Which of the following represents the distributive property?

2x+3 = 2(x+3)
2(x +3) = 2x +6
2x+3 = 3(x+2)
2(x+3) = x +6

Answer:

13 dollars for the first 10 for each additional one ( i think)

Step-by-step explanation:

Answer:

the answer is it is $10 for each one with an additional cost as $3 at the end

Based on the study, the team cannot conclude that sleep deprivation raises blood pressure (option b). The confidence interval (3.5, 16.8) indicates that there is a range of possible values for the difference between the mean blood pressure of the treatment group and the control group.

The confidence interval being wide does not invalidate the comparison (option c). A wide interval simply indicates a larger range of plausible values, which may result from variability within the data or a smaller sample size. It does not invalidate the validity of the comparison itself. However, it does highlight the uncertainty in estimating the true difference between the two groups.

It's important to note that this study alone does not establish a causal relationship between sleep deprivation and blood pressure (option d). While the theory suggests such a relationship, this particular study's results do not provide conclusive evidence for causation. Additionally, the generalization of the conclusions to all UMD students would require a more diverse and representative sample.

The true difference could be anywhere within that range. Since the interval includes zero, it suggests that there is no statistically significant difference between the two groups' mean blood pressure after two days. Therefore, the data do not provide sufficient evidence to support the theory that sleep deprivation leads to higher blood pressure.

Finally, the data do not support the idea that blood pressure causes sleep deprivation (option e). The study design does not focus on examining the causal direction in that manner, and the results do not provide evidence to support this hypothesis.

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The volume is measured at the bottom of the meniscus when the liquid level is between the two etched marks on the pipet.

1. Place the graduated pipet on a flat surface and make sure it is level

2. Draw the liquid up until the meniscus is level with the etched line that corresponds to the desired volume

3. Make sure the liquid is not touching the etched line above it

4. Observe the bottom of the meniscus and take the reading at the point where the liquid level is between the two etched lines

The volume is measured at the bottom of the meniscus when the liquid level is between the two etched marks on the pipet.

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Answer:

4 ft higher

Step-by-step explanation:

Since the ladder is 10 ft long and its top is 6 feet high(above the ground), we find the distance of its base from the wall since these three (the ladder, wall and ground) form a right angled triangle. Let d be the distance from the wall to the ladder.

So, by Pythagoras' theorem,

10² = 6² + d²  (the length of the ladder is the hypotenuse side)

d² = 10² - 6²

d² = 100 - 36

d² = 64

d = √64

d = 8 ft

Since the ladder is moved so that the base of the ladder travels toward the wall twice the distance that the top of the ladder moves up.

Now, let x be the distance the top of the ladder is moved, the new height of top of the ladder is 6 + x. Since the base moves twice the distance the top of the ladder moves up, the new distance for our base is 8 - 2x(It reduces since it gets closer to the wall).

Now, applying Pythagoras' theorem to the ladder with these new lengths, we have

10² = (6 + x)² + (8 - 2x)²

Expanding the brackets, we have

100 = 36 + 12x + x² + 64 - 32x + 4x²

collecting like terms, we have

100 = 4x² + x² + 12x - 32x + 64 + 36

100 = 5x² - 20x + 100

Subtracting 100 from both sides, we have

100 - 100 = 5x² - 20x + 100 - 100

5x² - 20x = 0

Factorizing, we have

5x(x - 4) = 0

5x = 0 or x - 4 = 0

x = 0 or x = 4

The top of the ladder is thus 4 ft higher

Both solutions satisfy the original equation. As a result, there are no extraneous solutions in this case.

To determine the number of extraneous solutions in the given equation, let's simplify it step by step:

Step 1: Let's find the common denominator for the two fractions on the left side of the equation. The common denominator is (2m + 3)(2m - 3).

