Prove by cases that 25k^2 + 15k is an even integer whenever 5k- 3 is an integer. ​

Answer:

x = [tex]\frac{-57}{4}[/tex]

Step-by-step explanation:

solve for the value of x .

80(x+15)=60

80(x + 15) = 60

80x + 1200 - 60 = 0

80x + 1140 = 0

80x = -1140

x = -1140/80

x = [tex]\frac{-57}{4}[/tex]

Answer:

-14.25

Step-by-step explanation:

Given: 80 (x+15) = 60

Solution: On opening the brackets, we get

> 80x + 80 * 15 = 60

> 80x + 1200 = 60

Then, taking 1200 to the other side of the equation,

80x = 60 - 1200

Therefore, 80x = -1140

Now, dividing both sides by 80, we get:

80x/80 = -1140/80

So, x= -14.25

Hope this helps!

The expressions equivalent to 6\cdot6\cdot6\cdot6\cdot66\cdot6\cdot6\cdot6\cdot66 are 1296\cdot396\cdot2376 and 839808\cdot36.

We can use the associative property of multiplication to group the factors in different ways while preserving their product. For example, we can group the first four 6's together and then multiply by the remaining 6 and 66:

6\cdot6\cdot6\cdot6\cdot66\cdot6\cdot6\cdot6\cdot66 = (6\cdot6\cdot6\cdot6)\cdot(6\cdot66)\cdot(6\cdot6\cdot66)

= 1296\cdot396\cdot2376

Alternatively, we can group the last two 6's together and then multiply by the remaining factors:

6\cdot6\cdot6\cdot6\cdot66\cdot6\cdot6\cdot6\cdot66 = (6\cdot6\cdot6\cdot6\cdot66)\cdot(6\cdot6)

= 839808\cdot36 is the equivalent expression.

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The equation that can be used to find the total number of students would be n = 19 + 5.

It is acknowledged that Clara provided 2 bottles per each pupil joining her on the excursion. Denote, by using ‘x’, the quantity of learners present; we can then inscribe the succeeding formula:

2x = 38

By resolving this mathematical principal, the total number of students attending the event is revealed.

x = 38 / 2

x = 19

Currently, 19 individuals are confirmed to embark upon the outing, due to five individuals failing to furnish required consent forms and will be absent. Subsequently, we may deduce the quantity of pupils (designated as ‘n’) in Clara’s class:

n = 19  (students going on the trip ) + 5 ( students not going )

n = 19 + 5

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The backpack forms an angle of approximately 15 degrees with the ground.

What is Pythagorean theorem?

The Pythagorean theorem is a fundamental concept in mathematics that relates to the lengths of the sides of a right triangle.

To find the angle that Luis's backpack forms with the ground, we can use the inverse tangent function.

The height of the backpack when the handle is extended is 3.5 feet, and the distance from the ground to Luis's hand is 3 feet. So the opposite side of the triangle is 3.5 - 3 = 0.5 feet, and the adjacent side is the distance from Luis's hand to the backpack, which we can call x.

Using the tangent function, we have:

tan(theta) = opposite/adjacent

tan(theta) = 0.5/x

To solve for x, we can use the Pythagorean theorem:

x² + 3² = (3.5)²

x² = 3.5² - 3²

x² = 3.25

x = sqrt(3.25)

x ≈ 1.8 feet

Now we can substitute x into our tangent equation and solve for theta:

tan(theta) = 0.5/1.8

theta = arctan(0.5/1.8)

theta ≈ 15 degrees

Therefore, the backpack forms an angle of approximately 15 degrees with the ground.

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|4x +9|=|6-5x|

4x+5x=6-9

9x=-3

x=-3/9

x=-1/3

Answer:

x = 15

x = - ⅓

Step-by-step explanation:

|4x+9| = |6-5x|

4x+9=6-5x or 4x+9=-6+5x

Lets do the first one first

4x+9=6-5x

9x=-3

x= -3/9 = -1/3

4x+9=-6+5x

15=x

So the two solutions are x = 15 and x = -⅓

We know that the length of AB on the scale drawing is 10 inches

Using the scale of 1 inch: 250 feet, we can find the length of AB on the scale drawing by multiplying the actual length of AB by the scale factor.

