Evaluate the limit
And show your work pleaseee

According to the information, the exact values of sin(2u), cos(2u) and tan(2u) are sin(2u) = -120/169, cos(2u) = -1, and tan(2u) = -120/119.

We can use the double-angle formulas to find the values of sin(2u), cos(2u), and tan(2u) in terms of sin(u) and cos(u).

First, we need to find cos(u) using the Pythagorean identity:

Since pi/2 < u < pi, we know that sin(u) is positive and cos(u) is negative. Therefore:

Now we can substitute sin(u) and cos(u) into the double-angle formulas:

Therefore, sin(2u) = -120/169, cos(2u) = -1, and tan(2u) = -120/119.

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Answer:

apples per pound: 0.25

pears per pound: 0.33

Apples give him the best value per pound

Step-by-step explanation:

to find apple per pound: 4/16= 0.25

to find pears per pound: 7/21= 0.33

The areas of the figures are 72 square centimeter and 81 square inches

Figure 3

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

The composite figure

The total area of the composite figure is the sum of the individual shapes

So, we have

Area = 9 * (12 - 4 - 4) + 12 * 3

Evaluate

Area = 72

Figure 4

The total area of the composite figure is the difference between the area of the trapezoid and the rectangle

So, we have

Area = 1/2 * (8 + 16) * 8 - 5 * 3

Evaluate

Area = 81

Hence. the area of the figure (3) is 81 square inches

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Answer:

Step-by-step explanation:

We are given four graphs, and we want to determine which one is a function.

When given graphs, we can use the vertical line test to determine if a graph is a function or not. If the vertical line hits only one point on the graph, it means that it is a function.

If the vertical line hits more than one point, it means it is not a function.

This is because a function must have one unique output for every input. In other words, if we substitute 3 into a function, it cannot equal both 2 AND 4.

We can draw a vertical line down each graph. Feel free to use an online annotating tool to do this, or simply use your finger to draw a vertical line down each graph.

On the first one, we can see that if we draw a vertical line down it, the vertical line hits only one point on the graph. This means that the graph shows a function.

On the second graph, if we draw a vertical line through it, we can see it hits more than one point on the graph. This means the graph is not a function.

On the third graph, if we draw a vertical line through it, we can see it also hits more than one point on the graph, so it is not a function.

The final graph is a vertical line itself. If we draw a vertical line through it, all of the points will be shared; the line hits all of the points in the graph, so it is not a function.

This means that the first graph is a function.

See below for annotations.

Answer:

Draw three overlapping circles to represent the sets A, B, and C.

Label the regions inside each circle with the name of the corresponding set (i.e., A, B, and C).

Shade the region that represents the set A - B, which includes all the elements that belong to A but not to B.

Finally, shade the region that represents the intersection of (A - B) and C, which includes all the elements that belong to both (A - B) and C.

The resulting diagram should show three overlapping circles, with the region for A shaded but with a part removed (to represent the exclusion of elements in B from A), and the region for (A - B) n C shaded within that.

Answer:

The answer is d.

Step-by-step explanation:

Let m = a number

Let n = a different number

[tex]log(m^n)=nlog(m)[/tex]

[tex]log(10^3)=3*log(10)[/tex]

[tex]log(10)=1[/tex]

[tex]log(10^3)=3[/tex]

The HCF of 420 and 272 is 4.

To find the HCF of 420 and 272 using Euclid's division algorithm, we can proceed as follows:

Step 1: Divide 420 by 272

420 = 272 x 1 + 148

Step 2: Divide 272 by 148

272 = 148 x 1 + 124

Step 3: Divide 148 by 124

148 = 124 x 1 + 24

Step 4: Divide 124 by 24

124 = 24 x 5 + 4

Step 5: Divide 24 by 4

24 = 4 x 6 + 0

So, we can see that the last non-zero remainder obtained by Euclid's division algorithm is 4. Therefore, the HCF of 420 and 272 is 4.

