find the length of IG

1. The limit of (sin3x)/(2x - sinx) as x approaches 0 is -27.

2. The limit of x^(-1)lnx as x approaches infinity is -1.

3. The limit of x/e^x as x approaches infinity is 0.

1. To find the limit of (sin3x)/(2x - sinx) as x approaches 0 using L'Hôpital's rule, we can differentiate the numerator and denominator separately and take the limit again:

Let's differentiate the numerator and denominator:

Numerator: d/dx (sin3x) = 3cos3x

Denominator: d/dx (2x - sinx) = 2 - cosx

Now, we can find the limit of the differentiated function as x approaches 0:

lim x->0 (3cos3x)/(2 - cosx)

Again, differentiating the numerator and denominator:

Numerator: d/dx (3cos3x) = -9sin3x

Denominator: d/dx (2 - cosx) = sinx

Taking the limit as x approaches 0:

lim x->0 (-9sin3x)/(sinx)

Now, substituting x = 0 into the function gives:

(-9sin0)/(sin0) = 0/0

Since we obtained an indeterminate form of 0/0, we can apply L'Hôpital's rule again.

Differentiating the numerator and denominator:

Numerator: d/dx (-9sin3x) = -27cos3x

Denominator: d/dx (sinx) = cosx

Taking the limit as x approaches 0:

lim x->0 (-27cos3x)/(cosx)

Now, substituting x = 0 into the function gives:

(-27cos0)/(cos0) = -27/1 = -27

Therefore, the limit of (sin3x)/(2x - sinx) as x approaches 0 is -27.

2. To find the limit of x^(-1)lnx as x approaches infinity using L'Hôpital's rule, we can differentiate the numerator and denominator separately and take the limit again:

Let's differentiate the numerator and denominator:

Numerator: d/dx (lnx) = 1/x

Denominator: d/dx (x^(-1)) = -x^(-2) = -1/x^2

Now, we can find the limit of the differentiated function as x approaches infinity:

lim x->∞ (1/x)/(-1/x^2)

Simplifying the expression:

lim x->∞ -x/x = -1

Therefore, the limit of x^(-1)lnx as x approaches infinity is -1.

3. To find the limit of x/e^x as x approaches infinity using L'Hôpital's rule, we can differentiate the numerator and denominator separately and take the limit again:

Let's differentiate the numerator and denominator:

Numerator: d/dx (x) = 1

Denominator: d/dx (e^x) = e^x

Now, we can find the limit of the differentiated function as x approaches infinity:

lim x->∞ (1)/(e^x)

Since the exponential function e^x grows much faster than any polynomial function, the denominator goes to infinity much faster than the numerator. Therefore, the limit of (1)/(e^x) as x approaches infinity is 0.

Thus, the limit of x/e^x as x approaches infinity is 0.

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A. To determine the least and greatest percentage of students who could be in favor of year-round school, we can use the error given in the survey, which is +/5%. Let's denote the actual percentage of students in favor of year-round school as x.

The least percentage can be found by subtracting 5% from the reported percentage of 32%:

32% - 5% = 27%

So, the least percentage of students in favor of year-round school is 27%.

The greatest percentage can be found by adding 5% to the reported percentage of 32%:

32% + 5% = 37%

Therefore, the greatest percentage of students in favor of year-round school is 37%.

Hence, the least percentage is 27% and the greatest percentage is 37%.

B. A classmate claiming that ⅓ of the student body is actually in favor of year-round school conflicts with the survey data. According to the survey, the reported percentage in favor of year-round school is 32%, which is not equal to 33.3% (⅓). Therefore, the classmate's claim contradicts the survey results.

It's important to note that the survey provides specific data regarding the percentages of students in favor and opposed to year-round school. The claim of ⅓ being in favor does not align with the survey's findings and should be evaluated separately from the survey data.