Step 2: Apply the common denominator to both fractions:

[(2m)(2m - 3)]/[(2m + 3)(2m - 3)] - [(2m)(2m + 3)]/[(2m + 3)(2m - 3)] = 1

Step 3: Simplify the numerators:

\([4m^2 - 6m - 4m^2 - 6m]/[(2m + 3)(2m - 3)] = 1\)

[-12m]/[(2m + 3)(2m - 3)] = 1

Step 4: Cancel out common factors:

-12m = (2m + 3)(2m - 3)

\(-12m = 4m^2 - 9\)

Step 5: Rearrange the equation:

\(4m^2 + 12m - 9 = 0\)

Step 6: Solve the quadratic equation using factoring, completing the square, or using the quadratic formula. Let's use the quadratic formula:

\(m = (-b ± √(b^2 - 4ac))/(2a)\)

For our equation, a = 4, b = 12, and c = -9. Substituting these values:

m = (-(12) ± √((12)^2 - 4(4)(-9)))/(2(4))

m = (-12 ± √(144 + 144))/(8)

m = (-12 ± √288)/8

m = (-12 ± 12√2)/8

Simplifying further:

m = (-3 ± 3√2)/2

So, we have two potential solutions for m:

m = (-3 + 3√2)/2  and  m = (-3 - 3√2)/2

Now we need to check if these solutions satisfy the original equation. Let's substitute these values back into the equation:

For m = (-3 + 3√2)/2:

[(2(-3 + 3√2))/(2(-3 + 3√2) + 3)] - [(2(-3 + 3√2))/(2(-3 + 3√2) - 3)] = 1

Simplifying this equation, we find that it holds true.

For m = (-3 - 3√2)/2:

[(2(-3 - 3√2))/(2(-3 - 3√2) + 3)] - [(2(-3 - 3√2))/(2(-3 - 3√2) - 3)] = 1

Simplifying this equation, we also find that it holds true.

Therefore, both solutions satisfy the original equation. As a result, there are no extraneous solutions in this case.

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Answer:

2x + y = -2, x + 2y = 2

4x + y - 2z = 0, 2x+ 3z = 9 , -6x - 2y + z = 0

x + y = 4 , x - y = 2

Step-by-step explanation:

Answer:

x=2 and y=−5

Step-by-step explanation:

Let's solve your system by elimination.

7x−y=19;2x−3y=19

Multiply the first equation by -3,and multiply the second equation by 1.

−3(7x−y=19)

1(2x−3y=19)

Becomes:

−21x+3y=−57

2x−3y=19

Add these equations to eliminate y:

−19x=−38

Then solve−19x=−38for x:

−19x=−38

(Divide both sides by -19)

x=2

Now that we've found x let's plug it back in to solve for y.

Write down an original equation:

7x−y=19

Substitute2forxin7x−y=19:

(7)(2)−y=19

−y+14=19(Simplify both sides of the equation)

−y+14+−14=19+−14(Add -14 to both sides)

−y=5

y=-5

Answer:

5

Step-by-step explanation:

The measure of the segment AB is 5 units.

Given that on a number line point A is at -5 and B is at 0 we need to find the measure of the segment AB,

To find the measure of the segment AB on a number line, we need to calculate the absolute difference between the coordinates of points A and B. In this case, point A is at -5, and point B is at 0.

The absolute difference between -5 and 0 is calculated as follows:

|(-5) - 0| = |-5| = 5

Therefore, the measure of the segment AB is 5 units.

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Check the picture below.

well, let's take a peek at the distances of each side.

\(~\hfill \stackrel{\textit{\large distance between 2 points}}{d = \sqrt{( x_2- x_1)^2 + ( y_2- y_1)^2}}~\hfill~ \\\\[-0.35em] ~\dotfill\\\\ A(\stackrel{x_1}{0}~,~\stackrel{y_1}{0})\qquad B(\stackrel{x_2}{4}~,~\stackrel{y_2}{3}) ~\hfill AB=\sqrt{[ 4- 0]^2 + [ 3- 0]^2} \\\\\\ ~\hfill AB=\sqrt{25}\implies \boxed{AB=5} \\\\\\ B(\stackrel{x_1}{4}~,~\stackrel{y_1}{3})\qquad C(\stackrel{x_2}{4}~,~\stackrel{y_2}{-3}) ~\hfill BC=\sqrt{[ 4- 4]^2 + [ -3- 3]^2} \\\\\\ ~\hfill BC=\sqrt{36}\implies \boxed{BC=6}\)

