Find the equation of the line.

Answer:

(a).   y'(1)=0  and    y'(2) = 3

(b).  [tex]$y'(t)=kb2t\cos(bt^2)$[/tex]

(c).  [tex]$ b = \frac{\pi}{2} \text{ and}\ k = \frac{3}{2\pi}$[/tex]

Step-by-step explanation:

(a). Let the curve is,

[tex]$y(t)=k \sin (bt^2)$[/tex]

So the stationary point or the critical point of the differential function of a single real variable , f(x) is the value [tex]x_{0}[/tex]  which lies in the domain of f where the derivative is 0.

Therefore,  y'(1)=0

Also given that the derivative of the function y(t) is 3 at t = 2.

Therefore, y'(2) = 3.

(b).

Given function,    [tex]$y(t)=k \sin (bt^2)$[/tex]

Differentiating the above equation with respect to x, we get

[tex]y'(t)=\frac{d}{dt}[k \sin (bt^2)]\\ y'(t)=k\frac{d}{dt}[\sin (bt^2)][/tex]

Applying chain rule,

[tex]y'(t)=k \cos (bt^2)(\frac{d}{dt}[bt^2])\\ y'(t)=k\cos(bt^2)(b2t)\\ y'(t)= kb2t\cos(bt^2)[/tex]  

(c).

Finding the exact values of k and b.

As per the above parts in (a) and (b), the initial conditions are

y'(1) = 0 and y'(2) = 3

And the equations were

[tex]$y(t)=k \sin (bt^2)$[/tex]

[tex]$y'(t)=kb2t\cos (bt^2)$[/tex]

Now putting the initial conditions in the equation y'(1)=0

[tex]$kb2(1)\cos(b(1)^2)=0$[/tex]

2kbcos(b) = 0

cos b = 0   (Since, k and b cannot be zero)

[tex]$b=\frac{\pi}{2}$[/tex]

And

y'(2) = 3

[tex]$\therefore kb2(2)\cos [b(2)^2]=3$[/tex]

[tex]$4kb\cos (4b)=3$[/tex]

[tex]$4k(\frac{\pi}{2})\cos(\frac{4 \pi}{2})=3$[/tex]

[tex]$2k\pi\cos 2 \pi=3$[/tex]

[tex]2k\pi(1) = 3$[/tex]  

[tex]$k=\frac{3}{2\pi}$[/tex]

[tex]$\therefore b = \frac{\pi}{2} \text{ and}\ k = \frac{3}{2\pi}$[/tex]

The y'(1) =0, y'(2) = 3, and the  [tex]\rm y'(t) = kb2t \ cos(bt^2)[/tex]  and value of b and k are [tex]\pi/2[/tex]  and  [tex]3/2\pi[/tex] respectively.

It is given that the curve  [tex]\rm y(t) = ksin(bt^2)[/tex]

It is required to find the critical point, first derivative, and smallest value of b.

It is defined as a special type of relationship and they have a predefined domain and range according to the function.

We have a curve:

[tex]\rm y(t) = ksin(bt^2)[/tex]

Given that the first critical point of y(t) for positive t occurs at t = 1

First, we have to find the first derivative of the function or curve:

[tex]\rm y'(t) = \frac{d}{dt} (ksin(bt^2))[/tex]

[tex]\rm y'(t) = k\times2bt\times cos(bt^2)[/tex]   [ using chain rule]

[tex]\rm y'(t) = kb2t \ cos(bt^2)[/tex]

y(0) = 0

y'(0) = 0

The critical point is the point where the derivative of the function becomes 0 at that point in the domain of a function.

From the critical point y'(1) = 0 ⇒  [tex]\rm kb2 \ cos(b) =0[/tex]

k and b can not be zero

[tex]\rm cos(b) = 0[/tex]

b = [tex]\rm \frac{\pi}{2}[/tex]

and y'(2) =3

[tex]\rm y'(2) = kb2\times 2 \times cos(b\times2^2) =3\\\\\rm 4kb \ cos(4b) =3[/tex](b =[tex]\rm \frac{\pi}{2}[/tex])

[tex]\rm 4k\frac{\pi}{2} \ cos(4\frac{\pi}{2} ) =3\\\\\rm2 \pi kcos(2\pi) = 3[/tex]

[tex]\rm2 \pi k\times1) = 3\\\rm k = \frac{3}{2\pi}[/tex]

Thus, y'(1) =0, y'(2) = 3, and the  [tex]\rm y'(t) = kb2t \ cos(bt^2)[/tex]  and value of b and k are [tex]\pi/2[/tex]  and  [tex]3/2\pi[/tex] respectively.