The actual length of AB is the sum of the height of the tower (800 ft) and the length of the support wire (1,700 ft), which is 2,500 ft.

Multiplying 2,500 ft by the scale factor of 1 inch: 250 feet, we get:

2,500 ft ÷ 250 ft/inch = 10 inches

Therefore, the length of AB on the scale drawing is 10 inches.

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Using the Pythagorean theorem, the height of the ramp in the given diagram is 8.9 ft

From the question, we are to determine how high the ramp is

From the Pythagorean theorem which states that "in a right triangle, the square of the longest side, that is hypotenuse, equals sum of squares of the two other sides".

In the given diagram,

We have a right triangle

The measure of the hypotenuse is 21 ft

One of of the side measures 19 ft

Now, we will calculate x

By the Pythagorean theorem, we can write that

h² + 19² = 21²

h² = 21² - 19²

h² = 441 - 361

h² = 80

h = √80

h = 8.94427 ft

h ≈ 8.9 ft

Hence, the ramp is 8.9 ft high

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Step-by-step explanation:

Using the distance formula:

d = √[(x2 - x1)² + (y2 - y1)²]

where A = (x1, y1) and B = (x2, y2), we can find the distance between points A and B as follows:

d = √[(-3 - (-7))² + (-1 - (-7))²]

d = √[4² + 6²]

d = √52

d ≈ 7.2

Therefore, the distance between points A and B is approximately 7.2 units, rounded to the nearest tenth.

The product (k/3) x 12 is greater than 12 for values of k greater than 3.

To determine the values of k for which the product (k/3) x 12 is greater than 12, follow these steps:

Step 1: Set up the inequality:
(k/3) x 12 > 12

Step 2: Simplify the inequality by dividing both sides by 12:
(k/3) > 1

Step 3: Multiply both sides by 3 to solve for k:
k > 3

So, the product (k/3) x 12 is greater than 12 for values of k greater than 3.

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R1 and R2 are relations on a set A, which means they define a set of ordered pairs of elements in A. The matrices that represent R1 and R2 can be thought of as a way to visualize these ordered pairs.

Each row and column of the matrix corresponds to an element in A. If there is a 1 in the ith row and jth column of the matrix for R1, then (i,j) is an ordered pair in R1. Similarly, if there is a 1 in the ith row and jth column of the matrix for R2, then (i,j) is an ordered pair in R2.

If there is a 0 in a particular position in the matrix, then the corresponding pair is not in the relation.

Let R1 and R2 be relations on a set A. These relations can be represented by matrices M1 and M2, respectively, with dimensions |A|x|A|, where |A| is the cardinality of set A. The elements of the matrices M1 and M2 are binary, indicating whether there is a relation between the corresponding elements of set A in R1 and R2, respectively. If there is a relation, the matrix element will be 1, and if there is no relation, the matrix element will be 0.

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The minimum height of the ladder needed to change the blown outdoor lightbulb is 5 meters.

To determine the minimum height of the ladder needed to change a blown outdoor lightbulb that is 5m up, we need to consider the following terms:

1. The bulb's height (5m)

2. Your reach when on the top rung of the ladder (1m)

3. The required space off the house for the ladder's base for stability (3m)

First, subtract your reach from the bulb's height: 5m - 1m = 4m. This means the ladder needs to reach at least 4 meters up the wall.

Next, we need to use the Pythagorean theorem to find the ladder's minimum height. The theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the ladder) is equal to the sum of the squares of the lengths of the other two sides (the distance from the house and the height up the wall).

Let's denote the ladder's height as L, the distance from the house as A (3m), and the height up the wall as B (4m).

