the velocity of a vehicle is given in fig. \( 1.8 \). a. determine the equation of \( v(t) \) for i. \( 0 \leq t \leq 3 \) s ii. \( 3

A mathematical statement known as an equation is created by joining two expressions with an equal sign. For instance, 3x - 5 = 16 is an equation. This equation may be solved, and the result shows that the value of the variable x is 7.

Given,

(6.82x - 2.53y - 5.72) - (-5.72 + 9.45y - 8.11 x)

Simplifying the above equation,

= 6.82x - 2.53y - 5.72 - (-5.72 + 9.45y - 8.11 x)

= 6.82x - 2.53y - 5.72 + 5.72 - 9.45y + 8.11 x

Simplify

6.82x - 2.53y - 5.72 + 5.72 - 9.45y + 8.11 x : 14.93x - 11.98y

Then we get,

= 14.93x - 11.98y

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Answer:

Step-by-step explanation:

12x2  +3-4-10

Answer:

Quadrant IV

Step-by-step explanation:

cos(∅) > 0 means cos (∅) is positive

tan (∅) < 0 means tan (∅) is negative

thus,

∅ lies in Quadrant IV

The value of g(x) = [x] At, 2 ≤ x < 3, g(x) = 2, It is also known as the greatest integer function.

A function on real numbers is called a step function if it can be written as a finite linear combination of indicator functions of intervals.

Informally speaking, a step function is a piecewise constant function having only finitely many pieces.

A step function f(x) = [x] is called a step function where the output is the greatest integer less than or equal to the input.

The function is g(x) = [x].

At, - 1 ≤ x < 0, g(x) = 0.

At, 1 ≤ x < 2, g(x) = 1.

At, 2 ≤ x < 3, g(x) = 2.

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a. The two lines intersect at the point (-1/2, 8, 3/2).

b. The equation of the plane that passes through the two lines is -2y - 4z + 20 = 0.

a) To find the point where the two lines intersect, we need to solve the system of equations:

x = -2 + t

y = 3 + 2t

z = 4 - t

x = 3 - t

y = 4 - 2t

z = t

Equating x from the two equations, we get:

-2 + t = 3 - t

2t = 5

t = 5/2

Substituting t in either equation, we get:

x = -2 + 5/2 = -1/2

y = 3 + 2(5/2) = 8

z = 4 - 5/2 = 3/2

Therefore, the two lines intersect at the point (-1/2, 8, 3/2).

b) To obtain the equation of the plane that passes through the two lines, we first find the direction vectors of the lines. These are given by the coefficients of t in the equations:

Line 1: (-2, 3, 4) + t(1, 2, -1)

Line 2: (3, 4, 0) + t(-1, -2, 1)

The direction vectors are (1, 2, -1) and (-1, -2, 1), respectively.

The normal vector of the plane can be found by taking the cross-product of these two direction vectors:

(1, 2, -1) × (-1, -2, 1) = (0, -2, -4)

Now we use the point (-1/2, 8, 3/2) that we found in part (a) and the normal vector (0, -2, -4) to write the equation of the plane in the point-normal form:

0(x + 1/2) - 2(y - 8) - 4(z - 3/2) = 0

Simplifying, we get:

-2y - 4z + 20 = 0

Therefore, the equation of the plane that passes through the two lines is -2y - 4z + 20 = 0.

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The question is -

a) determine the point where the two lines

x=−2+t, y=3+2t, z=4−t and x=3−t, y=4−2t, z=t intersect.

b) use your results from part (a) to obtain the equation of the plane that passes through the two lines.

1) Multiply both sides of the equation by 2:

2(A)=2[1/2h(b+B)

2A=2/2h(b+B)

2A=1h(b+B)

2A=h(b+B)

2) Divide both sides of the equation by (b+B):

(2A)/(b+B)=[h(b+B)]/(b+B)

2A/(b+B)=h

h=2A/(b+B)

Answer: [tex]h=2\frac{A}{(b+B)}[/tex]

hope this helped!!

Nine cuboidal boxes and eight cuboidal boxes can each hold 153 and 136 toy cars, respectively.

How is the average calculated?

The ratio of the sum of all provided observations to the total number of observations is the arithmetic mean, which is another name for the average formula. So, any sample of data may have its arithmetic mean determined using the average formula.

What does arithmetic mean?

The mean or arithmetic average are terms that are frequently used to refer to the arithmetic mean. It is determined by adding up all the numbers in a given data collection, then dividing that total by the number of items in the data set. For uniformly distributed integers, the middle number is the arithmetic mean (AM).

