5. A map is drawn to a scale of 1: 20 000. (a) The perimeter of a lake is 2.5 km. Find, in cm, the perimeter of the lake on the map. (b) The area of the lake on the map is 12.5 cm². Find, in km², the actual area of the lake.​

Using properties of square,

a. Equation for total area = 2x² - 16x + 64

b. Area will be minimum for the value of x = 0

c. Minimum area will be 64cm².

With four equal sides, a square is a quadrilateral. There are numerous square-shaped objects in our immediate environment. The equal sides and 90° internal angles of each square form serve as indicators of its identity.

Total length of wire = 32cm.

Now side of one square is given as x.

Perimeter of big square will be = 4x.

Now perimeter of small square = 32- 4x.

Side of small square = (32-4x)/4

= 4(8-x)/4

= 8-x

Area of big square = x × x

= x².

Area of small square = (8-x) ².

Now Total area, A(x) = x² + (8-x) ²

= x² + 64 + x² - 16x

= 2x² - 16x + 64

The side length that will minimize the area will be, x = 0.

Minimum area of the total figure will be = 64cm²

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Answer: 175

Step-by-step explanation:

Taking the square root of both sides of the inequality, we get:

||U + V|| ≤ ||U|| + ||V||

The triangle inequality theorem states that, the sum of the length of two sides must be greater than the length of the third side of that triangle.

To prove it, we start by squaring both sides of the inequality:

||U + V||² ≤ (||U|| + ||V||)²

Expanding the right-hand side of the inequality, we get:

||U + V||² ≤ ||U||² + 2||U|| ||V|| + ||V||²

Now, let's use the fact that ||U + V||² = (U + V) · (U + V), where · denotes the dot product:

(U + V) · (U + V) ≤ ||U||² + 2||U|| ||V|| + ||V||²

Now, we have:

||U||² + 2U · V + ||V||² ≤ ||U||² + 2||U|| ||V|| + ||V||²

Therefore, we have:

||U + V||² ≤ ||U||² + 2||U|| ||V|| + ||V||²

≤ ||U||² + 2U · V + ||V||²

= (||U|| + ||V||)²

Taking the square root of both sides of the inequality, we get:

||U + V|| ≤ ||U|| + ||V||

This is exactly what we wanted to prove. Therefore, we have shown that ||U + V|| ≤ ||U|| + ||V||.

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The linear equatiοn has infinite sοlutiοns.

When twο expressiοns are jοined by the equal sign ('='), a mathematical statement is created. In οrder tο be determined, there must be at least οne unknοwn variable. An equatiοn is sοmething like 3x - 8 = 16. This equatiοn can be sοlved tο yield x = 8.Any οne variable linear equatiοns can have zerο, οne οr infinite sοlutiοns.

Emma substitutes 3 fοr x in a οne-variable linear equatiοn and finds that it makes the equatiοn true. She then substitutes 5 fοr x in the same linear equatiοn and finds that 5 alsο makes the equatiοn true.

If Emma tested twο values 3 and 5 fοr the same οne variable linear equatiοn and they were bοth true sοlutiοns, this means that the equatiοn has mοre than οne sοlutiοn.

If a οne variable linear equatiοn has mοre than οne sοlutiοn, than the equatiοn has infinite sοlutiοns.

Hence, the linear equatiοn Emma is talking abοut has infinite number οf sοlutiοns.

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The measure of angle m<BAC = 110 and the measure of angle m<ABC = 35.

The angle sum property of the triangle state that the sum of interior angles of triangles is 180.

<CDB = 55 (Given)

By degree measure theorem

<CAB = 2 <CDB

<CAB = 2 * 55 = 110

Again AC = AB (Radius of the circle)

Then

<ACB = <ABC (angles opposite to equal sides)

let <ACB = m

In triangle ACB,

By angle sum property: <ACB + <ABC + <CAB =180

m + m + 100 = 180

2m = 180 - 110

2m = 70

m = 70/2 = 35

<ACB = <ABC = 35

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Answer:

JK is a line segment

Step-by-step explanation:

Answer:A

Step-by-step explanation: i have had this question before

Answer:

A

Step-by-step explanation:

A proportion is given:

[tex] \frac{4}{5} = \frac{ {x}^{2} }{20} [/tex]

Use the property of proportion to find x (cross-multiply):

[tex]5 \times {x}^{2} = 20 \times 4[/tex]

[tex]5 {x}^{2} = 80[/tex]

Divide both sides of the equation by 5 to make x^2 the subject:

[tex] {x}^{2} = 16[/tex]

[tex]x > 0[/tex]

[tex]x = \sqrt{16} = 4[/tex]

The length of Z is 5.65 which is the sum of 3+2.65=5.65

How to find the length?

