Write the integral and solve for the area under the curve:

The values of x and y at t=3 are -4.95 and 0.423 respectively.

The value of dx/dt and dy/dt are 0.71 and -2.97 respectively.

The value of the tangent slope at t=3 is 4.21.

The speed at t=3 is 3.05 units/sec.

Given the equation of x as the function of t is

x= 5 cos t

similarly, the equation of y as the function of t is

y= 3 sin t

At t=3 the value of x will be

x (at t=3) = 5 cos 3= 5(-0.989)= -4.95

At t=3 the value of y will be

y (at t=3) = 3 sin 3= 3(0.141)= 0.423

The derivative of the function of x with respect to t will be

dx/dt= d(5 cos t)/dt= 5d(cos t)/dt= -5 sin t

at t=3 the value of dx/dt will be

dx/dt (at t=3) = -5 sin 3= -5(0.141)= 0.71

The derivative of the function of y with respect to t will be

dy/dt= d(3 sin t)/dt= 3d(sin t)/dt= 3 cos t

at t=3 the value of dy/dt will be

dy/dt (at t=3) = 3 cos t= 3(-0.989)= -2.97

The tangent slope is dy/dx which can be calculated by

dy/dx= (dy/dt)(dt/dx)= (dy/dt)/(dx/dt)= 3 cos t/ -5 sin t=  (-3/5) cot t

at t=3 the value of tangent slope will be

dy/dx (at t=3) = (-3/5) cot 3= 4.21

The speed at t=3 will be

speed v= [tex]\sqrt{v_{x} ^{2} + v_{y} ^{2} }[/tex]

= √(dx/dt)²+(dy/dt)²

at t=3

= √(0.71)²+(-2.97)²

= √9.325

= 3.05 unit/sec

Therefore the values of x and y at t=3 are -4.95 and 0.423 respectively.

The value of dx/dt and dy/dt are 0.71 and -2.97 respectively.

The value of the tangent slope at t=3 is 4.21.

The speed at t=3 is 3.05 units/sec.

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Answer:

30

Step-by-step explanation:

1500/50=30

The weight of the chocolate frosting if it is 55% cocoa powder will be 272.25 grams.

The quantity of anything is stated as though it were a fraction of a hundred.

Mama bear wants to bake a cake with chocolate frosting for her twin's birthday party.

She uses a package containing 495 grams of cocoa powder for the frosting.

Then the weight of the chocolate frosting if it is 55% cocoa powder will be

⇒ 0.55 × 495

⇒ 272.25 grams

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If the Worker is retrenched, the probability that he is a plumber or a plasterer will be  0.0152.

The chances of an event occurring are defined by probability. Probability has several uses in games, in business to create probability-based forecasts,

The probability that the employee is a plumber;

[tex]\rm P(plumber)= \frac{No \ of \ plumber}{Total \ number \ of \ employ} \\\\ P(plumber)=\frac{12}{108} \\\\\ P(plumber)= 0.11[/tex]

The probability that the employee is a plasterer;

[tex]\rm P(plasterers )= \frac{No \ of \ plasterers }{Total \ number \ of \ employ} \\\\ P(plasterers )=\frac{15}{108} \\\\\ P(plasterers )= 0.13[/tex]

The probability that he is a plumber or a plasterer:

P = 0.13 × 0.11

P= 0.0152

Hence, if the worker is retrenched, the probability that he is a plumber or a plasterer will be  0.0152.

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The correct answer is option B which is the graph that represents the equation y = (1/2)x will be the second option in the figure. At every value of y, the value of x is double.

A graph is the representation of the data on the vertical and horizontal coordinates so we can see the trend of the data.

The graph of the equation y =1/2x is represented in the answer below we can see in the graph that at every value of y the value of x is double for example if we put y = 1 then the value of x will be 2.

y = 1/2 x

at x = 2

y = 1/2 x 2

x = 1

Therefore the correct answer is option B which is the graph that represents the equation y = (1/2)x will be the second option in the figure. At every value of y, the value of x is double.

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Answer:

D

Step-by-step explanation:

[tex]1200( {1.025}^{5} ) = 1357.69[/tex]

Differentiate the function/equation with respect to x and solve for the derivative, dy/dx. The value of dy/dx at the given point is the slope of the tangent line to the curve at that point. Then use the point-slope formula to get the equation of the tangent.

