Show that the function f(x)=sin3x + cos5x is periodic and it’s period.

The minimum value of z is -38

The optimization equation is given as

z=−3x+5y

The constraints are given as:

x+y≥−2

3x−y≤2

x−y≥−4

Next, we plot the constraints on a graph and determine the points of intersections

See attachment for the graph

From the attached graph, the points of intersections are

(-9, -13) and (-10, -12)

So, we have:

(-9, -13)

(-10, -12)

Substitute these values in the objective function

z=−3x+5y

This gives

z= −3 * -9 +5 * -13 = -38

z= −3 * -10 +5 * -12 = -30

-38 is less than -30

Hence, the minimum value of z is -38

So, the complete parameters are:

Optimization Equation:

z=−3x+5y

Constraints:

x+y≥−2

3x−y≤2

x−y≥−4

Vertices of the feasible region

(0, -2)

(-3, 1)

(3, 7)

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Due to length restrictions, we kindly invite to check the explanation herein for further details of the hyperbola.

Herein we have an hyperbola whose axis of symmetry is parallel to the y-axis and the major semiaxis length is in the y-direction. By analytical geometry, we know that eccentricities of hyperbolae are greater than 1.

a) The formula for eccentricity is:

e = √(a² + b²) / a      (1)

Where:

If we know that a = 4 and b = 3, then the eccentricity of the hyperbola is:

e = √(4² + 3²) / 4

e = 5 / 4

b) The coordinates of the two vertices of the hyperbola are:

V(x, y) = (h, k ± a)      (2)

Where (h, k) are the coordinates of the center of the hyperbola.

V₁ (x, y) = (0, 4), V₂ (x, y) = (0, - 4)

The coordinates of the foci of the hyperbola are:

F(x, y) = (h, k ± c), where c = √(a² + b²).     (3)

c = √(4² + 3²)

c = 5

F₁ (x, y) = (0, 5), F₂ (x, y) = (0, - 5)

The equations of the asymptotes of the hyperbola are:

y = ± (a / b) · x

y = ± (4 / 3) · x      (4)

And the equations of the directrices of the hyperbola are:

y = k ± (2 · a - c)

y = 0 ± (8 - 5)

y = ± 3     (5)

The graph is presented in the image attached below.

c) The parametric equations for the hyperbola are the following formulae:

y = ± a · cosh t   →   y = ± 4 · cosh t      (6)

x = b · sinh t   →   x = 3 · sinh t     (7)

d) First, we determine the slopes of the two tangent lines by implicit differentiation:

m = (16 · x) / (9 · y)

m = (16 · 2.3) / [9 · (± 4.807)]

m = ± 0.851

Second, we find the intercept of each tangent line:

(x, y) = (2, 4.807)

b = 4.807 - 0.851 · 2

b = 3.105

y = 0.851 · x + 3.105      (8)

(x, y) = (2, - 4.807)

b = - 4.807 - (- 0.851) · 2

b = - 3.105

y = - 0.851 · x - 3.105      (9)

e) The definite integral of the arc length of the hyperbola is presented below:

[tex]s = \int\limits^{2}_{1} {\sqrt{\left(\frac{dx}{dt} \right)^{2}+\left(\frac{dy}{dt} \right)^{2}}} \, dt[/tex]

If we know that dx / dt = a² · sinh² t and dy / dt = b² · cosh² t, then the definite integral for the arc length is:

[tex]s = \int\limits^2_1 {\sqrt{a^{2}\cdot \sinh ^{2}t +b^{2}\cdot \cosh^{2}t}} \, dt[/tex]      (10)

f) We apply the following substitutions on (1): x = r · cos θ, y = r · sin θ. Then, we have the polar form by algebraic handling:

r(θ) = (a · b) / (b² · sin² θ - a² · cos² θ)     (11)

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Answer:

15√3

Step-by-step explanation:

after grouping and adding them to get 15√9

then you apply Surds

Answer:

x=3/2 y=1/2

Step-by-step explanation:

