Therefore, the dimensions of the box with minimal surface area and volume 4096 cm³ are (8, 8, 64).
To find the dimensions of the box with minimal surface area, we need to minimize the surface area function subject to the constraint that the volume is 4096 cm³. The surface area function is:
S = 2xy + 2xz + 2yz
Using the volume constraint, we have:
xyz = 4096
We can solve for one of the variables, say z, in terms of the other two:
z = 4096/xy
Substituting into the surface area function, we get:
S = 2xy + 2x(4096/xy) + 2y(4096/xy)
= 2xy + 8192/x + 8192/y
To minimize this function, we take partial derivatives with respect to x and y and set them equal to zero:
∂S/∂x = 2y - 8192/x² = 0
∂S/∂y = 2x - 8192/y² = 0
Solving for x and y, we get:
x = y = ∛(4096/2) = 8
Substituting back into the volume constraint, we get:
z = 4096/(8×8) = 64
The dimensions of the box with minimal surface area and volume 4096 cm³: (8, 8, 64)
To know more about minimal surface area,
https://brainly.com/question/2273504
#SPJ11
determine whether the quantitative variable is discrete or continuous. distance an athlete can jump question content area bottom part 1 is the variable discrete or continuous?
The variable in this case is "distance an athlete can jump" for the quantitative variable.
This variable is a quantitative variable, meaning it can be measured numerically. The answer to whether it is discrete or continuous depends on how the measurement is taken. If the measurement is taken in whole numbers or distinct categories (e.g. in feet or meters), then it is a discrete variable. However, if the measurement can take on any value within a range (e.g. in inches or centimeters), then it is a continuous variable. Therefore, without knowing the specific unit of measurement, it is impossible to determine if this variable is discrete or continuous.
A quantitative variable is a type of variable used in statistics that can take on numerical values to reflect quantities or amounts. Mathematical procedures such as addition, subtraction, multiplication, and division can be used to quantify and express these quantities. The quantitative variables height, weight, age, temperature, and income are a few examples. According to whether the values can take on any value within a range (continuous) or only certain specified values (discrete), quantitative variables can be further categorised as either continuous or discrete. In many disciplines, including economics, social sciences, and natural sciences, the examination of quantitative variables is a crucial part of statistical modelling and data analysis.
Learn more about quantitative variable here:
https://brainly.com/question/10961513
#SPJ11
Express the confidence interval (0.068,0.142) in the form of p-E«p
The confidence interval (0.068,0.142) in the form of p-E«p is p - E < p < p + E, where p = 0.105 and E = 0.037.
To express the confidence interval (0.068, 0.142) in the form of p ± E, we first need to find the sample proportion p and the margin of error E.
The sample proportion p is the midpoint of the confidence interval, so we have:
p = (0.068 + 0.142) / 2 = 0.105
The margin of error E is half the width of the confidence interval, so we have:
E = (0.142 - 0.068) / 2 = 0.037
Therefore, we can express the confidence interval (0.068, 0.142) in the form of p ± E as:
p - E < p < p + E
0.105 - 0.037 < p < 0.105 + 0.037
0.068 < p < 0.142
So the confidence interval (0.068, 0.142) can be expressed as p - E < p < p + E, where p = 0.105 and E = 0.037.
Know more about confidence interval here:
https://brainly.com/question/15712887
#SPJ11
HELP ASAP PLSSSSS HELP NOW
What is the correct numerical expression for "9 times 4 added to the difference of 3 and 2?"
9 x 4 + (3 − 2)
9 x (4 + 3) − 2
9 + (4 x 3) ÷ 2
9 − 2 x 4 + 3
Hello !
9 times 4 added to the difference of 3 and 2
9 x 4 + ( 3 - 2)
9 x 4 + (3 - 2)
If A and B are 2 x 7 matrices, and C is a 8 x 2 matrix, which of the following are defined? A. A-C B. ATCT C. AT D. CA E. A - B OF. BA
In the given scenario, the operations A - C, ATCT, AT, CA, and A - B are defined, while BA is not defined
A - C: The operation A - C is defined when the matrices A and C have the same dimensions. Since A is a 2 x 7 matrix and C is an 8 x 2 matrix, their subtraction (A - C) is not defined due to incompatible dimensions.
ATCT: The operation ATCT is defined when both matrices A and C are compatible for matrix multiplication. Since A is a 2 x 7 matrix and C is an 8 x 2 matrix, their product (ATCT) is defined, resulting in a 7 x 8 matrix.
AT: The operation AT represents the transpose of matrix A, which is defined for any matrix. Therefore, AT is defined, resulting in a 7 x 2 matrix.