To find the length of the perpendicular, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs (base and perpendicular).

So, if the hypotenuse is 4 and the base is 3, we can find the length of the perpendicular (P) as follows:

4² = 3² + P²

16 = 9 + P²

P² = 16 - 9

P²= 7

P = sqrt(7)

Therefore, the length of the perpendicular is sqrt(7), which is approximately 2.65 units (rounded to two decimal places).

To find the length of the hypotenuse, we can again use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs (base and perpendicular).

So, if the base is 2.65 and the perpendicular is also 2.65, we can find the length of the hypotenuse (H) as follows:

H²= 2.65² + 2.65²

H² = 7.0225 + 7.0225

H²= 14.045

H = sqrt(14.045)

Therefore, the length of the hypotenuse is sqrt(14.045), which is approximately 3.75 units (rounded to two decimal places).

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The vector v can be represented as (-2, 4) and has a magnitude of 2√5.

A vector is a quantity that not only describes the magnitude but also describes the movement of an object or the position of an object with respect to another point or object. It is also known as Euclidean vector, geometric vector or spatial vector.

To plot the vector v with an initial point at (0, -1) and a terminal point at (-2, 3), we can draw a line segment from the initial point to the terminal point. The vector v is represented by the direction and magnitude of this line segment.

To analyze the vector v, we can calculate its components and magnitude:

Components: The components of v are the differences between the x-coordinates and y-coordinates of the terminal point and initial point, respectively. We have:

v = (-2 - 0, 3 - (-1)) = (-2, 4)

Magnitude: The magnitude of v is the length of the line segment connecting the initial point and terminal point. We can use the Pythagorean theorem to find the magnitude:

||v|| = √(-2-0)² + (3 - (-1))²) = √20 = 2√5

Therefore, the vector v can be represented as (-2, 4) and has a magnitude of 2√5.

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In this case, Bridget's commission rate is 5%, which can also be written as   [tex]0.05[/tex] (since 5% is equivalent to 5/100 or 0.05 as a decimal).  Bridget earned a commission of $ [tex]16[/tex] on the sale of the television.

To calculate the commission that Bridget earned, we can use the formula:

commission [tex]= (percentage/100) \times sale price[/tex]

where the sale price is the price of the television that Bridget sold, and the percentage is the commission rate, which is 5%.

Using this formula, we can calculate the commission as:

commission[tex]= (5/100) \times $320 = $16[/tex]

Therefore, Bridget earned a commission of $  [tex]16[/tex] on the sale of the television.

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Answer: 1second

Step-by-step explanation:

Option C) 0 <= x <= 12 and 16 <= y <= 48 would be reasonable constraints for the context.

What is the inequality equation?

An inequality equation is a mathematical statement that compares two expressions using an inequality symbol such as < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to).

The x-axis represents the number of hours since 9 AM, which can range from 0 to 12 since the shift ends at 9 PM.

The y-axis represents the number of patients, which can realistically range from 16 to 48 as hospitals can have varying levels of patient traffic.

The constraint includes the endpoints of the range, which makes it more accurate.

Hence, Option C) 0 <= x <= 12 and 16 <= y <= 48 would be reasonable constraints for the context.

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Answer:

30º

Step-by-step explanation:

LP-MO=0.5(∠LNP)

112º-MO=2(41º)

MO = 30º

Answer:

Step-by-step explanation:

a. He did not distribute the negative sign, which resulted in changing the signs of the terms inside the parentheses.

b. The correct simplification of the expression is 33-20x.

What is algebra?

