a tank contains 1000 l of brine with 10kg of dissolved salt. brine that contains 0.01 kg of salt per liter of water enters the tank at a rate of 15 l/min. the solution is kept thoroughly mixed and drains from the tank at the same rate.(4pts) a) how much salt is in the tank after t minutes? b)how much salt is in the tank after 30 minutes

The dimensions of the box that minimize the amount of material used are a base side length of 25 cm and a height of 25 cm.

Let x be the side length of the square base and h be the height of the box. Since the box has a square base, the volume of the box is V = x²h. We want to minimize the amount of material used, which is given by the surface area of the box, A = x² + 4xh.

Using the volume constraint, we can solve for h in terms of x: h = V / x² = 62,500 / x². Substituting this into the expression for A, we get A = x² + 4x(62,500 / x²) = x² + 250,000 / x.

To minimize A, we take its derivative with respect to x and set it equal to zero: dA/dx = 2x - 250,000 / x² = 0. Solving for x, we get x = 25 cm. Substituting this back into the expression for h, we get h = 25 cm.

Therefore, base side length is 25 cm and height is 25 cm.

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The value of x in the right triangle with acute angles 8x and 4x + 6 is 7

From the question, we have the following parameters that can be used in our computation:

The right triangle with acute angles 8x and 4x + 6

The sum of acute angles in a right triangle is 90

Using the above as a guide, we have the following:

8x + 4x + 6 = 90

So, we have

12x = 84

Divide by 12

x = 7

Hence, the value of x is 7

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Statements that are true:
- The survey was conducted on a random sample of American adults.
- The sample size is large enough to conduct a hypothesis test.
- The sampling distribution can be assumed to be approximately normally distributed due to the large sample size.

To determine whether conditions are met for conducting a hypothesis test, we need to consider the following factors:

1. Random sampling: The survey should be based on a random sample of the population. In this case, the survey was conducted among 500 randomly selected American adults, which satisfies this condition.

2. Sample size: The sample size should be large enough to make the results more reliable. With 500 participants, the sample size is reasonably large.

3. Normality: The sampling distribution should be approximately normally distributed. Given the large sample size, we can apply the Central Limit Theorem, which states that the sampling distribution of the proportion will be approximately normally distributed.

Based on these conditions, we can conclude that it is appropriate to conduct a hypothesis test for the given situation.

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In summary, using the IVT, we can say that there exists at least one root of the function f(x) = x² - 2x + 3 on the interval [3, 5]. However, we cannot say exactly where this root is located or how many roots there are.

The Intermediate Value Theorem (IVT) states that if a continuous function f(x) takes on values of opposite signs at two points a and b, then there exists at least one point c between a and b such that f(c) = 0.

In this case, we are given the function f(x) = x² - 2x + 3 on the interval [3, 5]. We can first check that f(x) is continuous on this interval, which it is since it is a polynomial function.

Next, we can evaluate f(3) and f(5) to see if they have opposite signs:

f(3) = 3² - 2(3) + 3 = 3

f(5) = 5² - 2(5) + 3 = 13

Since f(3) is positive and f(5) is positive, we know that f(x) does not cross the x-axis on the interval [3, 5]. However, we can still use the IVT to show that there exists at least one point c between 3 and 5 such that f(c) = 0.

To do this, we can consider the fact that the graph of f(x) is a parabola that opens upward (since the coefficient of x² is positive), and that the vertex of the parabola is located at the point (1, 2). This means that the minimum value of f(x) occurs at x = 1, and that f(x) is increasing on the interval [3, 5].

Therefore, since f(3) = 3 is less than the minimum value of f(x) on the interval [3, 5], and since f(5) = 13 is greater than the minimum value of f(x) on the interval [3, 5], there must exist at least one point c between 3 and 5 such that f(c) = 0 by the IVT.

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No, 4xy³ and -5x³y are not like terms because they cannot be added or subtracted

Algebraic expressions are defined as expressions that are composed of terms, their coefficients, their variables, constants and factors.

