a store has clearance items that have been marked down by 25%. they are having a sale, advertising an additional 55% off clearance items. what percent of the original price do you end up paying?

Answer:                69 cm

To find the perimeter of the triangle, add together the length of all sides.

A right triangle has three sides and we know that:

When we know two sides in a right triangle, we can find the third side using the Pythagorean theorem:

a² + b² = c²

The variables 'a' and 'b' represent the base and height. The variable 'c' represents the hypotenuse, the longest side.

Since we know the base and height, substitute them into the formula and solve for the hypotenuse.

a² + b² = c²                Start with the Pythagorean theorem.

24² + 16² = c²            Substitute the base and height.

576 + 16² = c²            Solve 24².

576 + 256 = c²          Solve 16².

832 = c²                     Add.

Now, let's start to isolate 'c', which means making it alone on one side of the equal sign.

√832 = √c²               Square root both sides.

√832 = c                   Square root is the opposite of ², so it cancels out.

c ≈ 28.8...                  Keep the variable on the left side.

The question says to find the approximate perimeter, so let's round 28.8 to the nearest whole number.

The hypotenuse, 'c', is about 29 cm.

Now, we know the three sides:

Add together all three sides.

Perimeter = base + height + hypotenuse

Perimeter = 24 cm + 16 cm + 29 cm

Perimeter = 69 cm

∴ The perimeter of the right triangle is approximately 69 cm.

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To find the center of enlargement, we need to determine the scale factor of enlargement between the original quadrilateral ABCD and the image quadrilateral A'B'C'D'.

We can do this by finding the ratio of the corresponding side lengths. Let the scale factor be k. Then,

AB' = k * AB

and

AA' = k * AA'

Using the distance formula, we can find the lengths of AB and AA',

AB = sqrt((6-(-1))^2 + (3-3)^2) = sqrt(50)

AA' = sqrt((11-3)^2 + (-5-(-1))^2) = sqrt(80)

Thus, k = AB' / AB = AA' / AA' = sqrt(80 / 50) = sqrt(8 / 5) = 1.7889 (approx)

Next, we can use the formula for the center of enlargement, which states that the center of enlargement is the point of intersection of the corresponding lines joining the original and the image points.

The line joining A(3,-1) and A'(11,-5) has the equation y = -x/2 + 5/2

The line joining B(3,6) and B'(6,8) has the equation y = (2/3)x + 6

\

Solving for the point of intersection, we get:

x/2 + 5/2 = (2/3)x + 6

=> x = 21.6

Substituting x in either of the equations, we get:

y = -x/2 + 5/2

=> y = -21.6/2 + 5/2

=> y = -5.55 (approx)

Therefore, the center of enlargement is approximately (21.6, -5.55).

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To find the center of enlargement, we need to determine the scale factor of enlargement between the original quadrilateral ABCD and the image quadrilateral A'B'C'D'.

We can do this by finding the ratio of the corresponding side lengths. Let the scale factor be k. Then,

AB' = k * AB

and

AA' = k * AA'

Using the distance formula, we can find the lengths of AB and AA',

AB = sqrt((6-(-1))^2 + (3-3)^2) = sqrt(50)

AA' = sqrt((11-3)^2 + (-5-(-1))^2) = sqrt(80)

Thus, k = AB' / AB = AA' / AA' = sqrt(80 / 50) = sqrt(8 / 5) = 1.7889 (approx)

Next, we can use the formula for the center of enlargement, which states that the center of enlargement is the point of intersection of the corresponding lines joining the original and the image points.

The line joining A(3,-1) and A'(11,-5) has the equation y = -x/2 + 5/2

The line joining B(3,6) and B'(6,8) has the equation y = (2/3)x + 6

\

Solving for the point of intersection, we get:

x/2 + 5/2 = (2/3)x + 6

=> x = 21.6

Substituting x in either of the equations, we get:

y = -x/2 + 5/2

=> y = -21.6/2 + 5/2

=> y = -5.55 (approx)

Therefore, the center of enlargement is approximately (21.6, -5.55).

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The given f(x,y) is to be evaluated at 6 different points. At (0,0), the  f(x,y) is 4, at (0,1)=  -1, (2,3)= -37. When evaluated at (1,y), the  f(x,y) simplifies to f(1,y) = 3 - 4y^2,  at (x,0), the  f(x,y) simplifies to f(x,0) = 4 - x^2. Finally, when evaluated at (t,1), the  f(x,y) simplifies to f(t,1) = 3 - t^2 - 4.

