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3 years ago

Find the length of the legs of a right triangle if the legs are the same length and

Mathematics

1 answer:

xenn [34]3 years ago

6 0

Answer:

I think you have to multiple them all around the triangle

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Define the sequence recursively using function notation. 2.5, 7.5, 22.5, 67.5, 202.5,...

Lubov Fominskaja [6]

Answer: Aₙ = Aₙ₋₁*3; where A₀ = 2.5

Step-by-step explanation:

Let's try with some known sequences:

Arithmetic.

The difference between consecutive terms is constant:

7.5 - 2.5 = 5

22.5 - 7.5 = 15

We can already see that it is not an arithmetic sequence.

Geometric:

The ratio between consecutive terms is constant.

7.5/2.5 = 3

22.5/7.5 = 3

67.5/22.5 = 3

...

We can already see that this is a geometric sequence.

Then we can write the n-th term as:

Aₙ = Aₙ₋₁*3, where A₀ = 2.5

5 0

3 years ago

Solve differential equation: y'''+4y''-16y'-64y=0 y(0)=0, y'(0)=26, y''(0)=-16

Ipatiy [6.2K]

Answer: The required solution of the given differential equation is

$y(x)=3e^{4x}-3e^{-4x}+2xe^{-4x}.$

Step-by-step explanation: We are given to solve the following differential equation :

$y^{\prime\prime\prime}+4y^{\prime\prime}-16y^\prime-64y=0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)\\\\\y(0)=0,~y^\prime(0)=26,~y^{\prim\prime}(0)=-16.$

Let, $y=e^{mx}$ be an auxiliary solution of equation (i).

Then, $y^\prime=me^{mx},~~y^{\prime\prime}=m^2e^{mx},~~y^{\prime\prime\prime}=m^3e^{mx}.$

Substituting these values in equation (i), we get

$m^3e^{mx}+4m^2e^{mx}-16me^{mx}-64e^{mx}=0\\\\\Rightarrow (m^3+4m^2-16m-64)e^{mx}=0\\\\\Rightarrow m^3+4m^2-16m-64=0,~~~~~~~~~[\textup{since }e^{mx}\neq 0]\\\\\Rightarrow m^2(m-4)+8m(m-4)+16(m-4)=0\\\\\Rightarrow (m-4)(m^2+8m+16)=0\\\\\Rightarrow (m-4)(m+4)^2=0\\\\\Rightarrow m-4=0,~~(m+4)^2=0\\\\\Rightarrow m=4,~m=-4,~-4.$

So, the general solution is given by

$y(x)=Ae^{4x}+Be^{-4x}+Cxe^{-4x}.$

Then, we have

$y^\prime=4Ae^{4x}-4Be^{-4x}-4Cxe^{-4x}+Ce^{-4x},\\\\y^{\prime\prime}=16Ae^{4x}+16Be^{4x}+16Cxe^{-4x}-4Ce^{-4x}-4Ce^{-4x}\\\\\Rightarrow y^{\prime\prime}=16Ae^{4x}+16Be^{4x}+16Cxe^{-4x}-8Ce^{-4x}.$

With the conditions given, we get

$y(0)=A+B+C\times 0\\\\\Rightarrow A+B=0\\\\\Rightarrow A=-B~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(ii)$

$y^\prime(0)=4A-4B+C\\\\\Rightarrow 4A-4B+C=26\\\\\Rightarrow 4(A+A)+C=26~~~~~~~~~~~~~~~~[\textup{using equation (i)}]\\\\\Rightarrow C=26-8A~~~~~~~~~~~~~~~~~~~~~~~~~~~(iii)$

and

$y^{\prime\prime}(0)=16A+16B-8C\\\\\Rightarrow 16A-16A-8C=-16~~~~~~~~~~~~[\textup{using equation (ii)}]\\\\\Rightarrow -8C=-16\\\\\Rightarrow C=2.$

From equation (iii), we get

$C=26-8A\\\\\Rightarrow 2=26-8A\\\\\Rightarrow 8A=24\\\\\Rightarrow A=3.$

From equation (ii), we get

$B=-3.$

Therefore, the required solution of the given differential equation is

$y(x)=3e^{4x}-3e^{-4x}+2xe^{-4x}.$

4 0

3 years ago

The area of the yellow region is 112 cm^2. find the value of x

mrs_skeptik [129]

<h3>Answer: x = 7</h3>

===========================================================

Explanation:

The largest rectangle (composed of the green and yellow sections combined) has area of 11*12 = 132 cm^2.

The yellow region takes up 112 of those 132 sq cm. This must mean the green region takes up 132-112 = 20 cm^2.

The horizontal portion of the green rectangle is 12-x cm. The vertical portion is 11-x cm. We can form the area of the green rectangle as an algebraic expression like so

area = length*width

area = (11-x)*(12-x)

area = 132 - 11x - 12x + x^2 .... apply the FOIL rule

area = x^2 - 23x + 132

Set this equal to the 20 cm^2 we found earlier.

x^2 - 23x + 132 = 20

x^2 - 23x + 132-20 = 0

x^2 - 23x + 112 = 0

We could factor or we could use the quadratic formula. I'll go with the second option.

We'll plug in a = 1, b = -23, c = 112

$x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}\\\\x = \frac{-(-23)\pm\sqrt{(-23)^2-4(1)(112)}}{2(1)}\\\\x = \frac{23\pm\sqrt{81}}{2}\\\\x = \frac{23\pm9}{2}\\\\x = \frac{23+9}{2}\ \text{ or } \ x = \frac{23-9}{2}\\\\x = \frac{32}{2}\ \text{ or } \ x = \frac{14}{2}\\\\x = 16\ \text{ or } \ x = 7\\\\$

One of these solutions isn't feasible. Note how if x = 16, then this exceeds both the 11 cm and 12 cm sides. So this x value is not possible.

However, x = 7 is possible.

If x = 7, then the horizontal portion of the green rectangle is 12-x = 12-7 = 5 cm. Also, the vertical portion of the green rectangle would be 11-x = 11-7 = 4 cm. The area then is length*width = 5*4 = 20 cm^2 which matches up with what we got earlier. So the answer is confirmed.

7 0

3 years ago

The diagram shows two regular shapes

Bond [772]

Answer:

x = 144°

Step-by-step explanation:

In the figure given , two regular polygons are given to us. Since the polygon has five sides , it is a regular Pentagon. We know that each angle of a regular Pentagon is 108° .

So , here as per figure two regular Pentagons are attached base to base . Now , we know that,

Measure of complete angle is 360° .
Measure of regular polygon's angle is 108°.

According to Question ,

⇒ 108° + 108° + x = 360°

⇒ 216° + x = 360°

⇒ x = 360° - 216°

⇒ x = 144°

<h3>Hence the value of x is 144° </h3>

7 0

3 years ago

First right answer gets brain

Maurinko [17]

It is a ray. Which is a... please thank me

6 0

3 years ago