These algebraic expressions are also identified with the presence of arithmetic operations, such as;

It is important to note that like terms are terms an algebraic expression have like variables but not always coefficients. These terms also can be added or subtracted.

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The quadratic equation is solved and the y intercept is A ( 0 , 4 ) and the roots of the given equation are complex numbers

Given data ,

Let the quadratic equation be represented as A

Now , the value of A is

y = x² + 3x + 4

On simplifying , we get

the y-intercept of this equation, we set x = 0 and solve for y:

y = 0² + 3(0) + 4

y = 0 + 0 + 4

y = 4

So, the y-intercept of the given quadratic equation is (0, 4)

And , the roots of the equation is

x = [ -b ± √ ( b² - 4ac ) ] / ( 2a )

x = (-3 ± √(3² - 4(1)(4))) / (2(1))

x = (-3 ± √(9 - 16)) / 2

x = (-3 ± √(-7)) / 2

So , the roots are complex numbers

Hence , the quadratic equation is solved

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The y-intercept provides a useful reference point for understanding the relationship between X and Y in the model.

In the equation [tex]Y ^[/tex] = 39 - .0035X, the y-intercept represents the value of Y when X is equal to 0. This is because when X is 0, the term .0035X becomes 0 and the equation simplifies to[tex]Y ^[/tex] = 39.

Therefore, the y-intercept in this equation is 39. This means that when X is equal to 0, the predicted value of Y is 39.

It's important to note that this does not necessarily mean that the actual value of Y is 39 when X is 0, as the equation is a linear regression model and there may be variability in the data. However, the y-intercept provides a useful reference point for understanding the relationship between X and Y in the model.

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The arc length of this 60 degree central angle in a circle with radius 5 units is 5π/3 units.

In geometry, a central angle is an angle whose vertex is at the center of a circle and whose rays intersect the circle at two distinct points, creating an arc between them.

A central angle is an angle whose vertex is at the center of a circle and whose rays intersect the circle at two distinct points, creating an arc between them.

Central angles can be classified into two types: minor central angles and major central angles. A minor central angle is an angle that intercepts a minor arc, while a major central angle is an angle that intercepts a major arc.

The arc measure is the degree measure of the arc between the two points where the central angle intersects the circle.

The arc length is the actual length of the arc itself, and it depends on both the radius of the circle and the degree measure of the arc. The formula is:

[tex]Arc length = (arc measure/360)[/tex] x [tex]2\pi r[/tex]

For example, if a central angle of a circle has a measure of 60 degrees and the radius of the circle is 5 units, then the arc measure is also 60 degrees, and the arc length can be calculated as:

[tex]Arc length = (60/360)[/tex] x [tex]2\pi (5)[/tex]

[tex]= (1/6)[/tex] x [tex]10\pi[/tex]

[tex]= 5\pi /3[/tex] units

So the arc length of this 60 degree central angle in a circle with radius 5 units is 5π/3 units.

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The increasing functions in this problem are given as follows:

B and D.

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The container holding 2. 66 cubic feet can hold about 0.10 cubic yards.

For the conversion of cubic feet to cubic yards, we can divide the volume by appropriate values. There are 3 feet in one yard, so there are (3 feet)³ = 27 cubic feet in one cubic yard.

Therefore, to convert 2.66 cubic feet to cubic yards, we can use the following conversion factor,

1 cubic yard = 27 cubic feet

2.66 cubic feet / 27 cubic feet per cubic yard = 0.0985 cubic yards

Rounding this answer to two decimal places, we get, the container can hold approximately 0.10 cubic yards.

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B) Note that in the prompt above, you can use translation to map the line y= 2x -4 onto the line y = 2x + 4.

While you can use axial symmetry in the y -  axis to map the line  y = x onto the line y = -x.

Axial symmetry is symmetry around an axis; an item is axially symmetric if it retains its appearance when rotated around an axis.

A baseball bat with no brand or other design, or a plain white tea saucer, for example, looks the same when rotated by any angle around the line traveling longitudinally through its center, indicating that it is axially symmetric.

A transformation in which the coordinate system's origin is shifted but the orientation of each axis remains constant

So for B) you can use a translation to map the line y = 2x -4 onto the line y = 2x + 4 by shifting the first line 4 units upwards along the y - axis....

mathematically, that would be:

y = 2x - 4 + 4

y = 2x


For C) you can use axial symmetry on the y -  axis to achhieve the mapping of y = x onto y = -x by reflection.

The polar opoppsite of y = x  is y = -x.

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Answer:

x+15 dollars

Step-by-step explanation:

x could be any number, but if you add 15 to x, it would be x+15. Since you don't know what x is, you can't do anything else.

The formula to find the present value of an ordinary annuity is:

PV = PMT x ((1 - (1 + r)^-n) / r)

Where PV is the present value, PMT is the payment amount, r is the interest rate per compounding period, and n is the total number of compounding periods.

In this case, the payment amount is $85, the interest rate per quarter is 8%/4 = 2%, and the total number of quarters is 10 x 4 = 40.

Plugging these values into the formula, we get:

PV = $85 x ((1 - (1 + 0.02)^-40) / 0.02)
PV = $85 x ((1 - 0.296) / 0.02)
PV = $85 x (0.704 / 0.02)
PV = $85 x 35.2
PV = $2,992

Rounding to the nearest cent, the answer is $2,992.00. None of the given answer choices match this result.

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The probability that a fruit picked at random weighs between 443 grams and 492 grams is approximately 0.3695 or 36.95%.

To find the probability that a fruit picked at random weighs between 443 grams and 492 grams, we need to standardize these values using the formula:

z = (x - μ) / σ

where x is the weight of the fruit, μ is the mean weight (438 grams), σ is the standard deviation (17 grams), and z is the standardized score.

For the lower end of the range (443 grams), we have:

z = [tex]\frac{(443 - 438)}{17} = 0.29[/tex]

For the upper end of the range (492 grams), we have:

z = [tex]\frac{(492 - 438)}{17} = 3.18[/tex]

Using a standard normal distribution table or calculator, we can find the probability that a standardized score falls between these values.

The probability of a z-score between 0.29 and 3.18 is approximately 0.3695.

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We have shown that (A ∪ B) - B' = A - B for any sets A and B.

To prove that (A ∪ B) - B' = A - B, we need to show that any element in the left-hand side is also in the right-hand side and vice versa.

First, let's consider an arbitrary element x in (A ∪ B) - B'. This means that x is in the union of A and B, but not in the complement of B. Therefore, x is either in A or in B, but not in B'. If x is in A, then x is also in A - B because it is not in B. If x is in B, then it cannot be in B' and thus is also in A - B. Hence, we have shown that any element in the left-hand side is also in the right-hand side.

Now, let's consider an arbitrary element y in A - B. This means that y is in A, but not in B. Since y is in A, it is also in (A ∪ B). Moreover, since y is not in B, it is not in B' and thus also in (A ∪ B) - B'. Therefore, we have shown that any element in the right-hand side is also in the left-hand side.

Thus, we have shown that (A ∪ B) - B' = A - B for any sets A and B.

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The exact value of the expression is: sin(182°) ≈ -0.1492 (rounded to four decimal places)

To write this expression as a trigonometric function of a single number:

We can use the addition formula for sine and cosine:

sin(a + b) = sin(a) cos(b) + cos(a) sin(b)
cos(a + b) = cos(a) cos(b) - sin(a) sin(b)

Using these expressions, we can rewrite the expression as follows:

sin(11° + 190°) + sin(19°)

Simplifying the first term using the identity sin(a + 180°) = -sin(a),

we get:

sin(201°) - sin(19°)

Now, using the subtraction formula for sine, we can write:

sin(a - b) = sin(a) cos(b) - cos(a) sin(b)

Therefore,

sin(201° - 19°) = sin(182°)

So the exact value of the formula:

sin(182°) ≈ -0.1492 (rounded to four decimal places)

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It will take God approximately 32 days to use up all the gas in his tanker and need a refill.