Explanation:

To evaluate the function f(x,y) = 4 - x^2 - 4y^2 at point (0,0), we simply substitute x=0 and y=0 in the function. Thus, f(0,0) = 4 - 0^2 - 4(0)^2 = 4.

Similarly, to evaluate the function at point (0,1), we substitute x=0 and y=1 in the function. Thus, f(0,1) = 4 - 0^2 - 4(1)^2 = -1.

To evaluate the function at point (2,3), we substitute x=2 and y=3 in the function. Thus, f(2,3) = 4 - (2)^2 - 4(3)^2 = -37.

When we substitute x=1 in the function f(x,y), we get f(1,y) = 4 - 1^2 - 4y^2 = 3 - 4y^2. Hence, the function simplifies to f(1,y) = 3 - 4y^2 when evaluated at point (1,y).

Similarly, when we substitute y=0 in the function f(x,y), we get f(x,0) = 4 - x^2. Hence, the function simplifies to f(x,0) = 4 - x^2 when evaluated at point (x,0).

Finally, when we substitute y=1 in the function f(x,y), we get f(t,1) = 4 - t^2 - 4(1)^2 = 3 - t^2 - 4. Hence, the function simplifies to f(t,1) = 3 - t^2 - 4 when evaluated at point (t,1).

Therefore, the function values at the given points are 4, -1, -37, 3 - 4y^2, 4 - x^2, and 3 - t^2 - 4 for points (0,0), (0,1), (2,3), (1,y), (x,0), and (t,1), respectively.

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The measure of angle F is 141°.

In Mathematics and Geometry, two (2) triangles are said to be similar when the ratio of their corresponding side lengths are equal and their corresponding angles are congruent.

Additionally, the sum of all of the interior angles of a triangle is always equal to 180 degrees. In this scenario, we can logically deduce that the sum of the given angles are supplementary angles:

m∠E + m∠D + m∠F = 180°

83° + 56° + m∠F = 180°

139° + m∠F = 180°

m∠F = 180° - 139°

m∠F = 141°

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The answer is "inelastic". When demand is inelastic, changes in price do not significantly affect the quantity of goods or services demanded.

This means that increases in price result in increases in total revenue, while decreases in price result in decreases in total revenue. Inelastic demand occurs when consumers are not very responsive to changes in price and do not have good alternatives to the product or service being offered. Examples of products with inelastic demand are necessities like food, medicine, and gasoline, where consumers will continue to purchase the product regardless of price changes because they need it for daily living. On the other hand, products with elastic demand, like luxury items or non-essential goods, will see a significant decrease in demand when prices increase.

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The amount of pizza left for the other three classes is 1/2.

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.

The total number of pizzas ordered is 7.

The total number of pizzas eaten by the three classes is:

2 1/2 + 1 1/3 + 2 2/3

= (5/2) + (4/3) + (8/3)

= 15/6 + 8/6 + 16/6

= 39/6

= 6 3/6

= 6 1/2

Therefore, the amount of pizza left for the other three classes is:

7 - 6 1/2 = 1/2

So, there is half of a pizza left for the other three classes.

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Answer:

1/8

Step-by-step explanation:

2^2 / 2^5

2^2 = 2 x 2= 4

2^5 = 2 x 2 x 2 x 2 x 2 = 32

4/32 = 1/8

Note :

2 power 5 means you need to multiply 2, 5 times itself

(i.e) 2 x 2 x 2 x 2 x 2

Thus, the area of surface obtained by rotating the curve y=x−−√3y=x3 about the y-axis for 1≤y≤4 is 36π√3 square units.

To find the area of the surface obtained by rotating the curve y=x−−√3y=x3 about the y-axis for 1≤y≤4, we can use the formula:
A = 2π ∫(1 to 4) x √(1+(dy/dx)^2) dy

First, we need to find dy/dx by taking the derivative of y=x−−√3y=x3:

dy/dx = 1/(2√3x^(1/2))

Substituting this into the formula, we get:

A = 2π ∫(1 to 4) x √(1+1/(12x)) dy

Simplifying the expression under the square root, we get:

A = 2π ∫(1 to 4) x √(12x+1)/12 dy

We can simplify this expression further by using a substitution u = 12x+1:

A = π ∫(13 to 49) √u du

Integrating this, we get:

A = π (2/3)(u^(3/2))|(13 to 49)

A = π (2/3)(49√49-13√13)

A = 36π√3 square units

Therefore, the area of the surface obtained by rotating the curve y=x−−√3y=x3 about the y-axis for 1≤y≤4 is 36π√3 square units.