According to the Pythagorean theorem, we have:
L² = A² + B²

Substitute the values for A and B:
L² = (3m)² + (4m)²
L² = 9m² + 16m²
L² = 25m²

Now, find the square root to get the minimum height of the ladder:
L = √25m²
L = 5m

So, the minimum height of the ladder needed to change the blown outdoor lightbulb is 5 meters.

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The rectangle WXYZ after a 180° rotation about the point N(2,-3) is W(3,-2), X(0,-1), Y(2,5), and Z(5,4).

To perform a 180° rotation about the point N(2,-3), we can follow these steps:

1. Translate the rectangle and the point N to the origin by subtracting their respective coordinates from each vertex and point.

2. Perform the rotation by multiplying the coordinates of each vertex and point by the 2x2 rotation matrix:

  [cos(180°) -sin(180°)]

  [sin(180°) cos(180°)]

  which simplifies to:

  [-1  0]

  [ 0 -1]

3. Translate the rectangle and the point N back to their original positions by adding their respective coordinates to each vertex and point.

Let's apply these steps to rectangle WXYZ and point N:

1. Translate the rectangle and point N to the origin:

W' = (-3 - 2, -4 + 3) = (-5, -1)

X' = (0 - 2, -5 + 3) = (-2, -2)

Y' = (-2 - 2, -11 + 3) = (-4, -8)

Z' = (-5 - 2, -10 + 3) = (-7, -7)

N' = (2 - 2, -3 + 3) = (0, 0)

2. Perform the rotation using the matrix:

  [-1  0]

  [ 0 -1]

W'' = [-1  0] * [-5, -1] = [5, 1]

     [0 -1]

X'' = [-1  0] * [-2, -2] = [2, 2]

     [0 -1]

Y'' = [-1  0] * [-4, -8] = [4, 8]

     [0 -1]

Z'' = [-1  0] * [-7, -7] = [7, 7]

     [0 -1]

N'' = [-1  0] * [0, 0] = [0, 0]

     [0 -1]

3. Translate the rectangle and point N back to their original positions:

W = [5 - 2, 1 - 3] = (3, -2)

X = [2 - 2, 2 - 3] = (0, -1)

Y = [4 - 2, 8 - 3] = (2, 5)

Z = [7 - 2, 7 - 3] = (5, 4)

N = [0 + 2, 0 + 3] = (2, 3)

Therefore, the rectangle WXYZ after a 180° rotation about the point N(2,-3) is W(3,-2), X(0,-1), Y(2,5), and Z(5,4).

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Starting with a 1-by-1 square, a sequence of squares J1, J2, J3, ... is created by attaching squares half as wide as the previous squares to the outer edges of each successive square. The area of J∞, the limit of this sequence, is 4/3.

To find the area of J1, we simply calculate the area of the original 1-by-1 square, which is 1.

To find the area of J2, we need to attach squares half as wide (and half as tall) to the middle of each side of J1. The area of each attached square is (1/2)² = 1/4, so the total area added to J1 is 4(1/4) = 1. Thus, the area of J2 is 1 + 4(1/4) = 2.

To find the area of J3, we need to attach squares half as wide as the squares added in the previous step to every outer edge of J2. The area of each attached square is (1/4)² = 1/16, so the total area added to J2 is 4(1/16) = 1/4. Thus, the area of J3 is 2 + 4(1/4) = 3.

We can continue this process to find the areas of J4, J5, and so on. In general, the area of Jn is equal to the area of the previous square plus the area added by the attached squares, which is 4(1/2^(n-1))^2 = 1/2^(2n-2). Therefore, the area of Jn is 1 + 1/4 + 1/16 + ... + 1/4^(n-1) = (4/3)(1 - 1/4^n).

As n approaches infinity, the area of Jn approaches the limit of (4/3)(1 - 0) = 4/3. Therefore, the area of J∞, the limit of the sequence, is 4/3.

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There are approximately 0.1480 standard deviations (or 3.7 screws) to the right of the mean when there are 1009 screws in the box.  When the automatic filling device delivers 1065 screws to the box, there are approximately 2.6 standard deviations (or 65 screws) to the left of the mean.

