Roger can finish his math homework in 6 hours. Trish can finish the same homework in 5 hours. What part of the homework will Roger and Trish finish if they work together for 1 hours?

The volume of the solid obtained by rotating the region over the function is given by the integral and V = 44.46 units³

Given data ,

The volume of the solid obtained by rotating the region underneath the graph of the function f(x)=√x² +9 over the interval [0,2] about the y-axis

The formula for Area under definite integral is

∫ₐᵇ f ( x ) = f ( b ) - f ( a )

Since we are rotating the region about the y-axis, the radius is simply the x-coordinate of each point on the curve, or r = x

The height h of each shell is equal to the difference between the y-coordinates of the curve at x and x + Δx, or h = f(x + Δx) - f(x)

Using these expressions for r and h, we can write the volume of each cylindrical shell as:

V(x) = 2πx[f(x + Δx) - f(x)]Δx

V = ∫₀² 2πx[f(x + Δx) - f(x)]dx

As Δx approaches zero, this integral becomes:

V = ∫₀² 2πx√(1 + (f'(x))²) dx

where f'(x) is the derivative of f(x), which is:

f'(x) = x/√(x² + 9)

Substituting this expression for f'(x) into the integral, we get:

V = ∫₀² 2πx√(1 + (x/√(x² + 9))²) dx

This integral can be evaluated using a substitution, u = x² + 9, du/dx = 2x, and dx = du/2x, to get:

V = ∫₉¹³ 2π(x² + 9)^(3/2)/2 dx

V = [4/5 π(x² + 9)^(5/2)]₉¹³

V = 44.46 units³

Hence , the volume of the solid is 44.46 units³

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Answer: The answer is 6.25

Step-by-step explanation:

Answer:

6.25

Step-by-step explanation:

Answer:

15

Step-by-step explanation:

Let's call the number we try to find "[tex]x[/tex]"

In the problem, we know that "26 is less than twice a number" is the same as "4". So we can write this as an equation:

[tex]2x - 26 = 4[/tex]

Now we can solve for x by isolating it on one side of the equation.

First, we isolate "2x" by flipping "-26" to the other side, which will now be "26".

[tex]2x = 4 + 26[/tex]

Then we add

[tex]2x = 30[/tex]

finally, we isolate the x by dividing each side with 2

2x ÷ 2 = 30 ÷ 2
x = 15

Answer:

\( A=\frac{d^2}{2} \).

Step-by-step explanation:

Area of Square by Applying Formula for the Diagonal

We can use the relationship between diagonal length d and the side a length of a square to find its area A. So, the formula for area of square using diagonal is \( A=\frac{d^2}{2} \).

The equation of the ellipse  derived as is

(x - h)^2 / a^2 + (y - k)^2 / b^2 = 1.

The equation of foci is

F₁ = (h - √(a^2 - b^2), k) , F₂ = (h + √(a^2 - b^2), k)

An ellipse is  described as a set of points in a plane such that the sum of the distances from each point to two fixed points, called foci, is constant.

We can write the equation of an ellipse as:

(x - h)^2 / a^2 + (y - k)^2 / b^2 = 1

where (h, k) = the center of the ellipse,

a = the semi-major axis,

and b = the semi-minor axis.

The foci of the ellipse are located along the major axis and are equidistant from the center, with a distance of : √(a^2 - b^2).

we know also the formula to find the foci of an ellipse is:

F₁ = (h - √(a^2 - b^2), k)

F₂ = (h + √(a^2 - b^2), k)

The sum of the distances from each point on the ellipse to the foci is constant. The equation can then be written as:

2a = √((x - h + √(a^2 - b^2))^2 + (y - k)^2) + √((x - h - √(a^2 - b^2))^2 + (y - k)^2)

Simplifying, we then can write  the equation of the ellipse as:

(x - h)^2 / a^2 + (y - k)^2 / b^2 = 1

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The total money raised is (15/7) * $88,000 = $188,571.43.

a) Let x be the total amount raised.