Learn more about the function here:

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Answer:

12/25

Step-by-step explanation:

There were a total of 48 males, so 48/100 = 12/25.

+---------------+------+--------+-------+

|           | Male | Female | Total |

+---------------+------+--------+-------+

| Prefers Dogs  | 36 | 20 | 56   |

+---------------+------+--------+-------+

| Prefers Cats  | 10   | 26 | 36   |

+---------------+------+--------+-------+

| No Preference | 2 | 6  | 8   |

+---------------+------+--------+-------+

| Total         | 48   | 52     | 100   |

+---------------+------+--------+-------+

Answer:

Look at the image below

Answer:

b=-a

Step-by-step explanation:

Answer:

last option

Step-by-step explanation:

Basically, every 1 cm is 8 / 2 = 4 feet so the width is 4 * 4 = 16 and the length is 6 * 4 = 24.

Answer:

option D

Step-by-step explanation:

Answer:

Option B.

Step-by-step explanation:

From the given venn diagram it is clear that

[tex]A={1,2}[/tex]

[tex]B={1,2,3,4,5,6}[/tex]

[tex]A\cap B={1,2}[/tex]

Since the intersection of A and B is non-empty, therefore by conditional probability

[tex]P(B|A)=\dfrac{P(A\cap B)}{P(A)}[/tex]

[tex]P(B|A)\cdot P(A)=P(A\cap B)[/tex]

[tex]P(A\cap B)=P(B|A)\cdot P(A)[/tex]

Therefore, the correct option is B.

Answer:

3. time = 6.4999  approximately 6.5 years

4. $2235.35

5.  $ 3950

Step-by-step explanation:

4. amount of interest =  final amount - principle= 7535.35 - 5300 = 2235.35

5. principle = final amount - interest earned = 4435.25 - 485.25 = 3950

Answer:

A.) Mean = 237.9

B.) Median = 240

C.) Mode = 199

D.) Midrange = 239.5

Step-by-step explanation:

The given data are :

293 255 264 240 190 295 199 184 293 205 199

The mean = (sum of X) / f

Where frequency f = 11

X = 293 + 255 + 264 + 240 + 190 + 295 + 199 + 184 + 293 + 205 + 199

X = 2617

Substitute X and f into the formula

Mean = 2617/11

Mean = 237.9 approximately

B.) To get the median, you need to first rearrange the data, then pick the middle number.

184 190 199 199 205 240 255 264 293 293 295

The median = 240

C.) The mode is the highest frequency. That is the most occuring number

Mode = the two most occuring numbers are 199 and 293

D.) Range = highest number - lowest number

But midrange = (highest number + lowest number ) ÷ 2

Highest number = 295

Lowest number = 184

Substitute into the formula

Midrange = (295 + 184)/2

Midrange = 479/2

Midrange = 239.5

Answer:

The number is 3

Step-by-step explanation:

The fraction is 11/36

let

x = no. added to the numerator

2x = no. added to denominator

We have,

x+11/2x+36=1/3

Cross multiply

3(x+11) = (2x + 36)

3x + 33 = 2x + 36

Collect like terms

3x - 2x = 36 - 33

x=3

The number is 3

Check:

3+11/6+36=1/3

14/42=1/3

Answer:

X rounded to the nearest degree can me calculated based on parallel lines or if there is a straight line bisected by a line, 180 degress is a staright line. There is not enough info nor a picture to give a direct answer.

