The area of an ellipse with axes of length
2a and 2b is ab.
The percent change in the area when a increases by
2.74% and b increases by 2.65% is

(i) \(\lim_{x \to \pi/3}\) f(x) = 1 + √3.

(ii) \(\lim_{x \to \frac{\pi}{4}} f(x)\) does not exist.

For λ = 2.5, the function f(x) will be continuous at x = 0.

Given the function is,

f(x) = 1.5, when x = 0

    = 1 -2 sin x, when x < π/4

    = 1 + 2 sin x, when x > π/4

    = 1 + (sin λx/sin 5x)

Now,

\(\lim_{x \to \pi/3}\) f(x) = \(\lim_{x \to \pi/3}\) (1 + 2 sin x) [Since, π/3 > π/4]

                    = 1 + 2 sin (π/3)

                    = 1 + 2√3/2 = 1 + √3

Hence, \(\lim_{x \to \pi/3}\) f(x) = 1 + √3.

Now left hand limit is,

\(\lim_{x \to \frac{\pi^+}{4}}\) f(x) = \(\lim_{x \to \frac{\pi^+}{4}}\) (1 + 2sin x) = 1 +2 sin(π/4) = 1 + 2*(1/√2) = 1 + √2

and right hand limit is,

\(\lim_{x \to \frac{\pi^-}{4}}\) f(x) = \(\lim_{x \to \frac{\pi^-}{4}}\) (1 - 2 sin x) = 1 - 2*(1/√2) = 1 - √2

Since left hand limit and right hand limit are not equal so value of \(\lim_{x \to \frac{\pi}{4}} f(x)\) does not exist.

\(\lim_{x \to 0}\) f(x) = \(\lim_{x \to 0}\) (1 + (sin λx/sin 5x)) = 1 + \(\lim_{x \to 0}\) ((sin λx/λx)/(sin 5x/5x))*(λ/5) = 1 + λ/5

Since f(x) is continuous at x = 0.

So, \(\lim_{x \to 0}\) f(x) = f(0)

1 + λ/5 = 1.5

λ/5 = 1.5 - 1 = 0.5

λ = 2.5

Hence the value of λ will be 2.5.

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The value of the algebric  expression \(2^{-3} \cdot 3^{-2}$ is $\frac{1}{72}\).

To compute the expression \(2^{-3} \cdot 3^{-2}\), we can simplify each term separately and then multiply the results.

First, let's simplify \(2^{-3}\). The exponent -3 indicates that we need to take the reciprocal of the base raised to the positive exponent 3. Therefore, \(2^{-3} = \frac{1}{2^3} = \frac{1}{8}\).

Next, let's simplify 3^{-2}. Similar to before, the exponent -2 means we need to take the reciprocal of the base raised to the positive exponent 2. So, \(3^{-2} = \frac{1}{3^2} = \frac{1}{9}\).

Now that we have simplified both terms, we can multiply them together: \(\frac{1}{8} \cdot \frac{1}{9}\). When multiplying fractions, we multiply the numerators together and the denominators together. So, \(\frac{1}{8} \cdot \frac{1}{9} = \frac{1 \cdot 1}{8 \cdot 9} = \frac{1}{72}\).

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The value of the given composite functions would be,

1) (h o k)(3) = 3

2) (k o h)(-4b) = -4b

When two functions are inverses, then each will reverse the effect of the other.

For the functions f and g, we can say that;

(f o g)(x) = x and (g o f)(x) = x

For the given example,

The functions h and k are inverse functions.

Thus, for any real input value 'x',

(h o k)(x) = x and (k o h)(x) = x

So, the value of the given composite functions would be,

1) (h o k)(3) = 3

2) (k o h)(-4b) = -4b

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The value of the length x is 2√70. Option C

To determine the value of the length of the triangle, we need to consider the different trigonometric identities, they are;

Now, using the Pythagorean theorem, we have;

17² = x² + 3²

Find the square values and substitute

289 = x² + 9

collect the like terms

x² = 289 - 9

Subtract the values

x² = 280

Find the square root of both sides

x = √280

Find the values

x = √4× 70

x = 2√70

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Answer: The mean difference is between 799586.3 and 803257.9.

Step-by-step explanation: To estimate the mean difference for confidence interval:

Find the statistic sample:

For the traffic count, mean = 1835.8 and s = 1382607.3

The confidence interval is 90%, so:

α = \(\frac{1-0.9}{2}\)

α = 0.05

The degrees of dreedom are:

df = n - 1

df = 10 - 1

df = 9

Using a t-ditribution table, the t-score for α = 0.05 and df = 9 is: t = 1.833.

Error will be:

E = \(t.\frac{s}{\sqrt{n} }\)

E = 1.833.(\(\frac{1382607.3}{\sqrt{10} }\))

E = 801422.1

The interval is: mean - E < μ < E + mean

1835.8 - 801422.1 < μ < 1835.8+801422.1

-799586.3 < μ < 803257.9

The estimate mean difference in trafic count between 6th and 13th using 90% level of confidence is between 799586.3 and 803257.9.

Answer:

a. 1/25

b. 25/1

Step-by-step explanation:

There are 2 black 7s in a deck (7 of spades and 7 of clubs).  So there are 50 cards that are not black 7s.

The odds in favor of drawing a black 7 is therefore 2/50 = 1/25.

The odds against drawing a black 7 is therefore 50/2 = 25/1.

Answer:

Step-by-step explanation:

Answer:

e-5=2f

take '-5' to the other side where '2f' is

e=2f+5

y = -3(x - 6)

y = -3x + 18

3x + y = 18

Step-by-step explanation:

All you would need to do is to use the circumference of a circle equation, C = 2pi(r) or C=pi(d). Since we are given the radius we will use C= 2pi(r).
C=2pi(4.4)

C=8.8pi inches

Answer:

The correct equation that initially models the value of Rob's boat is option A, y = 40,000(0.93)*.

Since the value of the boat decreased by 7% each year, the value of the boat after three years can be represented by:

y = 40,000(0.93)^3

where y is the value of the boat after three years, and 40,000 is the initial value of the boat.

Simplifying this equation, we get:

y = 40,000(0.7951)

y = 31,804

However, we know that the actual value of the boat after three years was $32,174.28. This means that our initial assumption that the boat decreased by 7% each year is incorrect and we need to adjust the equation accordingly.

To find the correct equation, we can use the formula for exponential decay:

y = a(1 - r)^t

where y is the final value, a is the initial value, r is the rate of decay (expressed as a decimal), and t is the time in years.

In this case, we know that the final value of the boat is $32,174.28, and that the boat was owned for three years. We also know that the value of the boat decreased by 7% each year.

So we can set up an equation:

32,174.28 = 40,000(0.93)^3

Simplifying this equation, we get:

32,174.28 = 31,804.00

This equation is approximately true, which means that the initial value of the boat was $40,000 and the correct equation that initially models the value of Rob's boat is:

y = 40,000(0.93)^t

where t is the time in years.

Answer:

22 1/2

Step-by-step explanation:

The amount that would be in the account when Alana is 21  is $2890

We have to know the formula for simple interest.

This is expressed as;

S.I = PRT/100

Given that the parameters of the formula are enumerated as;

From the information given, substitute the values, we have;

SI = 1000 × 9 × 21/100

Multiply the values,

Then, divide by the denominator, we have;

SI = $1890

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Answer:

c

Step-by-step explanation:

Answer:

c is the correct answer

Answer:

1,2,4,7,14, or 28

Step-by-step explanation:

I got it right

HYPOTENUSE RULE :

So, third side is 10cm.

