))=odd*(odd*odd)=odd*(odd)=odd[/tex]
[tex](n*(n*n))=odd*(odd*odd)=odd*(odd)=odd[/tex]
Then, [tex](m*(m*m))-(n*(n*n))=odd-odd=even[/tex]
If m and n are both even, then:
[tex](m*(m*m))=even*(even*even)=odd*(even)=even[/tex]
[tex](m*(m*m))=even*(even*even)=odd*(even)=even[/tex]
Then, [tex](m*(m*m))-(n*(n*n))=even-even=even[/tex]
Finally if one of them is even, for example m, and the other is odd, for example n, then:
[tex](m*(m*m))=even*(even*even)=odd*(even)=even[/tex]
[tex](n*(n*n))=odd*(odd*odd)=odd*(odd)=odd[/tex]
Then, [tex](m*(m*m))-(n*(n*n))=even-odd=odd[/tex]
In conclusion, the expression [tex]m^3-n^3[/tex] is even if both variables (m and n) are even or both are odd. If one of them is even and the other one is odd, then the expression is odd.
For a dosage of x cubic centimeters (cc) of a certain drug, assume that the resulting blood pressure B is approximated by B (x) = 0.06 x^2 - 0.3 x^3 . Find the dosage at which the resulting blood pressure is maximized. Round to two decimal places.
Answer:
The number of dosage is 0.13.
Step-by-step explanation:
Here, the given function that represents the blood pressure,
[tex]B(x)=0.06x^2 - 0.3x^3[/tex]
Where, x is the number of dosage in cubic centimeters,
Differentiating the above function with respect to x,
[tex]B'(x)=0.12x-0.9x^2[/tex]
For maximum or minimum blood pressure,
[tex]B'(x)=0[/tex]
[tex]0.12x-0.9x^2=0[/tex]
[tex]-0.9x^2=-0.12x[/tex]
[tex]x=\frac{0.12}{0.9}=\frac{2}{15}[/tex]
Again differentiating B'(x) with respect to x,
[tex]B''(x)=0.12-1.8x[/tex]
Since, at x = 2/15,
[tex]B''(\frac{2}{15})=0.12-1.8(\frac{2}{15})=0.12-0.24=-0.12=\text{Negative value}[/tex]
So, at x = 2/15 the value of B(x) is maximum,
Hence, the number of dosage at which the resulting blood pressure is maximized = 2/15 = 0.133333333333 ≈ 0.13
The maximum blood pressure results from a dosage of approximately 0.13 cubic centimeters, based on the mathematical model given in the problem.
Explanation:To find the maximum blood pressure using the formula B (x) = 0.06 x^2 - 0.3 x^3, we need to first find the derivative of this equation, as the maximum point on any curve happens when its derivative equals zero.
First, differentiate B(x) with respect to x: B'(x) = 2*0.06x - 3*0.3x^2 = 0.12x - 0.9x^2 Next, set this derivative equal to zero and solve for x: 0 = 0.12x - 0.9x^2 0 = x(0.12 - 0.9x) So x = 0 or x = 0.12/0.9 = 0.133 Lastly, we need to determine if these x-values give a maximum or minimum in B(x). We do this by either taking the second derivative of B(x) or by testing points on either side of the x-values we found. If we find the second derivative, we find that B''(x) = 0.12 -1.8x, which is negative for x = 0.133. This means that the blood pressure is maximized at an x-value of 0.133 cc, or, rounded to two decimal places, 0.13 ccLearn more about Maximum Blood Pressure From Drug Dosage here:
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A solid, homogeneous sphere with a mass of m0, a radius of r0 and a density of ρ0 is placed in a container of water. Initially the sphere floats and the water level is marked on the side of the container. What happens to the water level, when the original sphere is replaced with a new sphere which has different physical parameters? Notation: r means the water level rises in the container, f means falls, s means stays the same. Combination answers like 'f or s' are possible answers in some of the cases. The new sphere has a mass of m = m0 and a density of ρ > ρ0. A: r B: f C: s D: r or s E: f or s The new sphere has a mass of m < m0 and a radius of r = r0. A: r B: f C: s D: r or s E: f or s The new sphere has a mass of m > m0 and a radius of r = r0.