CA: The operation CA is defined when both matrices C and A are compatible for matrix multiplication. Since C is an 8 x 2 matrix and A is a 2 x 7 matrix, their product (CA) is defined, resulting in an 8 x 7 matrix.
A - B: The operation A - B is defined when both matrices A and B have the same dimensions. Since both A and B are 2 x 7 matrices, their subtraction (A - B) is defined and results in a 2 x 7 matrix.
BA: The operation BA is not defined since the number of columns in matrix B (7 columns) is not equal to the number of rows in matrix A (2 rows), which violates the compatibility requirement for matrix multiplication.
Learn more about matrix multiplication here:
https://brainly.com/question/13591897
#SPJ11
The following table represents the highest educational attainment of all adult residents in a certain town. If an adult is chosen randomly from the town, what is the probability that they have a high school degree or some college, but have no college degree? Round your answer to the nearest thousandth.
the chart is in the image please answer asap!!!!
Answer:
Step-by-step explanation:
Find the equation of the parabola with the following properties.
Express your answer in standard form.
Symmetric with respect to the line y=−2
Directrix is the line x=−1
p=3
Answer: Since the directrix is the line x = -1, the vertex of the parabola must lie on the axis of symmetry, which is the line y = -2. So, the vertex must be of the form (h, -2).
Since p = 3, the distance from the vertex to the focus is 3 units, and since the directrix is x = -1, the focus must be at a point 3 units to the right of the vertex, i.e., at (h + 3, -2).
The standard form of the equation of a parabola with vertex (h, k) and focus (h + p, k) is:
(y - k)^2 = 4p(x - h)
So, substituting the vertex and focus coordinates, we get:
(y + 2)^2 = 4(3)(x - h)
Simplifying, we get:
y^2 + 4y + 4 = 12(x - h)
y^2 + 4y + (4 - 12h) = 0
To put this equation in standard form, we complete the square on the left-hand side by adding and subtracting (4/2)^2 = 4:
y^2 + 4y + 4 - 12h - 4 = 0
(y + 2)^2 - 12h - 4 = 0
(y + 2)^2 = 12h + 4
Finally, rearranging, we get the equation of the parabola in standard form:
y = (1/12)(x - h)^2 - 2
where h is a constant that determines the horizontal position of the vertex.
The equation of the parabola is y^2 = 6x - 3 in standard form.
Since the directrix is the line x=-1, we know that the focus is the point (-1+p,0)=(2,0). And since the parabola is symmetric with respect to the line y=-2, we know that the vertex is the point (2,-2). Using the definition of a parabola, we know that the distance between any point on the parabola (x,y) and the focus (2,0) is equal to the distance between (x,y) and the directrix x=-1.
So, we have:
sqrt((x-2)^2 + (y-0)^2) = abs(x+1)
Simplifying, we get:
(x-2)^2 + y^2 = (x+1)^2
Expanding, we get:
x^2 - 4x + 4 + y^2 = x^2 + 2x + 1
Simplifying, we get:
y^2 = 6x - 3
Therefore, the equation of the parabola is y^2 = 6x - 3 in standard form.
To know more about parabola refer here:
https://brainly.com/question/31142122
#SPJ11
Answer the following questions a Find A b 3 whose eigenvalues are 1 and 4, and whose eigenvectors are *> respectively b Find B whose eigenvalues are 1 and 3
This gives us the equation -2x + y = 0 and 4x - 3y = 0, which has the solution x = y/2. Therefore, the eigenvector for λ = 3 is v2 = [1; 2].
a) To find matrix A with eigenvalues 1 and 4 and corresponding eigenvectors v1 and v2 respectively, we can use the formula A = PDP^-1 where P is the matrix of eigenvectors and D is the diagonal matrix of eigenvalues.
We know that v1 and v2 are eigenvectors with eigenvalues 1 and 4 respectively, so we can set up the following equations:
Av1 = 1v1 and Av2 = 4v2
Multiplying both sides of each equation by P^-1, we get:
PDv1 = v1 and PDv2 = 4v2
Therefore, P = [v1 v2] and D = [1 0; 0 4], which gives us the matrix A = PDP^-1.
b) To find matrix B with eigenvalues 1 and 3, we can use the same formula A = PDP^-1. However, we don't know the eigenvectors yet. To find them, we can use the characteristic polynomial of B, which is (1-λ)(3-λ) = 0. This gives us eigenvalues λ = 1 and λ = 3.
To find the eigenvectors for λ = 1, we need to solve the equation (B-λI)v = 0, which gives us:
(B-1I)v = 0
[0 1; 1 2][x; y] = [0; 0]
This gives us the equation x + y = 0, so the eigenvector for λ = 1 is v1 = [1; -1].