Given: There are two storage options for 288 toy cars: 16 and 18 cuboidal boxes.

supposing there were 288 toy cars stored in 16 toy cars.

There are 18 toy cars in a box of 288 (288 / 16) toy cars.

if 288 toy cars were housed in 18 toy cars.

There are 16 toy cars in a box that holds 288 toy cars.

Average: 18 + 16 / 2 = 34 / 2 = 17 toy cars.

Nine boxes of toy cars include 153 toy cars (9 * 17).

eight boxes of toy cars include 136 toy cars (8 * 17)

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According to the information, he gets 6 pieces in total after cutting his boards.

To find the number of pieces Nathan gets, we must divide the total length of the boards by the length of the pieces. Below is the procedure:

According to the above, from one table he gets 4 2-feet pieces, from the other he gets 2 4-feet pieces. Additionally, in total he would have 6 pieces.

Note: This question is incomplete because there is some information missing. Here is the complete information:

Nathan has two 8-foot boards. He cuts one board into 2-foot pieces, and he cuts the other board into 4-foot pieces.

How many pieces does he get?

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At y there is a minimum for single-slit diffraction and a maximum for double-slit interference, as noted in the question.

For single-slit diffraction, the first minimum occurs when

sin θ = λ/a.

and double-slit interference maxima occur when

(m) λ / d = sin θ --------- (2)

The first diffraction minimum for single-slit diffraction and the fourth double-slit interference maximum (m = 4) occur at the same position y, as seen in the figure below.

On the left of the screen, the dashed curve is due to single-slit interference. On the right of the screen, the dashed curve is due to double-slit interference. The positive direction is reflected on each of the two sides of the screen and the screen position is zero amplitude.

Since the single-slit diffraction minimum masks the fourth double-slit interference maxima, one must estimate where the fourth double-slit maxima occur using the spacing between the double-slit interference pattern appearing on the right-hand side of the display screen, as seen in the figure above.

Using Eq. 1 and 2, we have

sin θ / λ = 1/ a = (m) 1/ d

d/ a = (4)

= 4.

Therefore, At y there is a minimum for single-slit diffraction and a maximum for double-slit interference, as noted in the question.

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The correct question is:

Consider the setup of double-slit experiment in the schematic drawing below. One of the double-slit interference minima is located at the first single-slit diffraction minimum. Determine the ratio d/a ; i.e., the slit separation d compared to the slit width a. Use a small angle approximation; e.g., sin θ ≈ tan θ ≈ θ and cos θ ≈ 1.

1. d/a = 13/2

2. d/a = 6

3. d/a = 15/2

4. d/a = 5

5. d/a = 7

6. d/a = 9/2

7. d/a = 11/2

8. d/a = 17/2

9. d/a = 4

10. d/a = 8

The probability density function is of E(Y1) and E(Y2) is 1/6 and variance is 1/18.

To find the expected value E(Y1+Y2), we can use the linearity of expectation and the fact that the expected value of a constant is that constant:

E(Y1+Y2) = E(Y1) + E(Y2)

To find E(Y1) and E(Y2), we need to integrate y1 and y2 respectively over their joint probability density function:

E(Y1) = ∫∫ y1 f(y1,y2) dy1 dy2

= ∫0^1 ∫0^(1-y1) 2y1 dy2 dy1

= ∫0^1 2y1(1-y1)/2 dy1

= ∫0^1 y1-y1^2 dy1

= [y1^2/2 - y1^3/3] from 0 to 1

= 1/6

Similarly,

E(Y2) = ∫∫ y2 f(y1,y2) dy1 dy2

= ∫0^1 ∫0^(1-y2) 2y2 dy1 dy2

= ∫0^1 2y2(1-y2)/2 dy2

= ∫0^1 y2-y2^2 dy2

= [y2^2/2 - y2^3/3] from 0 to 1

= 1/6

Therefore, E(Y1+Y2) = E(Y1) + E(Y2) = 1/6 + 1/6 = 1/3.

