Evaluate the integral
1
2
sin ¹(x/4)) ² dx
| (si
- 1
2
[(sin ~'(x/4)) ² dx
√16-x²
2
√16-x²
11

The dimensions of a rectangle with area 1728 square meter whose perimeter is as small(minimum) as possible are 41.57m and 41.56m.

Let the dimension of the rectangle be x and y m.

According to the given question.

The area of the rectangle is 1728 square meter.

⇒ x × y = 1728

⇒ x = 1728/y

Since, the perimeter of the rectangle is the sum of the length of its boundary.

Therefore,

Perimeter of the recatngle with dimensions x and y is given by

Perimeter of the rectangle, P = 2(x + y)

⇒ P = 2(1728/y + y)

⇒ P = 3456/y + 2y

Differentiate the above equation with respect to y

⇒ [tex]P^{'} = -\frac{3456}{y^{2} } + 2[/tex]

For the minimum(small) perimeter equate the above equation to 0.

⇒ [tex]-\frac{3456}{y^{2} } + 2 = 0[/tex]

⇒ [tex]2y^{2} =3456[/tex]

⇒ [tex]y^{2} = \frac{3456}{2}[/tex]

⇒ y^2 = 1728

⇒ y = √1728

⇒ y = 41.56 m

Therefore,

x = 1728/41.56

⇒ x = 41.57 m

Hence, the dimensions of a rectangle with area 1728 square meter whose perimeter is as small(minimum) as possible are 41.57m and 41.56m.

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The number of gigabytes of data that Christopher can use while staying within his budget exists 1.3 gigabytes.

The number of gigabytes of data that Christopher can use while staying within his budget is 1.8 gigabytes.

Based on the information, we get

59 + (3 × g) = 62.90

59 + 3g = 62.90

simplifying the equation, we get

3g = 62.90 - 59

3g = 3.9

Divide both sides by 3, and we get

3g/3 = 3.9/3

g = 1.3

The number of gigabytes of data that Christopher can use while staying within his budget exists 1.3 gigabytes.

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It will take the diver 1.5seconds for her to hit the water

Linear equations are equation that has a leading degree of 2. Given the expression that expresses the distance covered by the diver as a function of time as shown;

d(t)= -16(2t - 3)(t+1)

where;

d is the height of the diver above the water and;

t is the time in seconds.

Given the following

d = 0 (the distance on the ground)

Substitute into the formula below;

-16(2t - 3)(t+1)= 0

Divide through by -16

(2t - 3)(t+1) = 0

Determine the time

2t - 3 = 0

t = 3/2

Similarly;

t +1 = 0
t = -1

Hence it will take the diver 1.5seconds for her to hit the water

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Answer:

Step-by-step explanation:

The graph of the straight line [tex]y = -\frac{1}{3}x -2[/tex] is plotted. The graph is shown below.

A straight line is of the form y = mx + c, where m is the slope and c is the y-intercept.

To plot the given straight line [tex]y = -\frac{1}{3}x -2[/tex], follow the following steps:

Step 1: Substitute x = 0 in the given equation to obtain the point where the line intersects the y-axis.

y = 0 - 2

y = -2

The point at the y-axis is (0, -2).

Step 2: Substitute y = 0 in the given equation to obtain the point where the line intersects the x-axis.

[tex]0 = -\frac{1}{3}x -2\\x = -6[/tex]

The point at the x-axis is (-6, 0).

Step 3: Draw a straight line passing through both points.

Thus, the straight line [tex]y = -\frac{1}{3}x -2[/tex] is plotted.

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Answer:

D It repeats

Step-by-step explanation:

square root of 15 is 3.87298334621

11/25= 0.44

3/20=.6

D = 0.33333333333

Answer:

Step-by-step explanation:

A. a = $1350 b. b= 89%

B. 1350(.89)^t = 675

    t = 5.9 years

Answer:  48 cm

Step-by-step explanation:

A perimeter(P) of a rectangle is equal to 2(L + W), or 2L + 2W, so we write the equation as:  

1st way:

W = 50%(16) = 0.5 x 16 = 8

   P = 2(L + W)

   P = 2(16 + 8)

   P = 2(24)

   P = 48 cm

2nd way:

   P = 2L + 2W

   P = 2(16) + 2[50%(16)]

   P = 2(16) + 2[0.5 X 16]

   P = 32 + 2(8)

   P = 32 + 16

   P = 48 cm

Using the binomial distribution, we have that you would expected to draw a face card 8 times.

It is the probability of exactly x successes on n repeated trials, with p probability of a success on each trial.

The expected value of the binomial distribution is:

E(X) = np

For this problem, the parameters are given as follows:

Then the expected value is found as follows:

E(X) = np = 35 x 0.2308 = 8

You would expected to draw a face card 8 times.