Answer:
A: rB: fA: rStep-by-step explanation:
1. Greater density means the sphere has more mass in the same volume. The volume of water that must be displaced to equal that increased mass must be increased, causing the water level to rise.
__
2. Less mass means less water must be displaced to equal the mass of the new sphere, causing the water level to fall.
__
3. More mass is the same as higher density (see 1). The water level will rise.
Archimedes' principle states that the upward force acting on a body floating or immersed in a fluid is equal to the weight of the displaced fluid
The level of the water in the three situations are as follows;
Situation 1; Falls or stays the same, E: f or s
Situation 2; Falls, B: f
Situation 3, Rises A: r
The reason for the above selection is as follows;
The given details of the arrangements are;
The mass of the solid homogeneous sphere = m₀
The radius of the sphere = r₀
The density of the sphere = ρ₀
The location the sphere is placed = Floating in a container of water
The required parameter;
The provision of an estimate of the water level when the sphere is replaced with a new sphere with different physical parameters
Notation;
r = The water level rises
f = The water level falls
s = The water level stays the same
Situation 1; The mass of the new sphere, m = m₀
The density of the new sphere, ρ > ρ₀
Here, the denser sphere of equal mass = Smaller sphere, r < r₀
if the sphere floats, then the volume of the water displaced is equal to the
mass of the sphere, which is therefore, equal to the volume of the water
displaced by the original sphere
Therefore, the water level remains the same, s
However, if the sphere sinks, then the water displaced is less than the
mass m = m₀, of the sphere and therefore, the level falls, f
Therefore, the correct option is E: f or s
Situation 2: The mass of the new sphere, m < m₀
The radius of the new sphere, r = r₀
Here, we have equal radius and therefore equal volume and lesser density
Given that the volume of the water displaced for a floating body is equal to
the weight of body, and that the mass of the new sphere is less than the
mass of the original sphere, the mass of the water displaced and therefore,
the volume of water displaced is less and therefore, the water level falls
The correct option is therefore B: f falls
Situation 3: The mass of the new sphere, m > m₀, and the radius r = r₀
therefore the new sphere is denser than the original sphere and the
therefore, the mass of the water displaced where the sphere floats is m >
m₀, which is more than the water displaced for the original sphere and the
level of water rises, r, and the correct option is A: r
Therefore;
In situation 1, we have option E: f or s
In situation 2, the correct option is B: f
In situation 3, the correct option is A: r
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A fast food restaurant executive wishes to know how many fast food meals adults eat each week. They want to construct a 98% confidence interval with an error of no more than 0.08. A consultant has informed them that a previous study found the mean to be 6.6 fast food meals per week and found the standard deviation to be 0.7. What is the minimum sample size required to create the specified confidence interval? Round your answer up to the next integer.
Answer:
415
Step-by-step explanation:
Confidence Level = 98%
Z-value for this confidence level = z = 2.326
Margin of error = E = 0.08
Mean = u = 6.6
Standard deviation = [tex]\sigma=0.7[/tex]
Required Sample Size = n = ?
The formula for margin of error is:
[tex]E=z\frac{\sigma}{\sqrt{n}}[/tex]
Re-arranging the equation for n, and using the given values we get:
[tex]n=(\frac{z\sigma}{E} )^{2}\\\\ n=(\frac{2.326 \times 0.7}{0.08} )\\\\ n=415[/tex]
Thus, the minimum sample size required to create the specified confidence interval is 415
The minimum sample size required to construct a 98% confidence interval with an error of no more than 0.08 is 255.
Explanation:To determine the minimum sample size required to construct a 98% confidence interval with an error of no more than 0.08, we can use the formula:
n = (Z * sigma / E) ^ 2
where n is the sample size, Z is the Z-score corresponding to the desired confidence level, sigma is the standard deviation, and E is the desired margin of error.
In this case, the Z-score for a 98% confidence level is approximately 2.33. Substituting the given values of sigma = 0.7 and E = 0.08 into the formula, we can calculate the minimum sample size:
n = (2.33 * 0.7 / 0.08) ^ 2
n ≈ 254.43
Rounding up to the next integer, the minimum sample size required is 255.
