Assume f(x) = -2x + 8 and g(x) = 3x. What is the value of (g o f)(3)

When writing rational expressions, you need to be aware that ...

  1/4x = (1/4)x ≠ 1/(4x)

Parentheses around the denominator are required, unless you're typesetting the expression and can use a fraction bar for grouping.

The derivative of the curve expression is ...

  y' = x - 1/(4x) . . . . . parentheses added to what you wrote

and the expression (1 -(y')^2) can be written ...

  1 -(y')^2 = x^2 +1/2 +1/(16x^2) . . . . . parentheses added to what you wrote

The first and last terms of this trinomial are both perfect squares, so you might suspect the whole trinomial is a perfect square. You recall that ...

  (a +b)^2 = a^2 + 2ab + b^2

This is a good "pattern" to remember. Using it is a matter of pattern recognition, as is the case with a lot of math.

Here, you have ...

  a = x

  b = 1/(4x)

In order for your trinomial to be a perfect square, the product 2ab must equal the middle term of your trinomial. (Spoiler: it does.)

  2ab = 2(x)(1/(4x)) = (2x)/(4x) = 1/2 . . . . . matches the middle term of 1 -(y')^2

Hence your trinomial can be written as the square ...

  1 -(y')^2 = (x +1/(4x))^2

_____

This is convenient because you want to integrate the square root of this. Your integral then becomes ...

[tex]\displaystyle\int\limits_{2}^{4}{\left(x+\frac{1}{4x}\right)\,dx[/tex]

Answer:

25m + 10b = 400

Step-by-step explanation:

Total amount that can be earned = $ 400

Amount earned on mowing a lawn = $ 25

Amount earned per hour on babysitting = $10

Amount earned for mowing one lawn is $ 25, so amount earned for mowing m lawns will be $ 25m

Amount earned for one hour of babysitting is $ 10, so amount earned for babysitting for b hours will be $ 10b

Total amount earned = Amount earned from mowing m lawns + Amount earned from babysitting for b hours

Using the values from above, we can set up the equation as:

400 = 25m + 10b

or

25m + 10b = 400

This equation represents the number of lawns they can mow, m, and hours they can babysit, b, to earn $400.

Compare [tex]\dfrac1{\sqrt{n^2+1}}[/tex] to [tex]\dfrac1{\sqrt{n^2}}=\dfrac1n[/tex]. Then in applying the LCT, we have

[tex]\displaystyle\lim_{n\to\infty}\left|\frac{\frac1{\sqrt{n^2+1}}}{\frac1n}\right|=\lim_{n\to\infty}\frac n{\sqrt{n^2+1}}=1[/tex]

Because this limit is finite, both

[tex]\displaystyle\sum_{n=1}^\infty\frac1{\sqrt{n^2+1}}[/tex]

and

[tex]\displaystyle\sum_{n=1}^\infty\frac1n[/tex]

behave the same way. The second series diverges, so

[tex]\displaystyle\sum_{n=0}^\infty\frac1{\sqrt{n^2+1}}=1+\sum_{n=1}^\infty\frac1n[/tex]

is divergent.

The limit comparison test is used to determine the convergence or divergence of a series by comparing it to a known convergent or divergent series.

The limit comparison test is used to determine the convergence or divergence of a series by comparing it to a known convergent or divergent series. The test states that if the limit of the ratio between the terms of the given series and the terms of a known convergent series is a finite positive number, then both series behave in the same way. Here is how you can use the limit comparison test step by step:

For example, if we have the series ∑(n^2)/(2^n), we can use the limit comparison test with the series ∑(1)/(2^n). Taking the limit of the ratio (n^2)/(2^n) / (1)/(2^n) as n approaches infinity, we get:

lim(n→∞) [(n^2)/(2^n)] / [(1)/(2^n)] = lim(n→∞) (n^2)/(1) = ∞

Since the limit is infinite, the result is inconclusive. Therefore, the given series does not converge or diverge using the limit comparison test.