Now, let's verify this using the Fundamental Theorem of Arithmetic.

The Fundamental Theorem of Arithmetic states that every positive integer greater than 1 can be expressed as a product of prime numbers in a unique way, up to the order of the factors.

The prime factorization of 420 is:

420 = 2^2 x 3 x 5 x 7

The prime factorization of 272 is:

272 = 2^4 x 17

To find the HCF using the prime factorizations, we need to take the product of the common prime factors with the smallest exponent. In this case, the only common prime factor is 2, and it appears with an exponent of 2 in 420 and an exponent of 4 in 272. So, the HCF is:

HCF(420, 272) = 2^2 = 4

This matches the result obtained using Euclid's division algorithm. Therefore, we have verified that the HCF of 420 and 272 is indeed 4.

The right answers  outoff these options are: 3,5,6,8

Length is a measure of the size or distance of an object or space in one dimension, usually referring to the longest dimension of an object. It is typically measured in units such as inches, meters, or feet.

In geometry, a triangle is a closed two-dimensional shape that is formed by connecting three non-collinear points with straight line segments. The line segments are called sides, and the points where they meet are called vertices. A triangle has three angles, which are formed by the intersection of its sides.

To determine the possible length of the third side of a triangle, we need to apply the triangle inequality theorem which states that the sum of the lengths of any two sides is always greater than the third side.

Using this theorem, we can see that the possible lengths for the third side are:

- 3 inches

- 5 inches

- 6 inches

- 8 inches

Therefore, the correct answers are:

3,5,6,8.

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Answer:

Area = length * width

Step-by-step explanation:

If all the sides of the room are equal, all you have to do is 20*20 = area

Answer:

Step-by-step explanation:

Infra-red waves commonly known as heat waves and light rays are travel in a speed [tex]3*10^{8} ms^{-1}[/tex] in a vacuum.

So the volume of the cylindrical pool is approximately 1884.96 cubic feet.

So the volume of the soccer ball is approximately 329.59 cubic inches.

So the volume of the small container is approximately 56.55 cubic inches.

So the volume of the tank is 63 cubic feet.

In mathematics, volume is a measure of the amount of space that a three-dimensional object occupies. It is typically expressed in cubic units, such as cubic meters, cubic centimeters, cubic feet, or cubic inches. The volume of a simple shape, such as a cube, rectangular prism, cylinder, or sphere, can be calculated using a specific formula based on its dimensions.

Here,

13. The diameter of the cylindrical pool is 20 feet, which means the radius is 10 feet (half of the diameter). The depth of the pool is 6 feet. The volume of a cylinder is given by the formula V = πr²h, where r is the radius and h is the height or depth. Substituting the given values, we get:

V = π(10 ft)²(6 ft)

≈ 1884.96 cubic feet

14. The soccer ball has a radius of 4.3 inches. The volume of a sphere is given by the formula V = (4/3)πr³, where r is the radius. Substituting the given value, we get:

V = (4/3)π(4.3 in)³

≈ 329.59 cubic inches

15. The conical container has a radius of 3 inches and a height of 6 inches. The volume of a cone is given by the formula V = (1/3)πr²h, where r is the radius and h is the height. Substituting the given values, we get:

V = (1/3)π(3 in)²(6 in)

≈ 56.55 cubic inches

16. The tank is in the shape of a pyramid, which means it has a rectangular base with length 3 feet and width 7 feet, and a height of 9 feet. The volume of a pyramid is given by the formula V = (1/3)Bh, where B is the area of the base and h is the height. The base of the tank is a rectangle, so its area is given by the formula A = lw, where l is the length and w is the width. Substituting the given values, we get:

A = (3 ft)(7 ft)

= 21 square feet

V = (1/3)(21 sq ft)(9 ft)

= 63 cubic feet

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Step-by-step explanation:

There are six sides...   3 pairs of equal sides

2 x ( 4x5  +  4x7   +  5x7)   = 166 cm^2

Answer:

166 cm²

Step-by-step explanation:

5x7=35

35x2=70

4x7=28

28x2=56

5x4=20

20x2=40

70+56+40= 166cm²

hope this helps!