Answer:12002/9987

Step-by-step explanation:

Answer:

3.2

Step-by-step explanation:

4 x 8 / 8 + 2

32/10

=3.2

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Answer:

Perimeter o the rectangle = 16x - 2

Step-by-step explanation:

Perimeter of a rectangle = 2(long+ wide)

Then:

Perimeter = 2((6x-4)+(2x+3))

Perimeter = 2(6x+ 2x + 3 - 4)

Perimeter = 2(8x - 1)

Perimeter = 2*8x + 2*-1

Perimeter = 16x - 2

A)  Rotate by 90° counterclockwise.

B) Reflection transformation about the line y = 5.

C) Parallel Lines are C₂D₂ and A₂B₂; EF and E₁F₁.

D) Perpendicular lines are;  D₁F₁ and  E₁F₁; DF and EF; DC and AB.

E) Line segments with slope of 0 are; AB, C₂D₂, A₂B₂, EF and E₁F₁

F) Line segments on the boat that have an undefined slope are; Lines A₁B₁, DF and DC.

G) Slope of Line ED = 1

A) The blue segment CD is seen on the graph as a perpendicular line with 2 units while the line segment C₂D₂ is seen as a horizontal line. Thus, to match CD with C₂D₂, we will rotate by 90° counterclockwise.

B) The transformations that would be used on the blue triangle to get it to match with the red triangle is a reflection transformation about the line y = 5.

C) The line segments that are parallel to each other are; C₂D₂ and A₂B₂; EF and E₁F₁.

D) The line segments that are perpendicular are;  D₁F₁ and  E₁F₁; DF and EF; DC and AB.

E) Horizontal lines that are parallel to the x-axis have zero slope. Thus, AB, C₂D₂, A₂B₂, EF and E₁F₁ all have zero slopes.

F) Undefined slope is the slope of a vertical line. Thus, Lines A₁B₁, DF and DC have undefined slopes.

G) Slope of ED = (5 - 4)/(-10 - (-11))

Slope of ED = 1/1

Slope = 1

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The dimensions (length and width) of the poster are 6 inches and 2 inches

Represent the length with x and the width with y

From the question, we have the following parameters

x = 5 + 0.5y

Area, A = 12

The area of a rectangle is represented as:

A = xy

Substitute the known values in the above equation

(5 + 0.5y) * y = 12

Expand the bracket

5y + 0.5y^2 = 12

Multiply through by 2

10y + y^2 = 24

Rewrite as:

y^2 + 10y - 24 = 0

Expand

y^2 + 12y - 2y - 24 = 0

Factorize the expression

(y - 2)(y + 12) = 0

Solve for y

y = 2 or y = -12

The dimension cannot be negative.

So, we have

y = 2

Substitute y = 2 in x = 5 + 0.5y

x = 5 + 0.5 * 2

Evaluate

x = 6

Hence, the dimensions (length and width) of the poster are 6 inches and 2 inches

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The angles and length of the triangle area as follows:

The sum of angles in a triangle is 180 degree.

Therefore,

∠DBC = 180 - 29 - 90 = 61°

Hence,

∠A = 180 - 29 - 61 - 55 = 35°

∠CBD = 61°

Using trigonometric ratios,

sin 55 = opposite / hypotenuse

sin 55 = AD / 40

AD = 40 sin 55

AD = 32.7660817716

AD = 32.77 units

cos 55 = adjacent / hypotenuse

cos 55 = BD / 40

BD = 40 cos 55

BD = 22.943057454

BD = 22.94 units

sin 29 = 22.94 / BC

BC = 22.94 sin 29

BC = 11.1215326885

BC = 11.12 units

cos 29 = CD / 11.12

CD = 11.12 cos 29

CD = 9.72577114339

CD = 9.73 units

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The equation for the reflected function is:

g(x) =  (1/2)*(3)^(-x)

Here we know that function g(x) is a reflection of f(x) across the y-axis.