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Step-by-step explanation:

3. B

because the figure to the left of the y-axis yields the one to the right when you reflect it across y = x.

4. C

Flip either of the figures vertically across x = -1 which is their intersection point and you'll get the other.

5. D

it can't be clockwise because the x value of the vertices would have been negative. so it is 270° counterclockwise resulting in ( -y , x )

6. A

Similar to #4 which except this time, you flip it horizontally like how you lay a book or pick it up.

Answer:

x=-22

Step-by-step explanation:

Let's substitute the number you are thinking of with x

(x+4)*3=(x-5)*2

3x+12=2x-10

x+12=-10

x=-22

Answer: c. a-15= 20

Step-by-step explanation:

The answer is c because a represents her age and 15 less than that would mean that you have to subtract 15 from her age which is represented by a, and then 15 less than a would equal 20, so a-15=20

Answer:

d. 9

Step-by-step explanation:

y= 0.22x^2 - 1.76x +4.75

x is the time in months so we should replace x by 10

y = 0.22 (10)^2 - 1.76(10) + 4.75

0.22 (100) - 17.6 + 4.75

22 - 17.6 + 4.75

9.15

So the number of laptops that will be sold during month 10 is 9. (you can't have 0.15 of a computer)

Answer:

d. 9

Step-by-step explanation:

The equation: y = 0.22x^2 - 1.76x + 4.75

x: Time in months

y: Number of laptops sold

Since we need to predict the number of laptops that will be sold during month 10, we can replace the x with 10.

The new equation will be: y = 22 - 17.6 + 4.75

Then y will be 9.15

Since there cannot be 0.15 of a laptop, we round the number 9.15 to a whole number, which is 9.

D) 9 is our final answer to this question.

Answer:

Step-by-step explanation:

let we suppose the ratios as x , 2x and 3x

we know that,

x + 2x + 3x = -----

6x= -------

x= -----/6

therefore x = ......

2x = 2 * x (value of x)

3x = 3 * x (value of x)

and the question is solved

Answer:

let the ratio be 1x,2x,3x

Now,

1x+2x+3x=180°(sum of angles of triangle are 180)

or,6x=180°

or,x=180/6

or,x=30°

Then,

1x=1*30

=30°

2x=2*30

=60°

3x=3*30

=90°

Answer:

Step-by-step explanation:

4x - 10 + x + 7 = 4x

5x - 3 = 4x

-3 = -x

x = 3

4(3) - 10 = 12 - 10 = 2

3 + 7 = 10

10+2 = 12

4(3)= 12

Natasha got the power right. The power is 295 hp.

t = 5.825∛w / p

where,

Therefore,

t = 13.4 s

p = 3590

13.4 = 5.825∛ 3590 / p

cube both sides

13.4³ = 5.825³ × 3590 / p

2406.104 = 197.645890625 × 3590 / p

cross multiply

2406.104 p = 709548.747344

p = 709548.747344 / 2406.104

p = 294.895294361

p = 295 hp

Therefore, Natasha got the power right

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Answer:

[tex]\dfrac{3}{2} \ln |x-4| - \dfrac{1}{2} \ln |x+2| + \text{C}[/tex]

Step-by-step explanation:

Fundamental Theorem of Calculus

[tex]\displaystyle \int \text{f}(x)\:\text{d}x=\text{F}(x)+\text{C} \iff \text{f}(x)=\dfrac{\text{d}}{\text{d}x}(\text{F}(x))[/tex]

If differentiating takes you from one function to another, then integrating the second function will take you back to the first with a constant of integration.

Given indefinite integral:

[tex]\displaystyle \int \dfrac{x+5}{(x-4)(x+2)}\:\:\text{d}x[/tex]

Take partial fractions of the given fraction by writing out the fraction as an identity:

[tex]\begin{aligned}\dfrac{x+5}{(x-4)(x+2)} & \equiv \dfrac{A}{x-4}+\dfrac{B}{x+2}\\\\\implies \dfrac{x+5}{(x-4)(x+2)} & \equiv \dfrac{A(x+2)}{(x-4)(x+2)}+\dfrac{B(x-4)}{(x-4)(x+2)}\\\\\implies x+5 & \equiv A(x+2)+B(x-4)\end{aligned}[/tex]

Calculate the values of A and B using substitution:

[tex]\textsf{when }x=4 \implies 9 = A(6)+B(0) \implies A=\dfrac{3}{2}[/tex]

[tex]\textsf{when }x=-2 \implies 3 = A(0)+B(-6) \implies B=-\dfrac{1}{2}[/tex]

Substitute the found values of A and B:

[tex]\displaystyle \int \dfrac{x+5}{(x-4)(x+2)}\:\:\text{d}x = \int \dfrac{3}{2(x-4)}-\dfrac{1}{2(x+2)}\:\:\text{d}x[/tex]

