Prove the following properties of an open set: 1. The empty set and the real numbers are open. 2. Any union of open sets is open. 3. The complement of an open set is closed. Also, prove the following properties of a closed set: 1. The empty set and the real numbers are closed. 3. Any intersection of a closed set is closed.

(a) Yes, there is a smallest real number a for which x^26 is big-O of ax. To find this value, we can use the limit definition of big-O notation.

We want to find a value of a such that x^26 is less than or equal to ax multiplied by some constant C, for all x greater than some value N. Mathematically, we can write this as:

x^26 <= Cax, for all x >= N

Dividing both sides by x and taking the limit as x approaches infinity, we get:

lim x->inf (x^25 / a) <= C

This limit exists only if a is greater than zero, so let's assume that. Then we can simplify the left-hand side of the inequality as:

lim x->inf x^25 / a = inf

So for any value of C, we can always find a value of N such that x^26 is less than or equal to ax multiplied by C, for all x greater than or equal to N. Therefore, we can say that x^26 is big-O of ax, for any positive real number a, and there is no smallest such value of a.

(b) No, there is no smallest integer number a for which x^26 is big-O of ax. The proof is similar to part (a), but we need to show that for any positive integer a, there exists a constant C such that x^26 is not less than or equal to ax multiplied by C, for infinitely many values of x.

To do this, we can choose x to be a power of 2, say x = 2^k. Then we have:

x^26 = (2^k)^26 = 2^(26k)

ax = a * 2^k

So we want to find a value of a and a constant C such that:

2^(26k) > Ca * 2^k, for infinitely many values of k

Dividing both sides by 2^k, we get:

2^(25k) > Ca, for infinitely many values of k

But this is true for any value of a greater than 2^(25), since 2^(25k) grows faster than Ca for large enough values of k. Therefore, for any integer value of a greater than 2^(25), there exist infinitely many values of k for which x^26 is not less than or equal to ax multiplied by some constant C. Hence, x^26 is not big-O of ax for any integer value of a less than or equal to 2^(25), and there is no smallest such value of a.

(a) No, there isn't a smallest real number 'a' for which x^26x is big-O of ax. This is because x^26x has a higher growth rate than ax for any real number 'a'. As 'x' becomes larger, the term x^26x will always grow faster than ax, no matter the value of 'a'.

(b) Yes, there is a smallest integer number 'a' for which x^26x is big-O of ax. The smallest integer 'a' would be 1, because if we let 'a' be any integer smaller than 1, ax will have a lower growth rate than x^26x. When 'a' is equal to 1, we have x^26x = O(x), which means x^26x grows at most as fast as x, and there's no smaller integer 'a' for which this is true.

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Answer:

  c.  -1

Step-by-step explanation:

You want the slope of the line parallel to UV, where U=(a +2, b +2) and V = (a +5, b -1).

The slope of UV is given by ...

  m = (y2 -y1)/(x2 -x1)

  m = ((b -1) -(b +2))/((a +5) -(a +2)) = -3/3 = -1

The parallel line will have the same slope.

The slope of the line parallel to UV is -1, choice C.

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The sequence an = (8n)/(19n-1) converges, and its limit is 8/19.

Hi! To determine whether the sequence an = (8n)/(19n-1) converges and find its limit, we can follow these steps:

Step 1: Identify the given sequence.
The given sequence is an = (8n)/(19n-1).

Step 2: Analyze the sequence for convergence.
To analyze the convergence of the sequence, we can look at the behavior of the sequence as n approaches infinity.

Step 3: Find the limit of the sequence as n approaches infinity.
To find the limit of the sequence as n approaches infinity, we can use the fact that the highest power of n in the numerator and denominator is the same (n). Therefore, we can divide both the numerator and the denominator by n to simplify the expression:

lim (n→∞) (8n)/(19n-1) = lim (n→∞) (8n/n) / (19n/n - 1/n)

Step 4: Simplify the expression.
After dividing by n, we get:

lim (n→∞) (8) / (19 - 1/n)

Step 5: Evaluate the limit as n approaches infinity.
As n approaches infinity, the term 1/n approaches 0. Therefore, the limit of the sequence is:

lim (n→∞) (8) / (19 - 0) = 8/19

So, the sequence an = (8n)/(19n-1) converges, and its limit is 8/19.

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The conditions that must be satisfied by b1, b2, b3, b4, and b5 for the overdetermined linear system to be consistent are b1 = 10/11r + 1/11s, b2 = 9/11r + 2/11s, b3 = 5/11r + 6/11s, b4 = r, and b5 = s.