To answer this question, we need to use the normal distribution formula.

a. To find how many units to the right of 1000 is 1009, we need to calculate the z-score:

z = (X - μ) / σ

where X = 1009, μ = 1000, and σ = 25.

z = (1009 - 1000) / 25 = 0.36

Using a standard normal distribution table or calculator, we can find that the probability of getting a z-score of 0.36 or higher is 0.3520.

To convert this probability to units to the right of the mean, we subtract it from 0.5 (which represents the area to the left of the mean):

units to the right = 0.5 - 0.3520 = 0.1480

Therefore, there are approximately 0.1480 standard deviations (or 3.7 screws) to the right of the mean when there are 1009 screws in the box.

b. To find the X value that is 2.6 units to the left of the mean, we can rearrange the formula:

X = μ - zσ

where z = -2.6 (since we want units to the left of the mean) and μ and σ are the same as before.

X = 1000 - (-2.6) * 25 = 1065

Therefore, when the automatic filling device delivers 1065 screws to the box, there are approximately 2.6 standard deviations (or 65 screws) to the left of the mean.

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Maria jumped 6 inches farther than Nate.

To see why, we can subtract Nate's jump height from Maria's jump height:

32 - 26 = 6

So Maria jumped 6 inches farther than Nate did.

To represent this problem mathematically, we can use the equation:

Maria's jump height - Nate's jump height = the difference in their jump heights

Or, using variables:

M - N = D

Where M represents Maria's jump height, N represents Nate's jump height, and D represents the difference between their jump heights. Plugging in the numbers from the problem, we get:

32 - 26 = D

Simplifying, we get:

6 = D

So D, the difference between their jump heights, is 6 inches.

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Answer:

Acute triangle

Step-by-step explanation:

angles that are less than 90° are called acute

This triangle has 3 acute angles, so it is an acute triangle

The solution to the problems are given below:

Giving each of the letters numbers based on their numerical position on the English alphabet, we can solve below:

n (14) + e (5) = 14 + 5 = 19

t (20) - j (10) = 20 - 10 = 10

d (4) x e (5) = 4 x 5 = 20

p (16) / b (2) = 16 / 2 = 8

It can be seen that with the letter e for example is the 5th letter of the alphabet and the value is used to compute the addition of the problem.

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The upper quartile of the time spent reading is 300.

We have,

The upper quartile, also known as the third quartile, is a measure of central tendency that divides a data set into four equal parts.

It represents the data point that separates the top 25% of the data from the bottom 75%.

Now,

From the box pot.

Median = 250

Lower quartile = 100

Upper quartile = 300

Thus,

The upper quartile is 300.

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Answer:
38. (A) True
39. (B) False

Step-by-step explanation:

The set of people working in the summer consists of both female an male since there is no determiner to show that the 80% of students are a specific gender. Therefore, the answer to the first question is True (A).

My expression for finding the probability of being female and working part time in summer only is:
[tex]\frac{84}{100} \\\\ \\ \\ \\[/tex][tex](\frac{1}{2} *\frac{80}{100})\\[/tex][tex]=\frac{42}{125}[/tex] which is also equal to 0.336. Therefore the second question is false.




Please forgive me if I'm wrong but I'm open to any correction or criticisms.

The company's profits for March through May were each -$797.

Let's start by adding the profits for January and February:

Profit for January + Profit for February = $4,758 + (-$3,642) = $1,116

We know that the company's profits for March through May were the same in each of these months, so let's call this common profit "X". Therefore, the total profit for these three months would be:

3 * X = 3X

Adding up the profits for all five months gives us the total profit for the year:

$1,116 + 3X = -$1,275

Subtracting $1,116 from both sides gives us:

3X = -$2,391

Dividing both sides by 3 gives us:

X = -$797

Therefore, the company's profits for March through May were each -$797.

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The figure with opposite sides are parallel and equal is parallelogram.

From the group A:

In first image opposite sides are parallel and equal.