To find the eigenvectors for λ = 3, we need to solve the equation (B-λI)v = 0, which gives us:
(B-3I)v = 0
[-2 1; 4 -3][x; y] = [0; 0]
Using these eigenvectors and the formula A = PDP^-1, we can find the matrix B = PDP^-1 where P = [v1 v2] and D = [1 0; 0 3].
To learn more about : eigenvector
https://brainly.com/question/17079851
#SPJ11
1Function Spaces Preserved by the Derivative: In Section 5.2, Exercises 11, we found the matrix [D] of the derivative operation D on the subspaces W = Span (B). (a) Use your answers in that section to find the matrices of the 2nd and 3rd derivatives, [D²] and [D³]; (b) Use these matrices to find the 2nd and 3rd derivatives of the indicated function f(x) using a matrix product. (c) Show that D is both one-to-one and onto on W by finding the rref of [D], and describing ker(D) and range(B).a. W = Span(B), where B {ex, ex}; f(x) = 5e* - 3e2x. =
b. W = Span(B), where B {ex sin(x), e cos(x)}; f(x) = 4e* sin(x) - 3e* cos(x). =
c. W = Span (B), where B ({e-3x sin(2x), e-3x cos(2x)}); = f(x) = 5e 3x sin(2x) - 9e-3x cos(2x).
d. W = Span(B), where B ({xesx, esx}); f(x) = -2xe 5x+7e5x. ==
We can be obtained by taking the derivative of each basis vector of B three times and writing the result in terms of B and use it to describe ker(D) and range(D)
(a) The matrix [D²] of the 2nd derivative operation D² on the subspace W = Span(B) can be obtained by taking the derivative of each basis vector of B twice and writing the result in terms of B. Similarly, the matrix [D³] of the 3rd derivative operation D³ on We can be obtained by taking the derivative of each basis vector of B three times and writing the result in terms of B.
(b) Using the matrices [D²] and [D³], we can find the 2nd and 3rd derivatives of the given functions by multiplying the matrix with the column vector representing the coefficients of the function in terms of the basis B.
(c) To show that D is both one-to-one and onto on W, we can find the reduced row echelon form (rref) of [D], and use it to describe ker(D) and range(D).
(d) Using the same method as in parts (a) and (b), we can find the matrices [D²] and [D³] for the subspace W = Span(B), where B = {xesx, esx}, and use them to find the 2nd and 3rd derivatives of the given function f(x).
Learn more about reduced row echelon here:
https://brainly.com/question/30763331
#SPJ11
Evaluate the line integral ∫CF⋅dr where F=〈−3sinx,2cosy,10xz〉F=〈−3sinx,2cosy,10xz〉 and C is the path given by r(t)=(t^3,3t^2,2t) for 0≤t≤1
The value of the line integral ∫CF⋅dr is (-3cos(1) + 4sin(3) + 5)/3.
To evaluate the line integral ∫CF⋅dr, we need to compute the dot product F⋅dr along the path C=r(t) from t=0 to t=1.
First, we need to find the differential of the vector-valued function r(t):
dr/dt = <3t^2, 6t, 2>
Then, we can compute F(r(t)) and evaluate the dot product F(r(t))⋅(dr/dt):
F(r(t)) = <-3sin(t^3), 2cos(3t^2), 10t^3>
F(r(t))⋅(dr/dt) = (-9t^2sin(t^3)) + (12t^2cos(3t^2)) + (20t^4)
Now, we can integrate this expression over the interval [0,1] to get the value of the line integral:
∫CF⋅dr = ∫(F(r(t))⋅dr/dt)dt from 0 to 1
= ∫((-9t^2sin(t^3)) + (12t^2cos(3t^2)) + (20t^4))dt from 0 to 1
= (-3cos(1) + 4sin(3) + 5)/3
Therefore, the value of the line integral ∫CF⋅dr is (-3cos(1) + 4sin(3) + 5)/3.
To know more about line integral refer here:
https://brainly.com/question/30763905?#
#SPJ11
Find the limit of the sequence if it converges; otherwise indicate divergence.an= (ln n)^5/√n
To determine if the sequence converges or diverges, we can use the limit test. We'll analyze the limit of the given function as n approaches infinity:
an = (ln n)^5 / √n
We'll find the limit as n approaches infinity:
lim (n→∞) [(ln n)^5 / √n]
To evaluate this limit, we can apply L'Hopital's Rule, which states that if the limit of the ratio of the derivatives of the numerator and denominator exists, then the limit of the ratio of the functions exists and is equal to the limit of the ratio of the derivatives.
