Quinn is a budding fashion designer known for incorporating geometry into her creations. Her newest design is a simple black blouse patterned with same-size right triangles in different funky colors. The longer leg of each right triangle will be twice as long as the shorter leg, and the hypotenuse of each triangle will be 6 inches long. To the nearest tenth of an inch, what will be the length of the shorter leg of each triangle?

A circumscribed angle is an angle whose sides are tangent to a circle. In other words, the angle is formed by two intersecting chords of the circle. The vertex of the angle is located outside of the circle, while the two endpoints of the angle lie on the circle.

Circumscribed angles have some important properties in geometry. For example, the measure of a circumscribed angle is half the measure of its intercepted arc (the arc of the circle that lies inside the angle). Additionally, if two angles intercept the same arc of a circle, they are congruent.

Circumscribed angles also appear frequently in trigonometry, where they are used to define the sine, cosine, and tangent functions. The sine of a circumscribed angle is defined as the ratio of the length of the opposite side of the angle to the length of the circle's radius. The cosine of a circumscribed angle is defined as the ratio of the length of the adjacent side of the angle to the length of the radius.

Finally, the tangent of a circumscribed angle is defined as the ratio of the length of the opposite side to the length of the adjacent side.

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The measures of central tendencies changed as mode remained the same, the median increased and the mean increased.

To determine how the data set would've changed if she entered the competition, we simply need to work on the mean, median and mode of the data.

Given data;

35, 22, 31, 23, 35, 22, 30, 35, 40

Rearranging this data;

22, 22, 23, 30, 31, 35, 35, 35, 40

The mean of this data will be

mean = 30.3

The mode = 35

The median = 31

When her weight is added, the measures of central tendencies change to;

mean = 31.5

median = 33

mode = 35

The median decreases, the mode remains the same and the mean increases

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Following shows (sin 7x)/sin x = 64(cos x)^6 - 80(cos x)^4 +24(cos x)^2 - 1:

12sin^3 x - 8sin^

To show that (sin 7x)/sin x is equal to 64(cos x)^6 - 80(cos x)^4 + 24(cos x)^2 - 1, we can use trigonometric identities and algebraic manipulation.

Let's start with the left-hand side of the equation:

(sin 7x)/sin x

Using the trigonometric identity for sin(A + B):

sin(A + B) = sin A cos B + cos A sin B

We can rewrite sin 7x as sin (6x + x):

sin (6x + x) = sin 6x cos x + cos 6x sin x

Now we can substitute sin 7x with sin 6x cos x + cos 6x sin x:

(sin 6x cos x + cos 6x sin x)/sin x

Next, we can simplify this expression by dividing both terms by sin x:

(sin 6x cos x)/sin x + (cos 6x sin x)/sin x

The sin x term cancels out, leaving us with:

sin 6x cos x + cos 6x

Now, we can use the double-angle identity for sin 2A:

sin 2A = 2sin A cos A

To rewrite sin 6x cos x, we can treat it as sin 2A with A = 3x:

sin 6x cos x = 2sin 3x cos 3x

Next, we can use the triple-angle identity for sin 3A:

sin 3A = 3sin A - 4sin^3 A

To rewrite sin 3x, we can treat it as sin A with A = x:

sin 3x = 3sin x - 4sin^3 x

Substituting this into our expression:

2sin 3x cos 3x = 2(3sin x - 4sin^3 x) cos 3x

Expanding further:

= 6sin x cos 3x - 8sin^3 x cos 3x

Now, we can use the double-angle identity for cos 2A:

cos 2A = cos^2 A - sin^2 A

To rewrite cos 3x, we can treat it as cos A with A = x:

cos 3x = cos^2 x - sin^2 x

Substituting this into our expression:

6sin x cos 3x - 8sin^3 x cos 3x = 6sin x (cos^2 x - sin^2 x) - 8sin^3 x (cos^2 x - sin^2 x)

Expanding further:

= 6sin x cos^2 x - 6sin x sin^2 x - 8sin^3 x cos^2 x + 8sin^3 x sin^2 x

Now, we can use the Pythagorean identity for sin^2 x + cos^2 x:

sin^2 x + cos^2 x = 1

To rewrite sin^2 x, we can subtract cos^2 x from both sides:

sin^2 x = 1 - cos^2 x

Substituting this back into our expression:

= 6sin x (cos^2 x - (1 - sin^2 x)) - 8sin^3 x cos^2 x + 8sin^3 x sin^2 x

= 6sin x (2sin^2 x) - 8sin^3 x cos^2 x + 8sin^3 x sin^2 x

= 12sin^3 x - 8sin^

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The probability that a randomly selected admission charge is less than $3. 50 is 0.23% or 0.0023.

