The price of sugar having been increased 40%, a family reduced its consumption of

sugar by 25%. By what percent the expenditure on sugar of the family will be increased
or decreased?
​

We have proved that (tanx + cotx)^2 = sec^2x + csc^2x. The given equation (tanx+cotx)^2=sec^2x+csc^2x can be simplified using trigonometric identities.

Expanding the left-hand side of the equation:
(tanx+cotx)^2 = tan^2x + 2tanx·cotx + cot^2x
Using the identity tanx·cotx = 1, we can simplify further:
(tanx+cotx)^2 = tan^2x + 2 + cot^2x
Next, we can use the identity tan^2x + 1 = sec^2x and cot^2x + 1 = csc^2x:
(tanx+cotx)^2 = (sec^2x - 1) + 2 + (csc^2x - 1)
Simplifying:
(tanx+cotx)^2 = sec^2x + csc^2x
Therefore, the given equation is true.
To prove that (tanx + cotx)^2 = sec^2x + csc^2x, we can follow these steps:
1. Express tanx, cotx, secx, and cscx in terms of sinx and cosx:
  - tanx = sinx/cosx
  - cotx = cosx/sinx
  - secx = 1/cosx
  - cscx = 1/sinx
2. Substitute these expressions into the given equation:
  - (sinx/cosx + cosx/sinx)^2 = (1/cosx)^2 + (1/sinx)^2
3. Find a common denominator and simplify the left side:
  - [(sin^2x + cos^2x)/(sinx*cosx)]^2
  - Since sin^2x + cos^2x = 1, this simplifies to (1/(sinx*cosx))^2
4. Simplify the right side:
  - (1/cosx)^2 + (1/sinx)^2 = (1/cos^2x) + (1/sin^2x)
5. Now, compare both sides of the equation:
  - (1/(sinx*cosx))^2 = (1/cos^2x) + (1/sin^2x)
Hence, we have proved that (tanx + cotx)^2 = sec^2x + csc^2x.

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To find the magnitude of the vector r(t),  Given the vector r(t) = <e^(2t), e^(-2t), te^(2t)>, we are asked to find several properties or operations related to this vector, such as its magnitude, the unit tangent vector, and the derivative with respect to t.

To find the magnitude of the vector r(t), we use the formula

||r(t)|| = sqrt((e^(2t))^2 + (e^(-2t))^2 + (te^(2t))^2).

Simplifying this expression, we have

||r(t)|| = sqrt(e^(4t) + e^(-4t) + t^2e^(4t)).

To find the unit tangent vector, we first differentiate r(t) with respect to t, which gives us

r'(t) = <2e^(2t), -2e^(-2t), 2te^(2t) + e^(2t)>.

Then, we divide r'(t) by its magnitude, yielding the unit tangent vector

T(t) = r'(t)/||r'(t)||.

Lastly, to find the derivative of r(t) with respect to t, we differentiate each component of r(t) separately. The derivative of r(t) is given by

r'(t) = <2e^(2t), -2e^(-2t), 2te^(2t) + e^(2t)>.

These calculations provide us with the magnitude, unit tangent vector, and derivative of the vector r(t) = <e^(2t), e^(-2t), te^(2t)> with respect to t.

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The triangles are congruent according to the SAS congruence criteria.

Two triangles are said to be congruent if pairs of their corresponding sides and their corresponding angles are equal.

Therefore, If two sides and the included angle of one triangle are equal to two sides and included angle of another triangle, then the triangles are congruent.

Therefore, the triangles are ABC and EBD are congruent by SAS principle.

They are congruent according to the SAS congruence criteria.

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1. The total cost for orders of 5 sandwiches and 7 sandwiches are $25 and $33

2. The total cost for orders of x sandwiches is 5 + 4x

3. The graph is attached

Given that

Delivery fee = $5

Charge per sandwich = $4.00

This means that

Total cost = Delivery fee + Charge per sandwich * sandwiches

So, we have

Total cost = 5 + 4 * sandwiches

For 5 sandwiches, we have

Total cost = 5 + 4 * 5 = 25

For 7 sandwiches, we have

Total cost = 5 + 4 * 7 = 33

In (1), we have

Total cost = 5 + 4 * sandwiches

For x sandwiches, we have

Total cost = 5 + 4x

This means that we plot the graph of  5 + 4x

See attachment

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So f is concave up on the interval [0, π] and concave down on the interval (π, 2π].

