Write a real-life situation that can be modeled by the expression 1 1/3 x 2.
1 1/3 is a mixed number

We have a = 2 and b = 3.

To find the exact value of sin(165°) using sum or difference formulas, we notice that 165° can be expressed as the sum of two angles whose sine and cosine values we know exactly, for example, 120° and 45°. The sum formula for sine is sin(A + B) = sin(A)cos(B) + cos(A)sin(B).

Therefore, sin(165°) = sin(120° + 45°) = sin(120°)cos(45°) + cos(120°)sin(45°).

From here, we use the known values:

sin(120°) = sin(180° - 60°) = sin(60°) = √3/2

cos(45°) = √2/2

cos(120°) = cos(180° - 60°) = -cos(60°) = -1/2

sin(45°) = √2/2

Plugging these values into our formula gives:
sin(165°) = ( √3/2)(√2/2) + (-1/2)(√2/2) = (√6 - √2)/4

This fraction can be rewritten to match the given form by factoring out √2 in the numerator:
sin(165°) = √2(√3 - 1)/4

Hence, in the form sin(165°) = √a(√b - 1)/4, we have a = 2 and b = 3.

1, 1, 3, 3 - I just did it on e2020 and got it correct

Function transformation involves changing the form of a function

The function that represents the transformation is: g(x) = 1/2(5^(x -3))

The parent function is given as:

f(x) = 5^x

When the function is vertically compressed by a factor of one-half, the function becomes

f'(x) = 1/2(5^x)

When the function is shifted 3 units right, it becomes

f"(x) = 1/2(5^(x -3))

Rewrite the function as:

g(x) = 1/2(5^(x -3))

Hence, the function that represents the transformation is: g(x) = 1/2(5^(x -3))

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The correct function representing the transformation is B) [tex]\( g(x) = \left(\frac{1}{2}\right)5^{x-2} - 2 \).[/tex]

1. **Vertical Compression by a Factor of One-Half:**

  To vertically compress the function by a factor of one-half, we multiply the function by [tex]\( \frac{1}{2} \)[/tex]. This affects the [tex]\( y \)[/tex]-coordinates of all points on the graph. So, the function becomes: [tex]\( \frac{1}{2} \cdot 5^x = \left(\frac{1}{2}\right)5^x \).[/tex]

2. **Shifted to the Left Three Units:**

  To shift the function to the left three units, we add 3 to the [tex]\( x \)[/tex] -coordinate. This affects the input values. So, the function becomes: [tex]\( \left(\frac{1}{2}\right)5^{x+3} \).[/tex]

3. **Shifted Down Two Units:**

  To shift the function down two units, we subtract 2 from the [tex]\( y \)-[/tex]coordinate. This affects the output values. So, the final transformed function becomes: [tex]\( \left(\frac{1}{2}\right)5^{x+3} - 2 \).[/tex]

Now, let's compare this transformation with the provided options:

A) [tex]\( g(x) = 5^{\left(\frac{n}{2}\right)+3} - 2 \)[/tex]

  This option doesn't match the transformation. It introduces a variable [tex]\( n \)[/tex] and doesn't involve the original function [tex]\( 5^x \)[/tex].

B) [tex]\( g(x) = \left(\frac{1}{2}\right)5^{x-2} - 2 \)[/tex]

  This option matches the transformation. It has the vertical compression by [tex]\( \frac{1}{2} \)[/tex], shifting to the left by three units, and shifting down by two units.

C) [tex]\( g(x) = \left(\frac{1}{2}\right)5^{x+5} - 2 \)[/tex]

  This option doesn't match the transformation. It shifts the function to the right by five units instead of to the left.

D) [tex]\( g(x) = 5^{\left(\frac{1}{2}\right)^{x-3}} - 2 \)[/tex]

  This option introduces a fractional exponent, which doesn't correspond to the given transformations. It also has an incorrect shift to the right by three units.

