Find the radius of a circle given that the circumference is 75.4 cm .

Elis [28]

75/10 is 7.5 as a fraction

5 0

2 years ago

We have two fair three-sided dice, indexed by i = 1, 2. Each die has sides labeled 1, 2, and 3. We roll the two dice independent

Bogdan [553]

Answer:

(a) P(X = 0) = 1/3

(b) P(X = 1) = 2/9

(c) P(X = −2) = 1/9

(d) P(X = 3) = 0

(a) P(Y = 0) = 0

(b) P(Y = 1) = 1/3

(c) P(Y = 2) = 1/3

Step-by-step explanation:

Given:

- Two 3-sided fair die.

- Random Variable X_1 denotes the number you get for rolling 1st die.

- Random Variable X_2 denotes the number you get for rolling 2nd die.

- Random Variable X = X_2 - X_1.

Solution:

- First we will develop a probability distribution of X such that it is defined by the difference of second and first roll of die.

- Possible outcomes of X : { - 2 , -1 , 0 ,1 , 2 }

- The corresponding probabilities for each outcome are:

( X = -2 ): { X_2 = 1 , X_1 = 3 }

P ( X = -2 ): P ( X_2 = 1 ) * P ( X_1 = 3 )

: ( 1 / 3 ) * ( 1 / 3 )

: ( 1 / 9 )

( X = -1 ): { X_2 = 1 , X_1 = 2 } + { X_2 = 2 , X_1 = 3 }

P ( X = -1 ): P ( X_2 = 1 ) * P ( X_1 = 3 ) + P ( X_2 = 2 ) * P ( X_1 = 3)

: ( 1 / 3 ) * ( 1 / 3 ) + ( 1 / 3 ) * ( 1 / 3 )

: ( 2 / 9 )

( X = 0 ): { X_2 = 1 , X_1 = 1 } + { X_2 = 2 , X_1 = 2 } + { X_2 = 3 , X_1 = 3 }

P ( X = -1 ):P ( X_2 = 1 )*P ( X_1 = 1 )+P( X_2 = 2 )*P ( X_1 = 2)+P( X_2 = 3 )*P ( X_1 = 3)

: ( 1 / 3 ) * ( 1 / 3 ) + ( 1 / 3 ) * ( 1 / 3 ) + ( 1 / 3 ) * ( 1 / 3 )

: ( 3 / 9 ) = ( 1 / 3 )

( X = 1 ): { X_2 = 2 , X_1 = 1 } + { X_2 = 3 , X_1 = 2 }

P ( X = 1 ): P ( X_2 = 2 ) * P ( X_1 = 1 ) + P ( X_2 = 3 ) * P ( X_1 = 2)

: ( 1 / 3 ) * ( 1 / 3 ) + ( 1 / 3 ) * ( 1 / 3 )

: ( 2 / 9 )

( X = 2 ): { X_2 = 1 , X_1 = 3 }

P ( X = 2 ): P ( X_2 = 3 ) * P ( X_1 = 1 )

: ( 1 / 3 ) * ( 1 / 3 )

: ( 1 / 9 )

- The distribution Y = X_2,

P(Y=0) = 0

P(Y=1) = 1/3

P(Y=2) = 1/ 3

- The probability for each number of 3 sided die is same = 1 / 3.

7 0

3 years ago

If the principal is $300 rate 3% time 4 years then what is the interest earned and the new balance

Tomtit [17]

Answer:

a) Interest earned = $36

New Balance = $336

b) Interest rate = 0.05 or 5%

New Balance = $517.5

c) time t = 5

New Balance = $612.5

d) Principal Amount = $675

New Balance = $783

Step-by-step explanation:

We are given:

a) Principal (P) = $300

Rate (r) = 3% or 0.03

Time (t)= 4 years

Interest earned = ?

The formula used is: $Simple \ Interest (I)= P\times r\times t$

Putting values and finding interest

$Simple \ Interest (I)= P\times r\times t\\Simple \ Interest (I)= 300\times 0.03\times 4\\Simple \ Interest (I)= 36$

So, Interest earned = $36

New Balance = Principal + Interest = 300+36 = $336

b) a) Principal (P) = $300

Rate (r) = ?