To determine where f is concave up or down, we need to find the second derivative of f and then analyze its sign. We start by finding the first derivative of f using the chain rule:
f'(x) = -sin(x)/(2 + cos(x))
Next, we find the second derivative of f by differentiating f'(x):
f''(x) = [-cos(x)(2+cos(x)) + sin^2(x)]/(2+cos(x))^2
To find where f is concave up or down, we need to analyze the sign of f''(x). Let's start with where f is concave up:
f''(x) > 0
[-cos(x)(2+cos(x)) + sin^2(x)]/(2+cos(x))^2 > 0
We can simplify the numerator by using the identity sin^2(x) + cos^2(x) = 1:
[1-cos^2(x)-cos(x)]/(2+cos(x))^2 > 0
(-cos^2(x)-cos(x)+1)/(2+cos(x))^2 > 0
Now we can factor the numerator:
[-(cos(x)+1)(cos(x)-1)]/(2+cos(x))^2 > 0
We have two factors: (cos(x)+1) and (cos(x)-1). We need to look at where each factor is positive or negative. Let's start with (cos(x)+1):
cos(x)+1 > 0
cos(x) > -1
This is true for all x, so (cos(x)+1) is always positive. Now let's look at (cos(x)-1):
cos(x)-1 > 0
cos(x) > 1
This is never true, so (cos(x)-1) is always negative. Therefore, the sign of f''(x) is determined by the sign of (cos(x)-1), which is negative for 0 ≤ x ≤ π and positive for π < x ≤ 2π.

So f is concave up on the interval [0, π] and concave down on the interval (π, 2π].

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The original price of the varsity jacket before the discount was $5.93.

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.

To find the original price of the varsity jacket before the discount, we can subtract the discount amount from the total price.
In this case, the discount amount is $112.72, and the total price paid is $118.65.

Therefore, the original price of the varsity jacket can be calculated as follows:

Original Price = Total Price - Discount Amount

Original Price = $118.65 - $112.72

Original Price = $5.93

Hence, the original price of the varsity jacket was $5.93.

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Answer:

Look below

Step-by-step explanation:

Well, earthquakes do really shake things up, don't they?

For the value h = -1, the vector y = [4,-1,-1] is in the plane spanned by v1 and v2.

To determine whether y is in the plane spanned by v1 and v2, we need to check if y can be expressed as a linear combination of v1 and v2.

Let's set up an equation using the vectors v1, v2, and y:

y = av1 + bv2,

where a and b are scalar coefficients to be determined.

Substituting the given values of v1, v2, and y, we have:

[4,-1,h] = a[1,2,-1] + b[-2,-1,1].

Expanding the right side of the equation, we get:

[4,-1,h] = [a,2a,-a] + [-2b,-b,b].

Combining like terms, we obtain:

[4,-1,h] = [a-2b,2a-b,-a+b].

Now, we can set up a system of equations by comparing the corresponding components:

a - 2b = 4,     (1)

2a - b = -1,    (2)

-a + b = h.     (3)

We can solve this system to find the values of a, b, and h.

From equation (2), we can solve for a in terms of b:

2a = b - 1,

a = (b - 1)/2.

Substituting this expression for a into equation (1), we have:

(b - 1)/2 - 2b = 4,

b - 1 - 4b = 8,

-3b = 9,

b = -3.

Substituting the value of b = -3 into equation (2), we find:

2a - (-3) = -1,

2a + 3 = -1,

2a = -4,

a = -2.

Finally, substituting the values of a = -2 and b = -3 into equation (3), we get:

-(-2) + (-3) = h,

2 - 3 = h,

h = -1.

Therefore, for h = -1, the vector y = [4,-1,-1] is in the plane spanned by v1 and v2.

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Based on the assumption of independence, the expected number of people in the box U.S. South and Plays Golf is 17.5.

What is number?

A number is a mathematical concept used to quantify and measure quantities. It represents a value or a magnitude in mathematics. Numbers can be used for counting, calculating, and comparing quantities.

To determine the expected number of people in the box U.S. South and Plays Golf under the assumption of independence between region and golf-playing, we need to calculate the expected value based on the row and column totals.

The expected value for a cell in a contingency table is calculated using the formula:

Expected Value = (Row Total * Column Total) / Overall Total

Let's apply this formula to the given table for the U.S. South and Plays Golf cell:

Row Total for U.S. South = 50 (from the row total)

Column Total for Plays Golf = 70 (from the column total)

Overall Total = 200 (from the total)

Expected Value = (50 * 70) / 200

Calculating this expression, we have:

Expected Value = 3500 / 200

Expected Value = 17.5

Therefore, based on the assumption of independence, the expected number of people in the box U.S. South and Plays Golf is 17.5.

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Answer:

Step-by-step explanation:

All the numbers are greater than 2 and the circle is solid so it's also equal

x≥2

The next number in the series is 125.