Charity A received (1/3)(x - A), Charity C received (1/4)(x - C). We have A = (1/3)(B+C) and C = (1/4)(A+B).

Solving these equations, we find B's share to be 7/15 of the total amount.

p) Since Charity B received $88,000, which is 7/15 of the total amount, the total money raised is (15/7) * $88,000 = $188,571.43.

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Answer:

1,3,5,7,9

Step-by-step explanation:

0 can't be, let's take 10 or 20 or 30 for example, not a prime number.

1 can be because let's take 11, 31 as example, they're prime numbers.

2, 4, 6, 8 can't be the answer because they have factors

The rest of off numbers, namely 3,5,7,9 can be. Let's take for example 23, 05, 67, 59

The synthetic division of the polynomial indicates that other factor of the polynomial is; x² - 3·x + 13

A polynomial consists of terms that have positive integer powers or index of variables joined together by addition and subtraction symbols.

The possible polynomial in the question is; x³ + 5·x² - 11·x + 104

The factor of the polynomial is; (x + 8)

The steps for the synthetic division includes on the left side of the vertical line, placing the opposite sign of the constant term of the factor.

The other steps of synthetic division can then be presented, summarily as follows;

Therefore;

-8 | 1[tex]{}[/tex]   5     -11   104

   |   [tex]{}[/tex]   -8    24  104          

     1   -3    [tex]{}[/tex] 13      0

The factors of the polynomial

The values in the above last row of the long division indicates that the other factor of the polynomial is; x² - 3·x + 13

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Therefore, the linear approximation sin⁻¹(x) ≈ x is accurate to within 0.1 when x is in the interval [-0.099, 0.099].

In mathematics, a function is a relationship between two sets, called the domain and range, such that for each element in the domain there is exactly one element in the range. In other words, a function assigns a unique output value to each input value. Functions are typically denoted using a rule or formula that describes how to calculate the output value based on the input value.

Here,

The linear approximation of sin⁻¹(x) at a = 0 is given by:

sin⁻¹(x) ≈ x

To verify this, we can take the derivative of sin⁻¹(x) and evaluate it at x = 0:

d/dx(sin⁻¹(x)) = 1/√(1-x²)

d/dx(sin⁻¹(x))|x=0 = 1

Since the derivative of sin⁻¹(x) evaluated at x = 0 is equal to 1, we can use the linear approximation sin⁻¹(x) ≈ x near x = 0.

To determine the values of x for which the linear approximation is accurate to within 0.1, we can use a graphing calculator or computer to plot the graphs of sin⁻¹(x) and x, and find the intervals where the difference between the two functions is less than or equal to 0.1.

Using a graphing calculator, we can plot the two functions and find the intervals where the difference is less than or equal to 0.1. The result is:

-0.099 ≤ x ≤ 0.099

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Answer:  

C is the answer

Given :   dilation with a scale factor of 3

To find :  Which statement is true

Step-by-step explanation:

Solution: A dilation is a transformation that produces an image that is the same shape as the original, but is a different size

if Dilation factor  N then image Size is multiplied  by  N

if Scale factor  < 1 then image get reduced

if scale factor > 1  then image get enlarged

scale factor = 1 lead to Same size image

Here is given that Scale factor of 3

=> image will be enlarged to 3 times the original image

Hence correct option is :

C)Each side in the pre-image is multiplied by three to find the corresponding side length in the image

Answer: C

Step-by-step explanation: Edge Unit Test

The x-coordinate of this point is -1, so cos(5π) = cos(900 degrees) = -1.

The y-coordinate of this point is -1, so sin(-9π/2) = sin(-810 degrees) = -1.

The y-coordinate of this point is 1, so sin(183π/2) = sin(16,470 degrees) = 1.

It should be noted that go find cos(5π), we first need to convert 5π to degrees. We know that π radians is equal to 180 degrees, so 5π radians is equal to 5π × (180/π) = 900 degrees. The x-coordinate of this point is -1, so cos(5π) = cos(900 degrees) = -1.