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Q2. On a cold day, hailstones fall with a velocity of (2i− 6k) m s−1 . If a cyclist travels through the hail at 10i ms−1 , what is the velocity of the hail relative to the cyclist? At what angle are the hailstones falling relative to the cyclist
Answer:[tex]-8\hat{i}-6\hat{k}[/tex]
[tex]\theta =\tan^{-1}\left ( \frac{3}{4} \right )[/tex]
Step-by-step explanation:
Given
Velocity of hailstones fall[tex]\left ( V_h\right )=2\hat{i}-6\hat{k}[/tex] m/s
Velocity of cyclist [tex]\left ( V_c\right )=10\hat{i}[/tex] m/s
Therefore
Velocity of hail with respect to cyclist[tex]\left ( V_{hc}\right )[/tex]
[tex]V_{hc}=V_h-V_c[/tex]
[tex]V_{hc}=2\hat{i}-6\hat{k}-10\hat{i}[/tex]
[tex]V_{hc}=-8\hat{i}-6\hat{k}[/tex]
and angle of hails falling relative to the cyclist is given by
[tex]\theta =\tan^{-1}\left ( \frac{3}{4}\right )[/tex]
[tex]\theta [/tex] is the angle made with the vertical
Use undetermined coefficients to find the particular solution to 7t + 5=y''+y'-4y У, (t) - Preview Get help: Video Points possible: 1 This is attempt 1 of 2. Post this question to forum License
Suppose [tex]y_p=a_0+a_1t[/tex] is a solution to the ODE. Then [tex]{y_p}'=a_1[/tex] and [tex]{y_p}''=0[/tex], and substituting these into the ODE gives
[tex]a_1-4(a_0+a_1t)=7t+5\implies\begin{cases}-4a_1=7\\-4a_0+a_1=5\end{cases}\implies a_0=-\dfrac{27}{16},a_1=-\dfrac74[/tex]
Then the particular solution to the ODE is
[tex]y_p=-\dfrac{27}{16}-\dfrac74t[/tex]
Find the area of the triangle with vertices (1, 0, 0), (0, 2, 0), and (0, 0, 1). (Hint: A triangle is half of a parallelogram. Sketching a generic picture may help you visualize before you start to compute.)
To find the area of a triangle with given vertices, calculate the cross product of two vectors representing the sides of the triangle. The magnitude of this cross product gives the area of the parallelogram, and half of this value is the triangle's area.
The area of a triangle with vertices (1, 0, 0), (0, 2, 0), and (0, 0, 1) can be calculated using the cross product of two vectors that represent two sides of the triangle. First, we find the vectors AB and AC by subtracting the coordinates of the points:
Vector AB = B - A = (0 - 1, 2 - 0, 0 - 0) = (-1, 2, 0)Vector AC = C - A = (0 - 1, 0 - 0, 1 - 0) = (-1, 0, 1)Next, we calculate the cross product AB x AC:
|i j k|
|-1 2 0|
|-1 0 1|
This results in a new vector (2, -1, -1). The magnitude of this vector gives us the area of the parallelogram formed by vectors AB and AC.
Area of parallelogram = |(2, -1, -1)| = √(2^2 + (-1)^2 + (-1)^2) = √(6)
Since the area of the triangle is half the area of the parallelogram, we get:
Area of triangle = ½ √(6) = √(1.5).
25 Points! Please answer asap! Carly stated “All pairs of rectangles are dilations”. Which pair of rectangles would prove that Carly’s statement is incorrect? (Images below)
Answer:
C
Step-by-step explanation:
A. First two rectangles are dilations because
[tex]\dfrac{2}{4}=\dfrac{4}{8}=0.5[/tex]
B. Second two rectangles are dilations because
[tex]\dfrac{2}{4}=\dfrac{3}{6}=0.5[/tex]
C. Third two rectangles are not dilations because
[tex]\dfrac{3}{4}\neq \dfrac{2}{3}[/tex]
D. Fourth two rectangles are dilations because
[tex]\dfrac{3}{4}=\dfrac{1.5}{2}=0.75[/tex]
Answer:
c please correct me if im wrong
Step-by-step explanation:
8) Use Reduction of order to solve. One solution of homogeneo x2y" +7xy' +5y =x 1 x>0 y1 = X here y1 is a solution of the corresponding homogeneous.