Time (t)= 3 years

Interest earned = 67.50

The formula used is: $Simple \ Interest (I)= P\times r\times t$

Putting values and finding rate

$Simple \ Interest (I)= P\times r\times t\\67.50= 450\times r\times 3\\67.50=1350\times r\\r=\frac{67.50}{1350}\\r=0.05 \ or \ 5\%$

So, Interest rate = 0.05 or 5%

New Balance = Principal + Interest = 450+67.50 = $517.5

c) Principal (P) = $500

Rate (r) = 4.5% or 0.045

Time (t)= ?

Interest earned = $112.50

The formula used is: $Simple \ Interest (I)= P\times r\times t$

Putting values and finding time

$Simple \ Interest (I)= P\times r\times t\\112.50= 500\times 0.045\times t\\112.50=22.5 \times t\\t=\frac{112.50}{22.5}\\t=5$

So, time t = 5

New Balance = Principal + Interest = 500+112.50 = $612.5

d) Principal (P) = ?

Rate (r) = 8% or 0.08

Time (t)= 2 years

Interest earned = 108.00

The formula used is: $Simple \ Interest (I)= P\times r\times t$

Putting values and finding Principal

$Simple \ Interest (I)= P\times r\times t\\108=P\times 0.08 \times 2\\108=P\times 0.16\\P=\frac{108}{0.16}\\P=675$

So, Principal Amount = $675

New Balance = Principal + Interest = 675+108 = $783

8 0

3 years ago

6. if one factor of x ²-7x+10 is x-2, what is the other factor?

uranmaximum [27]

Answer:

6. B. x + 8

7. D. ( 3x-2)(3x-2)

8. D. Perfect square trimomal

9.C. x - 2

10. A. 3x and 8x

Step-by-step explanation:

Semoga Bermanfaat

6 0

2 years ago

Find the directional derivative of the function at the given point in the direction of the vector v. G(r, s) = tan−1(rs), (1, 3)

alexandr1967 [171]

The <em>directional</em> derivative of $f$ at the given point in the direction indicated is $\frac{5}{2}$ .

<h3>How to calculate the directional derivative of a multivariate function</h3>

The <em>directional</em> derivative is represented by the following formula:

$\nabla_{\vec v} f = \nabla f (r_{o}, s_{o})\cdot \vec v$ (1)

Where:

$\nabla f (r_{o}, s_{o})$ - Gradient evaluated at the point $(r_{o}, s_{o})$ .
$\vec v$ - Directional vector.

The gradient of $f$ is calculated below:

$\nabla f (r_{o}, s_{o}) = \left[\begin{array}{cc}\frac{\partial f}{\partial r}(r_{o},s_{o}) \\\frac{\partial f}{\partial s}(r_{o},s_{o}) \end{array}\right]$ (2)

Where $\frac{\partial f}{\partial r}$ and $\frac{\partial f}{\partial s}$ are the <em>partial</em> derivatives with respect to $r$ and $s$ , respectively.

If we know that $(r_{o}, s_{o}) = (1, 3)$ , then the gradient is:

$\nabla f(r_{o}, s_{o}) = \left[\begin{array}{cc}\frac{s}{1+r^{2}\cdot s^{2}} \\\frac{r}{1+r^{2}\cdot s^{2}}\end{array}\right]$

$\nabla f (r_{o}, s_{o}) = \left[\begin{array}{cc}\frac{3}{1+1^{2}\cdot 3^{2}} \\\frac{1}{1+1^{2}\cdot 3^{2}} \end{array}\right]$

$\nabla f (r_{o}, s_{o}) = \left[\begin{array}{cc}\frac{3}{10} \\\frac{1}{10} \end{array}\right]$

If we know that $\vec v = 5\,\hat{i} + 10\,\hat{j}$ , then the directional derivative is:

$\nabla_{\vec v} f = \left[\begin{array}{cc}\frac{3}{10} \\\frac{1}{10} \end{array}\right] \cdot \left[\begin{array}{cc}5\\10\end{array}\right]$

$\nabla _{\vec v} f (r_{o}, s_{o}) = \frac{5}{2}$

The <em>directional</em> derivative of $f$ at the given point in the direction indicated is $\frac{5}{2}$ . $\blacksquare$

To learn more on directional derivative, we kindly invite to check this verified question: brainly.com/question/9964491

3 0

2 years ago