The pattern between the numbers 3, 20, 110, and 1715 can be found using a mathematical method. In order to get the following number from each, there is a sequence that must be applied.Let's take a look at the sequence that was used to generate these numbers:

The first number is multiplied by 2 and then increased by 14 to get the second number. For example:

3 x 2 + 14 = 20

Then, the second number is multiplied by 3 and 20, and 110 is added.

20 x 3 + 110 = 170

The third number is multiplied by 4 and then increased by 110.

110 x 4 + 110 = 550

Finally, the fourth number is multiplied by 5 and then increased by 110.

550 x 5 + 110 = 2825

Therefore, using the above formula, the next number in the sequence can be calculated:

1715 x 6 + 110 = 10400

As a result, the sequence of numbers 3, 20, 110, 1715, 10400 can be calculated using the mathematical formula stated above.

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Answer:

Step-by-step explanation:

To find the product of the given expression, we can use the rules of multiplication and apply them to each term within the parentheses:

(p^3)(2p^2 - 4p)(3p^2 - 1)

Expanding the expression, we multiply each term within the parentheses:

= p^3 * 2p^2 * 3p^2 - p^3 * 2p^2 * 1 - p^3 * 4p * 3p^2 + p^3 * 4p * 1

Simplifying further, we combine like terms and perform the multiplication:

= 6p^7 - 2p^5 - 12p^6 + 4p^4

Therefore, the product of (p^3)(2p^2 - 4p)(3p^2 - 1) is 6p^7 - 2p^5 - 12p^6 + 4p^4.

The product of the expressions (p^3)(2p^2 - 4p)(3p^2 - 1) is computed using the distributive property, resulting in 6p^7 - 12p^6 - 2p^5 + 4p^4.

To find the product of the given expressions: (p^3)(2p^2 - 4p)(3p^2 - 1), you will need to apply the distributive property, also known as the multiplication across addition and subtraction.

First, distribute p^3 across all terms inside the brackets of the second expression, then do the same with the result across the terms of the third expression.

Steps are as follows:

All together, it results in: 6p^7 - 12p^6 - 2p^5 + 4p^4. That is the product of the initial expressions.

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Answer:

Volume : 300

Surface Area : 280

Step-by-step explanation:

Volume : 6*5*10

Surface Area : 50+50+30+30+60+60

a) Passed at least one subject: 100 students

b) Passed exactly two subjects: 90 students

c) Passed Geography and failed Mathematics: 20 students

d) Passed all three subjects: 15 students

e) Failed Mathematics given that they passed History: 30 students.

To solve this problem, let's draw a Venn diagram to visualize the information given:

In the Venn diagram above, the circles represent the three subjects: Mathematics (M), History (H), and Geography (G). The numbers outside the circles represent the students who did not pass that particular subject, and the numbers inside the circles represent the students who passed the subject. The numbers in the overlapping regions represent the students who passed multiple subjects.

Now, let's answer the questions:

a) Passed at least one subject:

To find the number of students who passed at least one subject, we add the number of students in each circle (M, H, and G), subtract the students who passed two subjects (since they are counted twice), and add the students who passed all three subjects.

Total = M + H + G - (M ∩ H) - (M ∩ G) - (H ∩ G) + (M ∩ H ∩ G)

Total = 70 + 60 + 45 - 30 - 25 - 35 + 15

Total = 100

Therefore, 100 students passed at least one subject.

b) Passed exactly two subjects:

To find the number of students who passed exactly two subjects, we sum the students in the overlapping regions (M ∩ H, M ∩ G, and H ∩ G).

Total = (M ∩ H) + (M ∩ G) + (H ∩ G)

Total = 30 + 25 + 35

Total = 90

Therefore, 90 students passed exactly two subjects.

c) Passed Geography and failed Mathematics:

To find the number of students who passed Geography and failed Mathematics, we subtract the number of students in the intersection of M and G from the number of students who passed Geography.