Remember that the reflection is written as:

g(x) = f(-x)

Here we know that:

f(x) = (1/2)*(3)^x

Then, replacing that in the equation for g(x), we get:

g(x) =  f(-x) = (1/2)*(3)^(-x)

g(x) =  (1/2)*(3)^(-x)

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The equation of the line that passes through the given points is

y = 1/3x + 5.

The formula for the equation of a line passing through two points (x1, y1) and (x2, y2) is

[tex](y-y1) = \frac{(y2-y1)}{(x2-x1)} (x-x1)[/tex]

Where the fraction (y2 - y1)/(x2 - x1) is the slope of the line denoted by 'm'.

It is given that, a line passes through the points (-3,4) and (6,7).

So, the equation of the line is

[tex](y-y1) = \frac{(y2-y1)}{(x2-x1)} (x-x1)[/tex]

On substituting x1 = -3, y1 = 4, x2 = 6, and y2 = 7

(y - 4) = [(7 - 4)/(6 + 3)](x + 3)

⇒ (y - 4) = (3/9)(x + 3)

⇒ (y - 4) = 1/3(x + 3)

⇒ y- 4 = 1/3x + 1

⇒ y = 1/3x + 1 + 4

∴ y = 1/3x + 5

Thus, the equation of the line passing through the points (-3,4) and (6,7) is y = 1/3x + 5. Where the slope m = 1/3.

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The constant of proportionality between the cost and the number of pounds is: A. 0.59.

The constant of proportionality of a relationship between two variables X and Y can be defined as the ratio between of X and Y at a constant value. This means that the ratio or product of X and Y will give us a constant if both are in a proportional relationship.

Thus, the constant of proportionality for the  relationship between two variables X and Y would be calculated as:

Constant of proportionality (k) = Y/X. [this is the proportional between the two quantities, X and Y].

Given the table of values, using a pair of points on the table, say, (5, 2.95), we can calculate the constant of proportionality as:

X = 5

Y = 2.95

Constant of proportionality (k) = Y/X = 2.95/5

Constant of proportionality (k) = 0.59 pounds per dollar.

Therefore, the constant of proportionality between the cost and the number of pounds is: A. 0.59.

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Answer:

a=4.40

Step-by-step explanation:

To find value of a use pythagoras theorem:

c^2=b^2+a^2

Rearrange the equation:

a^2=c^2-b^2

Substitute the values:

a^2=(4.58)^2-(1.26)^2

After calculation:

a=4.40

Answer:last option -5± √17

2

Step-by-step explanation:

Quadratic equations represent any equation that can be rearranged in standard forma (ax² + bx + c( =0(a, b & c) are known.

Quadratic equations can always be solved using the quadratic formula, but sometimes factoring or isolating the variable is also posible.

In the square equation ax² + bx + c = 0

[tex]\boldsymbol{\sf{x=\dfrac{-b\pm\sqrt{\Delta} }{2a} \ ,\Delta=b^{2}-4ac } }[/tex]

Let's calculate the discriminant of the quadratic equation:

Since the discriminant is greater than zero, then the quadratic equation has two real roots.

[tex]\boldsymbol{\sf{x_{1}=\dfrac{-b-\sqrt{\Delta} }{ 2\cdot a}=\dfrac{-5-\sqrt{17} }{2\cdot1} }}[/tex]

[tex]\boldsymbol{\sf{x_{2}=\dfrac{-b+\sqrt{\Delta} }{ 2\cdot a}=\dfrac{-5+\sqrt{17} }{2\cdot1} }}[/tex]

Solution:

[tex]\boldsymbol{\sf{x=\dfrac{-5\pm\sqrt{17} }{2} }}[/tex]

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Answer: I believe the answer is 25.

Step-by-step explanation:

You convert 1/5 to a decimal.

1/5 = 0.2

then you divide

180 divided by 7.2 = 25

answer is 25.