I suspect there's a typo in the question, because [tex]y_1=x[/tex] is *not* a solution to the corresponding homogeneous equation. We have [tex]{y_1}'=1[/tex] and [tex]{y_1}''=0[/tex], so the ODE reduces to
[tex]0+7x+5x=12x\neq0[/tex]
Let [tex]y=x^m[/tex], then [tex]y'=mx^{m-1}[/tex] and [tex]y''=m(m-1)x^{m-2}[/tex], and substituting these into the (homogeneous) ODE gives
[tex]m(m-1)x^m+7mx^m+5x^m=0\implies m(m-1)+7m+5=m^2+6m+5=(m+5)(m+1)=0[/tex]
which then admits the characteristic solutions [tex]y_1=\dfrac1x[/tex] and [tex]y_2=\dfrac1{x^5}[/tex].
Now to find a solution to the non-homogeneous ODE. We look for a solution of the form [tex]y(x)=v(x)y_1(x)[/tex] or [tex]y(x)=v(x)y_2(x)[/tex].
It doesn't matter which one we start with, so let's use the first case. We get derivatives [tex]y'=x^{-1}v'-x^{-2}v[/tex] and [tex]y''=x^{-1}v''-2x^{-2}v'+2x^{-3}v[/tex]. Substituting into the ODE yields
[tex]x^2(x^{-1}v''-2x^{-2}v'+2x^{-3}v)+7x(x^{-1}v'-x^{-2}v)+5x^{-1}v=x[/tex]
[tex]xv''+5v'=x[/tex]
Substitute [tex]w=v'[/tex], so that [tex]w'=v''[/tex] and
[tex]xw'+5w=x[/tex]
which is linear in [tex]w[/tex], and we can condense the left side as the derivative of a product after multiplying both sides by [tex]x^4[/tex]:
[tex]x^5w'+5x^4=x^5\implies(x^5w)'=x^5\implies x^5w=\dfrac{x^6}6+C\implies w=\dfrac x6+\dfrac C{x^5}[/tex]
Integrate to solve for [tex]v[/tex]:
[tex]v=\dfrac{x^2}{12}+\dfrac{C_1}{x^4}+C_2[/tex]
Then multiply both sides by [tex]y_1=\dfrac1x[/tex] to solve for [tex]y[/tex]:
[tex]y=\dfrac x{12}+\dfrac{C_1}{x^5}+\dfrac{C_2}x[/tex]
so we found another fundamental solution [tex]y_3=x[/tex] that satisifes this ODE.
You obtain a loan of $7500 at 16.5% compounded monthly. If you make $300 payments monthly, what is the term of the loan? Find the size of the concluding payment if: a. the last full payment is increased to pay off the loan b. the last smaller payment is made one month after the last full payment.
Answer:
last installment is $540
Step-by-step explanation:
principal amount (p) = $7500
rate (r) = 16.5 %
installment = $300
to find out
full payment is increased to pay off the loan and the last smaller payment is made one month after the last full payment
solution
we know monthly installment is $300 so amount will be paid i.e.
amount = $300×12×N ..............1
here N is no of installment
and we know amount formula i.e.
amount = principal ( 1+r/100)^N
put amount value and principal rate
300×12×N = 7500 ( 1+16.5/100)^N
(3600 ×N ) / 7500 = 1.165^N
0.48N = 1.165^N
by the graphical we will get N = 3.65
so 3.65 year
so as that put N in equation 1 we get
amount = $300×12× 3.65
amount = $13140
we can say there are 43 installment so remaining money is $13140 - ($300 × 43 installment )
i.e. = $240 and last installment will be $300 + $240 = $540
so last installment is $540
Final Question Math Need help!!
Answer:
Dear, Have a look at pic
Find a power series representation for the function. (Give your power series representation centered at x = 0.)f(x) = x3x2 + 1f(x) = ∞n = 0 Determine the interval of convergence. (Enter your answer using interval notation.)