Answer:

1. y = 4n -1

2. y = 9n -2

Step-by-step explanation:

1. For some integer n, numbers that are congruent to 2 mod 4 are ... 4n+2. Then you have ...

y + 3 = 4n +2 . . . . next, subtract 3 from both sides

y = 4n -1 . . . . values of y are natural numbers for all natural numbers n

___

2. Similarly, ...

3 - y = 9n +5

3 - 9n - 5 = y . . . . . add y-9n-5

-9n -2 = y . . . . . . . . collect terms

For every integer n, there is also an integer -n. For our purpose, we're only interested in those values of n that make -9n be positive. Rather than require n be negative so -9n is positive, we can require n be positive and use the expression 9n for the multiple of 9.

We want y a natural number, so we can write this as ...

y = 9n -2 . . . . for natural numbers n

To find the natural number solutions for congruences, we use algebraic manipulation to simplify the equation and find values that satisfy the congruence. For the given congruences, the natural number solutions are y = 3, 7, 11, 15, ... and y = 7, 16, 25, ..., respectively.

1. To find the natural number solutions for the congruence y + 3 ≡ 2 mod 4, we need to find all values of y that satisfy the given congruence.

Subtracting 3 from both sides of the congruence, we have y ≡ -1 mod 4. Since we are looking for natural number solutions, we can rewrite this as y ≡ 3 mod 4.

The natural number solutions for this congruence are y = 3, 7, 11, 15, ...

2. For the congruence 3 - y ≡ 5 mod 9, we need to determine the natural number solutions for y that make the congruence true.

Subtracting 3 from both sides, we have -y ≡ 2 mod 9. Multiplying both sides by -1, we get y ≡ -2 mod 9. Rewriting this as y ≡ 7 mod 9, the natural number solutions for y are y = 7, 16, 25, ...

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Answer:

Hemisphere radius = 3 inch

Sphere volume = 4/3 • π • radius^3

Sphere volume = 4/3 • π • 27

Sphere volume = 113.0973355292 cubic inches

Cylinder Volume = π • radius^2 • height

Cylinder Volume = π • 9 * 10

Cylinder Volume = 282.7433388231 cubic inches

TOTAL VOLUME = 113.10 + 282.74 cubic inches

TOTAL VOLUME = 395.84 cubic inches

Step-by-step explanation:

Answer: 126π in.³

Step-by-step explanation:

this is an isosceles triangle two sides are the same

so there are 180° in a triangle

180-58= 122 now divide this by two because the last two angles are equal

122/2=61° =x°

The dimensions should be 6ft by 9ft.

Final answer:

The dimensions of the floor with an area of 54 square feet, where the width is 2/3 of the length, are 9 feet in length and 6 feet in width.

Explanation:

Pascal wants to find the dimensions of a floor where the area is 54 square feet and the width is 2/3 the length. To find the dimensions, we set up an equation where the length is L and the width is W, with W = (2/3)L. Since area is calculated by the formula Area = length times width, we get 54 = L times (2/3)L. Solving this equation by multiplying the length terms and then dividing both sides by 2/3, we get L² = 54 times (3/2), which simplifies to L² = 81. Taking the square root of both sides gives us L = 9 feet. Consequently, the width would be W = (2/3) times 9 = 6 feet. Therefore, the dimensions of the floor are 9 feet by 6 feet.

Answer:

3 - 2√5

-------------  

      2

Step-by-step explanation:

These two expressions have different denominators, and thus use of the LCD is necessary.  The LCD is 4.

Thus, we have:

6/4 - (4/4)√5, or

6 - 4√5

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

      4

This can be reduced to:

3 - 2√5

-------------   (answer)

      2

Answer:

  3.  P(R1|Q) = 3/19 ≈ 0.16

Step-by-step explanation:

The desired probability is the ratio of P(Q·R1) to P(Q). The probability P(Q) is not given, but there is sufficient information to find it.

  P(Q·R1) = P(Q|R1)·P(R1) = 0.40·0.15 = 0.06

  P(Q·R2) = P(Q|R2)·P(R2) = 0.20·0.55 = 0.11

  P(Q·R3) = P(Q|R3)·P(R3) = 0.70·0.30 = 0.21

Since R1 and R2 and R3 are mutually exclusive and have a joint probability of 1, this means ...