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Part A:

Best graphical representation to display data will be line graph.

Given,

Scores of basketball team during the first 13 games of the season

{85, 94, 101, 118, 107, 110, 114, 96, 117, 105, 121, 88, 125}.

Now,

Part B:

We can create line graph with a very simple technique.

Firstly,

On x - axis take the number of season the team has played. In our case the number is 13 so take 13 distinct points on the x - axis.

Secondly,

On y -axis take the scores of the team in each of the 13 seasons corresponding to their values at x - axis.

Then,

Plot the points on the graph for all 13 seasons .

Last step,

Join all the points in the graph with the help of ruler. This will form the required line graph of the question.

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A 99% confidence interval needs to be constructed using the sample proportion and sample size. The given information includes n = 600 and ˆp = 0.85.

To construct the confidence interval, first calculate the standard error using the formula:

SE = sqrt [ (p-hat * (1 - p-hat)) / n ]

Substituting the given values, we get:

SE = sqrt [ (0.85 * (1 - 0.85)) / 600 ] = 0.0203 (rounded to four decimal places)

Next, calculate the margin of error using the formula:

ME = z* (SE)

Here, for a 99% confidence interval, z* = 2.576 (from the standard normal distribution table).

Substituting the values, we get:

ME = 2.576 * (0.0203) = 0.0523 (rounded to four decimal places)

Finally, the confidence interval can be calculated as:

ˆp ± ME

Substituting the given values, we get:

0.85 ± 0.0523 = (0.7977, 0.9023)

Therefore, the 99% confidence interval for the population proportion is (0.7977, 0.9023). This means we are 99% confident that the true population proportion falls between 0.7977 and 0.9023

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To determine how many times the spinner will land on blue or green, we need to know the number of sections that are shaded blue or green.

Let's assume the spinner has 8 sections in total, with 3 sections shaded blue, 4 sections shaded green, and the remaining sections shaded red.

Out of 24 spins, the probability of landing on blue or green can be calculated as the sum of the individual probabilities. The probability of landing on blue is 3/8, and the probability of landing on green is 4/8 (since there are 4 green sections out of 8 in total).

Therefore, the expected number of times the spinner will land on blue or green in 24 spins is:

(3/8 + 4/8) * 24 = (7/8) * 24 = 21.

So, the spinner is expected to land on blue or green approximately 21 times out of the 24 spins.

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We can use the Maclaurin series expansion of sin(x) and plug in π/2 to get: sum

sin(π/2) = ∑n=0^[infinity] (-1)^n (π/2)^(2n+1)/(2n+1)!

Simplifying the right-hand side:

sin(π/2) = π/2 - π^3/2! + π^5/4! - π^7/6! + ...

Multiplying both sides by π/2 and rearranging:

π^2/4 = π/2 - π^3/3! + π^5/5! - π^7/7! + ...

Now, we can use the Maclaurin series expansion of cos(x) and plug in 0 to get:

cos(0) = ∑n=0^[infinity] (-1)^n x^(2n)/(2n)!

Simplifying the right-hand side:

cos(0) = 1 - x^2/2! + x^4/4! - x^6/6! + ...

Multiplying both sides by x^2/2 and rearranging:

π^2/8 = π^2/4 - π^4/4! + π^6/6! - π^8/8! + ...

Now we can substitute these series expansions into the original sum and simplify:

∑n=0^[infinity] (-1)^n π^2n/(6^2n (2n)!)

= π^2/2 - π^4/4! + π^6/6! - π^8/8! + ...

= 2π^2/4 - π^4/4! + π^6/6! - π^8/8! + ...

= (2π^2 - π^4/3! + π^6/5! - π^8/7! + ...) / 4

= (2π^2 - π^4/6 + π^6/120 - π^8/5040 + ...) / 4

So the sum of the series is (2π^2 - π^4/6 + π^6/120 - π^8/5040 + ...) / 4.

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The Pharmaceutical company can use the method of t-test which assumes that the data is normally distributed and that the variances between the two groups are equal. If these assumptions are not met, alternative tests such as the Mann-Whitney U-test may be more appropriate

The pharmaceutical company wants to test whether their new drug can help regrow hair in balding men compared to a placebo.