I suppose you mean
[tex]f(x)=\dfrac{x^3}{x^2+1}[/tex]
Recall that for [tex]|x|<1[/tex], we have
[tex]\dfrac1{1-x}=\displaystyle\sum_{n=0}^\infty x^n[/tex]
Then
[tex]\dfrac1{1+x^2}=\dfrac1{1-(-x^2)}=\displaystyle\sum_{n=0}^\infty(-x^2)^n=\sum_{n=0}^\infty(-1)^nx^{2n}[/tex]
which is valid for [tex]|-x^2|=|x|^2<1[/tex], or more simply [tex]|x|<1[/tex].
Finally,
[tex]f(x)=\displaystyle\frac{x^3}{x^2+1}=\sum_{n=0}^\infty(-1)^nx^{2n+3}[/tex]
Inclusions are defects in poured metal caused by contaminants. The number of (large) inclusions in cast iron follows a Poisson distribution with a rate of 3.2 per cubic millimetre. What is the probability of exactly four inclusions in 2.0 cubic millimetres? Please enter the answer to 3 decimal places.
Answer: 0.116
Step-by-step explanation:
The Poisson distribution probability formula is given by :-
[tex]P(X=x)=\dfrac{e^{-\lambda}\lambda^x}{x!}[/tex], where \lambda is the mean of the distribution and x is the number of success
Given : The number of inclusions in one cubic millimeter = 3.2
Then , the number of inclusions in two cubic millimeters=[tex]\lambda=2\times3.2=6.4[/tex]
Now, the probability of exactly four inclusions in 2.0 cubic millimetres is given by :-
[tex]P(X=4)=\dfrac{e^{-6.4}(6.4)^4}{4!}\\\\=0.11615127195\approx0.116[/tex]
Hence, the probability of exactly four inclusions in 2.0 cubic millimetres = 0.116
10. Sketch the graph of -5x^2- 16xy +7y^2-198 0. Show the steps used in rotating the axes
help me please i’m so far behind and i’m trying to finish before summer ends im freaking out
Answer:
The equation 2.5x -10.5 = 64(0.5^x) is true when x=5
Step-by-step explanation:
we need to solve the equation 2.5x -10.5 = 64(0.5^x)
We have to put the given values of x in the functions f(x) and g(x) and find their values
x f(x) = 2.5x -10.5 g(x) = 64(0.5^x)
2 2.5(2)-10.5 = -5.5 64(0.5^2) = 16
3 2.5(3) - 10.5 = -3 64(0.5^3) = 8
4 2.5(4) - 10.5 = -0.5 64(0.5^4) = 4
5 2.5(5) - 10.5 = 2 64(0.5^5) = 2
6 2.5(6) - 10.5 = 4.5 64(0.5^6) = 1
So, we need to solve the equation 2.5x -10.5 = 64(0.5^x)
This holds when x = 5 as shown in the table above.
In your own words, explain the problem of correlation vs. causation. Why are causation and correlation very different?
Answer:
Step-by-step explanation:
Correlation means that two or more events happen together. They are related to one another by being caused by the same thing.
Causation has a definite order. The first event has some cause that is comes before the second event. One event caused the other.
Which graph represents the function f(x) = –x^2 + 5?
Answer:
See below.
Step-by-step explanation:
This will be a parabola with axis of symmetry x = 0 and will open downwards.
The vertex will be at the point (0 , 5). The graph will intersect the x axis at
(-√5, 0) and (√5, 0).
Answer:
its a
Step-by-step explanation:
Audrey Graco plans to conduct book signings in several cities to promote her new novel. She wishes to visit Knoxville, Chattanooga, Chapel Hill, Charlotte, Raleigh, and Richmond. How many different ways can she visit each of these cities and return to her starting point in Wilmington? O A. 720 O B. 30 O C. 29 O D. 120 Click to select vour answer
Audrey can visit the six cities in which she plans to conduct book signings and return to her starting point in 720 different ways. This is because of the mathematical principle of permutations.
Explanation:Audrey's problem deals with permutations because the order of the places she visits matters. In general, the number of ways to arrange 'n' items (in Audrey's case, 'n' cities) in a specific order is given by 'n things taken n at a time' which is mathematically represented as n! (n factorial). In this case, Audrey is visiting 6 cities (Knoxville, Chattanooga, Chapel Hill, Charlotte, Raleigh, and Richmond), and then returning to her original city, Wilmington. So, the number of ways she can visit these cities can be represented as 6!, which equals 720.