Por lo tanto, el área del vidrio de la lupa es de aproximadamente 50.24 centímetros cuadrados.

What is el área?

"El área" is a Spanish phrase that translates to "the area" in English. "The area" is a mathematical term that refers to the measure of the size of a two-dimensional region, usually measured in square units such as square centimeters (cm²) or square meters (m²). The area of a figure can be calculated using various mathematical formulas, depending on the shape of the figure.

El área del vidrio de una lupa circular se puede calcular usando la fórmula del área del círculo, que es A = πr², donde "A" es el área, "π" es una constante aproximadamente igual a 3.14 y "r" es el radio de la lupa.

En este caso, el radio de la lupa es de 4 centímetros. Sustituyendo este valor en la fórmula, obtenemos:

A = πr²

A = 3.14 x 4²

A = 3.14 x 16

A = 50.24

Por lo tanto, el área del vidrio de la lupa es de aproximadamente 50.24 centímetros cuadrados.

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Answer:

x = 2

Step-by-step explanation:

11 = 3 + 4x

8 = 4x

x = 2

Let's Check

11 = 3 + 4(2)

11 = 3 + 8

11 = 11

So, x = 2 is the correct answer.

The Results:

a) Cost of the bouncy castle for 20 children = £149

b) Number of children who went to the party = 14.

a) To find the cost of the bouncy castle for 20 children, we can substitute the number of children, which is 20, into the formula provided:

Cost = £3.65 * number of children + £76

Cost = £3.65 * 20 + £76

Now we can estimate the cost:

Cost = £73 + £76

Cost = £149

Thus, the cost of the bouncy castle for 20 children = £149.

b) To find the number of children who went to the party based on the given cost of £127.10, we can rearrange the formula to solve for the number of children:

Cost = £3.65 * number of children + £76

£127.10 = £3.65 * number of children + £76

Subtracting £76 from both sides of the equation:

£127.10 - £76 = £3.65 * number of children

£51.10 = £3.65 * number of children

Now we can divide both sides of the equation by £3.65 to isolate the variable:

£51.10 / £3.65 = number of children

14 = number of children

Therefore, the number of children who went to the party = 14.

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Answer:

  y = (1/3)(e^(-9(x+2)/(x+1)) -1)

Step-by-step explanation:

You want the particular solution to the differential equation ...

  y' = 3(1+3y)/(1+x)^2   with  y(-2) = 0

The differential equation can be rewritten as ...

  [tex]\dfrac{dy}{3(1+3y)}=\dfrac{dx}{(1+x)^2}[/tex]

We can define u = 1+3y, then du = 3 and the left side becomes ...

  dy/(3(1+3y)) = (1/9)du/u

and its integral is ...

  ∫(1/9)du/u = (1/9)ln(u) = (1/9)ln(1 +3y)

The integral of the right side is ...

  [tex]\int{(1+x)^{-2}}\,dx=-(1+x)^{-1}+C[/tex]

Then the result of integrating both sides of this rewritten differential equation is ...

  [tex]\dfrac{1}{9}\ln{(1+3y)}=-\dfrac{1}{1+x}+C[/tex]

The boundary condition can be used to find C:

  [tex]\dfrac{\ln(1+3\cdot0)}{9}=-\dfrac{1}{1-2}+C\\\\0=1+C\\\\C=-1[/tex]

Solving this equation for y, we get ...

  [tex]\ln(1+3y)=-9\left(\dfrac{1}{1+x}+1\right)\\\\\\1+3y=e^{\left(-\dfrac{9(x+2)}{x+1}\right)}\\\\\\\boxed{y=\dfrac{e^{\left(-\dfrac{9(x+2)}{x+1}\right)}-1}{3}}[/tex]

We have shown that dⁿ⁺¹ / d xⁿ⁺¹ (xⁿ ln x) = n!/x using Leibniz's theorem.