Also, to find sin(-9π/2), we first need to convert -9π/2 to degrees. We know that π radians is equal to 180 degrees, so -9π/2 radians is equal to -9π/2 × (180/π) = -810 degrees. The y-coordinate of this point is -1, so sin(-9π/2) = sin(-810 degrees) = -1.

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By looking at the y-values of the points, we can see that the first four terms are.

2, -2, -6, -10

We know that the x-value of each of the points is the index of the term, so:

x = 1 is the first term

x = 2 is the second term.

And so on.

The y-value will be the actual value of the term.

Here we can see 4 points graphed, these are:

(1, 2), (2, -2), (3, -6), (4, -10)

Then the values of the first four terms of the sequence are:

2, -2, -6, -10

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The solutions to the equation over the interval [0°, 360°) are: x = 45°, 135°, 225°, 315°.

We can rewrite this equation as:

1 - sin² x = sin²x

Solving for sin x, we get:

sin x = ±✓(1/2)

Since sin x is positive in the first and second quadrants and negative in the third and fourth quadrants, we have:

sin x = ✓(1/2) = 1/✓(2) in the first and second quadrants, corresponding to x = 45° and 135°.

sin x = -✓(1/2) = -1/✓(2) in the third and fourth quadrants, corresponding to x = 225° and 315°.

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Answer:

The walking distance that is saved by cutting across the lot is approximately 18 feet.

The sine of angle θ is given as follows:

sin(θ) = -8/17.

For an angle [tex]\theta[/tex] the unit circle is a circle with radius 1 containing the following set of points: [tex](\cos{\theta}, \sin{\theta})[/tex].

The point in this problem has the coordinates given as follows:

(15/17, -8/17).

Hence the trigonometric ratios are given as follows:

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There are possible 4,782,969 different telephone numbers.

We will multiply the number of options for each digit to calculate the total number of possible telephone numbers,.

Using the multiplication principle to estimate the number of possible telephone numbers for each digit:  

n1 x n2 x n3 x ... x nk

Here:

the first three digits are fixed: 011

The 4th digit (the first digit of the local phone number) can be any digit from 1 - 9 (i.e. 9 options).

The remaining 6 digits (the second to seventh digit of the local phone number) can be any digit from 0 to 9, except 0 (9 options for each digit).

Therefore, the total number of possible telephone numbers:

9 x 9 x 9 x 9 x 9 x 9 x 9 = 4,782,969,

Therefore, 4,782,969 different telephone numbers are possible.

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a

1.5/4

b

0.5/4

c

0.5/4

because half of 3 is 1.5

Answer:

I'm pretty sure the answer is B(

The value of the points is,

(1/5, 7/5)  or (0.2, 1.4)

The given equation may be simplified as follows:

x² + 14xy + 49y² = 100

(x + 7y)(x + 7y) = 100

(x + 7y)² = 10²

x + 7y = 10

This is a straight line with the equation.

y = -(1/7)x + 10/7

The minimum distance from the origin to this line is provided by a straight line that passes through the origin and which is perpendicular to the straight line.

The slope of the perpendicular line is 7 because the product of the two slopes should be -1.

The perpendicular line is of the form

y = 7x + c.

Because the line passes through (0,0), therefore c = 0.

The line y = 7x intercepts the original line when

y = 7x = -(1/7)x + 10/7

Therefore

7x = -(1/7)x + 10/7

Multiply through by 7.

49x = -x + 10

50x = 10

x = 1/5

y = 7x = 7/5

Hence, The minimum distance is

d = √(x² + y²)

  = √[(1/5)² + (7/5)²]

  = √2

Thus, The point is (1/5, 7/5).

So, Solution are, (1/5, 7/5)  or (0.2, 1.4)

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Answer: x=4, sides = 12, 8, 6, 8

Step-by-step explain

Perimeter means the sum of all sides of a shape, in this example there are four sides add the sides and set it equal to 34.

X+8+x+4+2x-2+x²-2x=34

x²+2x+10=34

x²+2x-24

(x-4)(x+6)

x=4, x=-6

x can not be -6 because then the x+4 side would be -2, therefore it is extraneous and x=4 is the correct answer.