  P(Q) = P(Q·R1) +P(Q·R2) +P(Q·R3) = 0.06 +0.11 +0.21 = 0.38

Then the desired probability is ...

  P(R1|Q) = P(Q·R1)/P(Q) = 0.06/0.38

  P(R1|Q) = 3/19 ≈ 0.16

Answer:

Option D. minimum value at −38

Step-by-step explanation:

we have

[tex]x^{2}-12x-2[/tex]

Let

[tex]y=x^{2}-12x-2[/tex]

Complete the square

[tex]y+2=x^{2}-12x[/tex]

[tex]y+2+36=(x^{2}-12x+36)[/tex]

[tex]y+38=(x^{2}-12x+36)[/tex]

[tex]y+38=(x-6)^{2}[/tex]

[tex]y=(x-6)^{2}-38[/tex] ------> equation of a vertical parabola in vertex form

The vertex is the point [tex](6,-38)[/tex]

The parabola open upward-----> the vertex is a minimum

therefore

minimum value at −38

Solve for x in 5x - 3y = 18

x = 3(6 + y)/5

Substitute x = 3(6 + y)/5 into 3x + 3y = 30

9(6 + y)/5 + 3y = 30

Solve for y in 9(6 + y)/5 + 3y = 30

y = 4

Substitute y = 4 into x = 3(6 + y)/5

x = 6

Therefore,

x = 6

y = 4

The solution to the system of equations is x = 6 and y = 4.

Step 1: Add the two equations.

5x - 3y = 18

3x + 3y = 30

8x = 48

Step 2: Solve for x.

8x = 48

x = 6

Step 3: Substitute x = 6 into one of the original equations to solve for y.

5(6) - 3y = 18

30 - 3y = 18

-3y = -12

y = 4

Answer:

C. 6.683

Step-by-step explanation:

Answer:30

Step-by-step explanation: its 30 because you divide 127 1/2 or 127.5 by 4.25 or 4 1/4.

Answer:

  A.  216 cm³

Step-by-step explanation:

The volume of a cube is the cube of the side length:

  V = s³ = (6 cm)³ = 216 cm³

Answer:

60

Step-by-step explanation:

∑[i=1,5] 4i = 4·1 + 4·2 + 4·3 + 4·4 + 4·5

= 4 + 8 + 12 + 16 + 20

= 60

To determine the dimensions of the wall, we set up an equation using the given area and the fact that the wall's height is 2/3 of its length. Solving for length and height, we find that the length is 30 feet and the height is 20 feet.

The question asks to determine the length and height of a wall that Leo wants to paint, given that the area of the wall is 600 square feet, and the height is 2/3 of its length. To solve this, we will set up an equation to represent the relationship between the length and height and use the area to find the specific dimensions.

Let's use L to represent the length of the wall and H to represent the height. According to the given information, H = (2/3)L. The area of a rectangle is calculated by multiplying its length by its height, so we can set up the equation:

Area = L \H = 600 sq ft

Substituting the height with its relation to length, we get:

600 = L \(2/3)L

Dividing both sides by L and then by 2/3, we find that:

L² = 600 \(3/2)

L² = 900

L = 30 ft (since length must be positive)

Now, to find the height:

H = (2/3)\L = (2/3) \ 30 = 20 ft

So, the length of the wall is 30 feet and the height is 20 feet.

Answer:

The equation of the line is y = 3/4x - 1/4

Step-by-step explanation:

To find the equation of this line, start by using the two points with the slope formula to find the slope.

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

m = (2 - 1/2)/(3 - 1)

m = (3/2)/2

m = 3/4

Now that we have the slope, we can use that and either point in point-slope form to find the equation.

y - y1 = m(x - x1)

y - 2 = 3/4(x - 3)

y - 2 = 3/4x - 9/4

y = 3/4x - 1/4

Answer

the answer is 6/10 or 0.6

Step-by-step explanation:

6/10 (3/5) because you have to share the amount of bags into the ratio of friends which would give you 0.6 or 6/10 simplified to 3/5

Answer:

58 at the point (9,8)

7 at the point (1, 1)

Step-by-step explanation:

The maximum points will be found in the vertices of the region.