Algebra is a branch of mathematics that deals with mathematical operations and symbols used to represent numbers and quantities in equations and formulas. It involves the study of variables, expressions, equations, and functions.

a. Victor made an error in the second step where he distributed only the coefficient 4 to both terms inside the parentheses. However, he did not distribute the negative sign, which resulted in changing the signs of the terms inside the parentheses.

b. To correct Victor's work, we need to distribute both the coefficient 4 and the negative sign to all the terms inside the parentheses, which gives us:

9 - 4(5x - 6)

9 - 20x + 24 (distribute)

33 - 20x (combine like terms)

Therefore, the correct simplification of the expression is 33-20x.

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[tex]Step-by-step-explanation-& Answer[/tex]

Well if our friend Rick continues the pattern that is shown in the question then the answer is at the bottom

Lets do some addition:

(6+3) * 5 ( 5 x 2 = 10 )

First find the parenthesis, 6 + 3 = 9

Next multiply, 9 x 5 = 45

Thus the answer to your problem is:

[tex]\left[\begin{array}{ccc}A&N&S\\=&=&=\\4&5&!\end{array}\right][/tex]

The total area of the trapezoid is approximately 273.664 square yards. The Option A is correct.

The area of a trapezoid is found using the formula, A = ½ (a + b) h, where 'a' and 'b' are the bases (parallel sides) and 'h' is the height (the perpendicular distance between the bases) of the trapezoid.

To find the area of the figure, we need to divide it into two triangles and find the area of each triangle.

The area of the first triangle (with base 21.3 yards and height 12.8 yards) is:

(1/2) * base * height = (1/2) * 21.3 * 12.8 = 136.704 square yards.

The area of the second triangle (with base 6.4 yards and height 12.8 yards) is:

(1/2) * base * height = (1/2) * 6.4 * 12.8 = 41.216 square yards.

The area of the third triangle (with base 14.9 yards and height 12.8 yards) is:

(1/2) * base * height = (1/2) * 14.9 * 12.8 = 95.744 square yards.

Now, the total area of the figure is the sum of the areas of these three triangles:

= 136.704 + 41.216 + 95.744

= 273.664 square yards.

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Answer:

15 cm (shorter side)

50 cm (longer side)

Step-by-step explanation:

Let the length of the shorter side of the parallelogram be x cm. Then, the length of the longer side is (2x + 20) cm. Since a parallelogram has opposite sides equal in length, there will be two shorter sides and two longer sides.

The perimeter of the parallelogram is given as 130 cm. The sum of the lengths of all four sides is equal to the perimeter. Therefore, we can set up the following equation:

2x + 2(2x + 20) = 130

Now, we can solve for x:

2x + 4x + 40 = 130

6x + 40 = 130

6x = 130 - 40

6x = 90

x = 90/6

x = 15

Now that we have found the length of the shorter side (x), we can find the length of the longer side:

Longer side = 2x + 20 = 2(15) + 20 = 30 + 20 = 50

So, the dimensions of the parallelogram are 15 cm (shorter side) and 50 cm (longer side).

The dimensions of the parallelogram are 15 cm x 50 cm.

A parallelogram is a quadrilateral with two pairs of parallel sides. The opposite sides of a parallelogram are equal in length, and the opposite angles are equal in measure.

Given that, in a parallelogram, the longer side is 20 cm longer than twice the shorter side. the perimeter is 130 cm.

We need to find the dimension of the parallelogram,

Let the length of the shorter side of the parallelogram be x cm.

Then, the length of the longer side is (2x + 20) cm.

The perimeter of the parallelogram is given as 130 cm.

Therefore, we can set up the following equation:

2x + 2(2x + 20) = 130

Now, we can solve for x:

2x + 4x + 40 = 130

6x + 40 = 130

6x = 130 - 40

6x = 90

x = 90/6

x = 15

Longer side = 2x + 20 = 2(15) + 20 = 30 + 20 = 50

Hence, the dimensions of the parallelogram are 15 cm x 50 cm.

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So, the dimensions that minimize the production costs are approximately:

Radius (r): 3.824 cm

Height (h): 7.610 cm

To minimize the production cost of the cup-of-soup package, we need to find the dimensions of the cylindrical container that minimize the surface area, as the cost is directly proportional to the surface area of the materials used.