To find the probability that a randomly selected admission charge is less than $3.50, we will use the z-score formula and a standard normal table. The z-score formula is:

Z = (X - μ) / σ

Where X is the value we are interested in ($3.50), μ is the average admission charge ($5.81), and σ is the standard deviation ($0.81).

Z = (3.50 - 5.81) / 0.81 ≈ -2.84

Now, look up the z-score (-2.84) in a standard normal table, which gives us the probability of 0.0023. Therefore, the probability that a randomly selected admission charge is less than $3.50 is approximately 0.23% or 0.0023.

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Place value is the value of a digit based on its position within a number system. To solve the equation 19 + 50, we use place value to add the digits in the ones and tens place.

Place value is a fundamental concept in mathematics that helps in understanding the value of digits in a number system. In the decimal system, each digit's value depends on its position relative to the decimal point, with the position to the left representing higher values than those to the right.

To solve 19 + 50, we add the digits in the ones place (9 + 0 = 9) and the digits in the tens place (1 + 5 = 6) separately, then combine the results to get the final answer of 69. This is possible due to the place value concept. The digit 9 in the ones place of 19 represents a value of 9 units, while the digit 1 in the tens place represents 10 units.

Similarly, the digit 5 in the tens place of 50 represents 50 units, and the digit 0 in the ones place represents 0 units. By adding the digits based on their place value, we get the answer 69, where 9 is in the ones place and 6 is in the tens place.

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If Ali took 6 pencils from a new box of pencils and there are 18 pencils left in the box, then there were 24 pencils in the brand new box.

To see why, you can add the number of pencils Ali took to the number of pencils left in the box:

6 + 18 = 24

Therefore, there were 24 pencils in the brand new box before Ali took 6 of them.

Carmen needs (6/7) cups of strawberry frosting and (1 1/7) cups of lemon frosting to make 2 cups of the frosting mixture.

To determine the amount of strawberry frosting and lemon frosting that Carmen needs to make 2 cups of the frosting mixture, we need to use a proportion.

Let x be the amount of strawberry frosting needed in cups, and y be the amount of lemon frosting needed in cups.

From the given information, we know that Carmen mixed 1/4 cup of strawberry frosting with 1/3 cup of lemon frosting. Thus, the ratio of the amounts of strawberry frosting to lemon frosting is:

x/y = (1/4)/(1/3)

Simplifying this ratio, we get:

x/y = 3/4

We also know that the total amount of frosting needed is 2 cups, so:

x + y = 2

Using substitution, we can solve for x:

x + (4/3)x = 2

(7/3)x = 2

x = (6/7) cups

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The function will be continuous at x = 3 for any values of a and b.

f(x) = ax + b to be continuous at x = 3

A function is continuous at a point x = c if:

1. f(c) is defined

2. The limit of f(x) as x approaches c exists

3. The limit of f(x) as x approaches c is equal to f(c)

For f(x) = ax + b to be continuous at x = 3:

1. f(3) is defined:

f(3) = a(3) + b

2. The limit of f(x) as x approaches 3 exists.

3. The limit of f(x):

lim (x->3) (ax + b) = a(3) + b

There are no specific values for a and b that must be satisfied. The function will be continuous at x = 3 for any values of a and b.