The most appropriate Hypothesis testing  for these data would be a two-sample t-test. This test compares the means of two independent groups, in this case, the group receiving the drug and the group receiving the placebo. The t-test will determine if the difference in the means between the two groups is statistically significant or due to chance.

To conduct the two-sample t-test, the researchers will need to calculate the mean and standard deviation of the percentage change in hair coverage for each group. They will also need to determine the sample size, which in this case is 50 for each group. The t-test will then calculate a t-statistic and a corresponding p-value.

If the p-value is less than the predetermined level of significance, usually 0.05, the researchers can reject the null hypothesis that there is no difference in hair regrowth between the drug and placebo groups. This would suggest that the new drug is effective in helping regrow hair in balding men.

It is important to note that the t-test assumes that the data is normally distributed and that the variances between the two groups are equal. If these assumptions are not met, alternative tests such as the Mann-Whitney U-test may be more appropriate.

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Step-by-step explanation:

To find the critical points of f(x,y), we need to find where the partial derivatives of f(x,y) are equal to zero or do not exist.

First, we find the partial derivative of f(x,y) with respect to x:

f_x(x,y) = y(1 - 7x - 9y) - 7xy = y - 7xy - 9y^2

Next, we find the partial derivative of f(x,y) with respect to y:

f_y(x,y) = x(1 - 7x - 9y) - 9xy = x - 7xy - 9x^2

To find the critical points, we set both partial derivatives to zero and solve for x and y:

y - 7xy - 9y^2 = 0

x - 7xy - 9x^2 = 0

Factoring out y in the first equation, we get:

y(1 - 7x - 9y) = 0

This gives us two solutions: y = 0 or 1 - 7x - 9y = 0

If y = 0, then from the second equation, we have x = 0.

If 1 - 7x - 9y = 0, then we can solve for y to get:

y = (1 - 7x)/9

Substituting this value of y into the second equation gives:

x - 7x(1-7x)/9 - 9x^2 = 0

Simplifying this equation gives:

-56x^2/9 + 56x/9 + x = 0

x(-56x + 9 + 56) = 0

x(8x - 1) = 0

So, x = 0 or x = 1/8.

If x = 0, then from the equation y = (1 - 7x)/9, we have y = 1/9.

If x = 1/8, then from the equation y = (1 - 7x)/9, we have y = -1/9.

Therefore, the critical points of f(x,y) are:

(0, 0), (0, 1/9), and (1/8, -1/9).

To classify these critical points, we need to use the second partial derivative test.

Calculating the second partial derivatives:

f_{xx}(x,y) = -7y

f_{xy}(x,y) = 1 - 7x - 18y

f_{yy}(x,y) = -9x

At the critical point (0,0), we have:

f_{xx}(0,0) = 0

f_{xy}(0,0) = 1

f_{yy}(0,0) = 0

The determinant of the Hessian matrix is:

f_{xx}(0,0) * f_{yy}(0,0) - [f_{xy}(0,0)]^2 = 1

Since the determinant is positive and f_{xx}(0,0) = f_{yy}(0,0) = 0, we have a saddle point at (0,0).

At the critical point (0,1/9), we have:

f_{xx}(0,1/9) = -7/9 < 0

f_{xy}(0,1/9) = 1 > 0

f_{yy}(0,1/9) = 0

The determinant of the Hessian matrix is:

f_{xx}(0,1/9) * f_{yy}(0,1/9) - [f_{xy}(0,1/9)]^2 = -7/9

Since the determinant is negative and f_{xx}(0,1/9) < 0, we have a local maximum at (0,1/9).

At the critical point (1/8,-1/9), we have:

f_{xx}(1/8,-1/9) = 7/9 > 0

f_{xy}(1/8,-1/9) = -15/8 < 0

f_{yy}(1/8,-1/9) = 0

The determinant of the Hessian matrix is:

f_{xx}(1/8,-1/9) * f_{yy}(1/8,-1/9) - [f_{xy}(1/8,-1/9)]^2 = 105/64

Since the determinant is positive and f_{xx}(1/8,-1/9) > 0, we have a local minimum at (1/8,-1/9).