Let r be the radius of the base and h be the height of the cylinder. The volume V of the cylinder is given by:

V = πr^2h

We know that the volume is 350 cubic centimeters:

350 = πr^2h

Now, let's express the height h in terms of the radius r:

h = 350 / (πr^2)

The surface area A of the cylinder consists of the area of the sides, the bottom, and the top:

A = (side area) + (bottom area) + (top area)

The side area of the cylinder is given by 2πrh, the bottom area is given by πr^2, and the top area is given by πr^2. So, the surface area A can be expressed as:

A = 2πrh + πr^2 + πr^2

Now, let's substitute h from the previous equation to express A in terms of r only:

A = 2πr(350 / (πr^2)) + πr^2 + πr^2

A = 700/r + 2πr^2

Next, we'll find the critical points of the function A(r) to find the minimum production cost. To do this, we'll differentiate A(r) with respect to r and set the derivative equal to 0.

dA/dr = -700/r^2 + 4πr

Now, set dA/dr = 0 and solve for r:

0 = -700/r^2 + 4πr

Rearrange the equation to isolate r:

700/r^2 = 4πr

Now, multiply both sides by r^2:

700 = 4πr^3

Divide both sides by 4π:

r^3 = 700 / (4π)

Now, find the cube root of both sides:

r = (700 / (4π))^(1/3)

Now, we can find the height h using the equation we derived earlier:

h = 350 / (πr^2)

Plug in the value of r:

h = 350 / (π(700 / (4π))^(2/3))

These are the dimensions of the cylinder that minimize the production costs: radius r and height h.

To get the final answers for the radius r and height h, we need to calculate the numerical values:

r = (700 / (4π))^(1/3)

r ≈ (700 / (4 × 3.14159265))^(1/3)

r ≈ (700 / 12.56637061)^(1/3)

r ≈ 55.75840735^(1/3)

r ≈ 3.824

Now, let's calculate the height h:

h = 350 / (π(700 / (4π))^(2/3))

h = 350 / (π(3.824^2))

h ≈ 350 / (π × 14.623)

h ≈ 350 / 45.978

h ≈ 7.610

So, the dimensions that minimize the production costs are approximately:

Radius (r): 3.824 cm

Height (h): 7.610 cm

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Rolle’s Theorem can be applied to the closed interval.

What is the Rolle’s Theorem?

In essence, Rolle's theorem or Rolle's lemma argues that there must be at least one stationary point—a position where the first derivative is zero—between any two separate sites where a real-valued differentiable function achieves equal values. The theorem bears Michel Rolle's name.

Here, we have

Given: f(x) = -x² + 3x, [0,3]

We must determine whether Rolle’s Theorem can be applied to the closed interval.

f(x) is continuous in the closed interval  [0,3].

f(x) is differentiable in the open interval (0,3)

f(0) = 0 + 0 = 0

Rolle's theorem can be applied.

f'(x) = -2x + 3

f'(x) = 0

-2x + 3 = 0

3 = 2x

x = 2/3

2/3 lies inside the [0,3].

Hence, Rolle’s Theorem can be applied to the closed interval.

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Option D is equivalent to 30 - 29m, which is equivalent to option B (8 / m) only if m is not equal to zero.

What is the equivalent expression?

Equivalent expressions are expressions that perform the same function despite their appearance. If two algebraic expressions are equivalent, they have the same value when we use the same variable value.

In the given options, option A and B are not equivalent, option C is not equivalent to any other option, and option D is equivalent to option B.

Option A can be written as 5m + 35 using the distributive property of multiplication over addition.

Option B can be simplified as follows:

(15 - 7) / m = 8 / m

Therefore, option B is equivalent to 8 / m.

Option C cannot be simplified to any other expression in the given options.

Option D can be simplified as follows:

30 - (m * 29) = 30 - 29m

Therefore, option D is equivalent to 30 - 29m, which is equivalent to option B (8 / m) only if m is not equal to zero.

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Step-by-step explanation:

so you can do in this way , hope it will give you clue

Answer: Length = 28 m , Breadth = 12 m

Step-by-step explanation:

(p.s look at the document before looking at this explanation so that you understand.)