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If sales increase by 66,000, income will increase by 220,000.00

Net operating income 37,200.00

The margin of safety is 14.01%

Statement showing Computations  

particulars Amount Per unit

Sales 628,000.00 40.00

Variable Expenses 439,600.00 28.00

Contribution Margin 188,400.00 $ 12.00

Fixed Expenses 151,200.00

Net operating income 37,200.00

'

1)  BEP in unit sales = 151,200/12 12,600.00

.BEP in sales $ = 12,600 * 40 504,000.00

2)  Total Contribution margin at BEP = Fixed costs 151,200.00

3)a Target Profit $ 64,800.00

Fixed Expenses 151,200.00

Desired Contribution 216,000.00

3b. No of units to be sold = 216,000/12 18,000.00

Sales 720,000.00

Variable Expenses 504,000.00

Contribution Margin 216,000.00

Fixed Expenses 151,200.00

Net operating income 64,800.00

4)  Margin of safety = 628,000.00-540,000.00 88,000.00

MOS in % = 88,000/6280001 14.01%

5)CM Ratio = 188,400/628,000 0.3

If sales increase by 66,000, income will increase by 220,000.00

66,000/0.3

=220,000.

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dx/dy = -sin(t)/cos(t) when x = sin^3(t) and y = cos^3(t).

To find dx/dy, we first need to find dx/dt and dy/dt, and then we can use the chain rule.

Given x = sin^3(t) and y = cos^3(t),

dx/dt = d(sin^3(t))/dt = 3sin^2(t) * cos(t) (using the chain rule)
dy/dt = d(cos^3(t))/dt = -3cos^2(t) * sin(t) (using the chain rule)

Now, we can find dx/dy by dividing dx/dt by dy/dt:

dx/dy = (dx/dt) / (dy/dt) = (3sin^2(t) * cos(t)) / (-3cos^2(t) * sin(t))

Simplify the expression:

dx/dy = -sin(t)/cos(t)

So, dx/dy = -sin(t)/cos(t) when x = sin^3(t) and y = cos^3(t).

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The quotient of 6 and 1/4 is 24.

We have applied division operation to this question. Firstly, we will understand the meaning of a proper fraction. A fraction in which the numerator is less than the denominator is called a proper fraction. This means that the denominators will always be bigger than the numerators for appropriate fractions.

We can represent this condition in either of the two ways.

Denominator < Numerator

(Or)

Numerator > Denominator

We are given a numerical expression which is 6÷ 1/4 and we have to solve this.

To convert this division sign into a multiplication sign, we will take the reciprocal of 1/4.

The reciprocal of 1/4 is 4.

Therefore,

6÷ 1/4

= 6 × 4

= 24

Therefore, the quotient of 6 and 1/4 is 24.

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The expression that explains what happened to the price of the jacket is 0.85(1.4s), which represents a 40% increase in price followed by a 15% discount.

The expression 0.85(1.4s) represents the current selling price of the jacket, which includes two price changes.

To explain what happened to the price of the jacket, we can break down the expression into two steps:

1. The first change was an increase by 40%, which can be represented as multiplying the original price "s" by 1.4 (100% + 40% = 140% or 1.4). So, the price after the first change is 1.4s.

2. The second change was a discount of 15%, which can be represented as multiplying the price after the first change by 0.85 (100% - 15% = 85% or 0.85). So, the price after both changes is 0.85(1.4s).

So, the expression that explains what happened to the price of the jacket is 0.85(1.4s), which represents a 40% increase in price followed by a 15% discount.

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The remaining trigonometric functions are:

sin (θ) = 3/√34

cos (θ) = 5/√34

csc(θ) =  (√34)/3

sec (θ) = (√34)/5

cot(θ) = 5/3

Trigonometry is a branch of mathematics that deals with the relationship between the ratios of the sides of a right-angled triangle with its angles.

Since tan θ = 3 /5

Recall: tan = opposite/adjacent. Thus, opposite = 3, adjacent = 5

hypotenuse = √(3²+5²) = √34

Therefore, the remaining trigonometric functions are

sin (θ) = 3/√34

cos (θ) = 5/√34

csc(θ) =  (√34)/3     (csc (θ) = 1/sin (θ))

sec (θ) = (√34)/5     (sec(θ) = 1/cos (θ))

cot(θ) = 5/3       (cot(θ) = 1/tan (θ))

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The researcher can be 95% confident that the mean number of hours all the residents of the city are watching news on television is between 1.7538 and 1.8462 hours, with a margin of error of 0.0462 hours.

Based on the information provided, the researcher surveyed 220 residents of the city and found that the mean number of hours they spend watching news on television each day is 1.8, with a standard deviation of 0.35.