\(C(\stackrel{x_1}{4}~,~\stackrel{y_1}{-3})\qquad A(\stackrel{x_2}{0}~,~\stackrel{y_2}{0}) ~\hfill CA=\sqrt{[ 0- 4]^2 + [ 0- (-3)]^2} \\\\\\ ~\hfill CA=\sqrt{25}\implies \boxed{CA=5} \\\\\\ \textit{so it's an \underline{isosceles triangle}}\)

Answer:

s = 156.25 square meters

Step-by-step explanation:

s = 2lw + 2lh + 2wh

s = 2(5.5)(3) + 2(3)(7.25) + 2(5.5)(7.25)

s = 33 + 43.5 + 79.75

s = 156.25 square meters

Answer:

We know,

\( \quad \)\({\boxed{ \rm{Surface \: Area_{ \: (rectangular \: prism)} = 2(lb + bh + hl)}}}\)

where,

\( \: \)

\( {\longrightarrow { 2 ( 5.5 \times 3 + 3 \times 7.25 + 7.25 \times 5.5 ) }} \)

\( \longrightarrow \) 2 ( 16.5 + 21.75 + 39.875 )

\( \longrightarrow \) 2 ( 78.125 )

\( \longrightarrow \) 156.25

Thus, The Surface Area of rectangular prism is 156.25cm².

\( \rule{200pt}{2pt} \)

\(\footnotesize{\boxed{ \begin{array}{cc} \small\underline{ \sf{\pmb{ \blue{More \: Formula}}}} \\ \\ \bigstar \: \bf{CSA_{(cylinder)} = 2\pi \: rh}\\ \\ \bigstar \: \bf{Volume_{(cylinder)} = \pi {r}^{2} h}\\ \\ \bigstar \: \bf{TSA_{(cylinder)} = 2\pi \: r(r + h)}\\ \\ \bigstar \: \bf{CSA_{(cone)} = \pi \: r \: l}\\ \\ \bigstar \: \bf{TSA_{(cone)} = \pi \: r \: (l + r)}\\ \\ \bigstar \: \bf{Volume_{(sphere)} = \dfrac{4}{3}\pi {r}^{3} }\\ \\ \bigstar \: \bf{Volume_{(cube)} = {(side)}^{3} }\\ \\ \bigstar \: \bf{CSA_{(cube)} = 4 {(side)}^{2} }\\ \\ \bigstar \: \bf{TSA_{(cube)} = 6 {(side)}^{2} }\\ \\ \bigstar \: \bf{Volume_{(cuboid)} = lbh}\\ \\ \bigstar \: \bf{CSA_{(cuboid)} = 2(l + b)h}\\ \\ \bigstar \: \bf{TSA_{(cuboid)} = 2(lb +bh+hl )}\\ \: \end{array} }}\)

The volume of the right cylinder is 1017.88 m² = 324π m²

One of the most fundamental curvilinear geometric shapes, a cylinder has traditionally been a three-dimensional solid. It is regarded as a prism with a circle as its base in basic geometry. In several contemporary fields of geometry and topology, a cylinder can alternatively be characterized as an infinitely curved surface.

The properties of cylinder are :

It features two flat circular faces, two curved edges, and one curved surface.

The two circular flat bases are parallel to one another.

There isn't a vertex on it.

The radius of a circular base and the height of a cylinder determine its size.

The radius of the cylinder = 6 m

Height = 9 m

The volume of the right cylinder is π(radius)²height

= π * 6² * 9 = 1017.88 m² = 324π m²

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To minimize the overall number of coins the value would be V mod (100 + 50 + 25 + 10 + 5) so option d is correct choice.

           
This function returns the remainder when the V value is divided by the total value of the largest coin denomination available (100) plus the four next largest coin denominations (50, 25, 10, 5). As a result, a number of changes that cannot be implemented using only the largest denominations of coins must be implemented using smaller denominations, including pennies.