First, let's rewrite the expression as:
an = (ln n)^5 * n^(-1/2)
Now, let's find the derivatives of (ln n)^5 and n^(-1/2) with respect to n:
d/dn (ln n)^5 = 5(ln n)^4 * (1/n)
d/dn n^(-1/2) = (-1/2)n^(-3/2)
Now, let's find the limit of the ratio of the derivatives:
lim (n→∞) [(5(ln n)^4 * (1/n)) / (-1/2)n^(-3/2)]
We can simplify this expression:
lim (n→∞) [(10(ln n)^4) / n^(1/2)]
Now, we observe that as n approaches infinity, the denominator (n^(1/2)) grows much faster than the numerator (10(ln n)^4). Therefore, the limit of the expression goes to zero:
lim (n→∞) [(10(ln n)^4) / n^(1/2)] = 0
Since the limit is zero, the sequence converges to 0.
To know more about sequence, visit:
https://brainly.com/question/30262438
#SPJ11
Find the missing varable
5/17 = x/10
Thank you guys in advance I really need your helpp!!
After cross multiplying and simplifying the given equation 5/17 = x/10, the value if the missing variable x is equal to 50/17 or 5.88.
To solve the equation 5/17 = x/10 for x, we can use cross-multiplication. This means we can multiply both sides of the equation by 10 to isolate x on one side:
5/17 = x/10
10 * 5/17 = x
Simplifying the left-hand side of the equation:
50/17 = x
So x is equal to 50/17. This is the solution to the equation, and it represents the value of x that would make the equation true. When 5 is 17% of 10, the missing variable x is 5.88.
To learn more about missing variable click on,
https://brainly.com/question/24004472
#SPJ1
Can someone help me quickly
What is the volume of a cone whose diameter is 324pi cm2, and the length of the diameter of the base is 24cm?
The height of the given cone is 6.75 cm.
Given that, the volume of a cone is 324π cm² and the length of the diameter is 24 cm.
Here, radius of the cone = 24/2 = 12
We know that, the volume of the cone is 1/3 πr²h.
Now, 1/3 πr²h = 1/3 π×12²h
324π = 1/3 π×12²×h
324 = 1/3 ×144×h
324 = 48h
h=324/48
h=6.75 cm
Therefore, the height of the given cone is 6.75 cm.
To learn more about the volume of a cone visit:
https://brainly.com/question/1984638.
#SPJ1
find the standard equation of the sphere with the given characteristics. center: (−1, −6, 3) radius: 5
The standard equation of the sphere with the given characteristics, center (-1, -6, 3), and radius 5 is
[tex](x+1)^{2} +(y+6)^{2}+ (z-3)^{2} =25[/tex].
The standard equation of a sphere is [tex](x-h)^{2} +(y-k)^{2}+ (z-l)^{2} =r^{2}[/tex], where (h, k, l) is the center of the sphere and r is the radius.
Using this formula and the given information, we can write the standard equation of the sphere:
[tex](x-(-1))^{2}+ (y-(-6))^{2} +(z-3)^{2}= 5^{2}[/tex]
Simplifying, we get:
[tex](x+1)^{2} +(y+6)^{2}+ (z-3)^{2} =25[/tex].
Therefore, the standard equation of the sphere with center (-1, -6, 3) and radius 5 is [tex](x+1)^{2} +(y+6)^{2}+ (z-3)^{2} =25[/tex].
Learn more about the standard equation of the sphere here:
https://brainly.com/question/31706340
#SPJ11
Two trains depart from City Center in opposite directions. Train A heads west at 60 mi. /hr. Train B heads east at 75 mi. /hr
The two trains will be 900 miles apart after 6 hours.
The problem can be solved using the formula Distance = Rate x Time. The distance covered by Train A in 6 hours would be 60 x 6 = 360 miles. Similarly, the distance covered by Train B would be 75 x 6 = 450 miles. Adding these distances, we get a total distance of 810 miles. However, we need to take into account the fact that the trains are moving in opposite directions and are getting further apart. Thus, we need to add their distances to get the total distance between them, which is 900 miles. Therefore, the answer is that the two trains will be 900 miles apart after 6 hours.
Know more about Distance here:
https://brainly.com/question/28165272
#SPJ11
Which answer choice correctly solves the division problem and shows the quotient as a simplified fraction?
A.
B.
C.
D
Thus, option A is the correct answer choice which shows the quotient of the given division problem as a simplified fraction in 250 words.
To solve the given division problem and show the quotient as a simplified fraction, we need to follow the steps given below:
Step 1: We need to perform the division of 8/21 ÷ 6/7 by multiplying the dividend with the reciprocal of the divisor.8/21 ÷ 6/7 = 8/21 × 7/6Step 2: We simplify the obtained fraction by cancelling out the common factors.8/21 × 7/6= (2×2×2)/ (3×7) × (7/2×3) = 8/21 × 7/6 = 56/126
Step 3: We reduce the obtained fraction by dividing both the numerator and denominator by the highest common factor (HCF) of 56 and 126.HCF of 56 and 126 = 14
Therefore, the simplified fraction of the quotient is:56/126 = 4/9
Thus, option A is the correct answer choice which shows the quotient of the given division problem as a simplified fraction in 250 words.