1.

[tex]y = \dfrac{x^2}{x+2} \implies \dfrac{dy}{dx} = \dfrac{2x\times(x+2) - x\times1}{(x+2)^2} = \dfrac{x(x+4)}{(x+2)^2}[/tex]

When x = 2, the derivative is

[tex]\dfrac{dy}{dx}\bigg|_{x=2} = \dfrac{2(2+4)}{(2+2)^2} = \dfrac34[/tex]

Then the equation of the tangent line at (2, 1) is

[tex]y - 1 = \dfrac34 (x - 2) \implies \boxed{y = \dfrac{3x}4 - \dfrac12}[/tex]

2.

[tex]x^3 + 2y^2 = 10y \implies 3x^2 + 4y \dfrac{dy}{dx} = 10 \dfrac{dy}{dx} \implies \dfrac{dy}{dx} = \dfrac{3x^2}{10-4y}[/tex]

When x = 2 and y = 1, the derivative is

[tex]\dfrac{dy}{dx}\bigg|_{(x,y)=(2,1)} = \dfrac{3\times2^2}{10-4\times1} = 2[/tex]

Then the tangent at (2, 1) has equation

[tex]y - 1 = 2 (x - 2) \implies \boxed{y = 2x - 3}[/tex]

Answer:

Step-by-step explanation:

Let :

C = number of cats

D = number of dogs

Raul's pet store has a play area that can fit up to 30 cats and dogs.

C + D = 30

The pet store never has more than 8 cats in the play areas.

(there can never be 9 or more cats)

As or the number of dogs :

C + D = 30

C = 30 - D

Since we know that C < 9. To get the number of dogs allowed, we just plug in 30 - D for C.

30 - D < 9

30 - 9 < D

or

(dogs have to be 22 or more)

Step-by-step explanation:

b because you have to multiply

The mean and the standard deviation are 16.15 and 12.03, respectively

The table of values is given as:

Work hours           Frequency

0 - <10                        45

10 - <20                      33

20 - <30                     20

30 - <40                       7

40 - <50                      8

Start by rewriting the table using the midpoint of each class

x      f

5     45

15    33

25    20

35    7

45    8

The mean is then calculated as:

[tex]\bar x = \frac{\sum fx}{\sum f}[/tex]

This gives

[tex]\bar x = \frac{5 * 45 + 15 * 33 + 25 * 20 35 * 7 + 45 * 8}{45 + 33 + 20 + 7 + 8}[/tex]

Evaluate the sum and the products

[tex]\bar x = \frac{1825}{113}[/tex]

Divide

[tex]\bar x = 16.15[/tex]

The standard deviation is:

[tex]\sigma =\sqrt{\frac{\sum f(x - \bar x)^2}{\sum f}}[/tex]

This gives

[tex]\sigma= \sqrt{\frac{45 * (5 - 16.15)^2 + 33 * (15 - 16.15)^2 + 20 * (25 - 16.15)^2 + 7 * (35 - 16.15)^2 + 8 * (45 - 16.15)^2 }{45 + 33 + 20 + 7 + 8}}[/tex]

Evaluate the sum and the products

[tex]\sigma= \sqrt{\frac{16350.4425}{113}}[/tex]

Divide

[tex]\sigma= \sqrt{144.694181416}[/tex]

Take the square root

[tex]\sigma= 12.03[/tex]

Hence, the mean and the standard deviation are 16.15 and 12.03, respectively

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Answer:

correct answer is B.

Step-by-step explanation:

for more information see the attachment.

Answer:

no 2/B is the correct answer

Answer:

Two functions which satisfies the H(x) are f(x) = 2x - 4, (x)    = x⁵  and

f(x)  = 2x,  g(x)  = x⁵ - 2

Step-by-step explanation:

The given question is based on the decomposition of a given function into two functions

Decomposition of a function :- functional decomposition is the process of resolving a functional relationship into its constituent parts in such a way that the original function can be reconstructed from those parts by function composition

so the function given in the question is H(x) = 2x⁵ - 4

now by using the concept of decomposition of a function we can find two functions f and g

therefore,

f(x) = 2x - 4

and    g(x)    = x⁵

similarly we can find another pair which satisfies the function H(x)

f(x)  = 2x

and g(x)  = x⁵ - 2

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The zeroes of the quadratic function f(x) = 6x² + 12x – 7 will be given below. Then the correct option is A.

The missing options are given below.

The quadratic equation is given as ax² + bx + c = 0.

Then the solution is given as

[tex]\rm x = \dfrac{-b\pm \sqrt{b^2 - 4ac}}{2a}[/tex]

The quadratic function is given below.

f(x) = 6x² + 12x – 7

Then the zeroes of the quadratic function will be

[tex]\rm x = \dfrac{-12\pm \sqrt{12^2 - 4 * 6 * (-7)}}{2 * 6}\\\\\\\rm x = \dfrac{-12\pm \sqrt{312}}{12}\\\\\\\rm x = -1 \pm \sqrt{\dfrac{13}{6}}\\\\\\x = -1 - \sqrt{\dfrac{13}{6}} , -1 + \sqrt{\dfrac{13}{6}}[/tex]

Then the correct option is A.

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Answer:

h(x) = -2|x+2| + 2

Step-by-step explanation:

let me know if you want an explanation :))

From indices : when you have a fraction like 3⁶/5

we can also write it as (5√3)^6

and we were given (c√a )^b

meaning that anywhere the letters are, we put them as the numbers there.

therefore: a = 3 , b = 6 , c = 5

Hope this helps and you understood it.

you're super free to ask any more questions eh.

Good luck ✅.

Answer:

Hamish earns $20 per hour, and Harry earns $26 per hour.

Step-by-step explanation:

Harry earns

$1+(5/4)•x and also (7/5)•x-$2

$1+(5/4)•x = (7/5)•x-$2

$1+$2= (7/5)•x-(5/4)•x

3=(7•4•x-5•5•x)/20

3=3x/20

3•20=3x

X=20, Hamish earns $20

Substitute x in one of the expressions

1+(5/4)•20=1+25=26, Harry earns $26

The approximate difference in the concentration of hydrogen ions between the two solutions is 4.0×10⁻³

The pH of a solution is a measurement of the concentration of hydrogen ions in the solution, as well as its acidity or alkalinity. Normally, the pH scale ranges from 0 to 14.

pH value for basic solution = 11.2

pH value for acidic solution = 2.4

Concentration for a basic solution:

[tex]\rm = 10^{-11.2}[/tex]

[tex]= 6.309\times10^{-12}[/tex]

Concentration for an acidic solution:

[tex]\rm = 10^{-2.4}[/tex]

[tex]= 3.981\times10^{-3}[/tex]

Difference between the concentration:

[tex]= 3.981\times10^{-3}- 6.309\times10^{-12}[/tex]

[tex]= 3.98\times10^{-3}[/tex]

[tex]= 4\times10^{-3}[/tex]

Thus, the approximate difference in the concentration of hydrogen ions between the two solutions is 4.0×10⁻³

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Answer:

Step-by-step explanation:

This shows that the greatest common factor is 8, Hence Venita's error is that she incorrectly determined the GCF.

Given the following expression

32ab - 8

Find the factors of each terms

32ab = 8 * 4. * a * b

8 = 8 * 1

Since 8 is common to both factors, hence

32ab - 8 = 8(4ab -1)

This shows that the greatest common factor is 8, Hence Venita's error is that she incorrectly determined the GCF.

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Answer:

60

Step-by-step explanation:

_---&$'_-4$':/-&$$--(:'##4--&&4___&$$_666&4&&&&&

Let

[tex]I = \displaystyle \int e^{-2x} \cos(2x) \, dx[/]tex

Integrate by parts:

[tex]\displaystyle \int u \, dv = uv - \int v \, du[/tex]

with

[tex]u = e^{-2x} \implies du = -2 e^{-2x} \, dx \\\\ dv = \cos(2x) \, dx \implies v = \dfrac12 \sin(2x)[/tex]

Then

[tex]\displaystyle I = \frac12 e^{-2x} \sin(2x) + \int e^{-2x} \sin(2x) \, dx + C[/tex]

Integrate by parts again, this time with

[tex]u = e^{-2x} \implies du = -2 e^{-2x} \, dx \\\\ dv = \sin(2x) \, dx \implies v = -\dfrac12 \cos(2x)[/tex]

so that

[tex]\displaystyle I = \frac12 e^{-2x} \sin(2x) - \frac12 e^{-2x} \cos(2x) - \int e^{-2x} \cos(2x) \, dx + C\\\\ \implies I = \frac{\sin(2x)-\cos(2x)}{2e^{2x}} - I + C \\\\ \implies 2I = \frac{\sin(2x) - \cos(2x)}{2e^{2x}} + C \\\\ \implies I = \boxed{\frac{\sin(2x) - \cos(2x)}{4e^{2x}} + C}[/tex]

Step-by-step explanation:

[tex] \frac{5 {}^{ - 4x + 7} }{125 {}^{x} } [/tex]

[tex] \frac{5 {}^{ - 4x + 7} }{5 {}^{3x} } [/tex]

[tex]5 {}^{ - 4x + 7 - 3x} [/tex]

[tex]5 {}^{ - 7x + 7} [/tex]

3779.309 is the answer....

Answer:

B

Step-by-step explanation:

1/9 can be simplified to 1/3 as 1/3 x 1/3 = 1/9  and 49/25 can be simplified to 7/5 as 7/5 x 7/5 = 49/25.

Now looking at both the fraction 1/3 and 7/5

1/3 simplifies to 0.34

It is an irrational value as the 3 keeps repeating.

7/5 simplifies to 1.4

It is a rational value as the decimal value is not repeating.

Thus making the second option the answer.

Answer:

Step-by-step explanation:

Given is isosceles right triangle, since its interior angles are 45°, 45° and 90°.

The missing length is therefore same as the length of the other leg:

Note: This is not included in the answer choices.

Answer:

4 units

Step-by-step explanation:

We can use the converse theorem of the isosceles triangle to find the missing leg.

Converse Isosceles triangle theorem: If two angles are congruent in a triangle, then sides opposite to those angles are also congruent.

Therefore,

As there are two 45° angles, the opposite sides to those are also congruent.

∴ s = 4

The true statement is:

"The range of the function is all real numbers less than or equal to 9."

Here we have the quadratic function:

[tex]f(x) = -x^2 - 4x + 5[/tex]

And we want to see which of the given statements are true.

The first one is:

"The domain is al real numbers less than or equal to -2"

This is false, for all quadratic functions the domain is the set of all real numbers (unless the domain is defined).

The second statement is:

" The domain of the function is all real numbers less than or equal to 9."

Also false

Third one:

" The range of the function is all real numbers less than or equal to −2"

The range of a quadratic function with a negative leading coefficient will be the set of all the values smaller than the y-value of the vertex.

In this case, the quadratic function is:

[tex]f(x) = -x^2 - 4x + 5[/tex]

So the vertex is at:

[tex]x = 4/(2*-1) = -2\\[/tex]

Then the y-value of the vertex is:

[tex]f(-2) = -(-2)^2 - 4*(-2) + 5 = -4 + 8 + 5 = 9[/tex]

So the range is the set of all real numbers less than or equal to 9.

So the above statement is false, and the final one:

"The range of the function is all real numbers less than or equal to 9."

Is the true statement.

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Answer:

A. -5

Explanation:

For (3x + 15)/(6 - x) to be equal to zero. Set the equation to 0.

Doing so:

(3x + 15)/(6 - x) = 0

3x + 15 = 0(6 - x)

3x + 15 = 0

3x = -15

 x = -15/3

 x = -5

Therefore, the value of x is -5 to equal to zero.

Answer:

y = x + 4

when x is 5, y is 9

Step-by-step explanation:

the pattern from the table is that the y value is 4 more than the x value, so the equation would be y = x + 4.

if you plug in x = 5, then this would be y = 5 + 4 = 9

Answer:

12.5%

Step-by-step explanation:

1/8 × 100% = 12.5%

So 9f you multiply an eighth by one hundred percent and simplify accurately you'll get 12.5%

Answer:

1

Step-by-step explanation:

When you graph the functions they intersect at one point

(-3.368119, -31.10436)