If God filled his gas tanker with 19/5/9 tank of gas and uses 1 5/6 gallons of gas each day, we can calculate how many days it will take for him to need a refill.

First, we need to convert the mixed number 19/5/9 to an improper fraction:

19/5/9 = (19 * 9 + 5) / 9 = 176/9

So God has 176/9 tanks of gas in his tanker.

Next, we can calculate how much gas God uses each day:

1 5/6 = (6 * 1 + 5) / 6 = 11/6

So God uses 11/6 gallons of gas each day.

To find out how many days it will take for God to need a refill, we can divide the amount of gas in his tanker by the amount of gas he uses each day:

(176/9) / (11/6) = (176/9) * (6/11) = 32

Therefore, it will take God approximately 32 days to use up all the gas in his tanker and need a refill.

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The function that has the greatest x-intercept is the function h(x)

The x-intercept of a function is the x-value of the function when the y-value is 0, which is the set of points at which the graph of the function intersects the x-axis.

The x-intercept of each function are found as follows;

f(x) = 3·x - 9 = 0

x = 9/3 = 3

g(x) = |x + 3| = 0

(x + 3) > 0 and |x + 3| = x + 3 = 0

x = 0 - 3 = -3

|x + 3| < 0 and |x + 3| = -(x + 3) = 0

x = -3

h(x) = 2·x - 16 = 0

x = 16/2 = 8

x = 8

j(x) = -5·(x - 2)² = 0

The x-intercept is x = 2

The function that has the greatest x-intercept is therefore the function h(x) = 2·x - 16

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Answer:

Intercept Form.

You can directly solve for x by setting them to zero to which X= 3, X= -2

x-3 = 0  x+2 =0

x= 3  x= -2

Oni walked at a rate of 66.67 miles per minute on the way home.

We have,

Let's use the formula:

distance = rate x time

Let x be the rate at which Oni walked on the way home (in miles per minute).

On the way to her sister's house,

Oni walked at a rate 25% faster than x, or 1.25x miles per minute.

The distance to her sister's house is half a mile, so it took her:

Time to get there

= distance/rate

= 0.5 / 1.25x

= 0.4x minutes

On the way back home, she walked at a rate of x miles per minute, and it took her:

Time to get back

= distance/rate

= 0.5 / x

= 0.5x minutes

The total time for the round trip was 60 minutes, so we can set up an equation:

Time to get there + time to get back = 60

0.4x + 0.5x = 60

0.9x = 60

x = 66.67 (rounded to two decimal places)

Therefore,

Oni walked at a rate of 66.67 miles per minute on the way home.

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The measure of the missing angle which is named ∠PQR = 84°

The first step to solving the problem is to identify the nature of the triangle.

Note that the information states that:

Side PQ and QR are equal,
This means that it is an isosceles triangle because only isosceles triangles have two equal sides.

Another property of isosceles triangles that will help determine the m∠PQR is that the angles at the base of those equal sides are always equal.

Since that is true, then,

∠PQR = 180 - (QPR x 2 )

We know ∠QPR is 48°, so


∠PQR = 180 - (48x 2 )

∠PQR = 180 - 96

Thus,

∠PQR = 84°

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Full Question:

Although part of your question is missing, you might be referring to this full question:

See attached image.

The volume of the solid enclosed by the paraboloid z = x² + y² and the plane z = 9 is V = 36π[tex]V = 36π[/tex] cubic units.

The solid is enclosed by the paraboloid z = x² + y² and the plane z = 9 is a region in 3D space that has a finite volume. To find the volume of this solid, we can use a method called triple integration.