For the system to be consistent, there must be a solution that satisfies all the equations in the system. In an overdetermined system, there are more equations than variables, so not all solutions will satisfy all the equations. Therefore, the system will only be consistent if the equations are not contradictory, meaning there is a common solution to all of them.

In this system, there are two variables, x1 and x2, and five equations. We can write the system in matrix form as Ax = b, where A is the coefficient matrix, x is the variable vector, and b is the constant vector.

⎡1 -1⎤ ⎡x1⎤ ⎡b1⎤

⎢-3 1⎥ x ⎢x2⎥ = ⎢b2⎥

⎢1 -5⎥ ⎣ ⎦ ⎢b3⎥

⎣1 6 ⎦ ⎣b4⎦

⎣b5⎦

To check the consistency of the system, we can use row reduction to determine the echelon form of the augmented matrix [A|b]. If the echelon form has a row of zeros with a non-zero constant on the right-hand side, then the system is inconsistent. Otherwise, the system is consistent.

Performing row reduction on [A|b], we get:

⎡1 0 0 0 10/11r+1/11s⎤

⎢0 1 0 0 9/11r+2/11s ⎥

⎢0 0 1 0 5/11r+6/11s ⎥

⎣0 0 0 1 r ⎦

Since the echelon form does not have a row of zeros with a non-zero constant on the right-hand side, the system is consistent. Therefore, the conditions that must be satisfied by b1, b2, b3, b4, and b5 for the system to be consistent are b1 = 10/11r + 1/11s, b2 = 9/11r + 2/11s, b3 = 5/11r + 6/11s, b4 = r, and b5 = s.

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a) Our assumption that there exist integers x and y such that x² - 4y = 2 is false, and we can conclude that for all integers x and y, x² - 4y ≠ 2.

b)  46¹⁰ ≡ 1 (mod 7).

     345 mod 9 ≡ 1 (mod 9).

(a) Proof by contradiction:

Assume that there exist integers x and y such that x² - 4y = 2.

Then x² = 2 + 4y.

Since 2 is an even integer, 4y must also be an even integer, which means that y is an even integer.

Let y = 2k, where k is an integer.

Then x² = 2 + 8k.

If x² is an even integer, then x must also be an even integer (by the given fact).

Let x = 2m, where m is an integer.

Then (2m)² = 2 + 8k.

Simplifying this equation, we get:

4m² = 1 + 4k.

This equation implies that 4m² is an odd integer, which is a contradiction.

Therefore, our assumption that there exist integers x and y such that x² - 4y = 2 is false, and we can conclude that for all integers x and y, x² - 4y ≠ 2.

(b)

(i) 46¹⁰ mod 7:

We can use the property that [tex]a^{b+c} = (a^b)*(a^c)[/tex] to simplify the exponent:

46¹⁰ = (46⁵)²

To find 46⁵ mod 7, we can reduce the base modulo 7:

46 ≡ 4 (mod 7)

Then, we can use the property that (a*b) mod m = ((a mod m) * (b mod m)) mod m:

46⁵ ≡ 4⁵ (mod 7)

≡ (44444) mod 7

≡ (-1)(-1)(-1)(-1)(-1) mod 7

≡ -1 mod 7

≡ 6 (mod 7)

Substituting this value back into the original expression:

46¹⁰ ≡ (46⁵)²

≡ 6² (mod 7)

≡ 36 (mod 7)

≡ 1 (mod 7)

Therefore, 46¹⁰ ≡ 1 (mod 7).

(ii) 345 mod 9:

We can use the property that [tex]a^{b+c} = (a^b)*(a^c)[/tex] to simplify the exponent:

345 = (3100 + 410 + 5)

Therefore, we can break down 345 into its digits and calculate each digit modulo 9:

3100 mod 9 ≡ 0 (mod 9)

410 mod 9 ≡ 5 (mod 9)

5 mod 9 ≡ 5 (mod 9)

Then, we can use the property that (a+b) mod m = ((a mod m) + (b mod m)) mod m:

345 mod 9 ≡ (0 + 5 + 5) mod 9

≡ 10 mod 9

≡ 1 (mod 9)

Therefore, 345 mod 9 ≡ 1 (mod 9).

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The surface area of the sphere, is approximately 172 square inches.

It should be noted that the Volume of a sphere = (4/3)πr^3

where r is the radius of the sphere.