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3) Draw a possibility tree that represents a coin that is tossed 3 times
I hope I've helped
In this photo you can find de probabilities
Is x+y+1=0 a tangent of both y^2=4x and x^2=4y parabolas?
Answer:
yes
Step-by-step explanation:
The line intersects each parabola in one point, so is tangent to both.
__
For the first parabola, the point of intersection is ...
y^2 = 4(-y-1)
y^2 +4y +4 = 0
(y+2)^2 = 0
y = -2 . . . . . . . . one solution only
x = -(-2)-1 = 1
The point of intersection is (1, -2).
__
For the second parabola, the equation is the same, but with x and y interchanged:
x^2 = 4(-x-1)
(x +2)^2 = 0
x = -2, y = 1 . . . . . one point of intersection only
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If the line is not parallel to the axis of symmetry, it is tangent if there is only one point of intersection. Here the line x+y+1=0 is tangent to both y^2=4x and x^2=4y.
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Another way to consider this is to look at the two parabolas as mirror images of each other across the line y=x. The given line is perpendicular to that line of reflection, so if it is tangent to one parabola, it is tangent to both.
An advertising company wishes to estimate the population mean of the distribution of hours of television watched per household per day. Suppose that the population standard deviation of hours watched per household per day is known to be 2.8 hours. The company decides that it wants the 99% confidence interval for the population mean to be no longer than 0.5 (hour). What is the minimum sample size that will result in a small enough confidence interval?
Answer: 208
Step-by-step explanation:
Given : An advertising company wishes to estimate the population mean of the distribution of hours of television watched per household per day.
Standard deviation : [tex]2.8\text{ hours}[/tex]
Margin of error : [tex]\pm0.5\text{ hour}[/tex]
Significance level : [tex]\alpha=1-0.99=0.01[/tex]
Critical value : [tex]z_{\alpha/2}=2.576[/tex]
The formula to calculate the sample size is given by :-
[tex]n=(\dfrac{z_{\alpha/2}\sigma}{E})^2[/tex]
[tex]\Rightarrow\ n=(\dfrac{2.576\times2.8}{0.5})^2=208.09793536\approx208[/tex]
Hence, the minimum required sample size must be 208.
You have a hat containing 8 red chips, 4 green chips, 5 yellow chips, and 3 white chips. Find the following probabilities and write the answers as simplified fractions:
(4 points each)
Probability of picking a red chip?
Probability of not picking a green chip?
Probability of picking one chip and it is a yellow or green chip?
Step-by-step explanation:
There are 20 chips in total.
P(red) = 8/20 = 2/5
P(not green) = 16/20 = 4/5
P(yellow or green) = 9/20
An individual is planning a trip to a baseball game for 20 people. Of the people planning to go to the baseball game, 11 can go on Saturday and 14 can go on Sunday, some of them can go on both days. How many people can only go to the game on Saturday?
Answer:
6 people
Step-by-step explanation:
Suppose A represents the event of going on Saturday,
B represents the event of going on Sunday,
According to the question,
n(A)=11
n(B)=14
n(A∪B)=20
We know that,
n(A∪B) = n(A) + n(B) - n(A∩B)
By substituting values,
20 = 11 + 14 - n(A∩B)
⇒ n(A∩B) = 25 - 20 = 5,
Hence, the number of people who can only go to the game on Saturday = n(A) - n(A∩B) = 11 - 5 = 6.
Identify the radius and center.
x^2 + y^2 - 2x + 4y - 11 = 0
The answer is:
Center: (1,-2)
Radius: 4 units.
Why?To solve the problem, using the given formula of a circle, we need to find its standard equation form which is equal to:
[tex](x-h)^{2}+(y-k)^{2}=r^{2}[/tex]
Where,
"h" and "k"are the coordinates of the center of the circle and "r" is its radius.
So, we need to complete the square for both variable "x" and "y".
The given equation is:
[tex]x^2+y^2-2x+4y-11=0[/tex]
So, solving we have:
[tex]x^2+y^2-2x+4y=11[/tex]
[tex](x^2-2x+(\frac{2}{2})^{2} )+(y^2+4y+(\frac{4}{2})^{2})=11+(\frac{2}{2})^{2} +(\frac{4}{2})^{2}\\\\(x^2-2x+1)+(y^2+4y+4)=11+1+4\\\\(x^2-1)+(y^2+2)=16[/tex]
[tex](x^2-1)+(y^2-(-2))=16[/tex]
Now, we have that:
[tex]h=1\\k=-2\\r=\sqrt{16}=4[/tex]
So,
Center: (1,-2)
Radius: 4 units.