Step-by-step explanation:

[tex]y = \sqrt{4x + 6} [/tex]

[tex] {y}^{2} = 4x + 6[/tex]

[tex] {y}^{2} - 6 = 4x[/tex]

[tex] \frac{1}{4} ( {y}^{2} - 6) = x[/tex]

Swap x and y

[tex] \frac{ {x}^{2} - 6 }{4} = y[/tex]

Using equivalent angles, the solutions are given as follows:

a) [tex]x = \frac{15\pi}{23}[/tex].

b) [tex]x = \frac{9\pi}{62}, x = \frac{53\pi}{62}[/tex].

c) [tex]x = \frac{3\pi}{8}, \frac{11\pi}{8}[/tex]

Each angle on the second, third and fourth quadrants will have an equivalent on the first quadrant.

For item a, we have to find the equivalent angle on the 2nd quadrant, where the sine is also positive.

Hence:

[tex]\pi - \frac{8\pi}{23} = \frac{23\pi}{23} - \frac{8\pi}{23} = \frac{15\pi}{23}[/tex]

Hence [tex]x = \frac{15\pi}{23}[/tex].

For item b, if two angles are complementary, the sine of one is the cosine of the other.

Complementary angles add to 90º = 0.5pi, hence:

[tex]x + \frac{11\pi}{31} = \frac{\pi}{2}[/tex]

[tex]x = \frac{31\pi}{62} - \frac{22\pi}{62}[/tex]

[tex]x = \frac{9\pi}{62}[/tex]

The equivalent angle on the second quadrant is:

[tex]\pi - \frac{9\pi}{62} = \frac{62\pi}{62} - \frac{9\pi}{62} = \frac{53\pi}{62}[/tex]

Hence the solutions are:

[tex]x = \frac{9\pi}{62}, x = \frac{53\pi}{62}[/tex]

For item c, the angles are also complementary, hence:

[tex]x + \frac{\pi}{8} = \frac{\pi}{2}[/tex]

[tex]x = \frac{4\pi}{8} - \frac{\pi}{8}[/tex]

[tex]x = \frac{3\pi}{8}[/tex]

The tangent is also positive on the third quadrant, hence the equivalent angle is:

[tex]x = \pi + \frac{3\pi}{8} = \frac{8\pi}{8} + \frac{3\pi}{8} = \frac{11\pi}{8}[/tex]

Hence the solutions are:

[tex]x = \frac{3\pi}{8}, \frac{11\pi}{8}[/tex]

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Answer:

  $711.23

Step-by-step explanation:

We assume the entire closing cost went to reducing the principal of the loan. Then the amount borrowed was $147,192.

The amortization formula tells you the monthly payment.

  A = P(r/12)/(1 -(1 +r/12)^(-12t))

P is the principal, r is the annual rate, and t is the number of years.

The monthly payment is ...

  A = $147,192(0.041/12)/(1 -(1 +0.041/12)^-360) ≈ $711.23

Jeff's monthly payment is $711.23.

Answer:

Step-by-step explanation:

The sum of interior angles of a quadrilateral is 360°.

If we consider the measure of one of the unknown angles to be [tex]x[/tex]°, we can set up the following equation:

[tex]x + x + 75^\circ + 117^\circ = 360^\circ[/tex]

Now we can solve for [tex]x[/tex]:

⇒ [tex]2x + 192^\circ = 360^\circ[/tex]

⇒ [tex]2x = 360^\circ - 192^\circ[/tex]        [subtracting 192° from both sides]

⇒ [tex]2x = 168^\circ[/tex]

⇒ [tex]x = \frac{168^\circ}{2}[/tex]                      [dividing both sides by 2]

⇒ [tex]x = \bf84^\circ[/tex]

Therefore, the other two angles each have a measure of 84°.

WhenWhen we divide 8x⁴ - 6x³ + 2 by 2x² + 3 The result obtained is 4x² - 3x - 6 remainder 9x + 20

Quotient is the result obtained when division operation is carried out.