Setting Volume of sphere equal to Volume of prism, we get:

(4/3)πr^3 = lwh

Plugging in the given value of r = 3.7 in, we can solve for lwh:

(4/3)π(3.7)^3 = lwh

lwh ≈ 209.7 cubic inches

A = 4πr^2

A = 4π(3.7)^2

A ≈ 171.9 square inches

Rounding this to the nearest square inch, we get:

A ≈ 172 square inches

Therefore, the surface area of the sphere, is approximately 172 square inches.

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The relative frequency in the class interval with 15 observations is 0.20 or 20%.

The correct answers are: Measurements from a sample are called: statistics. The relative frequency in the class interval with 15 observations is: 0.20.

Statistics are measurements or data collected from a sample of a larger population. They are used to make inferences about the population.

To find the relative frequency of a class interval, you divide the frequency of that interval by the total number of observations. In this case, the relative frequency is:

relative frequency = frequency of interval / total number of observations

relative frequency = 15 / 75

relative frequency = 0.20

Therefore, the relative frequency in the class interval with 15 observations is 0.20 or 20%.

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The area of the region in the given quadrant i is 1/3 square units.

To find the area of the region in quadrant i bounded by y = 2x2 and y = 2x3, we need to first find the x-coordinates where these two curves intersect.

Setting 2x2 equal to 2x3, we get:
2x2 = 2x3

Dividing both sides by 2x2 (which is non-zero since we are only considering quadrant i), we get:
x3 = x2

So the curves intersect at the point (0,0) and (1,2).

To find the area of the region between these curves in quadrant i, we can integrate the difference between the two curves with respect to x, from x = 0 to x = 1:

∫[0,1] (2x3 - 2x2) dx
= [x4 - 2/3 x3] from 0 to 1
= (1 - 2/3) - (0 - 0)
= 1/3

Therefore, the area of the region in quadrant i bounded by y = 2x2 and y = 2x3 is 1/3 square units.

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What points are you describing?

When grouping the given numbers according to congruence mod 13, we find the following groups:

Group 1: {-63}(equivalent to -11 mod 13)

Group 2: {-54, -41}(equivalent to -2 mod 13)

Group 3: {11, 76}(equivalent to 11 mod 13)

Group 4: {13,130}(equivalent to 0 mod 13

Group 5: {80,132}(equivalent to 2 mod 13)

Group 6: {137}(equivalent to 7 mod 13)

Here, we have,

To group the given numbers according to congruence mod 13, we need to find the remainders of each number when divided by 13.

We can find the remainder of a number when divided by 13 by using the modulo operator (%). For example, the remainder of 17 when divided by 13 is 4 (17 % 13 = 4).

Using this method, we can find the remainders of all the given numbers as follows:

=> (-63) % 13= -11

=> -54 % 13 = -2

=> -41 % 13 = -2

=> 11 % 13 = 11

=> 13 %13 = 0

=> (76) % 13 = 11

=> (80) % 13 = 2

=>130 % 13 = 0

=>132 %13 = 2

=>137 % 13 = 7

Now, we can group the numbers according to their remainders as follows:

Group 1: {-63}(equivalent to -11 mod 13)

Group 2: {-54, -41}(equivalent to -2 mod 13)

Group 3: {11, 76}(equivalent to 11 mod 13)

Group 4: {13,130}(equivalent to 0 mod 13

Group 5: {80,132}(equivalent to 2 mod 13)

Group 6: {137}(equivalent to 7 mod 13)

The given numbers have been grouped according to congruence mod 13. Numbers in the same group are equivalent mod 13, i.e., they have the same remainder when divided by 13.

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Hence, x is orthogonal to any vector in W, and hence x is in W^⊥

For any scalar c, u^T (cv) = c(u^Tv)

True. This follows from the distributive property of matrix multiplication and the fact that scalar multiplication is commutative.

Let u and v be non-zero vectors: If the distance from u to v is equal to the distance from u to -v, then u and v are orthogonal.

True. This statement can be restated as saying that u lies on the perpendicular bisector of the line segment connecting v and -v. Since the perpendicular bisector is a line perpendicular to this line segment, it follows that u is orthogonal to both v and -v, and hence orthogonal to their sum, which is the zero vector.

For square matrix A, vectors in R(A) are orthogonal to vectors in N(A).