The researcher wants to know the margin of error at a 95% confidence level for the mean number of hours all residents of the city are watching news on television.

To calculate the margin of error, we need to use the formula:
Margin of error = Critical value x Standard error

The critical value for a 95% confidence level is 1.96, and the standard error can be calculated as:
Standard error = Standard deviation /square root of sample space

Substituting the values given:
Standard error = 0.35 / sqrt(220) = 0.0236

Therefore, the margin of error can be calculated as:
Margin of error = 1.96 x 0.0236 = 0.0462

So, the researcher can be 95% confident that the mean number of hours all the residents of the city are watching news on television is between 1.7538 and 1.8462 hours, with a margin of error of 0.0462 hours.

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Probability or p-vale that the sample mean would differ from the population mean by less than 221 miles in a sample of 56 tires is equals to zero if the manager is correct.

We have data of an operation manager at a tire manufacturing company.

Mean mileage of a tire, [tex] \mu[/tex]

= 37,014 miles

standard deviation, [tex] \sigma[/tex]

= 4617 miles.

Sample size, n = 56

We have to determine the probability that the sample mean would differ from the population mean by less than 221 miles. Using Z-score formula in normal distribution, [tex]\small z= \frac{ \bar x-\mu }{\frac{\sigma }{\sqrt{n}}},[/tex]

Plugging all known values in above formula, [tex]z = \frac{ 221 - 37,014} {\frac{4617}{ \sqrt{56}}}[/tex]

= 59.634

[tex]P( \bar x < 221) = P ( \frac{ \bar x-\mu }{\frac{\sigma }{\sqrt{n}}} < \frac{ 221 - 37,014} {\frac{4617}{ \sqrt{56}}}) \\ [/tex]

=> P ( z < 59.63) = P( \bar x < 221)

Using the Z-distribution table, probability value is equals to 0. Hence, required probability is zero.

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If John has to buy 10 "2-Liter" bottles of soda, then the inequality representing this situation is "10(1.50) + 7.50p ≤ 150" and greatest number of pizzas he can buy is 18, Correct option is (d).

Let "p" denote the number of "large-pizzas" that John can buy.

One "2-liter" bottle of soda cost is = $1.50,

So, the cost of the 10 bottles of soda is : 10 × $1.50 = $15,

one "large-pizza's cost is = $7.50,

So, the cost of p large pizzas is : $p × $7.50 = $7.50p,

The "total-cost" of the soda and pizza must be less than or equal to $150, so we can write the inequality as :

10(1.50) + 7.50p ≤ 150

Simplifying the left-hand side of the inequality,

We get,

15 + 7.50p ≤ 150

7.50p ≤ 135

p ≤ 18

Therefore, John can buy at most 18 large pizzas with his remaining budget, the correct option is (d).

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The given question is incomplete, the complete question is

John is planning an end of the school year party for his friends he has $150 to spend on soda and pizza.

Soda (2-liter) costs $1.50;

large pizza cost $7.50;

He knows he has to buy 10 "2-Liter" bottles of soda.

Choose the inequality and calculate the greatest number of pizzas he can buy.

(a) 10(1.50) + 7.50p ≥ 150; 54 pizzas

(b) 10(7.50) + 1.50p ≤ 150; 53 pizzas

(c) 10(7.50) + 1.50p ≥ 150; 19 pizzas

(d) 10(1.50) + 7.50p ≤ 150; 18 pizzas

The function 1/[tex]r^2[/tex] can be expressed in terms of the spherical Bessel functions of the first kind, which are a family of solutions to the spherical Bessel differential equation.

The expansion involves a combination of the delta function and the first two spherical Bessel functions, j_0(r) and j_1(r). Specifically, the expansion can be written as (1/2)*[pi * delta(r) + (1/r)*d/d(r)(r * j_0(r)) + (1/[tex]r^2[/tex])*d/d(r)[[tex]r^2[/tex] * j_1(r)]]. This expansion is valid for all values of r except for r=0, where the first term dominates. The spherical Bessel functions are commonly used in physics, particularly in the context of scattering problems and wave propagation.

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To find the quantity of water in the reservoir after 11 hours, we need to integrate the rate of flow with respect to time from 0 to 11. The quantity of water in the reservoir after 11 hours is 38,225 gallons.