For example when 6 is divided by 2, the result obtained is 3. Thus, the quotient is 3

To divide 8x⁴ - 6x³ + 2 by 2x² + 3, we shall apply the long division method. This is illustrated below:

                          4x² - 3x - 6        

             2x² + 3 | 8x⁴ - 6x³ + 2

                         -(8x⁴ + 12x²)

                         -6x³ - 12x² + 2

                        -(-6x³ - 9x)

                        -12x² + 9x + 2

                         -(-12x² - 18)

                          9x + 20            

Thus, the result obtained when we divide 8x⁴ - 6x³ + 2 by 2x² + 3 is 4x² - 3x - 6 remainder 9x + 20

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The two ordered pairs that the mid point is (4,-10) are (4,-20),(4,0) & (4,0),(4,-20).

Given the coordinates of mid point be (4,-10).

We are required to find the ordered pairs that the mid point is (4,-20).

Coordinates show positions of points or something else on a surface.

There are various combinations whose mid point is (4,-10).

First are (4,-20),(4,0).

Mid point =[(4+4)/2,(-20+0)/2]

=(4,-10)

Second are (4,0) , (4,-20)

Mid point=[(4+4)/2,(0-20)/2]

=(4,-10)

Third are (8,-20),(0,0)

Mid point=[(8+0)/2,(-20+0)/2]

=(4,-10)

Fourth are (0,-20),(8,0)

Mid point =[(0+8)/2,(-10+0)/2]

=(4,-10)

Hence the ordered pairs that the mid point is (4,-10) are (4,-20),(4,0) & (4,0),(4,-20)& (8,-20),(0,0)&(0,-20),(8,0)&(0,0),(8,-20).etc.

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1a. 98 cm ^2

1b. 76. 98 cm^2

2a. 42cm

2b. 7. 19 cm

3. a + b/2 (h)

4. 24 cm ^2

5. πr + 2r

6. 13. 2m

7. 45cm^2

8. 252 cm ^ 2

9. 450 cm^ 2

1a. The shape given is a rectangle

The formula for area of a rectangle is given as;

Area = length × width

Area = 7 × 14

Area = 98 cm ^2

1b The shape given is a semi circle

The formula for area of a semicircle is given as;

Area = 1/2 π r^2

radius = diameter/2 = 14/2 = 7cm

Area = 1/2 × 3.142 × 7 × 7

Area = 76. 98 cm^2

2a. The shape given is a rectangle

The formula for perimeter of a rectangle is given as;

Perimeter = 2 ( length + width)

Perimeter = 2 ( 14 + 7) = 2( 21)

Perimeter = 42cm

2b. The shape is a semicircle

Perimeter = π r + 2r

r= 1.4cm; diameter divided by 2

Perimeter = 3. 142(1.4) + 2(1.4)

Perimeter = 7. 19 cm

3. The formula for area of a trapezium is given as

Area = a + b/2 (h)

4. The area of the trapezium is given as;

Area = 9 + 7/2 (3)

Area = 16/2 (3)

Area = 8 × 3

Area = 24 cm ^2

5. Area of semicircle = 1/2 πr^2

Perimeter of a semicircle = πr + 2r

6. From the information given, we have the following

Area = 480 m^2

a = 20m

b = unknown

h = 13. 2m

Area = a+b/2 (h)

Substitute the values

480 = 20+b/2 (13. 2)

480 = 10+ b (13. 2)

480/13. 2 = 10 + b

10+ b = 480/ 13. 2

10 + b = 36. 36

b = 36.36 - 10

b = 26. 36m

7. The formula for area of a rhombus is given as

Area = p × q/2

Where p and q are the diagonals

Area = 7. 5 × 12/2

Area = 90/2

Area = 45cm^2

8. The formula for area for a quadrilateral is given as;

Area of quadrilateral = (½) × diagonal length × sum of the length of the perpendiculars

sum of the length = 13 + 8 = 21cm

Diagonal A= 24cm

Area = 1/2 × 24 × 21

Area = 252 cm ^ 2

9. Area of a pentagonal park = 1/2 × sum of parallel sides × height

Sum of parallel sides = 15 + 15 = 30 cm

height = 30cm

Area = 1/2 × 30 × 30

Area = 450 cm^ 2

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y=2

-5(2)²-3(2)-2

-5(4)-3(2)-2

-20-6-2

=12

Answer:

