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KATRIN_1 [288]
3 years ago
7

Find the volume of the solid generated when R​ (shaded region) is revolved about the given line. x=6−3sec y​, x=6​, y= π 3​, and

y=​0; about x= 6

Mathematics
1 answer:
Dmitrij [34]3 years ago
6 0

Answer:

V=9\pi\sqrt{3}

Step-by-step explanation:

In order to solve this problem we must start by graphing the given function and finding the differential area we will use to set our integral up. (See attached picture).

The formula we will use for this problem is the following:

V=\int\limits^b_a {\pi r^{2}} \, dy

where:

r=6-(6-3 sec(y))

r=3 sec(y)

a=0

b=\frac{\pi}{3}

so the volume becomes:

V=\int\limits^\frac{\pi}{3}_0 {\pi (3 sec(y))^{2}} \, dy

This can be simplified to:

V=\int\limits^\frac{\pi}{3}_0 {9\pi sec^{2}(y)} \, dy

and the integral can be rewritten like this:

V=9\pi\int\limits^\frac{\pi}{3}_0 {sec^{2}(y)} \, dy

which is a standard integral so we solve it to:

V=9\pi[tan y]\limits^\frac{\pi}{3}_0

so we get:

V=9\pi[tan \frac{\pi}{3} - tan 0]

which yields:

V=9\pi\sqrt{3}]

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Serhud [2]

Answer:

The area of the shape can be divided into the area of the rectangle, and the area of the semi-circle.

The area of the rectangle can be found by 11*4=44

The area of a semi-circle can be found with the formula \frac{1}{2} \pi r^2 where r is the radius.

Since we know the diameter of the semi-circle is 4,

the radius will be 4 ÷ 2 = 2.

Therefore, the area of the semi-circle is \frac{1}{2}\pi 2^2=\frac{1}{2}\pi*4=2\pi

Therefore, the area of the shape is 44+2\pi or 50.283cm^2 (3 decimal places)

3 0
2 years ago
Find the midpoint of NP⎯⎯⎯⎯⎯ given N(2a, 2b) and P(2a, 0).
geniusboy [140]

Answer:

(2a, b )

Step-by-step explanation:

Given the endpoints (x₁, y₁ ) and (x₂, y₂ ) then the midpoint is

[ \frac{1}{2}(x₁ + x₂ ), \frac{1}{2}(y₁ + y₂ ) ]

Here (x₁, y₁ ) = N(2a, 2b) and (x₂, y₂ ) = P(2a, 0), thus

midpoint = [ \frac{1}{2}(2a + 2a), \frac{1}{2}(2b + 0 ) ] = (2a, b )

4 0
3 years ago
At a warehouse, boxes of merchandise are placed on shelves in stacks that are 7 boxes high. If each box is 3 1/5 inches in heigh
lesya692 [45]
22.4; 7 * 3 1/5 = 22.4
6 0
3 years ago
Ss below. <br> help would be appreciated.
sergeinik [125]

Answer:

(3,0,0)

Step-by-step explanation:

3x + 5y + z = 9     --------------(I)

-4y - z = 0   -------------------(II)

     z = -4y

2x + 3y  - 4z  = 6    (III)

Substitute  z = -4y      in equation (I) and (III)

3x  +5y - 4y = 9

3x + y = 9      -----------------------(a)

2x + 3y - 4*(-4y) = 6

2x + 3y + 16y = 6

2x + 19y = 6 -------------------(b)

Multiply equation (a) by (-19) and then add. So, y will be eliminated and we can get the value of x

(a)*(-19)            -57x - 19y  = -171

(b)                    <u>   2x  + 19y =   6 </u>

                        -55x           = -165

x = -165/-55

x = 3

Plugin x = 3 in equation (a)

3*3 + y = 9

 9  + y = 9

         y = 9 - 9

y = 0

Plug in y = 0 in equation (II)

-4*0 - z = 0

          z = 0

5 0
2 years ago
Read 2 more answers
A= 49 sq. in.<br><br> Find the value of x. If necessary, round to the nearest tenth.
BlackZzzverrR [31]

Answer:

9.90 inches

Step-by-step explanation:

Area of a triangle is equal to base times height divided by 2.

bh/2

Base and height is x.

xx/2=49

Solve for x.

x^2=49 \times 2

x^2=98

x=\sqrt{98}

x=9.899495

8 0
3 years ago
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