Therefore the first step to solve the problem is to identify through the graph, the vertices of the figure.

The vertices found are:

(1, 10)

(1, 1)

(9, 5)

(9, 8)

We look for the values of x and y belonging to the region, which maximize the objective function [tex]f(x, y) = 2x + 5y[/tex]. Therefore we look for the vertices with the values of x and y higher.

(1, 10), (9, 5), (9, 8)

Now we substitute these points in the objective function and select the one that produces the highest value for f (x, y)

[tex]f(1, 10) = 2(1) + 5(10) = 52\\\\f(9, 5) = 2(9) + 5(5) = 43\\\\f(9, 8) = 2(9) + 5(8) = 58[/tex]

The point that maximizes the function is:

[tex](9, 8)[/tex] with [tex]f(9, 8) = 58[/tex]

Then the value that produces the minimum of f(x, y) is (1, 1)

[tex]f(1, 1) = 2(1) + 5(1) = 7[/tex]

You can solve for 'x' in the equation 'ax = 7' by dividing both sides of the equation by 'a'. The result is 'x = 7/a'. This is based on the principle of maintaining equation equality through similar operations on both sides.

To solve for

x

in the equation

ax = 7

, you need to isolate the variable, x. This is done by

dividing

both sides of the equation by 'a'. This leaves x = 7/a. Therefore, you can find 'x' by dividing both sides of the equation by 'a'.

This process is based on the principle of equality.

You can maintain the equality of the equation by doing the same mathematical operation to both sides. Consequently, by dividing both sides by 'a', you successfully solve for x without disrupting the balance of the equation.

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Answer:

b) 15

Step-by-step explanation:

Given that,

Total people = 100

% of people prefer vanilla = 25%

Number of people prefer vanilla out of 100 = 25% of 100

=> 25/100 * 100 = 25

% of people prefer swirl = 35%

Number of people prefer swirl out of 100 = 35% of 100

=> 35/100 * 100 = 35

% of people prefer chocolate = 40%

Number of people prefer chocolate out of 100 = 40% of 100

=> 40/100 * 100 = 40

The number of more people preferring chocolate over vanilla =  40 - 25

=> 15 so option b is correct

Answer:

The answer is B (15)

Step-by-step explanation:

[tex]P_{ELPJ} = EK+ES+LU+LK+PV+PU JS+JV \\ \Leftrightarrow P_{ELPJ} = 2EK + 2LU + 2PV + 2JS = 2 \times 2 + 2 \times 4 + 2 \times 1 + 2 \times 2 = 18[/tex]

Answer:

a = 12

Step-by-step explanation:

Multiply the equation by the inverse of the coefficient of "a".

4·(1/4)a = 4·3

1·a = 12 . . . . . . simplify

a = 12

Let

[tex]\theta=\sin^{-1}\dfrac x2\implies\sin\theta=\dfrac x2[/tex]

Recall that

[tex]\cos\theta=\sqrt{1-\sin^2\theta}=\sqrt{1-\dfrac{x^2}4}[/tex]

Then

[tex]\tan\theta=\dfrac{\sin\theta}{\cos\theta}\implies\tan\left(\sin^{-1}\dfrac x2\right)=\dfrac{\frac x2}{\sqrt{1-\frac{x^2}4}}[/tex]

[tex]\implies\tan\left(\sin^{-1}\dfrac x2\right)=\dfrac x{\sqrt{4-x^2}}[/tex]

Final answer:

To find tan(sin⁻¹(x/2)), we use the relationship between the sides of a right triangle, leading to the solution tan(θ) = x/√(4 - x²).

Explanation:

The question "what is tan(sin-1(x/2))" involves understanding inverse trigonometric functions and their properties. To solve this, we can create a right-angled triangle where the angle θ has a sine value of x/2. According to the Pythagorean theorem, if one side (opposite) is x and the hypotenuse is 2, the adjacent side can be calculated as √(4 - x²). Thus, tan(θ) = opposite/adjacent = x/√(4 - x²).

The lateral area and surface area of the cuboid are 72 square feet and 142 square feet, respectively. The lateral area and surface area of the prism are 42 square cm and 62 square cm, respectively.