A legitimate capability would consider the value of the various coin sections and use that data to decide the most effective method for rolling out the improvement with the fewest coins possible.

The capability would almost certainly include division and modulus activities to compute the remainder of the value of V when partitioned by the value of each coin group.

The result of the capability would be the number of pennies required to complete the exchange, as pennies are the smallest denomination of coin that anyone could hope to find.


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Using the FOIL method, the products are:

1.  x² + 6x - 16

2. x² - 6x - 72

3. x² + 13x + 36

4. x² + 8x + 12

5. x² + 15x + 54

6. x² + x - 20

7. x² - 9

8. x² + 20x + 100

9. x² + 9x - 36

10. x² + x - 12

11. x² + 7x + 10

12. x² - 7x + 12

The FOIL method is an acronym that stands for "First, Outside, Inside, and Last". This method simply describes the order you use in multiplying the terms of a given expression, that is, you have to multiply first terms, then outside terms, then inside terms, then last terms. The final step is to combine like terms together to arrive at your final answer.

Therefore, using the FOIL method, we have the following products:

1.  (x - 2)(x + 8)

x² + 8x -2x - 16

Combine like terms

x² + 6x - 16

2. (x + 6)(x - 12)

x² - 12x + 6x - 72

Combine like terms

x² - 6x - 72

3. (x + 9)(x + 4)

x² + 4x + 9x + 36

x² + 13x + 36

4. (x + 6)(x + 2)

x² + 2x + 6x + 12

Combine like terms

x² + 8x + 12

5. (x + 6)(x + 9)

x² + 9x + 6x + 54

Combine like terms

x² + 15x + 54

6. (x - 4)(x + 5)

x² + 5x - 4x - 20

x² + x - 20

7. (x + 3)(x - 3)

x² - 3x + 3x - 9

x² - 9

8. (x + 10)(x + 10)

x² + 10x + 10x + 100

x² + 20x + 100

9. (x - 3)(x + 12)

x² + 12x - 3x - 36

x² + 9x - 36

10. (x + 4)(x - 3)

x² - 3x + 4x - 12

x² + x - 12

11. (x + 5)(x + 2)

x² + 2x + 5x + 10

x² + 7x + 10

12. (x - 3)(x - 4)

x² - 4x - 3x + 12

x² - 7x + 12

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The best estimate for the population mean that Bryce will score a 90% or better is 0.625.

Based on the given information, Bryce played the same song on Guitar Hero 8 times and scored percentages of 89%, 82%, 90%, 88%, 89%, 91%, 85%, and 95%. He wants to know the population mean that he will score a 90% or better. Which would be the best estimate?

To find the population mean, we need to calculate the average of Bryce's scores.

Step 1: Add up all the scores: 89 + 82 + 90 + 88 + 89 + 91 + 85 + 95 = 709.

Step 2: Divide the sum by the total number of scores (8): 709 / 8 = 88.625.

Therefore, the best estimate for the population mean that Bryce will score a 90% or better is 0.625.

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Answer:

1. 12

2. 8

Step-by-step explanation:

\PLEASE MARK BRAINLIEST IT WILL HELP SM

Answer:

4-3=1

8+9=17

7 x 2= 14

Step-by-step explanation:

Answer:

1

17

14

Step-by-step explanation:

4 + -3 = 1

8 - -9=17

14/2=7

Two figures are similar if their corresponding sides are in proportion and their corresponding angles are equal. In this case, Figure A and Figure B are similar, with a similarity ratio of r.

a. The ratio of the perimeters of similar figures is equal to the ratio of their corresponding sides. Since Figure A and Figure B are similar with a ratio of r, the ratio of their perimeters is also r.

b. If the perimeter of Figure A is p and the linear scale factor is r, the perimeter of Figure B can be found by multiplying the perimeter of Figure A by the linear scale factor:

Perimeter of Figure B = p * r

c. The area of similar figures is equal to the square of the linear scale factor multiplied by the area of the original figure. So, if the area of Figure A is a and the linear scale factor is r, the area of Figure B can be calculated as:

Area of Figure B = a * r^2

These formulas can be used to find the ratios and calculate the perimeters and areas of similar figures. Make sure to substitute the appropriate values given in the problem to find the specific answers.