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Answer:

10

Step-by-step explanation:

\( {8}^{2} + {6}^{2} = {100}^{2 } \)

\( \sqrt{100} = 10\)

Answer: \((12,-3)\ \text{or}\ (3,-12)\)

Step-by-step explanation:

Given

The product of integers is -36

The difference of integers is 15

Suppose the integers are x and y

\(\Rightarrow x-y=15\\\Rightarrow x=15+y\\\Rightarrow xy=-36\)

Put the value of \(x\) in the product

\(\Rightarrow (15+y)y=-36\\\Rightarrow 15y+y^2+36=0\\\Rightarrow y^2+12y+3y+36=0\\\Rightarrow y(y+12)+3(y+12)=0\\\Rightarrow (y+12)(y+3)=0\\\Rightarrow y=-3\ \text{or}\ -12\)

\(\therefore x=12\ \text{or}\ 3\)

Pairs of integers are \((12,-3)\ \text{or}\ (3,-12)\)

Answer:

  You'll get an A and you can celebrate.

Step-by-step explanation:

The hypothesis for the first conditional is met, so we can assume the conclusion is true: you'll get an A.

That conclusion satisfies the hypothesis of the second conditional, so we can assume its conclusion is true: you can celebrate.

Taken together, we can assume ...

  You'll get an A and you can celebrate.

\(\underset{in~degrees}{\textit{sum of all interior angles}}\\\\ n\theta = 180(n-2) ~~ \begin{cases} n=\stackrel{number~of}{sides}\\ \theta = \stackrel{degrees}{angle}\\[-0.5em] \hrulefill\\ n=5 \end{cases}\implies 5\theta =180(5-2) \\\\\\ 5\theta =180(3)\implies 5\theta =540\implies \theta =\cfrac{540}{5}\implies \theta =108\)

The dimensions of the living room are 3 meters by 10 meters, or 10 meters by 3 meters.

What is Rectangle?

The internal angles of a rectangle, which has four sides, are all exactly 90 degrees. At each corner or vertex, the two sides come together at a straight angle. The rectangle differs from a square because its two opposite sides are of equal length.

Let's assume that the length of the rectangular living room is 'l' and its width is 'w'. We know that the area of the living room is 30 square meters, so we can write:

l x w = 30

We also know that the perimeter of the living room is 26 meters. The perimeter of a rectangle is the sum of the lengths of all four sides, so we can write:

2l + 2w = 26

Simplifying this equation, we get:

l + w = 13

We can now use the system of equations to solve for the dimensions of the living room. Solving for one variable in terms of the other, we can substitute the expression into the other equation:

w = 13 - l

l x (13 - l) = 30

Expanding the left side, we get:

13l - l² = 30

Rearranging the terms, we get a quadratic equation:

l² - 13l + 30 = 0

This equation can be factored as:

(l - 3)(l - 10) = 0

So the solutions for l are l = 3 and l = 10. Since we know that the living room is a rectangle, we know that the width w must be the other side of the rectangle. Thus, if l = 3, then w = 13 - l = 10, and if l = 10, then w = 13 - l = 3. Therefore, the dimensions of the living room are 3 meters by 10 meters, or 10 meters by 3 meters.

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The solution to the system of inequalities is (0.667, 5.333)

The system of inequalities is given as:

y > 2x + 4

x + y ≤ 6

Next, we plot the graph of the system using a graphing tool

From the graph, both inequalities intersect at

(0.667, 5.333)

Hence, the solution to the system of inequalities is (0.667, 5.333)

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The solution of the function on the interval is x = 0.524 radians.

The solution of the function on the interval is calculated as follows;

cot(x)= √3

1/tan(x) = √3

Simplify the expression as follows;

1 = √3tan(x)

tan(x) = 1/√3

The solutions of this equation in the interval [-2,2], is calculated as;

x = tan⁻¹(1/√3)

x = 0.524 radians

Another solution of x;

x = tan⁻¹(1/√3) + π = 3.67 radians

This value is not in the given interval, so there is only one solution to the equation cot(x) = √3 in the interval [-2,2], which is approximately x = 0.524 radians.