True. The range of a matrix A consists of all vectors b that can be expressed as b = Ax for some vector x, whereas the null space of A consists of all vectors x such that Ax = 0. If v is in R(A) and w is in N(A), then v = Ax for some x, and we have w^T v = w^T Ax = (A^T w)^T x = 0, since A^T w is in N(A) by the definition of the null space. Hence, v is orthogonal to w.

v^Tv = Ilvll^2.

True. This follows from the definition of the Euclidean norm, which is given by ||v|| = sqrt(v^T v). Hence, ||v||^2 = v^T v.

If vectors v1,....,vp span subspace W and if x is orthogonal to each vj for j = 1,.....,p, then x is in W^⊥.

True. Let v1,....,vp be a basis for W, and let x be orthogonal to each vj. Then, any vector w in W can be expressed as w = c1v1 + ... + cpvp for some scalars c1,....,cp. Since x is orthogonal to each vj, we have x^T w = c1 x^T v1 + ... + cp x^T vp = 0. Hence, x is orthogonal to any vector in W, and hence x is in W^⊥.

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The area lying outside the circle r=6sin and inside the circle r=3+3sin is approximately 21.205 square units.

To solve this problem, we need to first understand what the equations =6sin and =3 3sin represent. These are actually equations of circles in polar coordinates, where r=6sin represents a circle with radius 6 units and centered at the origin, and r=3+3sin represents a circle with radius 3 units and centered at (-3,0) in Cartesian coordinates.

The area lying outside the circle r=6sin and inside the circle r=3+3sin can be found by integrating the equation for the area of a polar region, which is:

A = 1/2 ∫ [f(θ)]^2 - [g(θ)]^2 dθ

where f(θ) and g(θ) are the equations for the outer and inner boundaries of the region, respectively.

In this case, we have:

A = 1/2 ∫ (6sin)^2 - (3+3sin)^2 dθ

A = 1/2 ∫ 36sin^2 - (9+18sin+9sin^2) dθ

A = 1/2 ∫ 27sin^2 - 18sin - 9 dθ

To solve this integral, we can use the half-angle identity for sine, which is:

sin^2 (θ/2) = (1-cos θ)/2

Substituting this identity into our integral, we get:

A = 1/2 ∫ [27(1-cos θ)/2] - 18sin - 9 dθ

A = 1/2 ∫ (13.5-13.5cos θ) - 18sin - 9 dθ

A = 1/2 ∫ -18sin - 22.5cos θ - 9 dθ

Integrating each term separately, we get:

A = -9sin θ - 22.5sin θ - 9θ + C

where C is the constant of integration. To find the bounds of integration, we need to find the values of θ where the two circles intersect. Setting the equations equal to each other, we get:

6sin = 3+3sin

3sin = 3

sin θ = 1

θ = π/2

So the bounds of integration are 0 and π/2. Substituting these values into the equation for the area, we get:

A = -9sin(π/2) - 22.5sin(π/2) - 9(π/2) + C - (-9sin 0 - 22.5sin 0 - 9(0) + C)

A = -13.5π/2

Therefore, the area lying outside the circle r=6sin and inside the circle r=3+3sin is approximately 21.205 square units.

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The volume of the prism is determined as 120 in³.

The volume of the triangular prism is calculated by applying the following formula as shown below;

V = ¹/₂bhl

where;

The volume of the prism is calculated as follows;

V = ¹/₂ x 6 in x 4 in x 10 in

V = 120 in³

,

Thus, the volume of the prism is a function of its base, height and length.

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The Longest Common Subsequence (LCS) for strings A=(a, c, t, g, a, t, t) and B=(c, g, a, t, g, a) is (c, t, g, a, t) and the Longest Common Substring (LCSb) is (t, g, a).


1. Create a matrix of size (m+1)x(n+1) where m and n are the lengths of A and B respectively.

2. Initialize the first row and column of the matrix with 0.

3. Iterate through the matrix, comparing characters from A and B.

4. If characters match, update the matrix value as matrix[i-1][j-1] + 1.

5. If characters don't match, update the matrix value as the max(matrix[i-1][j], matrix[i][j-1]).

6. The LCS can be reconstructed by backtracking from the bottom-right corner of the matrix.

7. For LCSb, find the maximum value in the matrix and its position, then backtrack to construct the substring.

This provides the LCS and LCSb as defined above.

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Answer:

no choices given but it is x^

Step-by-step explanation:

when the bases are the same and you are multiplying, add the powers.