To know more about fraction visit:
https://brainly.com/question/10354322
#SPJ11
Given y=(x+2)(2x2+3)3 find the equation of the tangent line to this function when x = 1. First find the point on this function and the slope of the tangent line to this function when x = 1. Next use these to find the equation of the tangent line to this function when x = 1. Finally, put this equation in slope intercept form. All work must be shown!!
Point on function when x = 1 is (1, _____)
Slope of tangent line when x = 1 is _____________
Equation of tangent line in slope intercept form is:
_______________________________________
The equation of the tangent line in slope-intercept form is y = 67x - 40.
To find the point on the function when x = 1, we simply substitute x = 1 into the given equation:
y = (1+2)(2(1)^2+3)^3 = 27
So the point on the function when x = 1 is (1,27).
To find the slope of the tangent line when x = 1, we take the derivative of the given function and evaluate it at x = 1:
y' = (2x^2+7x+6)(2x^2+3)^2 + 3(x+2)(4x^3+18x^2+18x)
y'(1) = (2(1)^2+7(1)+6)(2(1)^2+3)^2 + 3(1+2)(4(1)^3+18(1)^2+18(1))
= 67
So the slope of the tangent line when x = 1 is 67.
Using the point-slope form of the equation of a line, we can write the equation of the tangent line when x = 1 as:
y - 27 = 67(x - 1)
Simplifying, we get:
y = 67x - 40
So the equation of the tangent line in slope-intercept form is y = 67x - 40.
Learn more about tangent line here
https://brainly.com/question/30162650
#SPJ11
Contestar las siguientes preguntas.
(a) ¿55% de cuánto es 33?
(b) ¿Qué número es 15% de 80?
The number whose 55 percent is 33 is 60.
The number whose 15 percent is 80 is 80.
We have,
(a)
To find the number that is 55% of 33, we can set up the equation:
0.55x = 33
By dividing both sides of the equation by 0.55, we can solve for x:
x = 33 / 0.55 ≈ 60
So, 33 is 55% of 60.
(b)
To find the number that is 15% of 80, we can calculate 15% of 80:
15% of 80 = 0.15 x 80 = 12
Therefore, 12 is 15% of 80.
Thus,
The number whose 55 percent is 33 is 60.
The number whose 15 percent is 80 is 80.
Learn more about percentages here:
https://brainly.com/question/11403063
#SPJ1
The complete question.
Answer the following questions.(a) 55% of what is 33?(b) What number is 15% of 80?
Let x and y have joint pdf fxy(x,y) = 12xy(1-x). Show that the support of (U,V) can be described by v >1 and 0
Answer:
v > 0 and 0 < u-v < 1, which is equivalent to v > 0 and u > v.
Step-by-step explanation:
We have the joint pdf of (X,Y) as:
f(x,y) = 12xy(1-x)
To find the support of (U,V), we need to determine the range of values that (U,V) can take. Let U = X + Y and V = Y.
Solving for X and Y, we get:
X = U - V and Y = V
The Jacobian of the transformation is:
J = [tex]\frac{∂(x,y)}{∂(u,v)}[/tex] = det [[[tex]\frac{∂x}{∂u}[/tex], [tex]\frac{∂x}{∂v}[/tex]], [[tex]\frac{∂y}{∂u}[/tex], [tex]\frac{∂y}{∂v}[/tex]]]
= det [[1, -1], [0, 1]] = 1
So, the joint pdf of (U,V) is:
f(u,v) = f(x,y) |J| = 12(u-v)v(1-(u-v))([tex]\frac{1}{2}[/tex]) = 6v(u-v)(1-(u-v))
The support of (U,V) is the range of values of (u,v) for which f(u,v) is non-zero. Since f(u,v) is non-zero only if v > 0 and 0 < u-v < 1, the support of (U,V) can be described as:
v > 0 and 0 < u-v < 1, which is equivalent to v > 0 and u > v.
To know more about Jacobian refer here
https://brainly.com/question/31326284#
#SPJ11
1.formulate and write mathematically the four maxwell’s equations in integral form
This equation relates the circulation of the magnetic field around a closed loop (left-hand side) to the current flowing through that loop (first term on the right-hand side) and to the time-varying electric field
equations describe the behavior of electromagnetic fields and are fundamental to the study of electromagnetism. Here are the four Maxwell's equations in integral form:
1. Gauss's law for electric fields:
∮E⋅dA=Q/ε0
This equation relates the electric flux through a closed surface (left-hand side) to the charge enclosed within that surface (right-hand side).