What is the Leibniz's theorem?

Leibniz's theorem, also known as the product rule for derivatives, is a rule for finding the nth derivative of a product of two functions. It states that if u(x) and v(x) are functions of x, then the nth derivative of their product

To apply Leibniz's theorem, we need to express the function in terms of a product of two functions: u and v.

Let u = xⁿ and v = ln(x)

Then, du/dx = n*xⁿ⁻¹ and dv/dx = 1/x

Using the formula,

d/dx(xⁿ ln(x)) = u(dv/dx) + v(du/dx)

= xⁿ * (1/x) + ln(x) * n * xⁿ⁻¹

= xⁿ⁻¹ + n*xⁿ⁻¹ ln(x)

To find the nth derivative, we differentiate n times using the product rule.

First, we need to find the first few derivatives:

f(x) = xⁿ⁻¹ + n*xⁿ⁻¹ ln(x)

[tex]f'(x) = nx^{(n-2)} + (n-1)*x^{(n-2)} ln(x)[/tex]

[tex]f''(x) = n(n-2)x^{(n-3)} + 2(n-1)x^{(n-3)} ln(x) - (n-1)x^{(n-3)}/(x^2)[/tex]

[tex]f'''(x) = n(n-2)(n-3)x^{(n-4)} + 3(n-1)(n-2)x^{(n-4)} ln(x) - 3(n-1)x^{(n-4)}/(x^2) - 2(n-1)x^{(n-4)} ln(x)/(x^2)[/tex]

and so on...

The pattern becomes apparent and the nth derivative can be expressed as:

[tex]f^{(n)}(x) = n!/(x^{(n+1)})[/tex]

So,

dⁿ⁺¹ / d xⁿ⁺¹ (xⁿ ln x) = [tex]f^{(n)}(x) = n!/(x^{(n+1)})[/tex]

Hence, we have shown that dⁿ⁺¹ / d xⁿ⁺¹ (xⁿ ln x) = n!/x using Leibniz's theorem.

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The probability that exactly 2 students prefer salad is 0.302

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

n = 10

r = 2

p = 20%

The probability is then calculated as

P(x = 2) = nCr * p^x * (1 - p)^(n - x)

substitute the known values in the above equation, so, we have the following representation

P(x = 2) = 10C2 * (20%)^2 * (1 - 20%)^8

Evaluate

P(x = 2) = 0.302

Hence, the probability is 0.302

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The probability of selecting a 2 given that a blue disk is selected is 1/8, or 0.125.

1. Let's calculate the probability of selecting a blue disk

We can select a blue disk with number 2, 6 or 8 out of a total of 8 disks, it means that the probability of selecting a blue disk is

3/8

2. Let's calculate the probability of selecting the blue disk labeled with number 2:

We can select the blue disk labeled with number 2 out of three blue disks, it means that the probability of selecting the blue disk labeled with number 2 is  1/3

3. These are dependent events because the fact that we select the disk labeled with number 2, depends on selecting a blue disk before, thus:

Probability of selecting a 2 given that a blue disk is selected =

3/8 * 1/3 = 3/24 = 1/8

So we can state that the probability of selecting a 2 given that a blue disk is selected is 1/8, or 0.125.


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Full Question:

See the attached image.

Answer:

C

Step-by-step explanation:

if we have a number line that has fractions listed, it should be C, because if 6/2 makes 3, which would be D, 5/2 should be C.

The image coordinates of M′ are M′(0, 3).

What is translation?

A translation is a geometric transformation when each point in a figure, shape, or space is moved in a specific direction by the same amount. A translation can also be thought of as moving the origin of the coordinate system or as adding a constant vector to each point.

Here, we have

Given: Polygon JKLM is drawn with vertices J(−2, −5), K(−4, 0), L(−1, 2), M (0, −1).

When the shape is moved up by k units, then replace y with y + k.