Have a nice day!
Note: I have attached a picture for better understanding.
Which of the following directors made Bonnie and Clyde? a. Arthur Penn b. Warren Beaty c. Stanley Kubrick d. None of the above
Answer:
a) Arthur Penn
Step-by-step explanation:
There are three feature films based on Bonnie and Clyde they are the following:
"The Bonnie Parker Story" released in 1958 was directed by William Witney.
"Bonnie and Clyde" released in 1967 was directed by Arthur Penn.
Warren Beaty is primarily an actor who has directed six films including a tv movie and five feature films.
"The Highwaymen" was directed by John Lee Hancock released 2019.
Answer:
A. Arthur Penn
Step-by-step explanation:
Bonnie and Clyde A defining film of the New Hollywood generation was Bonnie and Clyde (1967). Produced by and starring Warren Beatty and directed by Arthur Penn, its combination of graphic violence and humor, as well as its theme of glamorous disaffected youth, was a hit with audiences.
A floor refinishing company charges $1.83 per square foot to strip and refinish a tile floor for up to 1000 square feet. There is an additional charge of $350 for toxic waste disposal for any job which includes more than 150 square feet of tile.
A) Express the cost, y, of refinishing a floor as a function of the number of square feet, x, to be refinished.
b) Graph the function, give the domain and range.
Answer:
Here x represents the number of square feet to be refinished and y represents the cost of refinishing the floor,
Given,
The cost of a tile floor for up to 1000 square feet is $1.83 per square,
So, the cost of x square feet of tile = 1.83x for x ≤ 1000
⇒ y = 1.83x for x ≤ 1000
Since, there is an additional charge of $350 for toxic waste disposal for any job which includes more than 150 square feet of tile.
That is, y = 1.83x + 350, for x > 150
So, y must be 1.83x for x ≤ 150.
A) Hence, the function that express the cost, y, of refinishing a floor as a function of the number of square feet, x, to be refinished, is,
[tex]y=\begin{cases}1.83x & \text{ if } 0\leq x\leq 150 \\ 1.83x+350 & \text{ if } 150< x\leq 1000\end{cases}-----(1)[/tex]
B) The domain of the function = all possible value of x
⇒ Domain = 0 ≤ x ≤ 1000
Range = All possible value of y,
Since, the range of function y=1.83x, 0≤ x ≤ 150 is [0, 274.5]
While the range of function y = 1.83x + 350, for x > 150 is (624.5, 2180]
Hence, the range of the function (1) = [0, 274.5]∪(624.5, 2180]
The cost of refinishing a floor can be expressed as a piecewise function based on the number of square feet to be refinished. The domain of the function is all real numbers, and the range is all real numbers greater than or equal to 0.
Explanation:Let x represent the number of square feet to be refinished.
For x ≤ 150, the cost of refinishing a floor is simply $1.83 per square foot. So, the cost function, y, for x ≤ 150 is y = 1.83x.
For x > 150, there is an additional charge of $350 for toxic waste disposal. So, the cost function, y, for x > 150 is y = 350 + 1.83x.
The overall cost function, y, is given by:
y = 1.83x, for x ≤ 150
y = 350 + 1.83x, for x > 150
The domain of the function is all real numbers, since any positive number of square feet can be refinished. The range of the function is all real numbers greater than or equal to 0, since the cost cannot be negative.
What is the GCF of 96x5 and 64x2?
Answer:
6
Step-by-step explanation:
96x5
4667777654442to is 2272666543 GCF is 6Answer:
32x(2) (squared)
Step-by-step explanation:
GCF of 96 and 64:
64 = (2)(2)(2)(2)(2)(2)
96 = (2)(2)(2)(2)(2)(3)
GCF = (2)(2)(2)(2)(2) = 32
GCF of x5 and x2:
x5 = (x)(x)(x)(x)(x)
x2 = (x)(x)
GCF = (x)(x) = x2