Total = G - (M ∩ G)

Total = 45 - 25

Total = 20

Therefore, 20 students passed Geography and failed Mathematics.

d) Passed all three subjects:

To find the number of students who passed all three subjects, we look at the overlapping region (M ∩ H ∩ G).

Total = (M ∩ H ∩ G)

Total = 15

Therefore, 15 students passed all three subjects.

e) Failed Mathematics given that they passed History:

To find the number of students who failed Mathematics given that they passed History, we subtract the number of students in the intersection of M and H from the number of students who passed History.

Total = H - (M ∩ H)

Total = 60 - 30

Total = 30

Therefore, 30 students failed Mathematics given that they passed History.

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The statement that is correct about the dot plots is that:

A) The distribution for Class M is approximately symmetric

A symmetric distribution in dot plots is defined as a distribution with a vertical line of symmetry in the center of the graphical representation, so that the mean is equal to the median.

A symmetric distribution is one that occurs when the values ​​of a variable occur at regular frequencies, and often the mean, median, and mode all occur at the same point. If we draw a line through the middle of the chart, we can see that the two planes mirror each other.  

Now, from the given two dot plots of class M and Class N, it is very clear that class M is very close to being symmetric about the data value 3.

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Answer:

34.63

Step-by-step explanation:

The 2 in 34.6285 marks the hundredths place

We can round by seeing what number is to the right of it which is 8.

Since 8 is larger than 5, we round up by 1.

Thus, 34.63 is our answer

The answer is:

34.63

Work/explanation:

Rounding to the nearest hundredth means rounding to 2 decimal places (DP).

So we have 2 decimal places, and the 3rd one matters too because it will determine the value of the 2nd DP.

The third DP is 8. Since it's greater than 5, then the value of the 2nd DP will be rounded up. I will add 1 to it.

Remember that we round up when the digit that we're rounding to is followed by another digit that is greater than or equal to 5.

So we round :

[tex]\bf{34.6285 =\!=\!\!\! > 34.63}[/tex]

The values in the domain of fᵒg are all real numbers.

Therefore, the correct answer is: x - 6.

To determine the domain of the composite function fᵒg, we need to find the values of x that are valid inputs for the composition.

The composite function fᵒg represents applying the function f to the output of the function g. In this case, g(x) is equal to -9.

So, we substitute -9 into the function f(x) = √6x:

f(g(x)) = f(-9) = √6(-9) = √(-54)

Since the square root of a negative number is not defined in the set of real numbers, the value √(-54) is undefined.

Therefore, -9 is not in the domain of fᵒg.

To find the values in the domain of fᵒg, we need to consider the values of x that make g(x) a valid input for f(x).

Since g(x) is a constant function equal to -9, it does not impose any restrictions on the domain of f(x).

The function f(x) = √6x is defined for all real numbers, as long as the expression inside the square root is non-negative.

So, any value of x would be in the domain of fᵒg.

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Step-by-step explanation:

87,959 ==> 88,000 is the nearest hundred

The answer is:

Work/explanation:

When rounding to the nearest hundredth, round to 2 decimal places (DP).

Which means we should round to 5.

5 is followed by 9, which is greater than or equal to 5. So, we drop 9 and add 1 to 5 :

87. 96

10 and 5 are appropriate x-axis and y-axis unit scales given the coordinates (50, 40). Option B.

To determine the appropriate x-axis and y-axis unit scales given the coordinates (50, 40), we need to consider the relationship between the units on the axes and the corresponding values in the coordinate system.

The x-axis represents the horizontal dimension, while the y-axis represents the vertical dimension. The x-coordinate (50) represents a position along the x-axis, and the y-coordinate (40) represents a position along the y-axis.

To determine the appropriate scales, we need to consider how many units on the x-axis are needed to span the distance from 0 to 50 and how many units on the y-axis are needed to span the distance from 0 to 40.