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The dot isn’t shaded so > or <

From -55, it’s showing small than

From -49, it’s showing greater than

x < -55 or x > -49

Answer:  x < -55 or x > -49

Step-by-step explanation:

x < -55 or x > -49

They both start with an opened circle, and also these two opened circle dots do not connect to each other. On the left side, the number is going from -55 to the smaller side, -57 or less, so we can say that x is smaller than -55. On the right side, the number goes from -49 to the bigger number, we can say that x is greater than -49.

Answer: The quotient is 3y² + 2y - 1.

Step-by-step explanation:

You can use long division to solve this expression by setting the dividend under the roof and the divisor outside as you would normally do when dividing numbers.

1) Look at the first term of the dividend, 15y³, and the first term of the divisor, 5y. Determine what value multiplied by 5y would give you 15y³. The answer is 3y², and given the number, you multiply it by the WHOLE divisor.

3y²(5y + 6) = 15y³ + 18y² <-- subtract this from the dividend.

2) Continue this process and make sure to drag down the next term for each new value you form after subtracting. Keep in mind of any negative values! When you reach up to -5y, determine what value multiplied by 5y would give you the value of -5y, which would be -1.

3) You should be left with a remainder of 0, leaving you with a quotient of 3y² + 2y - 1.

Answer:

ST = 8 m

Step-by-step explanation:

Since both triangles are congruent to each other, therefore their corresponding angles and corresponding side lengths would be equal to each other accordingly.

Side ST in ∆STU corresponds to side PQ in ∆PQR. ST = PQ.

Side PQ = 8 m

Therefore, side ST = PQ = 8 m

Answer:

0.74

Step-by-step explanation:

The middle age citizens are those in the age group of 18-55.

We solve this question using z score formula

z = (x-μ)/σ, where

x is the raw score

μ is the population mean

σ is the population standard deviation.

Mean 46 years

Standard deviation 13 years

For x = 18 years

z = 18 - 46/13

z = -2.15385

Probability value from Z-Table:

P(x = 18) = 0.015626

For x = 55 years

z = 55 - 46/13

= 0.69231

Probability value from Z-Table:

P(x = 55) = 0.75563

The probability that he/she is middle age is

P(x = 55) - P( x = 18)

= 0.75563 - 0.015626

= 0.740004

Approximately = 0.74

The given passage provides a proof that the Separation Axioms follow from the Replacement Schema.

The proof involves introducing a set F and showing that {a: e X : O(x)} is equal to F (X) for every X. Therefore, the conclusion is that the Separation Axioms can be derived from the Replacement Schema.In the given passage, the author presents a proof that demonstrates a relationship between the Separation Axioms and the Replacement Schema.

The proof involves the introduction of a set F and establishes that the set {a: e X : O(x)} is equivalent to F (X) for any given set X. This implies that the conditions of the Separation Axioms can be satisfied by applying the Replacement Schema. Essentially, the author is showing that the Replacement Schema can be used to derive or prove the Separation Axioms. By providing this proof, the passage establishes a connection between these two concepts in set theory.

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The image of the ramp with dimensions is missing, so i have attached it.

Answer:

680 in² of wood is needed.

Step-by-step explanation:

The way to find how much wood would be needed by devon would be to find the total surface area of the ramp.

From the attached image,

Let's find the area of the 2 triangles first;

A1 = 2(½bh) = bh = 15 x 8 = 120 in²

Area of the slant rectangular portion;

A2 = 17 x 14 = 238 in²

Area of the base;

A3 = 15 × 14 = 210 in²

Area of vertical rectangle;

A4 = 8 × 14 = 112 in²

Total Surface Area = A1 + A2 + A3 + A4 = 120 + 238 + 210 + 112 = 680 in²