To find the lateral area and surface area of the given cuboid and prism, we need to use the formulas for each.

For the cuboid:
1. Lateral Area: The lateral area of a cuboid can be found by adding the areas of its four side faces. Since the cuboid has a length of 7 feet, a breadth of 5 feet, and a height of 3 feet, we can calculate the lateral area as follows:
- Side 1: length * height = 7 feet * 3 feet = 21 square feet
- Side 2: breadth * height = 5 feet * 3 feet = 15 square feet
- Side 3: length * height = 7 feet * 3 feet = 21 square feet
- Side 4: breadth * height = 5 feet * 3 feet = 15 square feet
The total lateral area is the sum of these four areas: 21 + 15 + 21 + 15 = 72 square feet.

2. Surface Area: The surface area of a cuboid is found by adding the areas of all six faces. In addition to the lateral area, we need to include the areas of the top and bottom faces. Using the dimensions of the cuboid, we can calculate the surface area as follows:
- Top face: length * breadth = 7 feet * 5 feet = 35 square feet
- Bottom face: length * breadth = 7 feet * 5 feet = 35 square feet
The total surface area is the sum of these six areas: 72 (lateral area) + 35 (top face) + 35 (bottom face) = 142 square feet.

Now, for the prism:
1. Lateral Area: To find the lateral area of the prism, we need to find the perimeter of the base and multiply it by the height of the prism. Since the prism has a base with a length of 5 cm and a breadth of 2 cm, the perimeter is: 2 * (length + breadth) = 2 * (5 cm + 2 cm) = 2 * 7 cm = 14 cm.
The lateral area is obtained by multiplying the perimeter by the height: 14 cm * 3 cm = 42 square cm.

2. Surface Area: The surface area of the prism is the sum of the lateral area and twice the area of the base. Since the base is a rectangle with a length of 5 cm and a breadth of 2 cm, its area is: length * breadth = 5 cm * 2 cm = 10 square cm.
The surface area is then calculated as: 42 (lateral area) + 2 * 10 (area of the base) = 42 + 20 = 62 square cm.

Final answer:

To plot a graph of the amount Betsy earns against the time she works, we'll put the hours worked on the x-axis and the amount earned on the y-axis. Betsy is paid $11.50 per hour and the equation for Betsy's earnings is Y = 11.50X.

Explanation:

To plot a graph of the amount Betsy earns against the time she works, we'll put the hours worked on the x-axis and the amount earned on the y-axis. In this case, Betsy worked an 8 hour shift and was paid $92.00. So, the point on the graph would be (8, 92). To find how much Betsy is paid per hour, we can divide the total amount earned by the number of hours worked. In this case, $92.00 divided by 8 hours equals $11.50 per hour.

Using the equation Y = MX + B, where Y is the amount earned, M is the rate per hour, X is the number of hours worked, and B is the y-intercept, we can write the equation for Betsy as Y = 11.50X + 0, since she doesn't have a one-time fee.

Answer:

The volume would be 1 1/4 cubic inches.

Step-by-step explanation:

To find this, multiply all three numbers together because the formula for volume is length multiplied by width multiplied by height.

V = lwh

V = (1/4)(1 1/4)(4)

V = 1 1/4

Answer:

8. A'(-1/2, 1), B'(0, 0), C'(1, 1 1/2)

10. 20°, 60°, 100°

Step-by-step explanation:

8. Point B is at the origin, the center of dilation, so its coordinates remain unchanged. The coordinates of A and C are each multiplied by the scale factor, 1/2.

A' = (1/2)A = (1/2)(-1, 2) = (-1/2, 1)

C' = (1/2)C = (1/2)(2, 3) = (1, 3/2)

__

10. The total number of ratio units is 1+3+5 = 9. The total of angle measures in any triangle is 180°. So, each of the "ratio units" must stand for 180°/9 = 20°. Then the angles are ...

1 · 20° = 20°

3 · 20° = 60°

5 · 20° = 100°

It would be located at (1, 3). Hope this helps!

Answer:

1,3

Step-by-step explanation: