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3 years ago

Question: is 1 > 0.99999999...?Prove algebraically.

1 answer:

6 0

No. We claim that $1=0.\overline{9}$ and use algebra to prove the statement.

Let $x=0.\overline{9}$ . Multiply this by ten to get $10x=9.\overline{9}$ . Subtract the initial equation to give $9x=9$ and divide by $9$ to see that $x=1$ . Substituting into the original equation gives $1=0.\overline{9}$ , proving the desired statement.

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The triangle ABC and XYZ are similar in that order and AB:XY = 3:2. Find the ratio of the area of triangle ABC to that of triang

kotegsom [21]

Answer:

=9:4

Step-by-step explanation:

The relationship between area and side of triangle is

= (S1/S2)^2= (A1/A2)

=(3/2)^2=(A1/A2)

=9/4=(A1/A2)

=A1:A2=9:4

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3 years ago

What is 2/5% as a fraction?

Mariulka [41]

If it’s a fraction and want it simplified it’s 1/500

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3 years ago

B is the midpoint of AC, D is the midpoint of CE. BD=10, and AE =2x. Find the length of CE if CD =3x

Sonja [21]

There is a theorem that states that the line joining two midpoints in a triangle is always parallel to the third side and its length is half the length of the third side.

Since B is midpoint of AC and D is midpoint of CE, therefore, BD is parallel to AE and BD = 0.5 AE
AE = 2 BD = 2 x 10 = 2x
Therefore, x= 10

D is the midpoint of CE and CD = 3x, therefore CE = 6x where x = 10
Based on this, CE = 6 x 10 = 60 units of length

7 0

3 years ago

Use a linear system to write u = (12, 19, 31) as a linear combination of U1 = (1,1,2), u2 = (2,3,5), and uz = (3,5,8). Is w = (1

KiRa [710]

Answer:

Let $t\in {\mathbb R}$ ,

$\vec{u} = (t-2)\vec{u}_1 + (-2t +7)\vec{u}_2 + t\cdot \vec{u}_3$ .

$\vec{w}$ is also a linear combination of $\vec{u}_1$ , $\vec{u}_2$ , $\vec{u}_3$ .

Step-by-step explanation:

Write a linear system for $\vec{u} = x_1\cdot\vec{u}_1 + x_2\cdot \vec{u}_2 + x_3\cdot \vec{u}_3$ , with one equation for each component. The augmented matrix for the first linear system will be:

$\displaystyle \left[\begin{array}{ccc|c}1 & 2 & 3 & 12\\1 & 3 & 5 & 19\\2 & 5 & 8 & 31\end{array}\right]$ .

Transform this matrix to its reduced row-echelon form using Gaussian Elimination. Solve for each variable.

$\begin{aligned} &\left[\begin{array}{ccc|c}1 & 2 & 3 & 12\\1 & 3 & 5 & 19\\2 & 5 & 8 & 31\end{array}\right]\\ &\sim \left[\begin{array}{ccc|c}1 & 2 & 3 & 12\\0 & 1 & 2 & 7\\0 & 1 & 2 & 7\end{array}\right]\\&\sim \left[\begin{array}{ccc|c}1 & 2 & 3 & 12\\0 & 1 & 2 & 7\\0 & 0 & 0 & 0\end{array}\right]\\&\sim\left[\begin{array}{ccc|c}1 & 0 & -1 & -2\\0 & 1 & 2 & 7\\0 & 0 & 0 & 0\end{array}\right]\\&\left\{\begin{array}{l}x_1 = t-2\\x_2=- 2t+7\\x_3=t\end{array}\right.\end{aligned}$ .

Therefore,

$\vec{u} = (t-2)\vec{u}_1 + (-2t +7)\vec{u}_2 + t\cdot \vec{u}_3$ .

Set up a similar augmented matrix for $\vec{w} = x_1\cdot\vec{u}_1 + x_2\cdot \vec{u}_2 + x_3\cdot \vec{u}_3$ :

$\left[\begin{array}{ccc|c}1 & 2 & 3 & 1\\1 & 3 & 5 & 0\\2 & 5 & 8 & 1\end{array}\right]$ .

The second part of this question isn't concerned about the exact value of $x_1$ , $x_2$ , or $x_3$ . Therefore, before proceeding with Gaussian Elimination, start by checking the determinant of the coefficient matrix. If this determinant is nonzero, $\vec{w}$ will always be a unique linear combination of $\vec{u}_1$ , $\vec{u}_2$ , $\vec{u}_3$ now matter what value it takes.

In this case (also as seen in the first part of this question), the determinant of the coefficient matrix for $\vec{u}_1$ , $\vec{u}_2$ , and $\vec{u}_3$ is zero. Determining whether the linear combination is possible will require elimination.

$\begin{aligned} &\left[\begin{array}{ccc|c}1 & 2 & 3 & 1\\1 & 3 & 5 & 0\\2 & 5 & 8 & 1\end{array}\right]\\ &\sim \left[\begin{array}{ccc|c}1 & 2 & 3 & 1\\0 & 1 & 2 & -1\\0 & 1 & 2 & -1\end{array}\right]\\&\sim \left[\begin{array}{ccc|c}1 & 2 & 3 & 1\\0 & 1 & 2 & -1\\0 & 0 & 0 & 0\end{array}\right]\\&\sim\left[\begin{array}{ccc|c}1 & 0 & -1 & 3\\0 & 1 & 2 & -1\\0 & 0 & 0 & 0\end{array}\right]\\&\left\{\begin{array}{l}x_1 = t+3\\x_2=- 2t-1\\x_3=t\end{array}\right.\end{aligned}$ .

Similar to the first part of this question, this linear system is consistent. $\vec{w} = (t+3)\vec{u}_1 + (-2t -1)\vec{u}_2 + t\cdot \vec{u}_3$ . $\vec{w}$ is indeed a linear combination of $\vec{u}_1$ , $\vec{u}_2$ , $\vec{u}_3$ .

4 0

3 years ago

A force of 90 Newtons is applied to a 20 cm2 surface.

chubhunter [2.5K]

The given question is incomplete. The complete question is:

A force of 90 Newton's is applied to a $20cm^2$ surface. The force applied is then increased by 5 Newton's and the surface is increased by $5cm^2$ . What percentage does the outcome decrease by