To find the variance V(Y1+Y2), we can use the formula:

V(Y1+Y2) = E((Y1+Y2)^2) - [E(Y1+Y2)]^2

To find E((Y1+Y2)^2), we need to integrate (y1+y2)^2 over their joint probability density function:

E((Y1+Y2)^2) = ∫∫ (y1+y2)^2 f(y1,y2) dy1 dy2

= ∫0^1 ∫0^(1-y1) (y1+y2)^2 2 dy2 dy1

= ∫0^1 [(2/3)y1^3 + y1^2 + (1/3)y1] dy1

= 5/18

Therefore, V(Y1+Y2) = E((Y1+Y2)^2) - [E(Y1+Y2)]^2 = 5/18 - (1/3)^2 = 5/18 - 1/9 = 1/18

The variance is 1/18.

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Answer: 2

4 – (6-4) = 2

Answer: 2- (y= 6 , z=2)

Step-by-step explanation:

If z = 4 and y = 6, then we can substitute these values into the expression:

z - (y - z) = 4 - (6 - 4)

Expanding the subtraction in the parentheses, we get:

z - (y - z) = 4 - (6 - 4) = 4 - (2) = 4 - 2

Finally, solving for the subtraction:

z - (y - z) = 4 - 2 = 2

So, the value of the expression z - (y - z) when z = 4 and y = 6 is 2.

The outer unit normal vector at any point (x, y, z) on the surface of the ellipsoid is given by:

n = <8x, 18y, 72z> / sqrt((8x)² + (18y)² + (72z)²)

In mathematics, an expression is a combination of one or more values, variables, operators, and/or functions that are evaluated to produce a single result.

For example, 3 + 4 is an expression that evaluates to 7, and (x + 2) / (y - 1) is an expression that can be evaluated for different values of x and y.

Expressions can be simple or complex, and can involve arithmetic operations (addition, subtraction, multiplication, division), algebraic operations (such as factoring or expanding), trigonometric functions, logarithmic functions, and many other mathematical operations.

Expressions are often used in mathematical modeling and problem solving, and can be written in a variety of formats, including symbolic notation, numerical values, or verbal descriptions.

The equation you provided, 4x² + 9y² + 36z² = 36, represents an ellipsoid centered at the origin (0,0,0) with semi-axes of length 3, 2, and 1 in the x, y, and z directions, respectively.

The outer unit normal vector, n, at a given point on the surface of the ellipsoid is perpendicular to the tangent plane at that point. To find the normal vector at a given point (x, y, z) on the surface, we can take the gradient of the function 4x² + 9y² + 36z², which gives us:

∇(4x² + 9y² + 36z²) = <8x, 18y, 72z>

At any point on the surface of the ellipsoid, this gradient vector is proportional to the normal vector, and the constant of proportionality is the reciprocal of the length of the gradient vector:

n = k<8x, 18y, 72z>

To find the value of k, we need to normalize n so that its length is 1. That is,

|n| = sqrt((8x)² + (18y)² + (72z)²) = 1

Solving for k, we get:

k = 1 / sqrt((8x)² + (18y)² + (72z)²)

Thus, the outer unit normal vector at any point (x, y, z) on the surface of the ellipsoid is given by:

n = <8x, 18y, 72z> / sqrt((8x)² + (18y)² + (72z)²)

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The area of the figure is calculated as: 536 km².

The area of the figure given can be found by decomposing the figure into three rectangles.

Area of rectangle 1:

Length = 15 km

Width = 11 km

Area = length * width = 15 * 11 = 165 km²

Area of rectangle 2:

Length = 6 + 11 = 17 km

Width = 11 km

Area = 17 * 11 = 187 km²

Area of rectangle 3:

Length = 23 km

Width = 8 km

Area = 23 * 8 = 184 km²

The area of the figure = 165 + 187 + 184 = 536 km²

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Answer:

6 Shirts

Step-by-step explanation:

if you buy 1 pair of jeans you can buy 6 pairs of shirts since 6 x 11 = 66

66 + 40 = 106

The distance between the cities of Palo Alto and Beijing is approximately equal to 8990.334 kilometers.

In this problem we must determine the distance between the cities of Palo Alto and Beijing, whose astronomical coordinates are given in spherical coordinates. The distance is the sum of distance along the terrestrial arc, whose formula can be derived from circular arc formula:

s = (Δθ / 180°) · 2π · R + (Δλ / 180°) · 2π · R

Where:

Now we proceed to determined the distance between the two cities:

s = [(39.914° - 37.429°) / 180°] · 2π · (6367.5 km) + [[116.392° - (- 122.138°)] / 180°] · (6367.5 km)

s ≈ 8990.334 km

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Answer:

To find the volume of a rectangular prism, you need to multiply its length, width, and height.