[tex]\boxed{\begin{minipage}{5 cm}\underline{Terms multiplied by constants}\\\\$\displaystyle \int ax^n\:\text{d}x=a \int x^n \:\text{d}x$\end{minipage}}[/tex]

If the terms are multiplied by constants, take them outside the integral:

[tex]\implies \displaystyle \dfrac{3}{2} \int \dfrac{1}{x-4}- \dfrac{1}{2} \int \dfrac{1}{x+2}\:\:\text{d}x[/tex]

[tex]\boxed{\begin{minipage}{5 cm}\underline{Integrating}\\\\$\displaystyle \int \dfrac{f'(x)}{f(x)}\:\text{d}x=\ln |f(x)| \:\:(+\text{C})$\end{minipage}}[/tex]

[tex]\implies \dfrac{3}{2} \ln |x-4| - \dfrac{1}{2} \ln |x+2| + \text{C}[/tex]

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For an alternative approach, expand and complete the square in the denominator to write

[tex](x-4)(x+2) = x^2 - 2x - 8 = (x - 1)^2 - 9[/tex]

In the integral, substitute [tex]x - 1 = 3 \sin(u)[/tex] and [tex]dx=3\cos(u)\,du[/tex] to transform it to

[tex]\displaystyle \int \frac{x+5}{(x - 1)^2 - 9} \, dx = \int \frac{3\sin(u) + 6}{9 \sin^2(u) - 9} 3\cos(u) \, du \\\\ ~~~~~~~~~~~~ = - \int \frac{\sin(u) + 2}{\cos(u)} \, du \\\\ ~~~~~~~~~~~~ = - \int (\tan(u) + 2 \sec(u)) \, du[/tex]

Using the known antiderivatives

[tex]\displaystyle \int \tan(x) \, dx = - \ln|\cos(x)| + C[/tex]

[tex]\displaystyle \int \sec(x) \, dx = \ln|\sec(x) + \tan(x)| + C[/tex]

we get

[tex]\displaystyle \int \frac{x+5}{(x - 1)^2 - 9} \, dx = \ln|\cos(u)| - 2 \ln|\sec(u) + \tan(u)| + C \\\\ ~~~~~~~~~~~~ = - \ln\left|\frac{(\sec(u) + \tan(u))^2}{\cos(u)}\right|[/tex]

Now, for [tex]n\in\Bbb Z[/tex],

[tex]\sin(u) = \dfrac{x-1}3 \implies u = \sin^{-1}\left(\dfrac{x-1}3\right) + 2n\pi[/tex]

so that

[tex]\cos(u) = \sqrt{1 - \dfrac{(x-1)^2}9} = \dfrac{\sqrt{-(x-4)(x+2)}}3 \implies \sec(u) = \dfrac3{\sqrt{-(x-4)(x+2)}}[/tex]

and

[tex]\tan(u) = \dfrac{\sin(u)}{\cos(u)} = -\dfrac{x-1}{\sqrt{-(x-4)(x+2)}}[/tex]

Then the antiderivative we found is equivalent to

[tex]\displaystyle - \int \frac{x+5}{(x - 1)^2 - 9} \, dx = - \ln\left|-\frac{3(x+2)}{(x-4) \sqrt{-(x-4)(x+2)}}\right| + C[/tex]

and can be expanded as

[tex]\displaystyle - \int \frac{x+5}{(x - 1)^2 - 9} \, dx = -\ln\left| \frac{3(x+2)^{1/2}}{(x-4)^{3/2}}\right| + C \\\\ ~~~~~~~~~~~~ = - \ln\left|(x+2)^{1/2}\right| + \ln\left|(x-4)^{3/2}\right| + C \\\\ ~~~~~~~~~~~~ = \boxed{\frac32 \ln|x-4| - \frac12 \ln|x+2| + C}[/tex]

Answer:

36 units

Step-by-step explanation:

to find the distance take the absolute value of the difference, that is

| - 42 - (- 78) | = | - 42 + 78 | = | 36 | = 36

or

| - 78 - (- 42) | = | - 78 + 42 | = | - 36 | = 36

Answer:

-36

Step-by-step explanation:

-78 - (-42)

= -78 + 42

= -36

By evaluating the function, we conclude that f(10) = -7

Here we know that:

g(x) = 2x - 8

And f(x) = 5 - g(x).

Then we can write:

f(x) = 5 - (2x - 8) = 5 - 2x + 8 = -2x + 13

Now we want ot evaluate it in x = 10, this means replace the variable by the number 10.

f(10) = -2*10 + 13 = -20 + 13 = -7

Then, we conclude that f(10) = 7

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The height of the tree and the building are 7.6 m and 28.6 m respectively.

An equation is an expression that shows the relationship between two or more variables and numbers.

Two triangles are similar if they have the same shape and the ratio of their corresponding sides are in the same proportion.

Triangle BGD, CHD and AFD are similar triangles.

CH = 1.6, DH = 8, GH = 30,  GF = 105

FH = GF + GH = 105 + 30 = 135

FD = FH + DH = 135 + 8 = 143; GD = 30 + 8 = 38

Hence, using similar triangles:

BG/CH = GD / DH

BG / 1.6 = 38/8

BG = 7.6 m

Also:

building/CH = DF / DH

building / 1.6 = 143/8

building = 28.6 m

The height of the tree and the building are 7.6 m and 28.6 m respectively.

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Answer:

40

Step-by-step explanation:

7 - 3 + 9 x 8 / 2

Multiplication and division left - right.

7 - 3 + 72 / 2

7 - 3 + 36

Add and subtract left to right.

4 + 36 = 40

Answer: 144

Step-by-step explanation:

The length of DF is 16.

The horizontal distance from DF to E is 18.

So, the area is [tex]\frac{1}{2}(16)(18)=144[/tex]

The area of the triangle DEF is approximately equal to 144.014 square units.

Triangles can be generated on a Cartesian plane by marking three non-colinear points on there. Heron's formula offers the possibility of calculating the area of a triangle by only using the lengths of its three sides, whose formula is now introduced:

A = √ [s · (s - DE) · (s - EF) · (s - DF)]     (1)

s = (DE + EF + DF) / 2     (2)

Where s is the semiperimeter of the triangle.

First, we determine the lengths of the sides DE, EF and DF by Pythagorean theorem:

Side DE

DE = √ [[10 - (- 8)]² + (- 2 - 8)²]

DE ≈ 20.591

Side EF

EF = √ [(- 8 - 10)² + [- 8 - (- 2)]²]

EF ≈ 18.974

Side DF

DF = √[[- 8 - (- 8)]² + (- 8 - 8)²]

DF = 16

Then, the area of the triangle DEF is by Heron's formula:

s = (16 + 18.974 + 20.591) / 2

s = 27.783

A = √[27.783 · (27.783 - 20.591) · (27.783 - 18.974) · (27.783 - 16)]

A ≈ 144.014

The area of the triangle DEF is approximately equal to 144.014 square units.

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Answer:

Twice

Step-by-step explanation:

In a convex hexagon the sum of all interior angle is equal to twice of exterior angles.

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The correct choices based on the integers are: 1a, 2b, 3d, and 4c.

In the question, we are asked to match the scenarios to their corresponding boundaries.

Thus, the correct choices based on the integers are: 1a, 2b, 3d, and 4c.

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Answer:

Jeffrey has 225 % chance to go to Cafe Shirley.

Step-by-step explanation:

given,

chance of going to Cafe

Chris = 3/4

Nina =3/4

iana = 3/4

To find the total probability you should add all of them:

Total probability= 3/4 + 3/4 + 3/4

= 9/4 = 2.25

to convert to percent you should times 100

2.25 x 100 = 225%//

18 liters of 14% bleach contains 0.14×18 = 2.52 liters of bleach.

Adding [tex]x[/tex] liters of pure water to the solution increases the total volume to [tex]18+x[/tex] liters without changing the total amount of bleach.

To end up with a 10% bleach solution, we must add

[tex]\dfrac{2.52}{18+x} = 0.10 \implies 2.52 = 1.8 + 0.10x \implies 0.10x = 0.72 \implies x=\boxed{7.2}[/tex]

liters of water.

Answer:

b

Step-by-step explanation:

The number of servings of pasta were in the dish Lois made is 31.67.

Number of servings of pasta were in the dish Lois made = Number of cups of pasta Lois made / A serving of the pasta

= 6 1/3 ÷ 0.2

= 19/3 ÷ 0.2

= 19/3 × 1/0.2

= (19×1) / (3 × 0.2)

= 19/0.6

= 31.6666666666666

Approximately,

Number of servings of pasta were in the dish Lois made is 31.67 servings

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The Biased samples that are most likely to be biased are as follows;

The residents of every other house on her street.

The people at the computer store.

At the mall every 100th person from the town’s phone registry.

Every 10th person who walks down the main street in her town.

According to the statement

we have given that the:

Maria wants to take a survey to predict who will win the election for her town’s mayor.

So, For this purpose

Biased sample In epidemiology, a sample of a group that does not equally represent the members of the group.

The above people are selected randomly with no specific characteristics, so it is less likely to be biased.

These people are selected according to specific characteristics; Subscribing to the golf magazine (who tends to be rich and biased in politics).

So, it is more likely to be biased.

The Biased samples that are most likely to be biased are as follows

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Answer:

A: The residents of every house on her street

B: The subscribers to a golf magazine

C: The people at the computer store at the mall