In this case, the x-coordinate is 50, which means we need the x-axis to span a distance of 50 units. However, we don't have enough information to determine the scale for the x-axis accurately. Therefore, options A (50; 10) and D (50; 1) cannot be definitively chosen.

Similarly, the y-coordinate is 40, which means we need the y-axis to span a distance of 40 units. Considering the given options, option B (10; 5) would be a suitable scale, as it allows for the y-axis to span the necessary distance of 40 units.

In summary, given the coordinates (50, 40), the appropriate unit scales would be 10 units per increment on the x-axis and 5 units per increment on the y-axis (Option B).

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Answer:

Any equation that has the power of x at 1

Step-by-step explanation:

Since the function f(x) = -2x -3 +8 has an infinite range, so any other equation that only contains x^1 would work

Answer:

2π(5)(10) = 100π yd² = about 314 yd²

B is the correct answer.

These are two instances where the product of two numbers yields a negative result.

To demonstrate two products that both result in negative values, we can choose two numbers with opposite signs and multiply them together. Here are two examples:

Example 1:

Let's consider the numbers -3 and 4. When we multiply these numbers, we get:

(-3) * (4) = -12

The product -12 is a negative value.

Example 2:

Let's consider the numbers 5 and -2. When we multiply these numbers, we get:

(5) * (-2) = -10

Once again, the product -10 is a negative value.

In both examples, we have chosen two numbers with opposite signs, and the multiplication of these numbers results in a negative value. Therefore, these are two instances where the product of two numbers yields a negative result.

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Answer:

| x | -7

---------------

x | x² | -7x

-4 | -4x | 28

x² - 11x + 28 = (x - 4)(x - 7)

The correct option is A. [13(cos 22.6 - isin 22.6)] in which the modulus is 13 and the argument is 22.6 degrees.

Given the complex number z = 12 + 5i. We have to express this complex number in the polar form which is\[z=r\cos\theta + i\sin\theta\]where r is the modulus and θ is the argument of the complex number.

The modulus of the complex number is given by,|z|=√(12²+5²)=√(144+25)=√169=13

Therefore, the modulus of the complex number is 13.

Now, we need to find the argument of the complex number, which is given byθ=tan⁻¹(b/a)Where a and b are the real and imaginary parts of the complex number z.θ=tan⁻¹(5/12)So, θ=22.6 degrees. (approximate value)

Thus, the complex number z = 12 + 5i can be expressed as\[z=13\cos(22.6^{\circ}) + i\sin(22.6^{\circ})

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Answer:

Step-by-step explanation:

x-intercept is where the line cuts the x-axis. That is, when y=0.

Substitute y=0 and we get:

       [tex]0=\frac{1}{3} x-1[/tex]

       [tex]1=\frac{1}{3} x[/tex]

       [tex]x=3[/tex]

So x-intercept is the point (3,0).

Answer:

The answer is, x= 145

Step-by-step explanation:

Since line BD passes through the center E of the circle, then the angle must be a right angle or a 90 degree angle.

Hence angle DAB must be 90 degrees

or,

[tex]angle \ DAB = 0.3(2x+10) = 90\\90/0.3 = 2x+10\\300 = 2x+10\\300-10=2x\\290=2x\\\\x=145[/tex]

Hence the answer is, x= 145

Answer:

127

Step-by-step explanation:

Angle C+Angle D=Angle ABC

Since C+D+CBD=180 and ABC+CBD=180

subtract getting C+D-CBD=0 and C+D=CBD

so 67+60=127 which is your answer

*Just to clarify, when i said C and D, i meant angle C and angle D

A) The probability that a randomly selected student studies only mathematics would be 20/75.

B) The probability that a randomly selected student studies mathematics or physics would be (20 + X) / 75.

C) The probability that a randomly selected student does not study either of the two subjects would be (75 - (20 + X)) / 75.

D) The exact probability that a student studies mathematics and physics cannot be determined without knowing the number of students who study both subjects.