Assuming that the dimensions are in inches, we have:

Length = 16 inches + 1/4 inch = 16.25 inches

Width = 1/2 inch

Height = some unknown value (not given in the question)

So, if we know the height, we can calculate the volume of the rectangular prism. If the height is, for example, 10 inches, then:

Volume = Length x Width x Height

Volume = 16.25 inches x 0.5 inches x 10 inches

Volume = 81.25 cubic inches

Therefore, the volume of the rectangular prism depends on its height, which is not given in the question.

Answer:

e

Step-by-step explanation:

e

From the given data, the company can produce 500 units per month at a minimum cost of $140,000.

To minimize the total monthly cost of production while producing 500 units per month, we can use the method of Lagrange multipliers. Let L(x, y, λ) be the Lagrangian function:

L(x, y, λ) = 2x² + xy + 4y² + 1600 + λ(500 - x - y)

We need to find the values of x and y that minimize L(x, y, λ), subject to the constraint that 500 units are produced per month. Taking partial derivatives of L with respect to x, y, and λ and setting them to zero, we get:

∂L/∂x = 4x + y - λ = 0

∂L/∂y = x + 8y - λ = 0

∂L/∂λ = 500 - x - y = 0

Solving these equations simultaneously, we get:

x = 200

y = 150

λ = 26/3

Therefore, to minimize the total monthly cost of production while producing 500 units per month, the company should produce 200 units at Factory X and 150 units at Factory Y.

To find the minimum cost, we substitute the values of x and y into the joint cost function:

C(x, y) = 2x² + xy + 4y² + 1600

C(200, 150) = 2(200)² + (200)(150) + 4(150)² + 1600

C(200, 150) = 140,000

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The point (2,5) satisfies only one of the two inequalities in the system, it is not included in the solution area for the system.

To see if the point (2,5) is in the system's solution area, we need to see if it satisfies both inequalities in the system:

4x + 2y ≤ 16

4(2) + 2(5) ≤ 16

8 + 10 ≤ 16

18 ≤ 16

Because this inequality is false, point (2,5) fails to satisfy the first inequality.

x + y ≥ 4

2 + 5 ≥ 4

7 ≥ 4

Because this inequality holds, the point (2,5) satisfies the second inequality.

Because point (2,5) satisfies only one of the system's two inequalities, it is not included in the system's solution area. Landon cannot, therefore, spend $16 on two sandwiches and five bags of chips to feed his four teammates.

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The minimum value of the function P over the feasible region is -3, which occurs at the point (1,2).

The expression that established the relationship between the dependent variable and independent variable is referred to as a function. In the function as the value of the independent variable varies the value of the dependent variable also varies.

To find the minimum value of the function P = x - 2y over the feasible region determined by the vertices (1,2), (7,3), and (5,1), we need to evaluate the function at each of these points and find the lowest value.

P(1,2) = 1 - 2(2) = -3

P(7,3) = 7 - 2(3) = 1

P(5,1) = 5 - 2(1) = 3

Therefore, the minimum value of the function P over the feasible region is -3, which occurs at the point (1,2).

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The lateral surface area of the box is given as follows:

112.5 in².

The area of a rectangle is given by the multiplication of the width and the length of the triangle, as follows:

A = lw.

The lateral of the triangular box for this problem is composed by two rectangles, with dimensions given as follows:

Hence the lateral surface area of the box is obtained as follows:

5 x 10 + 5 x 12.5 = 112.5 in².

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The algorithm to write the binary representation of the positive integer "64" is explained below and the binary representation of 64 is 1000000 .

Step(i) : 64 divided by 2 gives a quotient of 32 with a remainder of 0.

Step(ii) : 32 divided by 2 gives a quotient of 16 with a remainder of 0.

Step(iii) : 16 divided by 2 gives a quotient of 8 with a remainder of 0.

Step(iv) : 8 divided by 2 gives a quotient of 4 with a remainder of 0.

Step(v) : 4 divided by 2 gives a quotient of 2 with a remainder of 0.

Step(vi) : 2 divided by 2 gives a quotient of 1 with a remainder of 0.

Step (vii) : 1 divided by 2 gives a quotient of 0 with a remainder of 1.

So , we observe that  the remainders, read from bottom to top, are 1000000.

Therefore, the binary representation of "64" is 1000000 in binary.

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The given question is incomplete , the complete question is

Find the binary representation of the positive integer "64" by working through an algorithm by hand .

The area of the triangle ΔABC is given by A = √275 units²

A triangle is a plane figure or polygon with three sides and three angles.