Therefore, the correct option representing the transformation is B) [tex]\( g(x) = \left(\frac{1}{2}\right)5^{x-2} - 2 \).[/tex]

Complete Question :

The parent function, f(x)=5^x has been vertically compressed by a factor of one-half shifted to the right three units and down two units.

Choose the correct function to represent the transformation

A) g(X)=5^(((n)/(2))+3)-2  

B) g(x)=((1)/(2))5^(x-2)-2  

C) g(x)=((1)/(2))5^(x+5)-2

D) g(x)=5^(((1)/(2))^(x-3))-2

The expected number of times the sequence 'qqqq' appears in 1000 random typings of 6 different letters is approximately 0.769 times.

This problem falls under the category of discrete probability in mathematics. The child randomly types one of 6 different letters.

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The total cost of producing 100 widgets will be $800. Then the correct option is D.

A connection between a number of variables results in a linear model when a graph is displayed. The variable will have a degree of one.

The linear equation is given as,

y = mx + c

Where m is the slope of the line and c is the y-intercept of the line.

Wilbur's widgets, a widget company, produce 100 widgets. its average fixed cost is $5 and its total variable cost is $300.

Then the equation is given as,

y = 5x + 300

Where 'x' is the number of widgets and 'y' is the total cost.

Then the total cost of producing 100 widgets will be given as,

y = 5 × 100 + 300

y = 500 + 300

y = $800

The total cost of producing 100 widgets will be $800. Then the correct option is D.

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The total number of points = 13.5 points

What is the total amount of points gained?

We are given the scoring method as:

Touchdown pass: 6

Complete pass : 0.5

Incomplete pass : -0.5

Interception : -1.5

Now, we are told that Jeremy's quarterback had the following:

2 touchdowns passes,

16 complete passes,

7 incomplete passes,

2 interceptions.

Thus:

Total points = (2 * 6) + (0.5 * 16) + (-0.5 * 7) + (-1.5 * 2)

Total points = 13.5 points

Answer:

She can trim 19 sweaters.

Step-by-step explanation:

Kayla  has a particular pattern in which she needs 3 ounces of white yarn.

That is in order to make a sweater of that particular pattern she requires white wool = 3 ounces

Kayla has total white wool = 57 ounces

one sweater requires = 3 ounces

number of sweater =[tex]\frac{total wool}{wool required for one sweater}[/tex]

                              =[tex]\frac{57}{3} = 19[/tex]

She can trim 19 sweaters with the Yarn she has on hand

Answer:

She can trim 19 sweaters.

Step-by-step explanation:

The width is 60 yards and the length, being 80 yards more, is 140 yards.

To find the dimensions of the rectangle when the perimeter is 400 yards and the length is 80 yards more than the width, we can set up two equations based on the formula for the perimeter of a rectangle, which is P = 2l + 2w, where P is the perimeter, l is the length, and w is the width.

Let the width be w. Then the length is w + 80 yards.

Using the perimeter formula:

400 = 2(w + 80) + 2w

400 = 2w + 160 + 2w

400 = 4w + 160

Subtract 160 from both sides:

240 = 4w

Divide by 4:

w = 60 yards

Now, calculate the length:

l = w + 80 = 60 + 80

l = 140 yards

The width is 60 yards, and the length is 140 yards.

The extreme max value is "[tex]f(x,y,z) = 36[/tex]" and min value is "[tex]f(x,y,z) = -36[/tex]".