∫(3200 + 5t) dt from 0 to 11
= [(3200 * 11) + (5/2 * 11^2)] - [(3200 * 0) + (5/2 * 0^2)]
= 35,200 + 302.5
= 35,502.5 gallons
Therefore, the quantity of water in the reservoir after 11 hours is 35,502.5 gallons.

To find the quantity of water in the reservoir after 11 hours with the rate of 3200 + 5t gal/hour, we need to first find the total amount of water that flows into the reservoir within that time.
Step 1: Identify the given rate of flow: 3200 + 5t gal/hour.
Step 2: Integrate the flow rate function with respect to time (t) to find the total quantity of water. The integral of the function will give us the quantity of water in gallons:
∫(3200 + 5t) dt = 3200t + (5/2)t^2 + C, where C is the constant of integration.
Since the reservoir is initially empty, the constant C will be 0.
Step 3: Substitute t=11 hours into the integrated function to find the total quantity of water:
Q(11) = 3200(11) + (5/2)(11)^2
Q(11) = 35200 + 3025
Step 4: Add the values to find the total quantity of water in gallons:
Q(11) = 38225 gallons
The quantity of water in the reservoir after 11 hours is 38,225 gallons.

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A family camping in a national forest used a 4-foot pole and a tarp to build a temporary shelter. One corner of the tarp was staked 5 feet from the bottom of the pole. The slope of the tarp from that corner to the top of the pole is 0.8 or 4/5.

We can draw a right triangle with the pole being the height, the distance from the pole to the stake being the base, and the slope of the tarp being the hypotenuse. The hypotenuse is the longest side of the triangle and is opposite to the right angle.

Using the Pythagorean theorem, we can find the length of the hypotenuse

hypotenuse² = height² + base²

hypotenuse² = 4² + 5²

hypotenuse² = 41

hypotenuse = √(41)

Therefore, the slope of the tarp is the ratio of the height to the base, which is

slope = height / base = 4 / 5 = 0.8

So the slope of the tarp from that corner to the top of the pole is 0.8 or 4/5.

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The equation which most accurately represents the word problem,  is (b) 68 = 2L + 2(9).

The word problem states that the perimeter of a rectangle is 68 inches, and the perimeter equals twice the length (L) plus twice the width (9). We can represent this relationship by using the equation as :

We know that, the perimeter of rectangle is : 2(length + width),

Substituting the value,

We get,

⇒ 68 = 2(L + 9);

⇒ 68 = 2L + 2(9); and this statement is represented by Option(b).

Therefore, the correct equation is (b) 68 = 2L + 2(9).

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The given question is incomplete, the complete question is

Select the equation that most accurately depicts the word problem.

"The perimeter of a rectangle is 68 inches. The perimeter equals twice the length of L inches, plus twice the width of 9 inches".

(a) 68 = 9(L + 2)

(b) 68 = 2L + 2(9)

(c) 68 = 2(L - 9)

(d) 68 = 9L + 2

(e) 68 = 2/L + 2/9

(f) 68 = L/2 + 2(9)

In a case whereby When finding the Quotient of 8,397 divided 12, Calida first divided 83 by 12, then she will be wrong, because the answer is 699.75.

In maths, a division can be described as the process of splitting a specific amount  which can be spread to equal parts instance of thisd is when we divide a group of 20 members into 4 groups and this can be done using the mathematical sign.

In the case of Calida above, the division can be made as

8,397 divided 12

=8,397 / 12

=699.75

Therefore we can say that the right answer to the querstion is 699.75

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Answer:

(5,725)1.045^5

Step-by-step explanation:

(5,725)1.045^5

5,725 is the original amt of $

1.045 is the % of interest

5 is the # of years

Solve this and round the nearest

cent.

Answer:

The cost of Ticket = £63Booking Fee = 3% = 63*(3/100) = 1.89 = £ 1.89The total price of the ticket = £63 + £1.89 = £64.89. Therefore, the total price of the ticket is £64.89.

The height of the triangle is 8 inches and the width is 7 inches.

Let's start by using the formula for the area of a triangle:

A = (1/2)bh

where A is the area of the triangle, b is the base, and h is the height.