Therefore, the critical points of f(x,y) are:

Therefore, the critical points of f(x,y) are: - A saddle point at (0,0)

Therefore, the critical points of f(x,y) are: - A saddle point at (0,0)- A local maximum at (0,1/9)

Therefore, the critical points of f(x,y) are: - A saddle point at (0,0)- A local maximum at (0,1/9)- A local minimum at (1/8,-1/9)

Step-by-step explanation:

14^2 + 8^2 = 260

third side=√260 = 16.12 ~ 16

The chance that a randomly selected defective component was made by Alice is approximately 0.33 or 33%.

a. The probability of a randomly selected individual being a male who smokes is 19/100 or 0.19.

b. The probability of a randomly selected individual being a male is 0.6.

c. The probability of a randomly selected individual smoking is (19+12)/100 or 0.31.

d. The probability that a randomly selected female is a smoker is 12/40 or 0.3.

e. The probability that a randomly selected smoker is male is 19/(19+12) or 0.61.

To find the chance that a randomly selected defective component was made by Alice, we can use Bayes' Theorem.

Let A, B, and C represent the events that a component is made by Alice, Betty, and Cleo respectively, and let D represent the event that a component is defective.

We want to find P(A|D), the probability that the defective component was made by Alice. Using Bayes' Theorem, we have:

P(A|D) = P(D|A) * P(A) / [P(D|A) * P(A) + P(D|B) * P(B) + P(D|C) * P(C)]

We know that P(A) = P(B) = P(C) = 1/3, P(D|A) = 0.05, P(D|B) = 0.03, and P(D|C) = 0.02. Substituting these values, we get:

P(A|D) = 0.05 * (1/3) / [0.05 * (1/3) + 0.03 * (1/3) + 0.02 * (1/3)]

       = 0.333 or approximately 0.33

Therefore, the chance that a randomly selected defective component was made by Alice is approximately 0.33 or 33%.

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Complete Question:

             Yes No Total

Male 19 41 60

Female 12 28 40

Total 31 69 100

a. What is the probability of a randomly selected individual being a male who smokes (or male and smoker)?

b. What is the probability of a randomly selected individual being a male?

c. What is the probability of a randomly selected individual smoking?

d. A person is selected at random and if the person is female, what is the probability that she is a smoker?

e. What is the probability that a randomly selected smoker is male?

A manufacturing firm has three machine operators; Alice, Betty, and Cleo. The operators produce a component. Alice has a 5% defective rate, Betty a 3% defective rate, and Cleo a 2% defective rate. The three operators produce equal numbers of components. Suppose a randomly selected component is found to be defective. What is the chance it was made by Alice?

When Jude draws three counters at random from the bag, the likelihood that she will take exactly one red counter is roughly 0.5357.

The total number of outcomes and the number of favorable outcomes must be calculated.

The total number of outcomes is the number of ways Jude can freely choose any three counters from the bag. Combinations can be used to calculate this.

Total possible outcomes = C(8, 3) = 8! / (3! * (8 - 3)!)

= 8! / (3! * 5!)

= (8 * 7 * 6) / (3 * 2 * 1)

= 56

Next, we need to determine the number of favorable outcomes, which is the number of ways Jude can select exactly one red counter and two yellow counters.

Number of favorable outcomes = C(3, 1) * C(5, 2) = (3! / (1! * (3 - 1)!)) * (5! / (2! * (5 - 2)!))

= (3 * 2 / (1 * 2)) * (5 * 4 / (2 * 1))

= 3 * 10

= 30

Finally, we can calculate the probability of Jude taking exactly one red counter by dividing the number of favorable outcomes by the total possible outcomes:

Probability = Number of favorable outcomes / Total possible outcomes

= 30 / 56

≈ 0.5357 (rounded to four decimal places)

Therefore, When Jude draws three counters at random from the bag, the likelihood that she will take exactly one red counter is roughly 0.5357.

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The Maclaurin series expansion of f(x) is: f(x) = f(0) + f'(0)x + f''(0)x^2/2! + f'''(0)x^3/3! + ...

To find the coefficients of the series, we need to evaluate the derivatives of f(x) at x = 0. Let's find the first few derivatives:

f(x) = (9x)arctan(5x^2)

f'(x) = 9arctan(5x^2) + 18x^2/(1 + 25x^4)

f''(x) = 90x/(1 + 25x^4) - 90x^3(1 - 5x^4)/(1 + 25x^4)^2

f'''(x) = 90(1 - 25x^4)/(1 + 25x^4)^2 - 270x^2(1 - 5x^4)/(1 + 25x^4)^2 - 270x^4(1 - 5x^4)/(1 + 25x^4)^2 + 360x^6(1 - 5x^4)/(1 + 25x^4)^3

By evaluating these derivatives at x = 0, we can find the coefficients of the Maclaurin series expansion of f(x).

However, calculating the derivatives and evaluating them at x = 0 can be quite involved and require significant algebraic manipulation. Therefore, it is best to use computational software or tools to calculate the coefficients of the Maclaurin series expansion accurately.

Once we have the coefficients, we can express the Maclaurin series of f(x) by substituting the coefficients into the general formula.

The Maclaurin series expansion allows us to approximate the function f(x) for values of x close to 0, providing a polynomial representation of the function.

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To find a basis for the subspace dimension w of R4, we need to find a set of linearly independent vectors that span w.

First, we can rewrite the given condition for w as follows:

w = {(s, 0, 0, 5s) + (0, 4t, 0, -t) + (0, 0, s, 0) + (0, 0, 0, -t) : s, t are real numbers}

Notice that each term in the above expression corresponds to one of the four standard basis vectors in R4. Therefore, the subspace w can be expressed as the span of the following four vectors:

v1 = (1, 0, 0, 5)
v2 = (0, 4, 0, -1)
v3 = (0, 0, 1, 0)
v4 = (0, 0, 0, -1)

To show that these vectors form a basis for w, we need to show that they are linearly independent and that they span w.

To show linear independence, suppose that a linear combination of these vectors is equal to the zero vector:

c1 v1 + c2 v2 + c3 v3 + c4 v4 = (0, 0, 0, 0)

Then we have the following system of equations:

c1 = 0
4c2 = 0
c3 = 0
5c1 - c2 = 0

Solving for the coefficients, we get c1 = c2 = c3 = c4 = 0, which shows that the vectors are linearly independent.

To show that they span w, we need to show that any vector in w can be expressed as a linear combination of these vectors. Let (s, 4t, s, 5s - t) be an arbitrary vector in w. Then we can write:

(s, 4t, s, 5s - t) = (s, 0, 0, 5s) + (0, 4t, 0, -t) + (0, 0, s, 0) + (0, 0, 0, -t)

which is a linear combination of the vectors v1, v2, v3, and v4. Therefore, these vectors span w.

Since we have found a set of four linearly independent vectors that span w, we can conclude that they form a basis for w. Thus, the dimension of the subspace w is 4.

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The length of the arc shown in red is 5π/4 metre

To find the length of the arc, we need to find the circumference of the circle, which we find with the following formula :

C = 2πr

where r  is the radius which is indicated in the image: .

so the circumference  is:

C = 2π(3)

C = 6π

This is the measure of the entire perimeter of the circle, it is the measure of the 360 ° arc.

Because we only want 30° of that 360 °, we divide the value of the circumference by 360 and multiply po 45:

30/360=0.125 of full circle,

L(arc)=0.125L=5π/4

The length of the arc is 5π/4 m

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Answer:

its correct

Step-by-step explanation:good job

in triangle ΔCDE:

CD ≈ √1466

∠C ≈ -17

∠D = 90

in triangle  ΔJKL:

JL ≈ √808

∠K = 102 - ∠L

∠L = ∠L

We have,

To solve for the missing values in the given triangles, we can use the angle sum property and the side length ratios in triangles.

so,

For ΔCDE:

Given:

∠E = 107 degrees

CE = 29

DE = 25

To find the CD, we can use the Pythagorean theorem because CDE is a right triangle:

CD² = CE² + DE²

CD² = 29² + 25²

CD²= 841 + 625

CD² = 1466

CD ≈ √1466

To find ∠C, we can use the fact that the sum of angles in a triangle is 180 degrees:

∠C = 180 - ∠E - ∠D

∠C = 180 - 107 - 90

∠C = 180 - 197

∠C ≈ -17 (The negative angle suggests that there might be an error or inconsistency in the given information)

To find ∠D, we can use the fact that the sum of angles in a triangle is 180 degrees:

∠D = 180 - ∠C - ∠E

∠D = 180 - (-17) - 107

∠D = 180 + 17 - 107

∠D = 90

And,

For ΔJKL:

Given:

∠J = 78 degrees

KJ = 18

KL = 22

To find JL, we can use the side length ratios in triangles:

JL² = KJ² + KL²

JL² = 18² + 22²

JL² = 324 + 484

JL² = 808

JL ≈ √808

To find ∠K, we can use the fact that the sum of angles in a triangle is 180 degrees:

∠K = 180 - ∠J - ∠L

∠K = 180 - 78 - ∠L

∠K = 180 - 78 - ∠L

∠K = 180 - 78 - ∠L

∠K = 180 - 78 - ∠L

∠K = 180 - 78 - ∠L

∠K = 102 - ∠L

To find ∠L, we can use the fact that the sum of angles in a triangle is 180 degrees:

∠L = 180 - ∠J - ∠K

∠L = 180 - 78 - (102 - ∠L)

∠L = 180 - 78 - 102 + ∠L

∠L = 180 - 180 + ∠L

∠L = ∠L

Therefore,

in ΔCDE:

CD ≈ √1466

∠C ≈ -17 (possible error or inconsistency in the given information)

∠D = 90

And,

in ΔJKL:

JL ≈ √808

∠K = 102 - ∠L

∠L = ∠L

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The required solution to the system of equations in Question 11 is x = -6 and y = -4, and in Question 12 is x = -7 and y = 9.

11:

3x + 2y = -26

4x - 5y = -4

To eliminate one variable, we can multiply the first equation by 4 and the second equation by 3, so the coefficients of x will be the same:

12x + 8y = -104 (Multiplying the first equation by 4)

12x - 15y = -12 (Multiplying the second equation by 3)

Now, subtract the second equation from the first equation to eliminate x:

(12x + 8y) - (12x - 15y) = -104 - (-12)

12x + 8y - 12x + 15y = -104 + 12

y = -4

Now, substitute the value of y back into one of the original equations, let's use the first equation:

3x + 2(-4) = -26

3x = -18

x = -6

Therefore, the solution to the system of equations in Question 11 is x = -6 and y = -4.

Similarly,

The solution to the system of equations in Question 12 is x = -7 and y = 9.

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The solution of the logarithmic equation is x = 2.

Given is a logarithmic equation ㏒ x + ㏒ (x+2) = ㏒ 8, we need to solve for x,

So,

The logarithmic equation is ㏒ x + ㏒ (x+2) = ㏒ 8,

Applying the log rule we get,

x(x+2) = 8

x²+2x = 8

x² + 2x - 8 = 0,

Solving for x,

x = 2 and x = -4,

Checking for extraneous solutions.

Verify solution x = 2 [true]

x = -4 [false]

Hence the solution of the logarithmic equation is x = 2.

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The cylinder x^2 + 5z^2 = 8 is parallel to the y-axis.
- The cylinder x^2 + 5y^2 = 8 is parallel to the z-axis.
- The cylinder z^2 + 5y^2 = 8 is parallel to the x-axis.

Let's analyze each given cylinder equation and determine which coordinate axes they are parallel to in ℝ³:

1. x^2 + 5z^2 = 8:
This cylinder is parallel to the y-axis because the equation does not involve the y-coordinate. The equation is in the form x^2 + cz^2 = k, where c and k are constants.

2. x^2 + 5y^2 = 8:
This cylinder is parallel to the z-axis because the equation does not involve the z-coordinate. The equation is in the form x^2 + cy^2 = k, where c and k are constants.

3. z^2 + 5y^2 = 8:
This cylinder is parallel to the x-axis because the equation does not involve the x-coordinate. The equation is in the form cz^2 + cy^2 = k, where c and k are constants.

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The value of c-b based on the given integral is given as 0

We recognize the limit as the definition of the integral.

The integral represents the area under the curve of the function [tex]f(x) = x^3[/tex]from 1 to 6.

Using the limit definition, we can rewrite the integral as:

[tex]\int^6_1x^3dx= \lim_{n\to\infty}\sum^n_{i=1}f\left(1+\frac{bi}{n}\right)\frac{c}{n}[/tex]

Comparing this with the general form for Riemann sums:

[tex]\int^b_ax^3dx= \lim_{n\to\infty}\sum^n_{i=1}f\left(a+\frac{(b-a)i}{n}\right)\frac{b-a}{n}[/tex]

We can identify [tex]a = 1,b = 6[/tex]

Then, we have [tex]1 + \frac{bi}{n} = 1 + \frac{5i}{n}[/tex] and [tex]\frac{c}{n} = \frac{5}{n}[/tex]

Hence, [tex]b = 5[/tex]and [tex]c = 5[/tex]

Thus, [tex]c - b = 5 - 5 = 0[/tex]

The limit of an integral refers to a value an integral approaches as the interval of integration approaches a certain point, often used in improper integrals.

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Thus, the extreme points of s are (0,0), (2,2), (2,0), and (0,2), and the extreme directions are [2 -4], [-1 1], [0 1], and [0 -1].

To find the extreme points and extreme directions of the polyhedral set s, we need to first write down the set in standard form. We can rewrite the constraints as:

2x1 - 4x2 <= -4
-x1 + x2 <= 2
x2 <= 4
x2 >= 0

The first two constraints can be written as a matrix inequality:
[2 -4; -1 1][x1; x2] <= [4; 2]

The last two constraints can be written as x2 <= 4 and x2 >= 0. Thus, the polyhedral set s can be written as:
s = {x in R^2 : [2 -4; -1 1][x1; x2] <= [4; 2], x2 <= 4, x2 >= 0}

To find the extreme points, we can solve the linear program:

maximize 0x1 + 0x2
subject to [2 -4; -1 1][x1; x2] <= [4; 2]
x2 <= 4
x2 >= 0

The objective function is just 0x1 + 0x2, so it doesn't matter what the values of x1 and x2 are. The constraints, however, determine the feasible region. The intersection of the constraints is a polygon with vertices at (0,0), (2,2), (2,0), and (0,2). These are the extreme points of s.

To find the extreme directions, we need to look at the gradients of the constraints at each extreme point. If the gradient is non-zero, then that constraint is active at that point and the corresponding direction is extreme. The gradients of the constraints are:

[2 -4] for the first constraint
[-1 1] for the second constraint
[0 1] for the third constraint
[0 -1] for the fourth constraint

At the point (0,0), the first two constraints are active and their gradients are non-zero. Thus, the extreme directions are along [2 -4] and [-1 1].

At the point (2,2), the first two constraints and the third constraint are active. The gradients of the first two constraints are non-zero, as before, and the gradient of the third constraint is [0 1]. Thus, the extreme directions are along [2 -4], [-1 1], and [0 1].

At the point (2,0), the first two constraints and the fourth constraint are active. The gradients of the first two constraints are non-zero, and the gradient of the fourth constraint is [0 -1]. Thus, the extreme directions are along [2 -4], [-1 1], and [0 -1].

At the point (0,2), the second constraint and the third constraint are active. The gradient of the second constraint is non-zero, as before, and the gradient of the third constraint is [0 1]. Thus, the extreme directions are along [-1 1] and [0 1].

Therefore, the extreme points of s are (0,0), (2,2), (2,0), and (0,2), and the extreme directions are [2 -4], [-1 1], [0 1], and [0 -1].

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x = 3, y = 8 and when x = 3/7, y = 49/6. The equation that relates x and y is y=k/x, where k is a constant of proportionality. By substituting the given values of x and y, we can solve for k and then use the equation to find the value of y for a given x.

Using the first set of given values, x = 2 and y = 12, we can set up the equation y = k/x and solve for k:

12 = k/2

k = 24

Thus, the equation that relates x and y is y = 24/x. To find y when x = 3, we plug in x = 3 into the equation and get:

y = 24/3 = 8

For the second set of values, x = 3/7 and y = 7/6, we again use the equation y = k/x and solve for k:

7/6 = k/(3/7)

k = 49/18

So the equation that relates x and y is y = (49/18)x. To find y when x = 3, we plug in x = 3 and get:

y = (49/18)3 = 49/6

Therefore, when x = 3, y = 8 and when x = 3/7, y = 49/6.

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A true statement for most of the data in this plot is c. Most of the data in the Girl's Heights plot are around 52. Therefore, option c. Most of the data in the Girl's Heights plot are around 52 is correct.

For the Boy's Heights plot, we can see that the majority of the dots are above 48 and below 54, with the most dots being above 52. Therefore, a true statement for most of the data in this plot is:

Most of the data in the Boy's Heights plot are around 52.

For the Girl's Heights plot, we can see that the majority of the dots are also above 48 and below 54, with the most dots being above 52 as well. Therefore, a true statement for most of the data in this plot is:

Most of the data in the Girl's Heights plot are around 52.

So, the correct option is:

Most of the data in each plot are around 52.

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The data set for which the median would be the best measure of center is (b) 67, 72, 75, 74, 68, 69, 70, 65. This is because the data set has no extreme values or outliers, and the values are relatively evenly distributed around the middle.

Therefore, the median, which is the middle value when the data is arranged in order, would accurately represent the typical value in this set.

The median would be the best measure of center for data set d.) 93, 95, 88, 38, 79, 85, 88, 90. This is because the median is less sensitive to extreme values (such as 38 in this case) compared to the mean, and it provides a more accurate representation of the central tendency of the data set.

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