A Triangle has three vertices and the sum of the interior angles add up to 180°

Let the Triangle be ΔABC , such that

∠A + ∠B + ∠C = 180°

The area of the triangle = ( 1/2 ) x Length x Base

For a right angle triangle

From the Pythagoras Theorem , The hypotenuse² = base² + height²

if a² + b² = c² , it is a right triangle

if a² + b² < c² , it is an obtuse triangle

if a² + b² > c² , it is an acute triangle

Given data ,

Let the area of the triangle ΔABC be A

Now , the measure of side BD = 5

The measure of side ED = 11

So , the measure of side EB = 11 - 5 = 6

The measure of side AD = 10

And , the measure of side CE = 4

For a right angle triangle

From the Pythagoras Theorem , The hypotenuse² = base² + height²

From the triangle ΔABD ,

The measure of side AB = √ ( AD )² - ( BD )²

The measure of side AB = √ ( 100 - 25 )

The measure of side AB = √75 units

And , from the triangle ΔBCE

The measure of side CB = √ ( EB )² - ( CE )²

The measure of side CB = √ ( 36 - 16 )

The measure of side CB = √20 units

And , from the triangle ΔABC

The measure of side AC = √ ( AB )² - ( CB )²

The measure of side AC = √ ( 75 - 20 )

The measure of side AC = √55 units

And , the area of triangle ΔABC = ( 1/2 ) x AC x BC

The area of triangle ΔABC = ( 1/2 ) x √55 x √20

The area of triangle ΔABC = ( 1/2 ) x √1100

The area of triangle ΔABC , A = ( 1/2 ) x 2√275

The area of triangle ΔABC , A = √275 units²

Hence , the area of triangle ΔABC is √275 units²

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Answer: X= -8

Step-by-step explanation: 4 minus 6 equals to -2. So for 4x + 6 to equal -2 you have to do 4x times -8 to get -8 then add it by 6 to get -2.

4 times 8 = -8 + 6 = -2

The recurrence relation for the number of ternary strings of length n that contain two consecutive zeros looks correct.

You can see that the first few terms match with your calculations:

a_1 = 0 (no possible strings)

a_2 = 1 (only one possible string: "00")

a_3 = 5 (possible strings: "000", "001", "002", "100", "200")

To explain your reasoning:

You can add "1" or "2" to the end of all the a_{n-1} strings, so there are 2a_{n-1} possible strings with n-1 length that end with "1" or "2".

You can add "01" or "02" to the end of all the a_{n-2} strings, so there are 2a_{n-2} possible strings with n-2 length that end with "01" or "02".

To count the number of strings of length n that contain two consecutive zeros, you need to add the strings that end with "00". There are 3^{n-2} possible strings of length n-2 that you can append "00" to.

Therefore, the recurrence relation is:

a_n = 2a_{n-1} + 2a_{n-2} + 3^{n-2}

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The correct representation on the number line is given as follows:

Option C.

A dolphin was swimming at sea level, hence the position of the dolphin is given as follows:

Position zero.

A ship anchor was sitting on the ocean floor 8 meters below the sea level, hence the position of the ship anchor is given as follows:

Position -8.

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The equation for the polynomial is P(x) = -1/64(x + 4)(x + 1)(x - 2)(x - 4)^2

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

A polynomial is represented as

P(x) = (x - zero)^multiplicity

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

P(x) = a(x + 4)(x + 1)(x - 2)(x - 4)^2

The y-intercept of the graph is (0, 2)

So, we have

a(0 + 4)(0 + 1)^3(0 - 2)(0 - 4)^2 = 2

Evaluate

-128a = 2

So, we have

a = -1/64

The equation becomes

P(x) = -1/64(x + 4)(x + 1)(x - 2)(x - 4)^2

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Field Trip Location Number of Students Percent

Dairy farm 27 17%

Robotics center 65 41%

Lion tamer at the circus 70 44%

Wildlife recovery center 58 37%

To calculate the percentage, we divide the number of students who preferred each location by the total number of students and then multiply by 100. We round to the nearest whole percent.

For the dairy farm: (27/160) x 100 ≈ 16.875 ≈ 17%

For the robotics center: (65/160) x 100 ≈ 40.625 ≈ 41%

For the lion tamer at the circus: (70/160) x 100 ≈ 43.75 ≈ 44%

For the wildlife recovery center: (58/160) x 100 ≈ 36.25 ≈ 37%