We are given that the area of the triangle is 28 square inches, so we can write:

28 = (1/2)bh

Next, we are given that the height h is six less than twice the width w. In other words:

h = 2w - 6

Now we can substitute this expression for h into the formula for the area:

28 = (1/2)bw(2w - 6)

Simplifying this equation, we get:

56 = bw(2w - 6)

28 = w(w - 3)

w^2 - 3w - 28 = 0

We can solve this quadratic equation using the quadratic formula:

w = [3 ± √ ([tex]3^2[/tex] - 4(1)(-28))] / 2

w = [3 ± √ (121)] / 2

w = (3 + 11) / 2 or w = (3 - 11) / 2

w = 7 or w = -4

Since a negative width doesn't make sense in this context, we can ignore the second solution and conclude that the width of the triangle is 7 inches.

Now we can use the expression for h in terms of w to find the height:

h = 2w - 6

h = 2(7) - 6

h = 8

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Answer:

$9.50

Step-by-step explanation:

Divide the total earnings by total hours.

The value of angle HCI is determined as 244⁰.

The value of angle HCI is calculated by applying intersecting chord theorem as follows;

The intersecting chord theorem, also known as the secant-secant theorem, states that when two chords intersect inside a circle, the products of the segments of one chord are equal to the products of the segments of the other chord.

From the diagram, the value arc CUH is equal to the value of angle HCI.

Thus, angle HCI = 244⁰

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Answer:

310.2 cubic units

Step-by-step explanation:

since we know the diameter of the sphere (8.4), the radius is [tex]8.4/2 = 4.2[/tex]

The volume of a sphere is [tex]\frac{4}{3}\pi r^3[/tex]

Plugging in 3.14 as [tex]\pi[/tex] and 4.2 as r, we get 310.2

function f(x,y) = 2 + y

The minimum value are f(√5, √5) = 2 + √5.

To find the minimum value of the function f(x,y) = 2 + y subject to the constraint xy=5 using Lagrange multipliers,

we first set up the Lagrangian function:

L(x,y,λ) = f(x,y) - λ(xy - 5)

Taking partial derivatives with respect to x, y, and λ, we get:

∂L/∂x = 0 = -λy
∂L/∂y = 1 - λx
∂L/∂λ = xy - 5

Solving for λ from the first equation and substituting into the second equation, we get:

x/y = 0/λ
1 - λx = 0
xy - 5 = 0

From the first equation, we see that either x = 0 or y = 0. But since xy = 5, neither x nor y can be zero.

Therefore, we have:

λ = 0
1 - λx = 0
xy - 5 = 0

Solving for x and y from the last two equations, we get:

x = 5/y
y = ±√5

We take the positive root for y since we are looking for a minimum value of the function.

Substituting y = √5 into x = 5/y, we get x = √5.

Therefore, the minimum value of f(x,y) = 2 + y subject to the constraint xy=5 is:
f(√5, √5) = 2 + √5.

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To find the derivative of y = cos(sin(14x-13)), we will use the chain rule.

Let's start by defining two functions:

u = sin(14x-13)
v = cos(u)

We can now apply the chain rule:

dy/dx = dv/du * du/dx

First, let's find dv/du:

dv/du = -sin(u)

Next, let's find du/dx:

du/dx = 14*cos(14x-13)

Now we can put it all together:

dy/dx = dv/du * du/dx
dy/dx = -sin(u) * 14*cos(14x-13)

But we still need to substitute u = sin(14x-13) back in:

dy/dx = -sin(sin(14x-13)) * 14*cos(14x-13)

So the derivative of y = cos(sin(14x-13)) is:

dy/dx = -14*sin(sin(14x-13)) * cos(14x-13)

To find the derivative of the function y = cos(sin(14x - 13)), we can use the chain rule. The chain rule states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function.

Let u = sin(14x - 13), so y = cos(u). Now we find the derivatives:

1. dy/du = -sin(u)
2. du/dx = 14cos(14x - 13)

Now, using the chain rule, we get:

dy/dx = dy/du × du/dx

dy/dx = -sin(u) × 14cos(14x - 13)

Since u = sin(14x - 13), we can substitute back in:

dy/dx = -sin(sin(14x - 13)) × 14cos(14x - 13)

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