The translation is as follows:

Here, J(-2, -5) → (-2, -5+4) → J'(-2, -1)

K(-4, 0) → (-4, 0+4) → K'(-4, 4)

L(-1, 2) → (-1, 2+4) → (-1, 6)

M(0, -1) → (0, -1+4) → (0, 3)

Hence, the image coordinates of M′ are M′(0, 3).

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The set of all elements that are in either A or B is the union of A and B, which is 1, 2, 3, 4, 5, 6, 8, 9, 10. As a result, A u B = 1, 2, 3, 4, 5, 6, 8, 9, 10

Set operations are done on two or more sets to produce a combination of components based on the operation. There are three primary sorts of operations done on sets in a set theory, such as: Intersection of sets () Union of sets () Set difference (-).

¡) (A n B) u C:

The value of the intersection of A and B is 5, 8. As a result, (A n B) = 5, 8. The union of this set with C is thus 1, 2, 3, 4, 5, 6, 8. As a result, (A n B) u C = 5, 8 u 1, 2, 3, 4, 5, 6, 8 = 1, 2, 3, 4, 5, 6, 8.

¡¡) (A-B) u C:

A-B is the set of elements in A that are not in B, which is 3, 6. The union of this set with C is thus 1, 2, 3, 4, 5, 6, 8. As a result, (A-B) u C = 3, 6 u 1, 2, 3, 4, 5, 6, 8 = 1, 2, 3, 4, 5, 6, 8.

¡¡¡) (A u B):

The set of all elements that are in either A or B is the union of A and B, which is 1, 2, 3, 4, 5, 6, 8, 9, 10. As a result, A u B = 1, 2, 3, 4, 5, 6, 8, 9, 10.

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¡) (A ∩ B) ∪ C  = {1,2,3,4,5,6,8}.

¡¡) (A-B) ∪ C = {1,2,3,4,5,6,8,10}.

¡¡¡) (A ∪ B)' is the set of elements that are in U but not in {1,2,3,4,5,6,8,9,10}. This is simply the set {7}.

What is probability?

Probability is a measure of the likelihood of an event occurring.

Recall that:

A ∩ B denotes the intersection of sets A and B, i.e., the set of elements that are in both A and B.

A ∪ B denotes the union of sets A and B, i.e., the set of elements that are in either A or B or both.

A' denotes the complement of set A, i.e., the set of elements that are not in A.

Using these notations, we can solve each part of the question as follows:

¡) (A ∩ B) ∪ C

First, we need to find A ∩ B, which is the set of elements that are in both A and B. We can see that A={3,5,6,8,10} and B={1,2,4,5,8,9}. So, A ∩ B={5,8}.

Next, we take the union of (A ∩ B) and C. We have C={1,2,3,4,5,6,8}, so the final set is:

(A ∩ B) ∪ C = {5,8} ∪ {1,2,3,4,5,6,8} = {1,2,3,4,5,6,8}

Therefore, (A ∩ B) ∪ C = {1,2,3,4,5,6,8}.

¡¡) (A-B) ∪ C

First, we need to find A-B, which is the set of elements that are in A but not in B. We can see that A={3,5,6,8,10} and B={1,2,4,5,8,9}. So, A-B={3,6,10}.

Next, we take the union of (A-B) and C. We have C={1,2,3,4,5,6,8}, so the final set is:

(A-B) ∪ C = {3,6,10} ∪ {1,2,3,4,5,6,8} = {1,2,3,4,5,6,8,10}

Therefore, (A-B) ∪ C = {1,2,3,4,5,6,8,10}.

¡¡¡) (A ∪ B)'

Recall that A ∪ B denotes the union of sets A and B, i.e., the set of elements that are in either A or B or both. So, (A ∪ B)' denotes the complement of A ∪ B, i.e., the set of elements that are not in A or B.

We can see that A={3,5,6,8,10} and B={1,2,4,5,8,9}. So, A ∪ B={1,2,3,4,5,6,8,9,10}. Therefore, (A ∪ B)' is the set of elements that are not in {1,2,3,4,5,6,8,9,10}.