The given series appears to follow a pattern of dividing each number by 2 and then subtracting 1 to obtain the next number. Let's analyze the pattern step by step:

Starting with 8, when we divide it by 2, we get 4. Then subtracting 1 from 4 gives us 3, which is not part of the given series. However, if we continue the pattern, dividing 4 by 2 gives us 2, and subtracting 1 from 2 gives us 1, which matches the next number in the series.

Moving forward, if we divide 2 by 2, we get 1, and subtracting 1 from 1 gives us 0. However, 0 is not part of the given series. If we continue the pattern, dividing 1 by 2 gives us 0.5, and subtracting 1 from 0.5 gives us -0.5. Again, -0.5 is not part of the series.

Therefore, it seems reasonable to consider that the pattern is only valid when the result of the division is a whole number. Following this pattern, if we divide 1 by 2, we obtain 0.5, and subtracting 1 from 0.5 gives us -0.5. However, if we divide 0.5 by 2, we get 0.25, and subtracting 1 from 0.25 gives us -0.75. Therefore, we conclude that the pattern breaks down after reaching 1.

Taking into account the valid results of the pattern, the next number in the series would be obtained by dividing 1 by 2, which gives us 0.5, and then subtracting 1 from 0.5, resulting in -0.5. However, since negative numbers are not part of the given series, we disregard this result.

Considering that the pattern breaks down after reaching 1, we cannot determine the next number based on the given pattern. Therefore, the given series does not follow a consistent pattern, making it impossible to identify the next number accurately.

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Answer: 12:00 am

Step-by-step explanation: 11:35 is night time, after the hour passes, it becomes morning 25+35= 60

12:00 am

a. The matrix A is in upper triangular form.

b. The determinant of matrix A is 50.

Determinants are regarded as a matrix scaling factor. They can be viewed as results of the matrices' expansion and contraction.

(a) To reduce matrix A to an upper triangular matrix using elementary row operations, we perform the following steps:

1. Swap rows R1 and R2:

A = [tex][\begin{array}{ccc}2&1&1&1\\2&-2&2&3\\0&-3&8&3\\1&2&-5&3\end{array}\right][/tex]

=   [tex][\begin{array}{ccc}2&-2&2&3\\2&1&1&1\\0&-3&8&3\\1&2&-5&3\end{array}\right][/tex]

2. Multiply R2 by 1/2:

[tex]A = [\begin{array}{ccc}2&-2&2&3\\1&1/2&1/2&1/2\\0&-3&8&3\\1&2&-5&3\end{array}\right][/tex]

3. Replace R3 with R3 - (-3)*R1:

[tex]A = [\begin{array}{ccc}2&-2&2&3\\1&1/2&1/2&1/2\\0&0&2&-6\\1&2&-5&3\end{array}\right][/tex]

4. Replace R4 with R4 - (1)*R1:

[tex]A = [\begin{array}{ccc}2&-2&2&3\\1&1/2&1/2&1/2\\0&0&2&-6\\0&4&-7&0\end{array}\right][/tex]

5. Multiply R2 by 2:

[tex]A = [\begin{array}{ccc}2&-2&2&3\\2&1&1&1\\0&0&2&-6\\0&4&-7&0\end{array}\right][/tex]

6. Replace R4 with R4 - (2)*R2:

[tex]A = [\begin{array}{ccc}2&-2&2&3\\2&1&1&1\\0&0&2&-6\\0&0&-9&-2\end{array}\right][/tex]

7. Multiply R3 by 1/2:

[tex]A = [\begin{array}{ccc}2&-2&2&3\\2&1&1&1\\0&0&1&-3\\0&0&-9&-2\end{array}\right][/tex]

8. Replace R4 with R4 - (-9)*R3:

A = [tex][\begin{array}{ccc}2&-2&2&3\\2&1&1&1\\0&0&1&-3\\0&0&0&25\end{array}\right][/tex]

Now, matrix A is in upper triangular form.

(b) To find the determinant of matrix A, we multiply the diagonal entries of the upper triangular form:

det(A) = 2 * 1 * 1 * 25 = 50

Therefore, the determinant of matrix A is 50.

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The given expression consists of several mathematical series representing different functions, such as exponential, sine, cosine, and arctangent functions.

For f(x) = 5e^x + e^(5x), e^x can be represented as an infinite series using the power series expansion: e^x = 1 + x + (x^2)/2! + (x^3)/3! + ...
Similarly, sine and cosine functions have their respective Maclaurin series expansions:
sin(x) = x - (x^3)/3! + (x^5)/5! - (x^7)/7! + ...
cos(x) = 1 - (x^2)/2! + (x^4)/4! - (x^6)/6! + ...
The arctangent function can be represented as an alternating series:
tan^(-1)(x) = x - x^3/3 + x^5/5 - x^7/7 + ...
Lastly, the binomial expansion of (1+x)^k has its own series representation:
(1+x)^k = 1 + kx + k(k-1)(x^2)/2! + k(k-1)(k-2)(x^3)/3! + ...
The expressions with R=[infinity] signify that these series are convergent for all real numbers x within their radius of convergence.

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The simplified expressions and their products can be written as follows:

A. 2/36 = 1/18

B. 20/40 = 1/2

C. 15/ 56 = 15/56

D. 6/36 = 1/6

To simplify the expression, we can start by multiplying the numerators and the denominators and then dividing by the greatest common factor.

In the first question, the numerators can be multiplied to give us 2 while the denominators can be multiplied to give us 36. On dividing these two, we will have 1/18. So, the simplified form of this expression is 1/18.

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The function given in accordance with the given specifications is; y = 7.2 cos[(2π)/5.5 (x -1/8π)) + 4.8

To determine the period of this function, we can find the length of the interval on which the graph completes one full cycle.

We can see from the graph from that the function completes one full cycle on the interval [0,1].

Period: π

Phase Shift: -1/8π

Vertical Shift: 4.8

The general equation for cos wave

y = A cos (ωx + Ф) + k

Now, we know;

ω = 2π/T

Ф = -1/8π

k = 4.8

y = 7.2 cos[(2π)/5.5 (x -1/8π)) + 4.8

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The new rankings that were published on the 15th of this month indicated that Mitchel was now the 120th top American amateur golfer.

To determine the correct expression for numbers in the given sentence, we need to follow standard conventions for writing numbers. In this case, the correct expression is "the 120th" as it represents the ordinal number for ranking.

In English, ordinal numbers are typically written using numerical digits followed by the appropriate suffix (e.g., 1st, 2nd, 3rd, 4th). Therefore, the expression "120th" is the correct representation of the number in ordinal form.

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To find the value of f'(2), we need to compute the derivative of the function[tex]f(x) = x^2 + 3x + 1[/tex] and evaluate it at x = 2.  Among the given options, one of them does not allow us to find the value of f'(2) for the function[tex]f(x) = x^2 + 3x + 1.\\[/tex]

To find the value of f'(2), we need to compute the derivative of the function [tex]f(x) = x^2 + 3x + 1[/tex]and evaluate it at x = 2. The derivative of f(x) with respect to x gives us the rate of change of the function at any given point.

The options that do not find the value of f'(2) are those that involve integrating the function or evaluating the function at a different point, rather than differentiating it. Integration involves finding the area under the curve, while evaluating the function at a different point provides the value of the function at that specific point.

To find f'(2) for the given function, we need to differentiate f(x) with respect to x. The derivative of[tex]x^2 + 3x + 1[/tex]is 2x + 3. Evaluating this derivative at x = 2 gives us f'(2) = 2(2) + 3 = 7.

f'(2)=7

Therefore, the option that does not find the value of f'(2) is the one that involves integration or evaluating the function at a different point, rather than differentiating it and evaluating it at x = 2.

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The value of x from each diagram is;

(6x - 24)° + (2x + 12)° = 180°

The sum of angle on a straight line is 180°

6x - 24 + 2x + 12 = 180

8x - 12= 180

8x = 180 - 12

8x = 168

x = 168/8

x = 21

(2x + 17)° = (3x - 4)°

Alternate angles are equal

2x + 17 = 3x - 4

2x - 3x = -4 - 17

-x = -21

x = 21

3x° + (x - 20)° = 180°

Alternate interior angles are equal to 180°

3x + x - 20 = 180

4x - 20 = 180

4x = 180 + 20

4x = 200

x = 40

(x + 24)° + 2x° = 180°

Alternate interior angles are equal to 180°

x + 24 + 2x = 180

3x + 24 = 180

3x = 180 - 24

3x = 156

x = 52

(2x - 10)° = (x + 40)°

Vertical opposite angles are equal

2x - 10 = x + 40

2x - x = 40 + 10

x = 30

(3x - 10)° + (2x + 30)° = 180°

The sum of angle on a straight line is 180°

3x - 10 + 2x + 30 = 180.

