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salantis [7]
3 years ago
6

Consider the region bounded by 4y=x^2 and 2y=x.

Mathematics
1 answer:
gayaneshka [121]3 years ago
7 0

Answer:

a) ⅓ units²

b) 4/15 pi units³

c) 2/3 pi units³

Step-by-step explanation:

4y = x²

2y = x

4y = (2y)²

4y = 4y²

4y² - 4y = 0

y(y-1) = 0

y = 0, 1

x = 0, 2

Area

Integrate: x²/4 - x/2

From 0 to 2

(x³/12 - x²/4)

(8/12 - 4/4) - 0

= -⅓

Area = ⅓

Volume:

Squares and then integrate

Integrate: [x²/4]² - [x/2]²

Integrate: x⁴/16 - x²/4

x⁵/80 - x³/12

Limits 0 to 2

(2⁵/80 - 2³/12) - 0

-4/15

Volume = 4/15 pi

About the x-axis

x² = 4y

x² = 4y²

Integrate the difference

Integrate: 4y² - 4y

4y³/3 - 2y²

Limits 0 to 1

(4/3 - 2) - 0

-2/3

Volume = ⅔ pi

You might be interested in
Consider the linear transformation T from V = P2 to W = P2 given by T(a0 + a1t + a2t2) = (2a0 + 3a1 + 3a2) + (6a0 + 4a1 + 4a2)t
Svet_ta [14]

Answer:

[T]EE=\left[\begin{array}{ccc}2&3&3\\6&4&4\\-2&3&4\end{array}\right]

Step-by-step explanation:

First we start by finding the dimension of the matrix [T]EE

The dimension is : Dim (W) x Dim (V) = 3 x 3

Because the dimension of P2 is the number of vectors in any basis of P2 and that number is 3

Then, we are looking for a 3 x 3 matrix.

To find [T]EE we must transform the vectors of the basis E and then that result express it in terms of basis E using coordinates and putting them into columns. The order in which we transform the vectors of basis E is very important.

The first vector of basis E is e1(t) = 1

We calculate T[e1(t)] = T(1)

In the equation : 1 = a0

T(1)=(2.1+3.0+3.0)+(6.1+4.0+4.0)t+(-2.1+3.0+4.0)t^{2}=2+6t-2t^{2}

[T(e1)]E=\left[\begin{array}{c}2&6&-2\\\end{array}\right]

And that is the first column of [T]EE

The second vector of basis E is e2(t) = t

We calculate T[e2(t)] = T(t)

in the equation : 1 = a1

T(t)=(2.0+3.1+3.0)+(6.0+4.1+4.0)t+(-2.0+3.1+4.0)t^{2}=3+4t+3t^{2}

[T(e2)]E=\left[\begin{array}{c}3&4&3\\\end{array}\right]

Finally, the third vector of basis E is e3(t)=t^{2}

T[e3(t)]=T(t^{2})

in the equation : a2 = 1

T(t^{2})=(2.0+3.0+3.1)+(6.0+4.0+4.1)t+(-2.0+3.0+4.1)t^{2}=3+4t+4t^{2}

Then

[T(t^{2})]E=\left[\begin{array}{c}3&4&4\\\end{array}\right]

And that is the third column of [T]EE

Let's write our matrix

[T]EE=\left[\begin{array}{ccc}2&3&3\\6&4&4\\-2&3&4\end{array}\right]

T(X) = AX

Where T(X) is to apply the transformation T to a vector of P2,A is the matrix [T]EE and X is the vector of coordinates in basis E of a vector from P2

For example, if X is the vector of coordinates from e1(t) = 1

X=\left[\begin{array}{c}1&0&0\\\end{array}\right]

AX=\left[\begin{array}{ccc}2&3&3\\6&4&4\\-2&3&4\end{array}\right]\left[\begin{array}{c}1&0&0\\\end{array}\right]=\left[\begin{array}{c}2&6&-2\\\end{array}\right]

Applying the coordinates 2,6 and -2 to the basis E we obtain

2+6t-2t^{2}

That was the original result of T[e1(t)]

8 0
3 years ago
Find the area of a circle circumscribing an equilateral triangle of side 15cm.<br> Take pi = 3.14.
miskamm [114]

Answer:

236 cm²

Step-by-step explanation:

Height of an equilateral triangle (h) = √3 /2 (l)

l = side of the equilateral triangle.

h = √3 /2 (15)

In an equilateral triangle the orthocenter, centroid, circumcenter and incenter are in the same spot

The center of the circle is the centroid and height match  with the median. The radius of the circumcircle is equal to two thirds the height.

Formula for the Radius of the circumcircle = 2/3 h

= 2/3 x √3 /2 (15)

= 5 √3 cm     (=radius)

Area of the circle = πr^

= 3.14 x( 5 √3 ) ^

=3.14 x(25*3)

=3.14 x 75

=235.5

=236 cm²

8 0
3 years ago
In a group of dog 25% are boys. How many girls are there in the class ?<br><br> Explain
rewona [7]

Answer:

Step-by-step explanation:

identify the known ratio and the unknown ratio.

Set up the proportion.

Cross-multiply and solve.

Check the answer by plugging the result into the unknown ratio.

^^^^^^^^^^^^^^^^^^^^^^^^^^^

is the steps to solve your answer

5 0
3 years ago
Which answer represents the expression in its expanded form?
Luden [163]
The answer is 6x^2-2x-20
5 0
3 years ago
Do Now-Find the size of a flat-screen tv given the length and
nika2105 [10]
500 area
90 perimeter
7 0
3 years ago
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