The universal set U={1,2,...,10}, so (A ∪ B)' is the set of elements that are in U but not in {1,2,3,4,5,6,8,9,10}. This is simply the set {7}.

Therefore, (A ∪ B)'={7}.

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Brainiest please.

Answer:

The answer is -3

Step-by-step explanation:

log(1/1000)=x

log 10(1/1000)=x

10^x=(1/1000)

10^x=10⁰/10³

10^x=10^-3

x= -3

Answer:

-3

Step-by-step explanation:

I did the test

Hope this helps :)

(PLS MARK BRAINLIEST)

Answer:

The average velocity for a time period is given by the change in position divided by the change in time. For the time period beginning when t=3 and lasting 0.01 seconds, we have:

The change in position is y2 - y1 = -0.603 feet, and the change in time is t2 - t1 = 0.01 seconds. Therefore, the average velocity for this time period is:

average velocity = change in position / change in time = (-0.603 feet) / (0.01 seconds) ≈ -60.3 feet/second

Similarly, for the time periods beginning when t=3 and lasting 0.005 seconds, 0.002 seconds, and 0.001 seconds, we have:

To estimate the instantaneous velocity when t=3, we can calculate the derivative of the position function with respect to time:

y = 33t - 20t²

dy/dt = 33 - 40t

At t=3, we have:

dy/dt = 33 - 40(3) = -87

Therefore, the estimated instantaneous velocity when t=3 is -87 feet/second.

The correct option for the given question is c) lim x→a¯ f(x) = f(a) when the function f(x) is said to have jump discontinuity at a point x=a.

Jump continuity is a concept in calculus that describes the behaviour of a function at a specific point where the function jumps from one value to another value without any intermediate values. In other words, a function is considered jump continuous at a point if the function approaches a finite limit from both the left and right sides of that point, but the function values on the left and right sides of the point are not equal.

According to the given information:

The correct notation for the left-hand limit as x approaches a from the left side is lim x→a¯, where the horizontal line above the "a" indicates approaching from the left side.

The statement "lim x→a¯ f(x) = f(a)" means that the limit of f(x) as x approaches a from the left side is equal to the value of f(a) at x = a. This indicates that the function f(x) has a jump discontinuity at x = a, where the function jumps from one value to another value at that specific point.

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Answer: 3 and 14

Step-by-step explanation: x ft = 3ft and x + 4 x = 10 (as you stated) x would be 10 so (10 + 4)ft= 14ft

The mode, mean, and median are relatively close together (mode=33, mean=32.125, median=32), we can conclude that this data set is symmetric and does not have any significant outliers.

The area of mathematics known as statistics is concerned with the gathering, examination, interpretation, presentation, and arrangement of data.

The mean, median, and mode are all measures of central tendency that describe the center of a data set in different ways.

The mean is calculated by dividing the total number of values in a data collection by their sum.

The median is the median of the data set when the values ​​are ordered.  It is not affected by outliers.

The value that appears most frequently in a data collection is the mode.

It may not be unique or exist in a data set.

For the given data set, the mode is likely to be the best measure of center as it is the value that occurs most frequently. In this case, the mode is 33, which occurs three times, whereas all other values occur only once or twice. Therefore, the mode is the most representative value for this data set.

Alternatively, we can also calculate the mean and median to see how they compare with the mode:

The mean is (33+33+33+31+31+34+32+31)/8 = 32.125

The median is the middle value when the values are arranged in order, which is 32.

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The surface area of the triangular prism is 511 in²

An exponential equation is an expression that shows how numbers and variables using mathematical operators.

The surface area of the prism is gotten by summing the individual area of each surface of the prism.

The surface area = (0.5 * 11 in * 7 in) + (0.5 * 11 in * 7 in) + (11 in * 14 in) + (9 in * 14 in) + (8 in * 14 in) = 511 in²

The surface area is 511 in²

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