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Answer:

Below

Step-by-step explanation:

y= 1/4 x + 1

The graph has a y-axis intercept of +1     (y = mx + b form)

 last one looks correct....at x = 4   the value is 2 which is correct

Answer: minimum: free

maximum: infinite

Answer:

expanding

commutative prop

factorisation

simplifying

Step-by-step explanation:

The distributive property states that a(b+c) is equal to ab+ac and (a+b)c is equal to ac+bc. The commutative property states that a+b=b+a.

For the given algebraic expression, the value width is 15 units.

P = 2L + 2w (the perimeter is the two lengths and two widths put together) (the perimeter is the two lengths and two widths added together)

P = 100 (the perimeter is stated to be 100 feet).

L = 2w + 5 (length is 5 more than twice the width) (length is 5 more than twice the width)

Let's enter in some of the variable values we have now to see what results we have so far.

100 = 2(2w + 5) + 2w

Distribute 100 as follows: 4w + 10w + 2w

Rephrase: 100 = 6w + 10

You only need to separate w from here. Once you have that number, plug it into the equation L = 2w + 5 to determine the value of L, which will tell you the perimeter's length and breadth.

w = 100-10 / 6

w = 90/6

w = 15

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The number of different ways in which it can be built in the mentioned lots, using only five models as a reference is: 120.

Factorial numbers are the products of all natural numbers before a particular number, it is usually represented with an exclamation mark at the end of the number (!) and they are usually used to represent the possible combinations of a certain number of elements.

In the case of the five models, this can be represented by 5! to identify the number of possible combinations that can be made with these five models, which are calculated as follows:

Taking into account the calculation made, houses on the five lots can be arranged in 120 different ways.

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Answer:

\(4<QR<14\)

5, 6, 7, 8, 9, 10, 11, 12, and 13.

Step-by-step explanation:

We can use the triangle inequality. By the triangle inequality, the sum of any two sides of a triangle must be greater than the remaining side.

In other words, the sum of the two shorter sides must be greater than the longest side.

We know that PQ is 5 and PR is 9.

For the first case, let's say that neither PQ nor PR is the longest side. Instead, QR is the longest side. Then their sum must be greater than QR. Thus:

\(5+9>QR\Rightarrow QR<14\)

However, for the second case, let's say that QR is not the longest side. In this case, PR will be the longest side. Therefore, QR plus PQ must be greater than PR:

\(QR+5>9\Rightarrow QR>4\)

Thus, we have a compound inequality:

\(4<QR<14\)

Therefore, the possible whole-number lengths of QR are:

5, 6, 7, 8, 9, 10, 11, 12, and 13.

Step-by-step explanation:

=2[5+2(8-6)]

=2[5+2(2)]

=2[5+4]

=2×9

=18

Answer: 18

Concept:

When encountering a question that needs to do operations with expressions, using the PEMDAS method will ease the process.

Solve:

Given

2 [5 + 2 (8 - 6)]

Simplify values in the parentheses

=2 [5 + 2 × 2]

Simplify with multiplication

=2 [5 + 4]

Simplify the parentheses

=2 × 9

=18

Hope this helps!! :)

Please let me know if you have any questions

The area of the side of the cone is 100.53 cm² and the area of the cricle base is 50.27 cm² while the total surface area is  150.8 cm².

From the diagram, we are given:

base radius (r) = 4cm

slant height (l) = 8cm

(a) To find the area of the side of the cone, we need to find the slant height and the base perimeter.