So,

lets keep the breadth of the field as x,

Breadth = x

Length = 2x + 4

Perimeter = 2 breadths + 2 lengths

          80  = 2 ( x ) + 2( 2x + 4 )

          80  = 2x + 4x + 8

          80  = 6x + 8

          6x  = 80 - 8

                = 72

           x = 72/ 6

              = 12 ( breadth )

     Length = 2 (12) + 4

                  = 28

Checking

28m + 28m + 12m + 12m = 80

Answer:

Step-by-step explanation:

What is a rectangular prism?

A right rectangular prism is a box-shaped object, that is, a 3-dimensional solid which has six rectangular faces. Rectangular prisms can also be oblique - leaning to one side - but the side faces are parallelograms, not rectangles. A right rectangular prism is also called a cuboid, box, or rectangular hexahedron. Moreover, "rectangular prism" and "right rectangular prism" are often used interchangeably.

The most common math problems related to this solid are of the type right rectangular prism calc find V or find A, where the letters stand for the Volume and Area, respectively. Let's see the necessary rectangular prism formula and learn how to solve those problems quickly and easily.

How do I find the volume of a rectangular prism?

The rectangular prism volume formula is:

volume = h × w × l,

where h is prism height, w is its width, and l is its length. To calculate the volume of a cardboard box:

Find the box length. For example, it can be equal to 18 in.

Determine its width. Let's say you measured 12 in.

Find out the rectangular prism height. Assume it's 15 in.

Calculate the cuboid volume. Using the rectangular prism volume formula above, we get volume = (18 × 12 × 15) in = 3240 in³.

How do I find the area of a rectangular prism?

The surface area of the cuboid consists of 6 faces - three pairs of parallel rectangles. To find the rectangular prism surface area, add the areas of all faces:

surface_area = 2 × (h × w) + 2 × (h × l) + 2 × (l × w) = 2 × (h × w + h × l + l × w),

where h is prism height, w is its width, and l is its length.

Let's see an example of how to solve the right rectangular prism calc - find A problem. We'll come back to our example with the box and calculate its surface area:

Calculate the rectangular prism surface area. First rectangle area is 15in × 12in = 180in², second 15in × 18in = 270in² and third one 18in × 12in = 216in². Add all three rectangles' areas - it's equal to 666 in² (what a number!) - and finally multiply by 2. The surface area of our cardboard box is 1332in².

Or save yourself some time and use our rectangular prism calculator.

Finally, let's attack the right rectangular prism calc find d (that is, the diagonal) type of problem.

How do I calculate the diagonal of a rectangular prism?

To determine the diagonal of a rectangular prism, apply the formula:

diagonal = √(l² + h² + w²)

where h is prism height, w is its width, and l is its length.

Do you have the feeling that you saw the formula before? Yes, that's possible because this equation resembles the famous one from the Pythagorean theorem.

the roots of the polynomial [tex]x^{5}[/tex] - [tex]7x^{3}[/tex] + 10x are 0, ±√5, and ±√2. Thus, the other factors of f(x) are (x+√5), (x-√5), (x+√2), and (x-√2).

What is polynomial?

A polynomial is a mathematical expression that consists of variables and coefficients, which are combined using arithmetic operations such as addition, subtraction, multiplication, and non-negative integer exponents.

If (x+1) is a factor of f(x), then f(-1) = 0. Thus, we can substitute -1 for x in the expression for f(x) and solve for k:

f(-1) = [tex](-1)^{6}[/tex] - 6[tex](-1)^{4}[/tex] + 17[tex](-1)^{2}[/tex] + k = 1 - 6 + 17 + k = 12 + k

Since f(-1) = 0, we have:

12 + k = 0

Solving for k, we find:

k = -12

So, when k = -12, (x+1) is a factor of f(x).

To find another factor of f(x), we can divide f(x) by (x+1) using polynomial long division or synthetic division. The result is:

f(x) = (x+1)([tex]x^{5}[/tex] - [tex]7x^{3}[/tex] + 10x)

So, another factor of f(x) is (x+a), where a is a root of the polynomial x^5 - 7x^3 + 10x. We can find the roots of this polynomial using a numerical method or by factoring it using the rational root theorem. By inspection, we can see that x=0 is a root, so we can factor out x:

x([tex]x^{4}[/tex] - [tex]7x^{2}[/tex] + 10) = 0

The quadratic factor can be factored further as:

x([tex]x^{2}[/tex] - 5)([tex]x^{2}[/tex] - 2) = 0

So, the roots of the polynomial [tex]x^{5}[/tex] - [tex]7x^{3}[/tex] + 10x are 0, ±√5, and ±√2. Thus, the other factors of f(x) are (x+√5), (x-√5), (x+√2), and (x-√2).