One pair of parallel sides = 4 units and the another pair of parallel sides = 8 units

So, it is parallelogram.

In second image opposite sides are parallel and all the sides are equal.

So, it is rhombus.

In third image opposite sides are parallel and equal.

One pair of parallel sides = 3 units and the another pair of parallel sides = 2 units

So, it is parallelogram.

From the group B:

In first image opposite sides are parallel and equal.

One pair of parallel sides = 4 units and the another pair of parallel sides = 7 units and all angles measures equal to 90°.

So, it is rectangle.

In second image one pair of opposite sides are parallel.

So it is trapezium.

In third image opposite sides are parallel and all sides equal.

All angles measures equal to 90°.

So it is square.

Therefore, the figure with opposite sides are parallel and equal is parallelogram.

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Triangle PQR is not a right triangle.

To determine whether triangle PQR is a right triangle, we need to check if any of its angles is a right angle (90 degrees). We can use the slope formula to find the slopes of the sides of the triangle and check if any of the slopes are negative reciprocals (perpendicular) to each other.

Let's calculate the slopes of the sides PQ, QR, and RP:

Slope of PQ = (y₂ - y₁) / (x₂ - x₁)

= (2 - 1) / (3 - 0)

= 1 / 3

Slope of QR = (y₂ - y₁) / (x₂ - x₁)

= (-4 - 2) / (5 - 3)

= -6 / 2

= -3

Slope of RP = (y₂ - y₁) / (x₂ - x₁)

= (1 - (-4)) / (0 - 5)

= 5 / (-5)

= -1

Now, let's check if any of the slopes are negative reciprocals of each other. We can compare the products of the slopes:

Product of PQ slope and QR slope = (1/3) * (-3) = -1

Product of QR slope and RP slope = (-3) * (-1) = 3

Product of RP slope and PQ slope = (-1) * (1/3) = -1/3

Since the product of the slopes of QR and RP is not equal to -1, triangle PQR is not a right triangle.

Therefore, triangle PQR is not a right triangle.

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Answer:107

Step-by-step explanation:8*17-29=107 so our answer is 107

Answer:

The maximum is when both X and y = 1.The maximum value of the function is 3.When both X and YY are equal to 0, the minimum value is 0

The absolute maximum value is 3 and the absolute minimum value is 0.

To find the absolute maximum and minimum values of f(x,y) = x^2 + 2y^2 on the square [0,1] x [0,1], we need to consider both the interior and the boundary of the square.

First, check for critical points in the interior by finding the partial derivatives and setting them equal to zero:

fx = 2x and fy = 4y

Setting them equal to zero, we have:
2x = 0 => x = 0
4y = 0 => y = 0

The only critical point in the interior is (0,0).

Next, evaluate f(x,y) on the boundary of the square [0,1] x [0,1]. The boundary consists of four segments: x=0, x=1, y=0, and y=1.

1. x=0: f(0,y) = 2y^2 (for y in [0,1])
2. x=1: f(1,y) = 1 + 2y^2 (for y in [0,1])
3. y=0: f(x,0) = x^2 (for x in [0,1])
4. y=1: f(x,1) = x^2 + 2 (for x in [0,1])

Now, compare the values of f at the critical point and boundary points to find the absolute maximum and minimum:

Absolute minimum: f(0,0) = 0
Absolute maximum: f(1,1) = 3

So the absolute maximum value is 3 and the absolute minimum value is 0.

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The length of side AB is also 12 units.

What is a rhombus?

A rhombus is a four-sided quadrilateral with all sides of equal length. It is also known as a diamond or a lozenge. In a rhombus, opposite sides are parallel, and opposite angles are equal. Additionally, the diagonals of a rhombus bisect each other at right angles, meaning they intersect at a 90-degree angle and divide each other into two equal segments.

Since a rhombus has all sides of equal length, we know that YZ = AB. Therefore, if YZ = 12, we have:

AB = YZ = 12

So the length of side AB is also 12 units.

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