Using the Pythagorean theorem, we can find the height of the cone:

height = √(l² -r²)

      = √(8² - 4²)

      = √48

      = 4√3 cm

The base perimeter is simply the circumference of the circle with radius 4 cm:

base perimeter = 2πr

              = 2π(4)

              = 8π cm

Now we can use the formula for the lateral surface area of a cone:

lateral surface area = 1/2 (base perimeter) (slant height)

                    = 1/2 (8π) (8)

                    = 32π cm²

                    = 100.53 cm²

Therefore, the area of the side of the cone is approximately 100.53 cm².

(b) The area of the circle base can be found using the formula:

base area = πr²

         = π(4)²

         = 16π cm²

         = 50.27 cm²

Therefore, the area of the circle base is approximately 50.27 cm².

(c) The total surface area of the cone is the sum of the lateral surface area and the base area:

total surface area = lateral surface area + base area

                  = 32π + 16π

                  = 48π cm²

                  ≈ 150.8 cm²

Therefore, the total surface area of the cone is approximately 150.8 cm².

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To solve this problem, you can use the following formula for simple interest:

where P is the initial amount, r is the rate of interest in decimal form, and t is the time in years.

Substituting A=1920, r=0.08, t=6, and solving for P, you get:

Simplifying, you get:

Answer:B

Step-by-step explanation:

Option B is correct,  the inequality y≥-2x-2 is represented in the second graph or graph B.

Graph is a mathematical representation of a network and it describes the relationship between lines and points.

The given inequality is y≥-2x-2

y greater than or equal to minus two times of x minus two.

y≥-2x-2

The line passes through the points (-2, 2) and (0, -2).

As there is given greater than or equal to the line should be solid not dotted.

Greater than or equal to means we have to shade away from the origin.

The second graph satisfy all these properties.

y≥-2x-2 is represented in the second graph.

Hence, the inequality y≥-2x-2 is represented in the second graph.

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To evaluate each definite integral without using the Fundamental Theorem of Calculus.

Integrals -

(a) ∫(3 to -3) \(x^2 dx = 0\)

(b) ∫(0 to -3) \(x^2 dx = 9\)

(c) ∫(3 to 0) \(-8x^2 dx = -72\)

(d) ∫(3 to -3) \(5x^2 dx = 0\)

What is Integrals?

In mathematics, the integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise from combining infinitesimally small data. The process of finding integrals is called integration.

To evaluate the definite integrals without using the Fundamental Theorem of Calculus, we can make use of the given information:

∫(0 to x) \(x^2 dx = 9\)

(a) To evaluate the integral ∫(3 to -3) \(x^2 dx\):

We can rewrite this integral as the sum of two integrals: ∫(3 to 0) \(x^2 dx\) + ∫(0 to -3)\(x^2 dx.\)

Using the given information, we have:

∫(3 to 0) \(x^2 dx\) = -∫(0 to 3)\(x^2 dx\)\(x^2 dx\) = -9

∫(0 to -3) \(x^2 dx\) = 9

Thus, ∫(3 to -3) \(x^2 dx\) = ∫(3 to 0) \(x^2 dx\) + ∫(0 to -3) \(x^2 dx\) = -9 + 9 = 0.

(b) To evaluate the integral ∫(0 to -3) \(x^2 dx\):

We already have ∫(0 to -3) \(x^2 dx\) = 9.

(c) To evaluate the integral ∫(3 to 0) \(-8\)\(x^2 dx\):

Since the given information is in terms of positive values of x, we need to reverse the limits of integration and change the sign:

∫(3 to 0) \(-8x^2 dx\) = -∫(0 to 3) \(-8x^2 dx\) = -8 * ∫(0 to 3) \(x^2 dx\) = -8 * 9 = -72.

(d) To evaluate the integral ∫(3 to -3) \(5x^2 dx\):

Similar to part (a), we can split this integral into two parts:

∫(3 to 0) \(5x^2 dx\)  + ∫(0 to -3) \(5x^2 dx\).

Using the given information, we have:

∫(3 to 0)\(5x^2 dx\) = -∫(0 to 3) \(5x^2 dx\) = -5 * 9 = -45

∫(0 to -3)\(5x^2 dx\) = 5 * 9 = 45

Thus, ∫(3 to -3) \(5x^2 dx\)= ∫(3 to 0) \(5x^2 dx\) \(5x^2 dx\) + ∫(0 to -3)\(5x^2 dx\) = -45 + 45 = 0.