The given series is not a geometric series as the ratio between consecutive terms is not constant. However, it is an arithmetic series with a common difference of 4 in the denominator and 1 in the numerator.

To find a formula for Sn, we need to first find a general term for the series. We can see that the numerator of each term is increasing by 1, starting from 2. Therefore, the nth term of the numerator is n + 1.

For the denominator, we can see that it is increasing by 4, starting from 5. Therefore, the nth term of the denominator is 4n + 1.

Hence, the general term of the series can be written as (n + 1)/(4n + 1).

To find the formula for Sn, we can use the formula for the sum of an arithmetic series:

Sn = n/2[2a + (n-1)d]

where a is the first term, d is a common difference, and n is the number of terms.

In our case, a = 2/5, d = 4/9, and n is not given. However, we can use the formula for the nth term of an arithmetic series to find n:

(n + 1)/(4n + 1) = 6/21
Solving for n, we get n = 5.

Plugging in the values, we get:

S5 = 5/2[2(2/5) + 4/9(5-1)] = 1.23

Therefore, the formula for Sn is Sn = (n + 1)/(4n + 1) and the sum of the first 5 terms is 1.23.

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Answer: 27, 29

Step-by-step explanation:

Let's say that the 2 numbers are x and x+2

That means that: x+x+2=56

Simplify: 2x+2=56

Solve: 2x=54

x=27

27,29 are the 2 numbers

To evaluate the integral ∫7x/(1 − x^4) dx, we first need to perform partial fraction decomposition to separate it into simpler fractions. Using algebraic manipulation.

we can rewrite the integrand as: 7x/(1 − x^4) = A/(1 + x) + B/(1 − x) + C/(1 + x^2) + D/(1 − x^2), where A, B, C, and D are constants to be determined. Then, we can multiply both sides by the common denominator (1 − x^4) and solve for the constants by equating coefficients of like terms.

After performing partial fraction decomposition, we get: ∫7x/(1 − x^4) dx = ∫A/(1 + x) dx + ∫B/(1 − x) dx + ∫C/(1 + x^2) dx + ∫D/(1 − x^2) dx, Integrating each of these simpler fractions individually, we get: ∫A/(1 + x) dx = A ln|1 + x| + c1
∫B/(1 − x) dx = −B ln|1 − x| + c2
∫C/(1 + x^2) dx = C arctan(x) + c3
∫D/(1 − x^2) dx = D ln|(1 + x)/(1 − x)| + c4.

where c1, c2, c3, and c4 are constants of integration, Therefore, the final answer to the given integral is: ∫7x/(1 − x^4) dx = A ln|1 + x| − B ln|1 − x| + C arctan(x) + D ln|(1 + x)/(1 − x)| + C, where A, B, C, and D are the constants obtained from partial fraction decomposition, and C is the constant of integration.

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Answer: ( 0,-5)

Step-by-step explanation:

y-intercept The value of y at the point where a curve crosses the y-axis.

The correct answer is b. It would become wider with fewer values. This is because as the sample size decreases, the variability of the sample mean increases, leading to a wider confidence interval.

The distribution used for the confidence interval calculation (whether z or t) is not impacted by the sample size, only the size of the sample itself affects the confidence interval.

In a situation where the sample size was decreased from 39 to 29, the impact on the confidence interval would be (b) It would become wider with fewer values.

A smaller sample size generally leads to a wider confidence interval, as the decreased sample size provides less information about the overall distribution.

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Answer:

a) kinda but not really b) no c) yes

Step-by-step explanation:

a) It's somewhat possible. The mean is the numbers added together divided but the amount so it would be (3(purple)+2(blue)+2(red)+green)/8. It doesn't completely work because they are not numbers.

b)Their median is not possible. It needs to be in order from largest to greatest and that's not possible with words

c) The mode is the most common thing in a set of data. Since this can be applied to words, purple would be the mode.

The sum of the given series is 72 / (72 + π^2).

We can use the formula for the sum of an infinite geometric series:

S = a / (1 - r)

where S is the sum, a is the first term, and r is the common ratio. In this case, the first term is (-1)^0 π^0 / (6^0 (2*0)!), which simplifies to 1, and the common ratio is (-1) π^2 / (6^2 (2*1)!), which simplifies to -π^2 / 72. Thus, we have:

S = 1 / (1 + π^2 / 72)

Now, we can simplify the denominator by multiplying the numerator and denominator by 72:

S = 72 / (72 + π^2)

Therefore, the sum of the given series is 72 / (72 + π^2).