2. Gauss's law for magnetic fields:
∮B⋅dA=0
This equation states that the magnetic flux through any closed surface is always zero, which means that there are no magnetic monopoles.
3. Faraday's law of electromagnetic induction:
∮E⋅dl=−dΦB/dt
This equation relates a changing magnetic field (the time derivative of magnetic flux ΦB) to an induced electric field (left-hand side).
4. Ampere's law with Maxwell's correction:
∮B⋅dl=μ0(I+ε0dΦE/dt)
To learn more about : magnetic field
https://brainly.com/question/14411049
#SPJ11
Maxwell's equations describe the fundamental principles of electromagnetism. These equations are comprised of four integral forms: Gauss's law, Gauss's law for magnetism, Faraday's law of induction, and Ampere's law with Maxwell's correction.
Gauss's law states that the electric flux through a closed surface is equal to the charge enclosed within the surface. Gauss's law for magnetism states that there are no magnetic monopoles, and that the magnetic flux through a closed surface is always zero. Faraday's law of induction states that a changing magnetic field induces an electric field. Ampere's law with Maxwell's correction states that a changing electric field can induce a magnetic field. Formulating these four equations in integral form involves expressing them using calculus and integrating over a surface or volume.
1. Gauss's Law for Electric Fields:
∮E⋅dA = (1/ε₀) ∫ρ dV
This equation relates the electric flux through a closed surface to the enclosed electric charge.
2. Gauss's Law for Magnetic Fields:
∮B⋅dA = 0
This equation states that the magnetic flux through a closed surface is zero, as there are no magnetic monopoles.
3. Faraday's Law of Electromagnetic Induction:
∮E⋅dl = -d(∫B⋅dA)/dt
This equation shows the relationship between a changing magnetic field and the induced electric field that creates a voltage.
4. Ampère's Law with Maxwell's Addition:
∮B⋅dl = μ₀ (I + ε₀ d(∫E⋅dA)/dt)
This equation connects the magnetic field around a closed loop to the current passing through the loop and the changing electric field.
Learn more Maxwell's equations about here: brainly.com/question/31958772
#SPJ11
the time until a person is served in a cafeteria is t, which follows an exponential distribution with mean of β = 4 minutes. what is the probability that a person has to wait more than 10 minutes
The probability that a person has to wait more than 10 minutes is approximately 0.0821 or 8.21%.
We know that the probability density function of the exponential distribution with mean β is given by:
f(t) = (1/β) * exp(-t/β)
where t is the time and exp(x) is the exponential function with base e raised to the power x.
To find the probability that a person has to wait more than 10 minutes, we need to integrate the probability density function from t = 10 to infinity:
P(t > 10) = ∫[10,∞] f(t) dt
Substituting the value of β = 4, we get:
P(t > 10) = ∫[10,∞] (1/4) * exp(-t/4) dt
Using integration by substitution, let u = -t/4, then du = -1/4 dt:
P(t > 10) = ∫[-10/4,0] e^u du
P(t > 10) = [-e^u]_(-10/4)^0
P(t > 10) = [-e^0 + e^(-10/4)]
P(t > 10) = [1 - e^(-5/2)]
Therefore, the probability that a person has to wait more than 10 minutes is approximately 0.0821 or 8.21%.
To know more about probability refer here:
https://brainly.com/question/30034780
#SPJ11
The sum of two numbers is 55 the smaller number is 21 less than the larger number what are the numbers?
We know that the sum of two numbers is 55. This means that if we add two numbers together, we get 55.
We also know that the smaller number is 21 less than the larger number. This means that the smaller number is 21 units smaller than the larger number.
To find the two numbers, we can use these two pieces of information together.
We can start by writing an equation using the given information:
x + (x - 21) = 55
Here, x represents the larger number and (x - 21) represents the smaller number.
We can simplify this equation by adding x and (x - 21) on one side and then subtracting 21 from both sides:
2x + 21 - 21 = 55 - 21
Simplifying this equation, we get:
2x = 34
Dividing both sides by 2, we get:
x = 17
Therefore, the two numbers are 21 and 17.
Learn more about numbers Visit : brainly.com/question/25734188
#SPJ11
Occasionally an airline will lose a bag. a small airline has found it loses an average of 2 bags each day. find the probability that, on a given day,
We can use the Poisson distribution to solve this problem.
Let X be the number of bags lost by the airline in a given day. Then, X follows a Poisson distribution with parameter λ = 2, since the airline loses an average of 2 bags each day.
The probability of losing exactly k bags on a given day is given by the Poisson probability mass function:
P(X = k) = e^(-λ) (λ^k) / k!
Substituting λ = 2, we get:
P(X = k) = e^(-2) (2^k) / k!
We can use this formula to calculate the probabilities for the requested scenarios:
(a) Probability of losing no bags on a given day (k = 0):
P(X = 0) = e^(-2) (2^0) / 0! = e^(-2) ≈ 0.1353
(b) Probability of losing at least 3 bags on a given day (k ≥ 3):
P(X ≥ 3) = 1 - P(X ≤ 2)
We can calculate P(X ≤ 2) as follows:
P(X ≤ 2) = P(X = 0) + P(X = 1) + P(X = 2)
= e^(-2) (2^0) / 0! + e^(-2) (2^1) / 1! + e^(-2) (2^2) / 2!
≈ 0.4060
Therefore,
P(X ≥ 3) = 1 - P(X ≤ 2) ≈ 0.5940
(c) Probability of losing exactly 1 bag on each of the next 3 days:
Since the number of bags lost on each day is independent, the probability of losing exactly 1 bag on each of the next 3 days is given by the product of the individual probabilities:
P(X = 1)^3 = [e^(-2) (2^1) / 1!]^3 = e^(-6) (2^3) / 1!^3 ≈ 0.0048
To Know more about Poisson distribution refer here
https://brainly.com/question/31316745#
#SPJ11
find the área.......
Answer: 42,120
Step-by-step explanation:
Area is calculated by multiplying the length of a shape by its width-
suppose f 3 = 2 and f ′ 3 = −3. let g(x) = f(x) sin(x) and h(x) = cos(x) f(x) . find the following. (a) g ′ 3 (b) h ′ 3
The chain rule is a formula in calculus that describes how to compute the derivative of a composite function.
We can use the product rule and the chain rule to find the derivatives of g(x) and h(x):
(a) Using the product rule and the chain rule, we have:
g'(x) = f'(x)sin(x) + f(x)cos(x)
At x=3, we know that f(3) = 2 and f'(3) = -3, so:
g'(3) = f'(3)sin(3) + f(3)cos(3) = (-3)sin(3) + 2cos(3)
Therefore, g'(3) = -3sin(3) + 2cos(3).
(b) Using the product rule and the chain rule, we have:
h'(x) = f'(x)cos(x) - f(x)sin(x)
At x=3, we know that f(3) = 2 and f'(3) = -3, so:
h'(3) = f'(3)cos(3) - f(3)sin(3) = (-3)cos(3) - 2sin(3)
Therefore, h'(3) = -3cos(3) - 2sin(3).
To learn more about calculus visit:
brainly.com/question/31801938
#SPJ11
NEED HELP ASAP PLEASE!
It's A
An independent event means that the probability of events A and B occurring is equal to event A's probability multiplied by event B.
What is the range of possible lengths for the third side of a triangle that has side lengths of 7 and 10? Please show your answer in this format: a < n < b. The a and b will be the numbers you need to add in for this answer. If your answer is correct but you were marked wrong please let your teacher know.
The range of possible values for the third side of the triangle is:
3 < n < 17
How to find the range of possible lengths?For any triangle we can define the triangular inequality, it says that the sum of any two sides must be longer than the remaining side.
So if the lengths of the sides are A, B, and C, that inequality says that:
A + B > C
A + C > B
B + C > A
In this case, we can define:
A = 7
B = 10
C = n
Then the triangular inequality becomes:
7 + 10 > n
7 + n > 10
10 + n > 7
Solving these 3, we will get:
17 > n
n > 3
n > -3
Then the range of possible values for the last side is:
3 < n < 17
Learn more about triangles:
https://brainly.com/question/1058720
#SPJ1
Answer:
The range of possible lengths for the third side of the triangle is greater than 3 units and less than 17 units in other words 3 < n < 17
For his exercise today, bill plans to run and swim some laps. The table below shows how long ( in minutes) it takes him to run each lap and swim to each lap
The inequality describing this problem is given as follows:
4r + 2s > 30.
How to define the inequality?The variables for this problem are given as follows:
Variable r: number of laps run.Variable s: number of laps swam.Bill will practice for more than 30 minutes, hence the inequality is given as follows:
4r + 2s > 30.
(the sign > is used as the sign is the more than symbol in inequality).
As we have more than and not at least in the sentence, the symbol used does not contain the equal sign, meaning that the interval is open.
Missing InformationThe problem is given by the image presented at the end of the answer.
More can be learned about inequalities at brainly.com/question/25275758
#SPJ1
evaluate ∫413x 5x√ dx. enter your answer as an exact fraction if necessary.
∫^16_9 (-x^1/2-5)dx
provide your answer below:
The value of the second integral is -109/3.
For the first integral, we can use the power rule and the constant multiple rule of integration:
∫413x 5x√ dx = [tex]4/3 \times 13x^{3/2 }\times 2/3 \times 5x3/2+1/2 + C[/tex]
= 40[tex]x^{5/2[/tex] / 15 + C
= 8[tex]x^{5/2[/tex] / 3 + C
where C is the constant of integration.
For the second integral, we can use the power rule and the constant multiple rule of integration:
∫[tex]^{16}_9 (-x^1/2-5)dx = (-2/3 \times x^(3/2) - 5x)^{16_9}[/tex]
= [tex](-2/3 \times 16^{(3/2)} - 5 \times 16) - (-2/3 \times 9^{(3/2)} - 5 \times 9)[/tex]
= (-2/3 × 64 - 80) - (-2/3 × 27 - 45)
= (-128/3 - 80) - (-54/3 - 45)
= -208/3 + 99/3
= -109/3
Therefore, the value of the second integral is -109/3.
for such more question on integral
https://brainly.com/question/27746495
#SPJ11
To evaluate ∫413x 5x√ dx, we can use integration by substitution. Let u = 5x√, then du/dx = 5/2x^1/2 and dx = 2/5u^2/5 du.
Substituting these into the integral, we get:
∫413x 5x√ dx = ∫4u u(2/5u^2/5) du
Simplifying:
∫413x 5x√ dx = 8/5 ∫u^7/5 du
Integrating:
∫413x 5x√ dx = 8/5 * (5/12)u^(12/5) + C
Substituting back in for u:
∫413x 5x√ dx = 2/3 x^(3/2) * (5x√)^(2/5) + C
Simplifying:
∫413x 5x√ dx = 2/3 x^(3/2) * (5x)^(2/5) + C
Now, to evaluate ∫^16_9 (-x^1/2-5)dx, we can use the power rule of integration:
∫^16_9 (-x^1/2-5)dx = [-2/3x^(3/2) - 5x] from 9 to 16
Substituting in the limits:
∫^16_9 (-x^1/2-5)dx = [-2/3(16)^(3/2) - 5(16)] - [-2/3(9)^(3/2) - 5(9)]
Simplifying:
∫^16_9 (-x^1/2-5)dx = [(-32/3) - 80] - [(-18/3) - 45]
∫^16_9 (-x^1/2-5)dx = -112/3
Therefore, the answer to the second integral is -112/3.
To evaluate the given integral ∫^16_9 (-x^(1/2) - 5) dx, we'll find the antiderivative of the function and then apply the Fundamental Theorem of Calculus.
Learn more abour fraction here : brainly.com/question/10354322
#SPJ11
evaluate the line integral l=∫c[x2ydx (x2−y2)dy] over the given curves c where (a) c is the arc of the parabola y=x2 from (0,0) to (2,4):
The value of the line integral over the given curve c is 16/5.
We are given the line integral:
css
Copy code
l = ∫c [tex][x^2*y*dx + (x^2-y^2)*dy][/tex]
We will evaluate this integral over the given curve c, which is the arc of the parabola y=x^2 from (0,0) to (2,4).
We can parameterize this curve c as:
makefile
Copy code
x = t
y =[tex]t^2[/tex]
where t goes from 0 to 2.
Using this parameterization, we can express the differential elements dx and dy in terms of dt:
css
Copy code
dx = dt
dy = 2t*dt
Substituting these expressions into the line integral, we get:
css
Copy code
l = [tex]∫c [x^2*y*dx + (x^2-y^2)*dy][/tex]
= [tex]∫0^2 [t^2*(t^2)*dt + (t^2-(t^2)^2)*2t*dt][/tex]
= [tex]∫0^2 [t^4 + 2t^3*(1-t)*dt][/tex]
= [tex][t^5/5 + t^4*(1-t)^2] from 0 to 2[/tex]
= 16/5
Therefore, the value of the line integral over the given curve c is 16/5.
For such more questions on line integral
https://brainly.com/question/28381095
#SPJ11
The concept that allows us to draw conclusions about the population based strictly on sample data without having anyknowledge about the distribution of the underlying population
Inferential statistics allows researchers to draw conclusions about a population based on sample data, without knowing the complete distribution of the underlying population.
How does inferential statistics work?Inferential statistics is a concept in statistics that allows us to draw conclusions about a population based on a sample of data, without having complete knowledge about the distribution of the underlying population.
It involves using probability theory to estimate population parameters based on sample statistics.
This approach is useful in research when it is not feasible or practical to study an entire population.
Instead, a smaller, representative sample can be taken to draw conclusions about the larger population.
Inferential statistics allows researchers to make informed decisions and predictions based on data that is not fully known, ultimately leading to more accurate and reliable results.
Learn more about Inferential statistics
brainly.com/question/30761414
#SPJ11