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The probability that the fire department arrives at the scene in 8 minutes or less is approximately 0.9772 or 97.72%.

What is the standard deviation?

The standard deviation is a measure of the amount of variation or dispersion in a set of data. It measures how much the data deviates from the mean value.

We are given that the times the fire department takes to arrive at the scene of an emergency are normally distributed with a mean of 6 minutes and a standard deviation of 1 minute.

Let X be the random variable representing the time taken by the fire department to arrive at the scene of an emergency. Then, we have:

X ~ N(6, 1)

We want to find the probability that the fire department arrives at the scene in 8 minutes or less. Mathematically, we want to find:

P(X ≤ 8)

To find this probability, we can standardize the random variable X by subtracting the mean and dividing by the standard deviation:

Z = (X - μ) / σ

Substituting the values of μ and σ, we get:

Z = (X - 6) / 1

Z represents the standard normal distribution with a mean of 0 and a standard deviation of 1. Therefore, we can use a standard normal distribution table or calculator to find the probability that Z is less than or equal to the standardized value of 8:

P(Z ≤ (8 - 6) / 1) ≈ P(Z ≤ 2)

Looking up the probability of Z being less than or equal to 2 in a standard normal distribution table, we find that it is approximately 0.9772.

Therefore, the probability that the fire department arrives at the scene in 8 minutes or less is approximately 0.9772 or 97.72%.

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Answer:

whooper write an equation of the line in slope-intercept form for each of the following write an equation of the line in slope-intercept form for each of the following write an equation of the line in slope-intercept form for each of the following write an equation of the line in slope-intercept form for each of the following

Step-by-step explanation:

The Null hypothesis is that the population mean depression score of all college students is less than or equal to 38 (H₀: µ ≤ 38) while the Alternative hypothesis is that the population mean depression score of all college students is greater than 38 (Ha: µ > 38).

Null hypothesis: The population mean depression score of all college students is less than or equal to 38 (H₀: µ ≤ 38).

Alternative hypothesis: The population mean depression score of all college students is greater than 38 (Ha: µ > 38).

Step 2: Test statistic

We will use a one-sample t-test since the population standard deviation is known and the sample size is greater than 30. The test statistic is calculated as:

t = (x - µ) / (σ / √(n))

where x is the sample mean, µ is the hypothesized population mean, σ is the population standard deviation, and n is the sample size.

Plugging in the numbers, we get:

t = (40 - 38) / (16 / √(64))

t = 2

Step 3: P-value

Using a t-table with 63 degrees of freedom (df = n - 1), we find the p-value for a one-tailed test with a t-value of 2 is approximately 0.025.

Step 4: Decision

Since the p-value (0.025) is less than the alpha level (0.025), we reject the null hypothesis.

Step 5: Conclusion

There is sufficient evidence to suggest that the population mean depression score of all college students is greater than 38.

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[tex] \:\:\:\:\:\:\:\:\:\:\:\star\:\sf \underline{Circumference\: of\: circle = 2πr}\\[/tex]

As per question, the radius of a circle is 13 in and we are asked to find out it's circumference in terms of π. Here-

Now that we are given the value of radius of the circle, so we can substitute it into the formula and solve for circumference-

[tex] \:\:\:\:\:\:\longrightarrow \sf \underline{Circumference_{( circle)} = 2\pi r}\\[/tex]

[tex] \:\:\:\:\:\:\longrightarrow \sf Circumference_{( circle)} = 2 \times \pi \times 13\\[/tex]

[tex] \:\:\:\:\:\:\longrightarrow \sf \underline{Circumference_{( circle)} = 26 \pi \:inch}\\[/tex]

Answer:

2 triangles, 180 for each triangle and 360 degrees in a square

Answer:

A.998

Step-by-step explanation:

256 times 4 =1024-26=998 the facts that she uses 16 beads on each necklace is useless