To determine the requested probabilities, we will use the information provided about the group of 75 students.

a) Study only mate:

We know that there are 20 students studying mathematics and physics, so the number of students studying only mathematics would be the total number of students studying mathematics (20) minus the number of students studying both subjects. Since no information is provided on the number of students studying both subjects, we will assume that none of the students study both subjects. Therefore, the number of students studying only mathematics would be 20 - 0 = 20.

The probability that a randomly selected student studies only mathematics would be 20/75.

b) Study math or physics:

To determine this probability, we need to add the number of students who study mathematics and the number of students who study physics, and then subtract the number of students who study both subjects (we again assume that none of the students study both subjects).

Number of students studying mathematics = 20

Number of students studying physics = X (not given)

Number of students studying both subjects = 0 (assumed)

Therefore, the number of students studying mathematics or physics would be 20 + X - 0 = 20 + X.

The probability that a randomly selected student studies mathematics or physics would be (20 + X) / 75.

c) That he does not study either:

The number of students not studying either subject would be the complement of the number of students studying mathematics or physics. So it would be 75 - (20 + X).

The probability that a randomly selected student does not study either of the two subjects would be (75 - (20 + X)) / 75.

d) To study math and physics:

Since no information is provided on the number of students studying both subjects, we cannot determine the exact probability that a student will study mathematics and physics.

In summary:

a) The probability that a randomly selected student studies only mathematics would be 20/75.

b) The probability that a randomly selected student studies mathematics or physics would be (20 + X) / 75.

c) The probability that a randomly selected student does not study either of the two subjects would be (75 - (20 + X)) / 75.

d) The exact probability that a student studies mathematics and physics cannot be determined without knowing the number of students who study both subjects.

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The predicted sales in month 5 is -2778.

Obtaining the linear equation which models the data :

b = slope = (y2-y1)/(x2-x1)

b = (926-7408)/(4-1)

b = -2160.67

c = intercept ;

taking the points (x = 2 and y = 3704)

Inserting into the general equation:

3704 = -2160.67(2) + c

3704 = -4321.33 + c

c = 3704 + 4321.33

c = 8025.33

General equation becomes : y = -2160.67x + 8025.33

To obtain sales in month 5:

y = -2160.67(5) + 8025.33

y = -2778

Hence, the predicted sales in month 5 is -2778.

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Answer:

D. A pair of intersecting lines

Step-by-step explanation:

A conic section is a fancy name for a curve that you get when you slice a double cone with a plane. Imagine you have two ice cream cones stuck together at the tips, and you cut them with a knife. Depending on how you cut them, you can get different shapes. These shapes are called conic sections, and they include circles, ellipses, parabolas and hyperbolas. If you cut them right at the tip, you get a point. If you cut them slightly above the tip, you get a line. If you cut them at an angle, you get two lines that cross each other. That's what happened in your question. The plane cut the cone at an angle, so the curve is two intersecting lines. That means the correct answer is D. A pair of intersecting lines.

I hope this helps you ace your math question.

Answer:

a. 36.65 in

b. 14.14 km²

Step-by-step explanation:

Solution Given:

a.

Arc Length = 2πr(θ/360)

where,

Here Given: θ=150° and r= 14 in

Substituting value

Arc length=2π*14*(150/360) =36.65 in

b.

Area of the sector of a circle = (θ/360°) * πr².

where,

Here  θ = 45° and r= 6km

Substituting value

Area of the sector of a circle = (45/360)*π*6²=14.14 km²

Answer:

[tex]\textsf{a)} \quad \overset{\frown}{AC}=36.65\; \sf inches[/tex]

[tex]\textsf{b)} \quad \text{Area of sector $ABC$}=14.14 \; \sf km^2[/tex]

Step-by-step explanation:

The formula to find the arc length of a sector of a circle when the central angle is measured in degrees is:

[tex]\boxed{\begin{minipage}{6.4 cm}\underline{Arc length}\\\\Arc length $= \pi r\left(\dfrac{\theta}{180^{\circ}}\right)$\\\\where:\\ \phantom{ww}$\bullet$ $r$ is the radius. \\ \phantom{ww}$\bullet$ $\theta$ is the angle measured in degrees.\\\end{minipage}}[/tex]

From inspection of the given diagram:

Substitute the given values into the formula:

[tex]\begin{aligned}\overset{\frown}{AC}&= \pi (14)\left(\dfrac{150^{\circ}}{180^{\circ}}\right)\\\\\overset{\frown}{AC}&= \pi (14)\left(\dfrac{5}{6}}\right)\\\\\overset{\frown}{AC}&=\dfrac{35}{3}\pi\\\\\overset{\frown}{AC}&=36.65\; \sf inches\;(nearest\;hundredth)\end{aligned}[/tex]

Therefore, the arc length of AC is 36.65 inches, rounded to the nearest hundredth.

[tex]\hrulefill[/tex]

The formula to find the area of a sector of a circle when the central angle is measured in degrees is:

[tex]\boxed{\begin{minipage}{6.4 cm}\underline{Area of a sector}\\\\$A=\left(\dfrac{\theta}{360^{\circ}}\right) \pi r^2$\\\\where:\\ \phantom{ww}$\bullet$ $r$ is the radius. \\ \phantom{ww}$\bullet$ $\theta$ is the angle measured in degrees.\\\end{minipage}}[/tex]

From inspection of the given diagram:

Substitute the given values into the formula:

[tex]\begin{aligned}\text{Area of sector $ABC$}&=\left(\dfrac{45^{\circ}}{360^{\circ}}\right) \pi (6)^2\\\\&=\left(\dfrac{1}{8}\right) \pi (36)\\\\&=\dfrac{9}{2}\pi \\\\&=14.14\; \sf km^2\;(nearest\;hundredth)\end{aligned}[/tex]

Therefore, the area of sector ABC is 14.14 km², rounded to the nearest hundredth.

Answer:

-2097152 is the 20th term

Step-by-step explanation:

Write geometric sequence as an explicit formula

[tex]-4,8,-16,32\rightarrow-4(-2)^0,-4(-2)^1,-4(-2)^2,4(-2)^3\\a_n=a_1r^{n-1}\rightarrow a_n=-4(-2)^{n-1}[/tex]

Find the n=20th term

[tex]a_{20}=-4(-2)^{20-1}=-4(-2)^{19}=4(-524288)=-2097152[/tex]

The expression is completed as (x-4)(x -7)

From the information given, we have that the polynomial is given as;

x² - 11x + 28

Using the factorization method, we have;

First, find the product of the coefficient of x squared and the constant value

Then, we have;

1(28) = 28

Now, find the pair factors of the product that adds up to -11, we have;

-7x and -4x

Substitute the values, we have;

x² - 7x - 4x + 28

Group in pairs, we get;

x(x-7) - 4(x - 7)

Then, we have the expressions as;

(x-4)(x - 7)

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The exponential function that represents the number of members contacted after x days is given as follows:

[tex]N(x) = 3(3)^x[/tex]

An exponential function has the definition presented according to the equation as follows:

[tex]y = ab^x[/tex]

In which the parameters are given as follows:

The parameter values for this problem are given as follows:

Hence the function is given as follows:

[tex]N(x) = 3(3)^x[/tex]

The problem asks for the exponential function that represents the number of members contacted after x days.

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Answer:

2147483648

Step-by-step explanation:

Write the geometric sequence as an explicit formula

[tex]8,\,32,\,128\rightarrow8(4)^0,8(4)^1,8(4)^2\rightarrow a_n=a_1r^{n-1}\rightarrow a_n=8(4)^{n-1}[/tex]

Find the n=15th term

[tex]a_{15}=8(4)^{15-1}=8(4)^{14}=8(268435456)=2147483648[/tex]