Therefore:

(a) ∫(3 to -3) \(x^2 dx\) = 0

(b) ∫(0 to -3)\(x^2 dx\) = 9

(c) ∫(3 to 0) \(-8x^2 dx\) = -72

(d) ∫(3 to -3) \(5x^2 dx\) = 0

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The expected value of Z, E(Z), is αμ + μ * √(1 - α²).

The correlation coefficient between U and Z, ρUZ, is α.

These results provide insights into the relationship between the independent random variables U and V, and the newly defined random variable Z.

The expected value of a random variable represents the average value we would expect to obtain if we were to repeat the experiment many times. To find E(Z), we need to calculate the average value of Z.

We have Z = αU + V * √(1 - α²). Since U and V are independent, their expected values are simply their means, which are μ.

Now, let's calculate E(Z):

E(Z) = E(αU + V * √(1 - α²))

= αE(U) + E(V * √(1 - α²)) (by linearity of expectation)

= αμ + E(V) * √(1 - α²) (since E(U) = μ)

= αμ + μ * √(1 - α²) (since U and V have the same variance, their variances are both σ²)

Therefore, the expected value of Z, E(Z), is equal to αμ + μ * √(1 - α²).

Correlation coefficient between U and Z (ρUZ):

The correlation coefficient measures the linear relationship between two random variables. In this case, we want to find the correlation coefficient between U and Z, denoted as ρUZ.

The correlation coefficient ρUZ is defined as the covariance between U and Z divided by the product of their standard deviations.

To calculate ρUZ, we need to find the covariance between U and Z. The covariance between two random variables U and Z is given by:

Cov(U, Z) = E[(U - E(U))(Z - E(Z))]

Since U and V are independent, the covariance between U and V is zero. Therefore, we have:

Cov(U, Z) = Cov(U, αU + V * √(1 - α²))

= Cov(U, αU) + Cov(U, V * √(1 - α²))

= αCov(U, U) + 0

= αVar(U) (since Cov(U, U) = Var(U))

Using the given information that U has variance σ², we have:

Cov(U, Z) = ασ²

Next, let's calculate the standard deviations of U and Z.

Standard deviation of U (σU) = √(Var(U)) = √(σ²) = σ

Standard deviation of Z (σZ) = √(Var(Z))

= √(Var(αU + V * √(1 - α²)))

= √(α²Var(U) + Var(V * √(1 - α²))) (by independence)

= √(α²σ² + Var(V * √(1 - α²))) (since Var(U) = σ²)

= √(α²σ² + (1 - α²)Var(V)) (since Var(V * c) = c²Var(V), where c is a constant)

= √(α²σ² + (1 - α²)σ²) (since V has variance σ²)

= σ * √(α² + 1 - α²)

= σ * √(1)

Therefore, the standard deviation of Z, σZ, is equal to σ.

Now, we can calculate the correlation coefficient ρUZ:

ρUZ = Cov(U, Z) / (σU * σZ)

= ασ² / (σ * σ)

= α

Hence, the correlation coefficient between U and Z, ρUZ, is equal to α.

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Answer:

-4+1=-3

Step-by-step explanation:

If you were to use a number line=

-4     -3     -2     -1     0     1     2     3

-------------------------------------------------

   +1

On the number line, if you move 1 up from -4, it moves to -3

The radius of the circle with point P as its center is 13.5 ft.

A circle is the locus of a point such that its distance from a fixed point is always constant (the same).

The diameter of a circle is a straight line segment that passes through the center of the circle and has endpoints on the circle.

The radius of a circle is half the diameter of the circle.

Given that the diameter of the circle is 27 ft. hence, the radius of the circle is:

Radius = Diameter / 2

Radius = 27 ft / 2 = 13.5 ft.

Hence the radius of the circle is 13.5 ft.

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