12km/h

Step-by-step explanation:

s=d/t

900m is 0.9km and 4min and 30sec is 0.075h

0.9/0.075=12 which gives us 12km/h

Answer:

Step-by-step explanation:

4 min 30 seconds = 270 seconds

900m = 270 seconds

100m = 30 seconds

1000m= 300 seconds (5min)

12,000m=3600seconds (60min)

Answer = 12km/h

The number 0.22 represents the population proportion and the correct way to represent it is p = 0.22.

Statistics is a mathematical tool defined as the study of collecting data, analysis, understanding, representation, and organization. Statistics is described as the procedure of collecting data, classifying it, displaying that in a way that makes it easy to understand, and analyzing it even further.

It is given that:

It is given that:

The Department of Motor Vehicles reports that the proportion of all vehicles registered in California that are imported is 0.22.

The number 0.22 represent the population proportion.

The population proportion can be represented by the letter p:

p = 0.22

Thus, the number 0.22 represents the population proportion, and the correct way to represent it is p = 0.22.

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The value of cos(L) in the triangle is Five-thirteenths

Right triangles are triangles whose one of its angle has a measure of 90 degrees

The value of a cosine function is calculated as:

cos(L) = Adjacent/Hypotenuse

The hypotenuse is calculated as

Hypotenuse^2 = Opposite^2 + Adjacent^2

So, we have:

Hypotenuse^2 = 12^2 + 5^2

Evaluate

Hypotenuse^2 = 169

Take the square root of both sides

Hypotenuse = 13

So, we have

Adjacent = 5

Hypotenuse = 13

Recall that

cos(L) = Adjacent/Hypotenuse

This gives

cos(L) = 5/13

Hence, the value of cos(L) in the triangle is Five-thirteenths

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Using the z-distribution, a sample of 142,282 should be taken, which is not practical as it is too large of a sample.

The confidence interval is:

[tex]\overline{x} \pm z\frac{\sigma}{\sqrt{n}}[/tex]

The margin of error is:

[tex]M = z\frac{\sigma}{\sqrt{n}}[/tex]

In which:

Assuming an uniform distribution, the standard deviation is given by:

[tex]S = \sqrt{\frac{(4 - 0)^2}{12}} = 1.1547[/tex]

In this problem, we have a 95% confidence level, hence[tex]\alpha = 0.95[/tex], z is the value of Z that has a p-value of [tex]\frac{1+0.95}{2} = 0.975[/tex], so the critical value is z = 1.96.

The sample size is found solving for n when the margin of error is of M = 0.006, hence:

[tex]M = z\frac{\sigma}{\sqrt{n}}[/tex]

[tex]0.006 = 1.96\frac{1.1547}{\sqrt{n}}[/tex]

[tex]0.006\sqrt{n} = 1.96 \times 1.1547[/tex]

[tex]\sqrt{n} = \frac{1.96 \times 1.1547}{0.006}[/tex]

[tex](\sqrt{n})^2 = \left(\frac{1.96 \times 1.1547}{0.006}\right)^2[/tex]

n =  142,282.

A sample of 142,282 should be taken, which is not practical as it is too large of a sample.

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The answers to the service charge rate are:

This is the number of calls for the help desk

160000/3200 = 50per call

For the network center

735000/9800

= 75 per device that was monitored

For the Electronic mail

100000/10000 =

10 for the mail account

For local voice support

124600/8900

= 14 per phone extension

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Answer:

40 + 10 + 30 = 50 + 30 = 80

Step-by-step explanation:

40 + 10 + 30 = 50 + 30 = 80

Answer:

10(4+3+1)

Step-by-step explanation:

Expand 40+30+10 by the distributive property:: 10(4+3+1)