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The tangential component of acceleration is:
Bold a subscript T left parenthesis t right parenthesis equals 18 t cubed divided by square root of 1 plus 9 t to the power of 4 plus 1

To find the tangential component of acceleration:

We first need to find the velocity vector.

Taking the derivative of the position vector gives us:
bold v left parenthesis t right parenthesis equals bold i plus 3 t squared bold j plus bold k
The tangential component of acceleration is the component of acceleration that is parallel to the velocity vector.

Taking the derivative of the velocity vector gives us:
bold a left parenthesis t right parenthesis equals 0 bold i plus 6 t bold j plus 0 bold k
So the tangential component of acceleration is:
bold a subscript T left parenthesis t right parenthesis equals bold a left parenthesis t right parenthesis dot bold v left parenthesis t right parenthesis divided by the magnitude of bold v left parenthesis t right parenthesis
Since the velocity vector is:
bold v left parenthesis t right parenthesis equals bold i plus 3 t squared bold j plus bold k
The dot product of bold a and bold v is:
bold a left parenthesis t right parenthesis dot bold v left parenthesis t right parenthesis equals 0 times 1 plus 6 t times 3 t squared plus 0 times 1 equals 18 t cubed
The magnitude of the velocity vector is:
magnitude of bold v left parenthesis t right parenthesis equals square root of 1 plus 9 t to the power of 4 plus 1
So the tangential component of acceleration is:
bold a subscript T left parenthesis t right parenthesis equals 18 t cubed divided by square root of 1 plus 9 t to the power of 4 plus 1

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Answer:

its 6/6

Step-by-step explanation:

Answer: C

Step-by-step explanation:

Because all of the numbers are lower than 7 on a 1 to 6 dice.

Please post the full question by replying to my answer

(maybe you can like it so that I'll know you have updated the question)

Each point (x, y) on the graph of h(x) becomes the point (x - 3, y - 3) on v(x).

Each point (x, y) on the graph of h(x) becomes the point (x + 3, y + 3) on w(x).

In Mathematics and Geometry, the translation a geometric figure or graph to the left simply means subtracting a digit from the value on the x-coordinate of the pre-image;

g(x) = f(x + N)

On the other hand, the translation a geometric figure to the right simply means adding a digit to the value on the x-coordinate (x-axis) of the pre-image;

g(x) = f(x - N)

Since the parent function is v(x) = h(x + 3), it ultimately implies that the coordinates of the image would created by translating the parent function to the left by 3 units.

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The solution to the equation 25,000 = 10,000(1.05)5x correct to 3 decimal places is x = 4.017.

To solve this equation, we can first divide both sides by 10,000 to get:

2.5 = 1.05^(5x)

Next, we can take the natural logarithm of both sides:

ln(2.5) = ln(1.05^(5x))

Using the logarithmic identity ln(a^b) = b*ln(a), we can simplify the right side of the equation:

ln(2.5) = 5x*ln(1.05)

Finally, we can solve for x by dividing both sides by 5ln(1.05) and rounding to 3 decimal places:

x = ln(2.5) / (5*ln(1.05)) = 4.017

Therefore, the solution to the equation is x = 4.017, correct to 3 decimal places. This means that after 5 years of an initial investment of $10,000 at an annual interest rate of 5%, the investment will be worth $25,000.

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one can expect about 16.67 feet of snowfall in any given month of January in this remote region of Alaska.

To find the expected amount of snowfall in any given month of January in Alaska, you need to calculate the expected value (E) of the continuous random variable with the given probability density function f(x) = 6/125(5x - x^2), where 0 ≤ x ≤ 5.

The expected value (E) is found using the following formula:

E(X) = ∫[x * f(x)]dx, with integration limits from 0 to 5.

For this problem, we need to evaluate:

E(X) = ∫[x * (6/125)(5x - x^2)]dx from 0 to 5.

Upon integrating, you get:

E(X) = (6/125) * [5/3 * x^3 - x^4/4] evaluated from 0 to 5.

Now, substitute the limits:

E(X) = (6/125) * [5/3 * (5^3) - (5^4)/4 - (0)]

E(X) = (6/125) * [5/3 * 125 - 625/4]

E(X) = (6/125) * [625/3 - 625/4]

E(X) = (6/125) * (625/12)

E(X) = 50/3 ≈ 16.67 feet

So, one can expect about 16.67 feet of snowfall in any given month of January in this remote region of Alaska.

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