TJ’s drawer has many loose socks: 5 gray, 5 black, and 6 white. If two socks are randomly pulled out without replacement, what is the probability that he pulls 2 white socks?

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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The equation of the tangent line to f(x) is y = -4x + 4

The equation of the tangent line to f(x) = 1/x at a point x = a is given by:

y - f(a) = f'(a) * (x - a)

where f'(x) is the derivative of f(x) with respect to x.

We can find a "nice" point near 0.504 by choosing a = 0.5, which is close to 0.504 and makes the calculation easy.

The derivative of f(x) = 1/x is given by:

[tex]f'(x) = -1/x^2[/tex]

At x = 0.5, we have:

f(0.5) = 1/0.5 = 2

[tex]f'(0.5) = -1/(0.5)^2 = -4[/tex]

Plugging these values into the equation of the tangent line, we get:

y - 2 = -4 * (x - 0.5)

Simplifying, we get:

y = -4x + 4

Now we can use this tangent line to approximate (1/0.504) as follows:

(1/0.504) ≈ y(0.504)

Plugging x = 0.504 into the equation of the tangent line, we get:

y(0.504) = -4(0.504) + 4 = 1.784

Therefore, (1/0.504) ≈ 1.784

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The actual percent loss on the original $7000 investment is 46%. This means that the investment consultant's statement of a 10% overall increase is not accurate.

To calculate the actual percent gain or loss on the original $7000 investment, we can use the following formula:

Actual percent gain or loss = (Ending value - Beginning value) / Beginning value * 100%

At the end of the first year, the investment decreased by 70% of its original value, which means its value was only 30% of $7000, or $2100.

At the end of the second year, the investment increased by 80% of the value it had at the end of the first year. So, its value at the end of the second year was:

Value at end of second year = $2100 + 80% of $2100
Value at end of second year = $2100 + $1680
Value at end of second year = $3780

Therefore, the actual percent gain or loss on the original $7000 investment is:

Actual percent gain or loss = ($3780 - $7000) / $7000 * 100%
Actual percent gain or loss = -46%

So, the actual percent loss on the original $7000 investment is 46%.

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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.

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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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

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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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 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)

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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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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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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The drink mix that is needed to make one cup of drink is 21/20

Ali uses 21/2 scoops of drink mix to make 10 cups off drinks

The amount of drink mix needed to make one cup can be calculated as follows

21/2= 10

x= 1

cross multiply both sides

10x= 21/2

Divide by the coefficient of x which is 10

x= 21/2 ÷ 10

x= 21/2 × 1/10

x= 21/20

Hence the drink mix needed to make one cup is 21/20

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The equation of the perpendicular line to y = -3x + 4 and passing through (-6, 2) is y = (1/3)x + 4.

There are a lot of perpendicular lines that we may see in reality. The corners of a chalkboard, a window, the sides of a set square, and the Red Cross symbol are a few examples.

Knowing that the slopes of perpendicular lines are the negative reciprocals of one another is necessary to determine the equation of a line perpendicular to a given line.

The given line has a slope of -3, so the slope of a line perpendicular to it would be the negative reciprocal of -3, which is 1/3.

We also know that the line passes through the point (-6, 2).

The equation of the line passing through (-6, 2) and perpendicular to y = -3x + 4 can be found using the point-slope form of a line:

y - 2 = (1/3)(x + 6)

Simplifying this equation, we get:

y = (1/3)x + 4

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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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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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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 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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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.

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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Trevor spent a mean amount of $0.85 per day on snacks during

Week 1.

We have,

To determine during which week Trevor spent a mean amount of $0.85 per day on snacks, we need to calculate the mean (average) amount spent per day for each week and find the week that matches $0.85.

Let's calculate the mean amount spent per day for each week:

Week 1:

= (0.50 + 0.75 + 0.50 + 1.00 + 1.25 + 1.25 + 0.75 + 0.25 + 1.00) / 9

= $0.86 (approximately)

Week 2:

= (1.00 + 0.50 + 0.75 + 0.25 + 0.25 + 1.25 + 0.25) / 7

= $0.61 (approximately)

Week 3:

= (0.75 + 1.00) / 2

= $0.88

Week 4:

= (1.25 + 1.00 + 1.00 + 1.25 + 0.50) / 5

= $1.00

From the calculations, we can see that the mean amount spent per day for Week 1 is closest to $0.85.

Therefore,

Trevor spent a mean amount of $0.85 per day on snacks during

Week 1.

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To understand how to approach problems involving similar triangles, parallel lines, and proportional parts.

When you have two similar triangles, their corresponding sides are proportional. This means that the ratio between the sides of one triangle is the same as the ratio between the sides of the other triangle.

If you have parallel lines cut by a transversal, corresponding angles will be congruent, which can help establish similarity between triangles.

For example, if you have two similar triangles ABC and DEF, where AB/DE = BC/EF = AC/DF, then you can use these ratios to solve for unknown side lengths or other values.

To answer questions 1-16, identify the given information and determine whether the triangles are similar. If they are, use the proportional relationships to solve for the unknowns.

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How do you solve these problems on Unit 6: similar triangles homework 5: parallel lines and proportional parts?

7. X-5/6 = 14/2x+3

16. X-7/35 = 4/x-3

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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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 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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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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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.

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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