5x + 20 = 180

5x = 180 - 20

5x = 160

x = 160/5

x = 32

(x - 15)° + 2x° = 90°

Complementary angles

x - 15 + 2x = 90

3x - 15 = 90

3x = 90 + 15

3x = 105

x = 105 /3

x = 35

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Answer:

£735,000

Step-by-step explanation:

we can keep the first three numbers and then round the rest (284). since the first number (2) is less than 5, we round to 0. thus, 284 -> 000 and 735,284 -> 735,000. note that the first 3 numbers do not include a zero, so we don't have to do anything fancy here.

The ordered pair that could be the image point is (a) (2.5,1)

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

Point C = (5, 2)

This means that

C = (5, 2)

The point is dilated using the origin as a center of the origin

So, the image point is

C = k(5, 2)

Let k = 1/2

So, we have

C = 1/2(5, 2)

Evaluate

C' = (2.5, 1)

Hence, the ordered pair that could be the image point is (a) (2.5,1)

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Question

Triangle ABC is dilated with the origin as the center of the dilation. Which ordered pair could represent the image of point C(5, 2) after the dilation?

A (2.5,1)

B (5,−2)

C (7.5,4.5)

D (−1,−4)

Answer:

10/9

Step-by-step explanation:

10/9

Just flip-flop the numerator and denominator

There are 120 ways to arrange the six books on a shelf if one of the books is a dictionary and it must be on one of the ends.

To find the number of ways the books can be arranged, we first consider the position of the dictionary.

Since the dictionary must be on one of the ends, it has two possible positions: the left end or the right end.

Once we fix the position of the dictionary, we have five remaining books to arrange on the shelf.

These five books can be arranged in 5! (5 factorial) ways, which means there are 5 × 4 × 3 × 2 × 1 = 120 possible arrangements.

Since the position of the dictionary was fixed at the beginning, each of the 120 arrangements corresponds to a unique arrangement of all six books on the shelf with the dictionary on one of the ends.

Therefore, the total number of ways to arrange the six books on the shelf, given that one of the books is a dictionary and it must be on one of the ends, is 120.

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The size of a sample designed for dual-purpose testing should be determined based on several factors, including the specific objectives of the testing, the desired level of confidence, the acceptable margin of error, and the variability of the population being tested.

Dual-purpose testing typically involves testing for multiple purposes or criteria simultaneously, such as both quality control and compliance assessment.

In such cases, it is important to consider the requirements and constraints of each purpose and determine an appropriate sample size that satisfies both objectives.

To determine the sample size, statistical techniques can be used, such as power analysis or sample size calculation formulas.

These methods take into account the significance level, desired power, effect size, and variability of the variables under consideration. The specific formulas or approaches used may vary depending on the nature of the data and the statistical test being performed.

It is essential to strike a balance between the sample size and the resources available for testing.

A larger sample size generally provides more precise results but may require more time, effort, and resources. Conversely, a smaller sample size may be more practical but could result in less precise estimates.

Ultimately, the appropriate sample size for dual-purpose testing should be determined through careful consideration of the specific objectives, statistical requirements, available resources, and the trade-off between precision and practicality.

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The set of arrows that best represents the change in momentum for balls A and B is the set of arrows labeled "C". This is because the arrows are pointing in opposite directions, indicating that the momentum of ball A is decreasing while the momentum of ball B is increasing.

This is consistent with the principle of conservation of momentum, which states that the total momentum of a closed system remains constant unless acted upon by an external force.

B) The arrow that indicates the direction of the impulse applied to ball A by ball B is the arrow labeled "D". This is because the impulse is defined as the change in momentum, and the change in momentum of ball A is in the direction of the arrow labeled "D". This means that ball B exerted an impulse on ball A in the direction of the arrow labeled "D" during the collision. The magnitude of the impulse is equal to the change in momentum of ball A, which is equal to the area under the curve of the force-time graph during the collision.

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The required image of the dog would be 26.6 cm tall on the screen.

To find the height of the image of the dog on the screen, we need to calculate 140% of its original height when printed on a sheet of paper.

Let's start by converting 140% to decimal form:

140% = 140/100 = 1.4

Next, we multiply the original height of the image (19 cm) by 1.4 to find the height on the screen:

19 cm * 1.4 = 26.6 cm

Therefore, the image of the dog would be 26.6 cm tall on the screen.

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Answer:

Step-by-step explanation:

Step-by-step explanation:

The solution to the system is the point that is on BOTH lines....this is the point where they intersect (cross) each other

for this system, from the graph, this is point ( 3,2)  

2 + 1 = 3

-7 - (-4) can be written as -7 + 4 = -3

-1 - (-7) can be written as -1 + 7 = 6

Answer: The addition problems and their simplified expressions are:

2 + 1 = 3

-7 - (-4) = -7 + 4 = -3

-1 - (-7) = -1 + 7 = 6