Answer:

its D

Step-by-step explanation:

Answer:

[tex]x=20[/tex]

[tex]y=10\,\sqrt{3}[/tex]

Step-by-step explanation:

In the triangle we have the following trigonometric equations:

[tex]tan(60^o)=\frac{opposite}{adjacent} \\\sqrt{3} =\frac{y}{10} \\y=10\,\sqrt{3}[/tex]

and also:

[tex]cos(60^o)=\frac{adjacent}{hypotenuse}\\\frac{1}{2} =\frac{10}{x}\\x=20[/tex]

Answer:  243 cm²

Step-by-step explanation:  If the diameter is 18, the radius is 9. Each square is  9×9, so 81 cm² for each.   Multiply:  81×3 = 243

Or take the length times width  to get area  27×9= 243

Answer:

Simply subtract the sum of the the three angles given from 360° in order to get the measure of the fourth angle!

Step-by-step explanation:

Answer:

(x - 7)² + (y - 4)² = 49

Step-by-step explanation:

Given

Equation: x² + y² = 49

Required:

New Equation when translated 7 units right and 4 units up

Taking it one step at a time.

When the equation is translated 7 units right, this implies a negative unit along the x axis.

The equation becomes

(x - 7)² + y² = 49

When the equation is translated 4 units up, this implies a negative unit along the y axis.

(x - 7)² + (y - 4)² = 49

The expression can be further simplified but it's best left in the form of

(x - 7)² + (y - 4)² = 49

Answer:

[tex]\boxed{\sqrt{37} }[/tex]

Step-by-step explanation:

Use Pythagorean theorem.

Create a right triangle, with legs of length 6 and 1 units. Find the length of hypotenuse.

[tex]6^2+1^2 =c^2 \\36+1=c^2 \\37=c^2 \\c=\sqrt{37}[/tex]

Answer:

When you use Pythagorean theorem, the answer is √37.

Step-by-step explanation:

Step-by-step explanation:

The coordinates of the feasible region are:(In clockwise direction)

(200, 200)

(300, 200)

(500, 0)

(300, 0)

Answer:

The Answer is the first option

Step-by-step explanation:

If you take the 3x-4>5

than: 3x>5+4

3x>9

x>9/3

x>3

Answer:

[tex]5.\overline{55}[/tex] or  [tex]5\frac{5}{9}[/tex]

Step-by-step explanation:

[tex]\frac{5}{r} =\frac{9}{10}[/tex]

We multiply the cross values of 5 and 10

5 × 10 = 50

Now we divide by the remaining number, 9

[tex]50 \div 9 = 5.555555...[/tex]

So, r = [tex]5.\overline{55}[/tex]

As a fraction, this is [tex]5\frac{5}{9}[/tex]

Hope this helped!

Answer:

5 5/9 = r

Step-by-step explanation:

5/r = 9/10​

Using cross products

5 * 10 = 9 * r

50 = 9r

Divide by 9

50/9 = r

5 5/9 = r

Replace f(x) with 0 and solve for x. We do this because the x intercepts always occur when y = 0. Keep in mind that y = f(x).

f(x)=x^3-9x^2+20x

0=x^3-9x^2+20x

x^3-9x^2+20x = 0

x(x^2-9x+20) = 0 .... factor out GCF x

x(x-5)(x-4) = 0 ... factor the stuff inside

x = 0 or x-5 = 0 or x-4 = 0 ... zero product property

x = 0 or x = 5 or x = 4

Answer:

675 – (15-12) ³+3

675-3³ +3

675-27+3

651

Step-by-step explanation:

The value of the given expression is 666 which is the correct answer that would be an option (C).

PEMDAS rule states that the order of operation starts with the calculation enclosed in brackets or the parentheses first. Exponents (degrees or square roots) are then operated on, followed by multiplication and division operations, and then addition and subtraction.

We have been given the expression below as:

675 – (15-12)³ ÷ 3

Using the PEMDAS rule to determine the evaluation of the given expression  

⇒ 675 – (15-12)³ ÷ 3

Apply the subtraction operation,

⇒ 675 - (3)³ ÷ 3

⇒ 675 - 27 ÷ 3

⇒ 675 - 27 / 3

Apply the division operation,

⇒ 675 - 9

Apply the subtraction operation,

⇒ 666

Therefore, the value of the given expression would be 666,

Learn more about the PEMDAS rule here :

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Answer:

if its according to the graph but not according to the function then

domain = {x:-4<x<8}

range = {y:y<16}

Answer:

1

Step-by-step explanation:

There is only one 3 on the cube

Answer: 320 students

Step-by-step explanation:

From the question, we are informed that a school counselor surveyed 90 randomly selected students about thé langages they speak and thé students surveyed 16 speak more than one langage fluently. This means that 16/90 speak more than one language.

When 1800 students are surveyed, the number of students that can be expected to speak more than one langage fluently will be:

= 16/90 × 1800

= 16 × 20

= 320 students

Answer:

C) Yes, since the slopes are the same and the y-intercepts are different.

Step-by-step explanation:

You need to have both equations in slope-intercept form.

4x - 6y = -6

-6y = -4x - 6

y = 2/3x + 1

The two equations are

y = 2/3x + 1

y = 2/3x - 17

For lines to be parallel, the slopes must be equal.  The y-intercepts don't matter.

Since the lines have the same slopes but different y-intercepts the answer is C.

Answer:

C

Step-by-step explanation:

Use Pythagorean theorem

X^2 + 44^2 = 55^2

X^2 = 55^2 - 44^2

= 3025 - 1936

X^2 = 1089

X = sqrt (1089)

X = 33

a) [tex]\frac{8}{2}[/tex]

b) [tex]\frac{9}{4}[/tex]

c) [tex]\frac{3}{-1}[/tex]

Answer:

Sine angle of <ACB = 38.68°

Step-by-step explanation:

Hello,

To solve this problem, we need a good representation of the sides and the angle.

See attached document for better illustration.

Assuming it's a right angled triangle,

AC = hypothenus

AB = opposite

BC = adjacent

AC = 40

BC = 25

AB = 25

From trigonometric ratios

Sinθ = opposite/ hypothenus

Sinθ = AB / AC

Sinθ = 25 / 40

Sinθ = 0.625

θ = sin⁻¹0.625

θ = 38.68°

Sine angle of <ACB = 38.68°

Answer:

a) [tex]\boxed{4m^2+20m+25}[/tex]

b) [tex]\boxed{m^2-m+\frac{1}{4} }[/tex]

Step-by-step explanation:

a) [tex](2m+5)^2[/tex]

Using formula [tex](a+b)^2 = a^2+2ab+b^2[/tex]

=> [tex](2m)^2 + 2(2m)(5)+(5)^2[/tex]

=> [tex]4m^2+20m+25[/tex]

b) [tex](m-\frac{1}{2} )^2[/tex]

Using Formula [tex](a-b)^2 = a^2-2ab+b^2[/tex]

=> [tex](m)^2 - 2(m)(\frac{1}{2} ) + (\frac{1}{2} )^2[/tex]

=> [tex]m^2-m+\frac{1}{4}[/tex]

Answer:

a) [tex]\boxed{4m^2 + 20m + 25}[/tex]

b) [tex]\boxed{m^2 - m + \frac{1}{4} }[/tex]

Step-by-step explanation:

[tex](2m + 5)^2[/tex]

[tex](2m + 5) (2m + 5)[/tex]

Use FOIL method.

[tex]2m(2m + 5)+5(2m + 5)[/tex]

[tex]4m^2 + 10m + 10m + 25[/tex]

[tex]4m^2 + 20m + 25[/tex]

[tex](m - \frac{1}{2} )^2[/tex]

[tex](m- \frac{1}{2})(m- \frac{1}{2})[/tex]

Use FOIL method.

[tex]m(m- \frac{1}{2})- \frac{1}{2}(m- \frac{1}{2})[/tex]

[tex]m^2- \frac{1}{2} m- \frac{1}{2}m+ \frac{1}{4}[/tex]

[tex]m^2-m+ \frac{1}{4}[/tex]

Answer:

72 N

Step-by-step explanation:

From the question.

Force is directly proportional to acceleration, with the mass of the object constant.

F α a

F = ka

F₁/a₁ = F₂/a₂....................... Equation 1

Where F₁ = Initial force, a₁ = initial acceleration. F₂ = final force, a₂ = Final acceleration.

make F₂ the subject of the equation

F₂ = (F₁/a₁)a₂................ Equation 2

Given; F₁ = 64 N, a₁ = 8 m/s², a₂ = 9 m/s²

Substitute these values into equation 2

F₂ = (64/8)(9)

F₂ = 72 N

Answer:

2

Step-by-step explanation:

green

Answer: 2

Step-by-step explanation: