In a factory, a square sheet of metal has of a circle cut from it, as shown. The center of the circle is on one vertex of the square. What is the perimeter of the shaded region to the nearest hundredth?

I think how you approach this depends on your knowledge of calculus.

If you don't know how to compute definite integrals yet, but you do know that they represent signed areas under curves, then you can plot both curves |x - 4| and √(36 - x ²), then recognize that the areas represented by these integrals are areas of geometric shapes. (See attached images)

First integral: if you plot |x - 4| on the interval [3, 6], you'll see that the integral corresponds to the area of two triangles. One of them has base = height = 1, and the other has base = height = 2. Then

[tex]\displaystyle\int_3^6|x-4|\,\mathrm dx=\frac12\times1\times1+\frac12\times2\times2=\frac52[/tex]

Second integral: if [tex]y=\sqrt{36-x^2}[/tex], then [tex]x^2+y^2=6^2[/tex], meaning this curve is the upper half of a circle with radius 6. On the interval [-6, 0], the area amounts to 1/4 of the total area of such a circle, so that

[tex]\displaystyle\int_{-6}^0\sqrt{36-x^2}\,\mathrm dx=\frac{\pi\times6^2}4=9\pi[/tex]

* * *

If you already know a few things about calculus and integration, you can compute these areas directly.

First integral:

[tex]\displaystyle\int_3^6|x-4|\,\mathrm dx=\int_3^4(4-x)\,\mathrm dx+\int_4^6(x-4)\,\mathrm dx[/tex]

[tex]\displaystyle\int_3^6|x-4|\,\mathrm dx=\left(4x-\frac{x^2}2\right)\bigg|_{x=3}^{x=4}+\left(\frac{x^2}2-4x\right)\bigg|_{x=4}^{x=6}[/tex]

[tex]\displaystyle\int_3^6|x-4|\,\mathrm dx=\left(\left(4\times4-\frac{4^2}2\right)-\left(4\times3-\frac{3^2}2\right)\right)+\left(\left(\frac{6^2}2-4\times6\right)-\left(\frac{4^2}2-4\times4\right)\right)[/tex]

[tex]\displaystyle\int_3^6|x-4|\,\mathrm dx=\frac52[/tex]

Second integral:

Substitute x = 6 sin(t ) and dx = 6 cos(t ) dt, then

[tex]\displaystyle\int_{-6}^0\sqrt{36-x^2}\,\mathrm dx=\int_{-\frac\pi2}^0\sqrt{6^2-(6\sin(t))^2} (6\cos(t))\,\mathrm dt[/tex]

[tex]\displaystyle\int_{-6}^0\sqrt{36-x^2}\,\mathrm dx=36\int_{-\frac\pi2}^0\cos (t) \sqrt{1-\sin^2(t)} \,\mathrm dt[/tex]

[tex]\displaystyle\int_{-6}^0\sqrt{36-x^2}\,\mathrm dx=36\int_{-\frac\pi2}^0\cos (t) \sqrt{\cos^2(t)} \,\mathrm dt[/tex]

[tex]\displaystyle\int_{-6}^0\sqrt{36-x^2}\,\mathrm dx=36\int_{-\frac\pi2}^0\cos (t) |\cos(t)| \,\mathrm dt[/tex]

For t ∈ [-π/2, 0], cos(t ) > 0, so |cos(t )| = cos(t ) :

[tex]\displaystyle\int_{-6}^0\sqrt{36-x^2}\,\mathrm dx=36\int_{-\frac\pi2}^0\cos^2(t)\,\mathrm dt[/tex]

Recall the half-angle identity,

cos²(t ) = (1 + cos(2t )) / 2

[tex]\displaystyle\int_{-6}^0\sqrt{36-x^2}\,\mathrm dx=18\int_{-\frac\pi2}^0(1+\cos(2t))\,\mathrm dt[/tex]

[tex]\displaystyle\int_{-6}^0\sqrt{36-x^2}\,\mathrm dx=18\left(t+\frac{\sin(2t)}2\right)\bigg|_{t=-\frac\pi2}^{t=0}[/tex]

[tex]\displaystyle\int_{-6}^0\sqrt{36-x^2}\,\mathrm dx=18\left(\left(0+\frac{\sin(2\times0)}2\right)-\left(-\frac\pi2+\frac{\sin\left(2\times\left(-\frac\pi2\right)\right)}2\right)\right)[/tex]

[tex]\displaystyle\int_{-6}^0\sqrt{36-x^2}\,\mathrm dx=9\pi[/tex]

Answer:

[tex]A=150\ cm^2[/tex]

Step-by-step explanation:

The Hypotenuse of triangle, H = 25 cm

Base = 15 cm

Height = 20 cm

We need to find the area of the triangle. The area of a right-angled triangle is given by :

[tex]A=\dfrac{1}{2}\times b\times h\\\\A=\dfrac{1}{2}\times 15\times 20\\A=150\ cm^2[/tex]

So, the area of the triangle is equal to [tex]150\ cm^2[/tex].

Answer:

11 3/4

31 5/8

Step-by-step explanation:

1. 54 1/2 - 42 3/4

= 109/2 -171/4

= 218-171/4

= 47/4

= 11 3/4

2. 45 3/8 - 11 3/4

= 363/8 -55/4

= 363-110/8

= 253/8

= 31 5/8

Answer:

6n + 20

Step-by-step explanation:

10 + 2n + 10 +4n

6n +20

Answer:

[tex]10 + 2n + 10 + 4n \\ 6n + 20 \\ 2(3n + 10)[/tex]

Answer:

2,4,5 got it right on egde2021

Step-by-step explanation:

Answer:

True

Step-by-step explanation:

Answer:

0.9976 = 99.76% probability that he/she actually is a future terrorist

Step-by-step explanation:

Conditional Probability

We use the conditional probability formula to solve this question. It is

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

In which

P(B|A) is the probability of event B happening, given that A happened.

[tex]P(A \cap B)[/tex] is the probability of both A and B happening.

P(A) is the probability of A happening.

In this question:

Event A: Person identified as a future terrorist

Event B: Person is actually a future terrorist.

Probability that a person is identified as a future terrorist:

96% of [tex]\frac{1000}{200000000}[/tex]

100 - 99.8 = 0.2% of [tex]\frac{200000000-1000}{200000000}[/tex]

So

[tex]P(A) = 0.96\frac{1000}{200000000} + 0.002\frac{200000000-1000}{200000000} = 0.002[/tex]

Probability of being identified and being a future terrorist:

100 - 99.8 = 0.2% of [tex]\frac{200000000-1000}{200000000}[/tex]

[tex]P(A \cap B) = 0.002\frac{200000000-1000}{200000000} = 0.00199999[/tex]

What is the probability that he/she actually is a future terrorist

[tex]P(B|A) = \frac{P(A \cap B)}{P(A)} = \frac{0.00199999}{0.00200479} = 0.9976[/tex]

0.9976 = 99.76% probability that he/she actually is a future terrorist

9514 1404 393

Answer:

Step-by-step explanation:

a) The fraction is ...

  2(5 1/4)/(15 2/3) = (21/2)/(47/3) = 63/94

Two streamers will require 63/94 of the original material.

__

b) $4800 is 70% of Sara's salary, so that salary is ...

  4800 = 0.70s

  4800/0.70 = s = 6857.14

Sara's salary is about $6857.14.

Answer: 1/3 of 24 tickets is 8

Step-by-step explanation:

1/3 of 24 is 8.

The way I usually find this is by multiplying different numbers by three and seeing if it equals (in this case) 24.

8 x 3 = 24

Because eight goes into twenty-four three times, eight is one-third of twenty-four.

Hope this helps✨

Answer:

ab=a×b

cb=c×b

ad=a×d

cd=c×d

Answer:

well the factor: Add and subtract the second term to the expression and factor by grouping.     (a+c)(b+d)

Step-by-step explanation:

Hope this is helpful! God Bless you !

Answer:

x = 3/2

Step-by-step explanation:

Log x^4-logx^3=log 3x-log2x solve​

log(x^4/x^3) = log(3x/2x)

Log cancels out

x^4/x^3 = 3x/2x

x^3 * x/x^3 = 3/2

x^3/x^3 * x = 3/2

x = 3/2

Hence the required value of x is 3/2

Answer: It would take her another 8 weeks to get a total of $249.

Step-by-step explanation:

Answer:

D) {-5,1}

R) {-4,7}

x (-31/11,0)

y (0,31/6)

Hope this helped! Have a nice day! Plz mark as brainliest!!! :D

-XxDeathshotxX

Answer: -2 is the slope

(0,0) (1,4) (rise over run) *Y2 - Y1/X2 - X1*

X1= 0

Y1= 4

X2= 1

Y2= 2

2-4= -2

1-0= 1

-2/1= -2

the slope is -2

Answer:

mean.

Step-by-step explanation:

since this is a math question i'm assuming you want a mathematical term. the mean of a set of data represents the entire set of data.

Answer: The above answer is correct.

Step-by-step explanation: I got this correct on Edmentum.

Answers

[tex]A' = \left[\begin{array}{cc}5.830\\-0.098\end{array}\right][/tex], [tex]\theta = \frac{2\pi}{3}[/tex]

[tex]A' = \left[\begin{array}{cc}-5.657\\-1.414\end{array}\right][/tex], [tex]\theta = \frac{7\pi}{4}[/tex]

[tex]A' = \left[\begin{array}{cc}-2.83\\5.098\end{array}\right][/tex], [tex]\theta = \frac{4\pi}{3}[/tex]

[tex]A' = \left[\begin{array}{cc}5\\-3\end{array}\right][/tex], [tex]\theta = \frac{\pi}{2}[/tex]

This exercise consist in finding the resulting Vector Matrix for each Angle of Rotation and a given Vector Matrix. The result is found by multiplying the given Vector Matrix for a Rotation Matrix, defined below as follows:

[tex]A' = R\cdot A[/tex] (1)

Where:

[tex]A[/tex] - Given vector matrix.

[tex]R[/tex] - Rotation matrix.

[tex]A'[/tex] - Resulting vector matrix.

For 2-dimension Vector Matrices, The Rotation Matrix is defined by the following entity:

[tex]R = \left[\begin{array}{cc}\cos \theta&-\sin \theta\\\sin \theta&\cos \theta\\\end{array}\right][/tex] (2)

Where [tex]\theta[/tex] is the Angle of Rotation, in radians.

Let be [tex]A = \left[\begin{array}{cc}x\\y\end{array}\right][/tex], the resulting vector matrix is found by (1) and (2):

[tex]A' = \left[\begin{array}{cc}\cos \theta&-\sin \theta\\\sin \theta&\cos \theta\end{array}\right] \cdot \left[\begin{array}{cc}x\\y\end{array}\right][/tex]

[tex]A' = \left[\begin{array}{cc}x\cdot \cos \theta - y\cdot \sin \theta\\x\cdot \sin \theta +y\cdot \cos \theta \end{array}\right][/tex]

If we know that [tex]A = \left[\begin{array}{cc}-3\\-5\end{array}\right][/tex], then the resulting vector matrix for each angle is, respectively:

[tex]\theta = \frac{\pi}{4}[/tex]

[tex]A' = \left[\begin{array}{cc}1.414\\-5.657\end{array}\right][/tex]

[tex]\theta = \frac{\pi}{2}[/tex]

[tex]A' = \left[\begin{array}{cc}5\\-3\end{array}\right][/tex]

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

[tex]A' = \left[\begin{array}{cc}5.830\\-0.098\end{array}\right][/tex]

[tex]\theta = \frac{4\pi}{3}[/tex]

[tex]A' = \left[\begin{array}{cc}-2.83\\5.098\end{array}\right][/tex]

[tex]\theta = \frac{5\pi}{3}[/tex]

[tex]A' = \left[\begin{array}{cc}-5.83\\0.098\end{array}\right][/tex]

[tex]\theta = \frac{7\pi}{4}[/tex]

[tex]A' = \left[\begin{array}{cc}-5.657\\-1.414\end{array}\right][/tex]

Therefore, we have the following answers:

[tex]A' = \left[\begin{array}{cc}5.830\\-0.098\end{array}\right][/tex], [tex]\theta = \frac{2\pi}{3}[/tex]

[tex]A' = \left[\begin{array}{cc}-5.657\\-1.414\end{array}\right][/tex], [tex]\theta = \frac{7\pi}{4}[/tex]

[tex]A' = \left[\begin{array}{cc}-2.83\\5.098\end{array}\right][/tex], [tex]\theta = \frac{4\pi}{3}[/tex]

[tex]A' = \left[\begin{array}{cc}5\\-3\end{array}\right][/tex], [tex]\theta = \frac{\pi}{2}[/tex]

Related question: https://brainly.com/question/11815899

Answer:

4 5/6 + 5 7/8= 10 17/24

Step-by-step explanation:

pretty sure I answered this exact question like minutes ago

Answer: 257/24, or 10 and 17/24.

Step-by-step explanation:

First, we need to turn the fractions into improper fractions and then find the common denominator.

4 and 5/6 = 29/6

5 and 7/8 = 47/8

The common denominator is 24. Now, we convert the fractions and add. Hope this helps!

Answer:

m = 2

The slope of the line through (1, 3) and (-2, -3) is 2!

The formula to find the slope is attached below.

Hope I was of help. Have a good new year!

Answer:

we should find the value of h(3) and do invers

Answer:

The slope is -4 and the y-intercept is 8.

Step-by-step explanation:

y+4x=8

y=8-4x

y=-4x+8

y=mx+b where m=slope and b=y-intercept.

Hello!

The equation of this line is written in slope-intercept form (y=mx+b)

Where

m is the slope

b is the y-intercept

m is 4

b is 8

Hope it helps!

If you have any questions, feel free to ask! :)

~An excited gal

[tex]MagicalNature[/tex] here to help

Answer:

I don't know the answer but have a site that helps me with math, its called

Step-by-step explanation:

Just check it out

Answer:

20

Step-by-step explanation:

Because this angle is 90 and we have 70