We need to determine the limits of integration for each variable. Since the paraboloid is symmetric about the z-axis, we can integrate over one quadrant and multiply by four to get the total volume. In this case, we can integrate from 0 to 3 for both x and y, and from x² + y² to 9 for z.

The triple integral for the volume is then: [tex]V = 4 * ∫∫∫ z dz dy dx[/tex] Limits: 0 to 3 for x 0 to 3 for y x² + y² to 9 for z. Solving this integral gives us:[tex]V = 36π[/tex]

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(a) 248 to decimal
Since it's already in decimal form, there's no need for conversion.
Answer: 248

(b) 2416 to decimal (assuming it's a hexadecimal number)
Step 1: Identify the place values of the hexadecimal number (from right to left): 1, 16, 256
Step 2: Multiply the digits by their place values and sum them up: (2 * 256) + (4 * 16) + (1 * 1) = 512 + 64 + 1 = 577
Answer: 577

(c) 2c16 to decimal (assuming it's a hexadecimal number)
Step 1: Identify the place values of the hexadecimal number (from right to left): 1, 16, 256
Step 2: Convert the letter "c" to its decimal equivalent: C = 12
Step 3: Multiply the digits by their place values and sum them up: (2 * 256) + (12 * 16) + (1 * 1) = 512 + 192 + 1 = 705
Answer: 705

(d) 00110011101000112 to hexadecimal
Step 1: Group the binary digits into sets of four from right to left: 0011 0011 1010 0011
Step 2: Convert each group of four binary digits into their corresponding hexadecimal values:
      0011 = 3
      0011 = 3
      1010 = A
      0011 = 3
Step 3: Combine the hexadecimal values: 33A3
Answer: 33A3

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The calculated probabilities are approximately:
(a) P(Z > -1.75) = 0.9599
(b) P(Z ≤ 1.82) = 0.9656
(c) P(Z < -1.09) = 0.1379

We have,

To calculate probabilities using the standard normal distribution, with the given values

(a) P(Z > -1.75), (b) P(Z ≤ 1.82), and (c) P(Z < -1.09):

1. Identify the Z-score for each probability:
  (a) Z > -1.75
  (b) Z ≤ 1.82
  (c) Z < -1.09

2. Use a standard normal distribution table, calculator, or software (such as ALEKS) to find the probability associated with each Z-score:
  (a) P(Z > -1.75) = 1 - P(Z ≤ -1.75)
  (b) P(Z ≤ 1.82) = P(Z ≤ 1.82)
  (c) P(Z < -1.09) = P(Z ≤ -1.09)

3. Look up the probabilities in the standard normal distribution table or calculate them using a calculator or software:
  (a) P(Z > -1.75) = 1 - 0.0401 = 0.9599 (approx.)
  (b) P(Z ≤ 1.82) = 0.9656 (approx.)
  (c) P(Z < -1.09) = 0.1379 (approx.)

Thus,
The calculated probabilities are approximately:
(a) P(Z > -1.75) = 0.9599
(b) P(Z ≤ 1.82) = 0.9656
(c) P(Z < -1.09) = 0.1379

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We would expect about 4 students to respond that rock is their favorite music if we randomly select 20 students from the population.

We need to first calculate the proportion of students in the population who prefer rock music.

We can do this by adding up the number of students who prefer rock music in both samples, and then dividing by the total sample size:

Number of students who prefer rock music = 19 + 23 = 42

Total sample size = 100 + 100 = 200

Proportion of students who prefer rock music = 42 / 200 = 0.21

Next, we can use the proportion to estimate the number of students who would prefer rock music in a sample of 20.

To do this, we multiply the proportion by the sample size:

Expected number of students who prefer rock music in a sample of 20

= 0.21 x 20 = 4.2

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To prove that the product of two odd integers is odd, we can follow these steps and justifications:

1. Let x and y be two odd integers.
(We start by assuming x and y are odd integers.)

2. x = 2a + 1 and y = 2b + 1, where a and b are integers.
(Since x and y are odd, they can be expressed in this form, as the sum of an even integer (2a or 2b) and 1.)

3. Find the product of x and y: xy = (2a + 1)(2b + 1).
(To show that their product is odd, we multiply x and y.)

4. Expand the product: xy = 4ab + 2a + 2b + 1.
(Using the distributive property to multiply and simplify.)

5. Factor out a 2: xy = 2(2ab + a + b) + 1.
(We factor out a 2 from the even terms to emphasize the structure of the expression.)

6. Let c = 2ab + a + b, where c is an integer.
(We introduce a new variable, c, to represent the sum of the even terms.)

7. Therefore, xy = 2c + 1, where c is an integer.
(Substituting c back into the expression for xy.)

8. The product xy is an odd integer.
(Since xy is in the form of an even integer (2c) plus 1, it is an odd integer.)

In conclusion, the product of two odd integers (x and y) is also an odd integer, as we have proven through these steps and justifications.

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The volume of the solid that lies within both the cylinder x2+y2=9 and the sphere x2+y2+z2=49 using cylindrical coordinates is: ∫∫∫ dV = 2∫0^2π ∫0^3 r√(49-r2) dr dθ = 2(229/3)π = 458/3π.

To find the volume of the solid that lies within both the cylinder x2+y2=9 and the sphere x2+y2+z2=49 using cylindrical coordinates, we first need to rewrite the equations in terms of cylindrical coordinates.

Recall that in cylindrical coordinates, a point is specified by (r,θ,z), where r is the distance from the origin to the point in the xy-plane, θ is the angle between the positive x-axis and the line segment connecting the origin to the point, and z is the vertical distance from the point to the xy-plane.

For the cylinder x2+y2=9, we have r2 = x2 + y2 = 9, so r = 3.

For the sphere x2+y2+z2=49, we have x2 + y2 = 49 - z2, so r2 = 49 - z2.

Therefore, the solid lies within the cylinder x2+y2=9 and the sphere x2+y2+z2=49 if and only if 0 ≤ r ≤ 3 and -√(49-r2) ≤ z ≤ √(49-r2).

To find the volume of the solid, we integrate over the region:

∫∫∫ dV = ∫0^2π ∫0^3 ∫-√(49-r2)^√(49-r2) r dz dr dθ

= ∫0^2π ∫0^3 r(√(49-r2) + √(49-r2)) dr dθ

= ∫0^2π ∫0^3 2r√(49-r2) dr dθ

= 2∫0^2π ∫0^3 r√(49-r2) dr dθ

(We can drop the factor of 2 since the integrand is even in r.)

To evaluate the integral, we use the substitution u = 49 - r2, du/dr = -2r:

∫0^3 r√(49-r2) dr = -∫49^40 1/2 √u du

= -(1/3)u3/2 |49^4

= (1/3)(49√49 - 4√4)

= (1/3)(49(7) - 4(2))

= 229/3

Therefore, the volume of the solid that lies within both the cylinder x2+y2=9 and the sphere x2+y2+z2=49 using cylindrical coordinates is:

∫∫∫ dV = 2∫0^2π ∫0^3 r√(49-r2) dr dθ = 2(229/3)π = 458/3π.

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The method of assigning probabilities that assumes the experimental outcomes are equally likely is referred to as the classical method. This method is also known as the a priori method. The classical method assumes that all outcomes in the sample space are equally likely to occur, meaning that each outcome has an equal probability of occurring.

The method of assigning probabilities that assumes the experimental outcomes are equally likely is referred to as the classical method (option d). In the classical method, each outcome in an experiment has an equal chance of occurring, and the probability of a particular event happening is determined by the number of favorable outcomes divided by the total number of possible outcomes. This approach is most suitable for situations where there is limited or no prior information about the likelihood of different outcomes, and it relies on the principle of indifference or symmetry.

For example, when tossing a fair coin, the classical method assumes that the probability of getting a heads or tails is 0.5 or 50%. Similarly, when rolling a fair dice, the probability of getting any of the six faces is assumed to be 1/6 or 16.67%. The classical method is commonly used in theoretical probability, which involves predicting the likelihood of outcomes based on assumptions and mathematical calculations rather than experimentation.
In contrast, the objective method (option a) relies on observed data from previous similar experiments or events to determine probabilities, while the subjective method (option b) relies on an individual's personal beliefs, intuition, or opinions to estimate probabilities. The experimental method (option c), on the other hand, refers to the process of conducting experiments and collecting data to determine probabilities.
The classical method is a straightforward and widely applicable approach to probability, especially when dealing with simple problems involving fair games, combinatorics, or situations with symmetrical outcomes. However, it may not always provide the most accurate or realistic probability estimates when dealing with complex, real-world scenarios where outcomes are not equally likely or where prior information is available.

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By showing that a statement is true for a base case and proving that it is true for k+1, assuming that it is true for k, we can show that it is true for all integers in the range of interest.

To show that a formula is true for all integers 1 ≤ k ≤ n, we can use mathematical induction. The process of mathematical induction has two steps: the base case and the induction step.

Base case: Show that the formula is true for k = 1.

Induction step: Assume that the formula is true for some integer k ≥ 1, and use this assumption to prove that the formula is also true for k + 1.

If we can successfully complete both steps, then we have shown that the formula is true for all integers 1 ≤ k ≤ n.

Let's illustrate this with an example. Suppose we want to show that the formula 1 + 2 + 3 + ... + n = n(n+1)/2 is true for all integers 1 ≤ k ≤ n.

Base case: When k = 1, the formula becomes 1 = 1(1+1)/2, which is true.

Induction step: Assume that the formula is true for some integer k ≥ 1. That is,

1 + 2 + 3 + ... + k = k(k+1)/2

We need to prove that the formula is also true for k + 1. That is,

1 + 2 + 3 + ... + (k+1) = (k+1)(k+2)/2

To do this, we can add (k+1) to both sides of the equation in our assumption:

1 + 2 + 3 + ... + k + (k+1) = k(k+1)/2 + (k+1)

Simplifying the right-hand side, we get:

1 + 2 + 3 + ... + k + (k+1) = (k+1)(k/2 + 1/2)

We can rewrite k/2 + 1/2 as (k+2)/2:

1 + 2 + 3 + ... + k + (k+1) = (k+1)(k+2)/2

This is the same as the formula we wanted to prove for k + 1. Therefore, by mathematical induction, we have shown that the formula is true for all integers 1 ≤ k ≤ n.

In summary, mathematical induction is a powerful tool for proving statements about a range of integers. By showing that a statement is true for a base case and proving that it is true for k+1, assuming that it is true for k, we can show that it is true for all integers in the range of interest.

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The value of E(Z)=[tex]=e^{\frac{2}{3} }[/tex]

To find the pdf of Z, we need to use the transformation formula for pdfs:

[tex]f_Z(z) = f_X((g)^{(-1)}z ) * |(\frac{d}{dz}) (g)^{-1} (z)|,[/tex]

where [tex]g(x) = e^x[/tex] and [tex](g)^{-1} (z) = ln(z)[/tex] since [tex](e)^{(ln(z)} = z[/tex].

So, we have:

[tex]f_Z(z) = f_X(ln(z)) * |\frac{d}{dz} ln(z)|[/tex]

[tex]=\frac{1}{3z} (for 0 < z < e^2)[/tex]

To find E[Z], we can use the definition of expected value:

[tex]E(Z) = \int\limits {0^{e^{2} } } z f_Z(z) dz \,[/tex]

[tex]E(Z) = \int\limits {0^{e^{2} } } z (\frac{1}{3z} ) dz \,[/tex]

[tex]= (\frac{1}{3} ) \int\limits {0^{e^{2} } } dz \,[/tex]

[tex]= (\frac{1}{3} ) {e^{2} -0 }[/tex]

[tex]=e^{\frac{2}{3} }[/tex]

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