According to the question,

let,

By using Lagrange multipliers,

→ [tex]\bigtriangledown f= \lambda \bigtriangledown g[/tex]...(equation 1)

∴ [tex]<8,8,4>=<8\lambda x, 8 \lambda y, 8 \lambda z >[/tex]

then,

→ [tex](x = \frac{1}{\lambda}, y=\frac{1}{\lambda}, z =\frac{1}{2 \lambda} )[/tex]

As we know,

→             [tex]4x^2 +4y^2+4z^2=36[/tex]

By substituting the values, we get

→ [tex]4(\frac{1}{\lambda} )^2+ 4(\frac{1}{\lambda} )^2+4(\frac{1}{2\lambda} )^2 =36[/tex]

→                 [tex]\frac{4}{\lambda^2} +\frac{4}{\lambda^2} +\frac{1}{\lambda^2} =36[/tex]

→                                  [tex]\frac{9}{\lambda^2}=36[/tex]

                                     [tex]\frac{1}{\lambda}= \pm \frac{6}{3}[/tex]

                                     [tex]\frac{1}{\lambda} = \pm 2[/tex]

∴ x = 2, y = 2, z = 1

then,

→ [tex]f(x,y,z) = 36[/tex] (Extreme max value)

and,

∴ x = -2, y = -2, z = -1

then,

→ [tex]f(x,y,z) = -36[/tex] (min value)

Thus the above answer is correct.

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The function shows exponential growth a stretch of the function f(x) = (1/3)^x .

When the expression of a function is such that it involves the input to be present as the exponent (power) of some constant, then such a function is called an exponential function. Their usual form is specified below. They are written in several equivalent forms.

Given that the exponential function f(x) = 3

The initial value of the function is 2.

The base of the function is 3.

When the intial value is where x=0

where x=0, the value of f(0)=3

The growth is 1/3 but it is decay, that might be correct, or not because it is decay.

It is a stretch of the function f(x)=(1/3) power of x

when x=3, then f(3)=1/9

The function shows exponential growth a stretch of the function f(x) = [tex](1/3)^x[/tex].

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The volume of the solid region. the solid between the planes z = 3x + 2y + 1 and z = x + y, and above the triangle with vertices (1, 0, 0), (2, 2, 0), and (0, 1, 0) in the xy-plane is 6 cubic unit.

It is defined as a three-dimensional space enclosed by an object or thing.

It is given that

The solid between the planes z = 3x + 2y + 1 and z = x + y, and above the

triangle with vertices (1, 0, 0), (2, 2, 0), and (0, 1, 0) in the xy-plane.

r(u) = (1, 2)

r(v) = (-2, -1)

|r(u)xr(v)| = 3

The volume of the solid:

∬S(3x + 2y + 1 − x − y)ds = ∫₀¹ ∫(u, 0)(2x + y +1)3 dv du

After solving:

= 3(4/3 - 5 /6 + 3/2)

= 6

Thus, the volume of the solid region. the solid between the planes z = 3x + 2y + 1 and z = x + y, and above the triangle with vertices (1, 0, 0), (2, 2, 0), and (0, 1, 0) in the xy-plane is 6 cubic unit.

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Final answer:

The volume of the solid region between the planes z = 3x + 2y + 1 and z = x + y, above the triangular region in the xy-plane, can be found using a double integral over the triangular area defined by vertices (1, 0, 0), (2, 2, 0), and (0, 1, 0).

Explanation:

To find the volume of the solid region between the planes z = 3x + 2y + 1 and z = x + y, and above the triangular region in the xy-plane, we can use a double integral over the triangular region. The volume is given by the difference between the two planes integrated over the area of the triangle formed by the vertices (1, 0, 0), (2, 2, 0), and (0, 1, 0).

First, define the vertices of the triangle in the xy-plane: A(1, 0), B(2, 2), and C(0, 1).

Second, find the equations of the lines forming the sides of the triangle, which helps in defining the limits of the integral.

Lastly, set up the double integral:

∫∫ (3x + 2y + 1) - (x + y) dA,

after solving, we get volume = 6

where dA is the differential area element in the xy-plane, and the limits of integration are determined by the triangular region.

The integration can be simplified by determining a suitable ordering for dx and dy, often choosing to integrate in x first and then y, or vice versa, based on the geometry of the triangle. The exact bounds for x and y need to be worked out from the equations of the lines defining the triangle's edges.

The result of this integral will give the volume of the solid region bounded by the two planes and above the triangle.