Now to find side lengths just plug in 4 for x

4+8=12

4+4=8

2*4-2=6

4*4-2*4=8

Answer:

a) In 2 hours

b) 16 km

Step-by-step explanation:

Let's first find out how far Abel has traveled in 2 hours. We can use the formula:

distance = speed x time

distance = 4 km/h x 2 h = 8 km

So when Jack starts cycling, Abel has already covered 8 km.

Now, let's say it takes Jack t hours to catch up to Abel. During this time, Abel has continued walking for t + 2 hours. So we can write:

distance covered by Abel = distance covered by Jack

4 km/h x (t + 2 h) = 8 km/h x t

Simplifying this equation, we get:

4t + 8 = 8t

4t = 8

t = 2

So Jack will overtake Abel after 2 hours of cycling. To find out how far they will have traveled, we can use the formula:

distance = speed x time

distance = 8 km/h x 2 h = 16 km

Therefore, when Jack overtakes Abel, they will have traveled 16 km in total.

Answer:

complex

Step-by-step explanation:

The given sentence "Although Kevin has money, he is not spending it." is a complex sentence.

This is because it contains one independent clause, "he is not spending it," which can stand alone as a complete sentence. The other part of the sentence, "Although Kevin has money," is a dependent clause because it cannot stand alone as a complete sentence.

The dependent clause "Although Kevin has money" introduces a contrast or concession to the independent clause that follows it. Therefore, this sentence is an example of a complex sentence that uses a dependent clause to add meaning and complexity to the independent clause.

The first 5 terms of the sequence are 1/3, 2/3, 4/3, 8/3 ans 16/3

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

The sequence is generated by using gn = 2(gn-1)

Such that g1 = 1/3 and n is a whole number greater than 1.

Using the above as a guide, we have the following:

The sequence means that we multiply the current term by 2 to get the next term

So, we have the following representations

a(2) = 2 * 1/3 = 2/3

a(3) = 2 * 2/3 = 4/3

a(4) = 2 * 4/3 = 8/3

a(5) = 2 * 8/3 = 16/3

Hence, the first 5 terms of the sequence are 1/3, 2/3, 4/3, 8/3 ans 16/3

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The value of x in the equation x² = -20 is x  = 2i√5

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

x² = -20

To start with, we take the square root of both sides of the equation

So, we have

√x² = √-20

When the square roots are evaluated, we have

x  = √-20

Express -20 as 4 * -5

So, we have

x  = √4 * -5

This gives

x  = 2√-5

The expression √-5 is a complex number

So, we have

x  = 2i√5

Hence, the value of x in x² = -20 is x  = 2i√5

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Answer:

B. 1

Step-by-step explanation:

Let's use the formula for finding the mean of a set of data:

mean = (sum of all data values) / (number of data values)

We know that the mean is 2.2, and we have all the other data values except for one. Let's call the missing value "x". The total number of data values is 10 (since there are 10 scores listed in the frequency table). So we can set up an equation:

2.2 = (sum of all 10 data values, including x) / 10

Multiplying both sides by 10, we get:

22 = sum of all 10 data values, including x

We can then subtract the sum of the known data values from both sides to find the missing value:

22 - (01 + 12 + 23 + 31 + 42 + 51) = x

22 - (0 + 2 + 6 + 3 + 8 + 5) = x

x = 22 - 24 = -2

However, a quiz score cannot be negative, so we made a mistake somewhere. Checking the frequency table, we see that there are no scores of 0 or 1, so the missing score must be 0 or 1. Let's try both possibilities:

If the missing score is 0:

2.2 = (01 + 12 + 23 + 31 + 42 + 51 + 0*1) / 10

2.2 = 18 / 10

2.2 = 1.8

This is not true, so the missing score cannot be 0.

If the missing score is 1:

2.2 = (01 + 12 + 23 + 31 + 42 + 51 + 1*?) / 10

2.2 = (18 + ?) / 10

22 = 18 + ?

? = 4

So the missing score is 1, and the answer is B.

Answer:

Answer is B

Total scores/total frequency =mean

54/24=2,25

If you added a score of 1

55/25=2,2

Step-by-step explanation:

hope that helps you

To prove: The number of left cosets of a subgroup is equal to the number of its right cosets.

Proof: Let H be a subgroup of a group G. Let g be an element of G. Consider the map f: H→ gH defined by f(h) = gh for all h in H. We claim that f is a bijection from H to gH.

First, we show that f is injective. Suppose f(h1) = f(h2) for some h1, h2 in H. Then gh1 = gh2, which implies that h1 = h2, by left cancellation law. Therefore, f is injective.

Next, we show that f is surjective. Let gh be an element of gH. Then h is an element of G, since gH is a subset of G. Since H is a subgroup of G, it follows that gh is an element of gH. Therefore, f(h) = gh, which shows that f is surjective.

Hence, f is a bijection from H to gH. Therefore, the number of left cosets of H is equal to the number of right cosets of H.

Let x be the length of each of the equal sides of the isosceles triangle. Then the length of the base is x - 4. The perimeter of the triangle is the sum of the lengths of the three sides, which is:

x + x + (x - 4) = 3x - 4

We know that the perimeter is 32, so we can set 3x - 4 equal to 32 and solve for x:

3x - 4 = 32

Adding 4 to both sides gives:

3x = 36

Dividing by 3 gives:

x = 12

Therefore, one of the equal sides of the triangle is 12 inches long, and the length of the base is 12 - 4 = 8 inches.

So the three sides of the triangle are 12 inches, 12 inches, and 8 inches.

The correct equation to solve the problem is (C) 3x - 4 = 32.

An equation for L, in terms of t, representing the amount Kadeesha owes her friend after t weeks is L = 8t + 120.

The table of values is as follows;

Number of Weeks               Amount of Money Owed

0                                                    $120

1                                                    $128

2                                                    $136

3                                                    $144

In order to write a linear equation to describe this situation, we would assign variables to the amount of money that Kadeesha borrowed originally and the number of weeks respectively, and then translate the word problem into a linear equation as follows:

Since Kadeesha originally borrowed $120 and agreed to pay back her friend $8 per week, a linear equation which can be used to model the total amount of money that Kadeesha owed is given by;

L = 8t + 120

When t = 0, the value of L is given by;

L = 8(0) + 120

L = $120.

When t = 1, the value of L is given by;

L = 8(1) + 120

L = $128.

When t = 2, the value of L is given by;

L = 8(2) + 120

L = $136.

When t = 3, the value of L is given by;

L = 8(3) + 120

L = $144.

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To find the multiplicative inverse of 2i, we need to find a complex number z such that:

(2i) * z = 1

Multiplying both sides by -i, we get:

z = -i/2

Therefore, the multiplicative inverse of 2i is -i/2.

A ski lodge inspects 80 skis and discovers four that are faulty. The skis that are likely to be defective are 21.

It is assumed that there are 80 skis, four of which are faulty.

As a result, the total number of outcomes is 80.

Positive outcomes = 4

Let [tex]E_{1}[/tex] represent the event of selecting a faulty. As a result, the likelihood of selecting a defective ski is expressed mathematically as follows.

[tex]E_{1}[/tex] = Favorable case of defective / Total number of skis

= 4 / 80

= 1 / 20

=0.05

Skis will be discovered in 5% of cases.

So, the event of choosing a defective is 4 / 80.

If the total number of skis is 420, then the number of defective skis will be:

4 / 80 × 420

= 1/20 × 420

= 21

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Correct question:

A ski lodge inspects 80 skis and finds 4 to be defective. The lodge has 420 skis. Predict the number of skis that are likely to be defective.

Answer:

$2,440 + $1,220 = $3,660

Step-by-step explanation:

If Amanda makes $61,000 a year, and she contributes 4% of her salary to her 401(k), that would be:

$61,000 x 0.04 = $2,440

Since her employer has a 50% match on the first 4% of her salary, they would contribute:

$2,440 x 0.5 = $1,220