Linear relations I need help on these two

The probabilities of interest when rolling a pair of fair six-sided dice are: P(A) the probability of the first die being a 5 is 1/6, P(B) the probability of the second die being a 3 is 1/6, P(A and B) the probability of both events occurring is 1/36, and P(B|A) the conditional probability of B given A is equal to P(B), indicating that events A and B are independent.

When rolling a pair of fair six-sided dice, each die is independent of the other. Therefore, the sample space consists of 6 x 6 = 36 equally likely outcomes. The probability that the first die is a 5, event A, is P(A) = 1/6 because there are 6 possible outcomes for the first die and only one of them is a 5.

The probability that the second die is a 3, event B, is also P(B) = 1/6 for the same reason as A, as there are 6 possible outcomes and only one of them is a 3. Now, the probability that the first die is a 5 and the second die is a 3, both events occurring simultaneously (A and B), is P(A and B) = P(A) times P(B) which equals 1/36.

The conditional probability that the second die is a 3 given that the first die is a 5, P(B|A), is simply the probability of B since events A and B are independent, so P(B|A) = P(B) = 1/6. This implies that knowing the outcome of the first die does not change the probability of the second die rolling a 3.

Given this independence, P(B|A) = P(B), and thus events A and B are indeed independent events.

[tex](a) \( P(A) = \frac{1}{6} \)\\(b) \( P(B) = \frac{1}{6} \)\\(c) \( P(A \text{ and } B) = \frac{1}{36} \)\\(d) \( P(B|A) = \frac{1}{6} \)\\(e) No, \( P(B|A) \neq P(B) \); events A and B are dependent.[/tex]

(a) P(A), the probability that the first die is a 5:

There are 6 outcomes where the first die is a 5.

[tex]\[ P(A) = \frac{\text{Number of outcomes where first die is a 5}}{\text{Total number of outcomes}} = \frac{6}{36} = \frac{1}{6} \][/tex]

(b) P(B), the probability that the second die is a 3:

There are 6 outcomes where the second die is a 3.

[tex]\[ P(B) = \frac{\text{Number of outcomes where second die is a 3}}{\text{Total number of outcomes}} = \frac{6}{36} = \frac{1}{6} \](c) \( P(A \text{ and } B) \), the probability that the first die is a 5 and the second die is a 3:[/tex]

There is only 1 outcome where the first die is a 5 and the second die is a 3, which is (5,3).

[tex]\[ P(A \text{ and } B) = \frac{\text{Number of outcomes where both A and B occur}}{\text{Total number of outcomes}} = \frac{1}{36} \][/tex]

(d) P(B|A)  the conditional probability that the second die is a 3 given that the first die is a 5:

Given that the first die is a 5, there are 6 possible outcomes for the second die, and 1 of those outcomes is a 3 (the outcome (5,3)).

[tex]\[ P(B|A) = \frac{\text{Number of outcomes where B occurs given A}}{\text{Number of outcomes where A occurs}} = \frac{1}{6} \][/tex]

(e) [tex]\[ P(B|A) = \frac{1}{6} \neq \frac{1}{6} = P(B) \]\\Since \( P(B|A) \neq P(B) \), the events A and B are dependent, not independent.[/tex]

Answer:

1/6 chance

Step-by-step explanation:

10+5+3+3+3+6=30 draws

out of 30 draws 5 blue

5/30 reduced is 1/6.

Answer:

The answer to your question is     (3, -6)

Step-by-step explanation:

Data

           x  -  2y  =  15                  Equation l

         2x +  4y  = -18                  Equation ll

Solve the system of equations by elimination

-Multiply equation l by 2

         2x  -  4y  =  30

         2x  + 4y  = -18

         4x   + 0    = 12

                    4x = 12

                      x = 12/4

                      x = 3

-Substitute x in equation l

          3  -  2y = 15

-Solve for y

               -2y = 15 - 3

               - 2y = 12

                   y = 12/-2

                   y = -6

-Solution

              (3, -6)

Answer:

C.  x = 3, y = -6

Step-by-step explanation

Answer is correct from Plato

Answer:

orthogonal

Step-by-step explanation:

If the dot product of the two vectors is 0, then they must be orthogonal.

[tex]u*v=9(0)-9(0) = 0[/tex]

Answer: c

Step-by-step explanation: edge 2021

Given:

The system of equation:

[tex]x+y = -4[/tex] -------- (1)

[tex]y = x^{2} -6x[/tex] --------- (2)

To find the values of x and y.

We will use substitution method.

From (1) we get,

[tex]y = -4-x[/tex]

We will put the value of y in (1) and we get,

[tex]-x-4 = x^{2} -6x[/tex]

or, [tex]x^{2} -5x+4= 0[/tex]

Now we will apply middle term factor method.

  [tex]x^{2} -(4+1)x+4 = 0[/tex]

     [tex]x^{2} -4x-x +4 = 0[/tex]

[tex]x(x-4)-1(x-4) = 0[/tex]

       [tex](x-4)(x-1)= 0[/tex]

so, x = 4 and 1

Now,

Substitute x = 4 in (1) we get,

[tex]y = -4-4 = -8[/tex]

And putting x = 1 in (1) we get,

[tex]y = -4-1 = -5[/tex]

Hence, the solution of the given system of equation is (4,-8) and (1,-5)

Thus, Option A is the correct answer.

Answer:

Step-by-step explanation:

Answer:

[tex]200+48n\leq 3500[/tex]

[tex]n\leq 66.7[/tex]

Step-by-step explanation:

Let the number of cartons he can safely put on the elevator at one time=n

Total weight of Carton=48n

Since the weight of the man and the cartons combined must not be more than 3500 pounds,

Therefore,an inequality that represents the situation is:

[tex]200+48n\leq 3500[/tex]

We can solve for n if required

[tex]200-200+48n\leq 3500-200\\48n\leq 3200\\\text{Divide both sides by 48}\\n\leq66.7[/tex]

The delivery man can safely carry 66 Cartons.

Answer:

The height of the cross section if 2 feet

Step-by-step explanation:

To solve this problem recall the formula for the area of a trapezoid of bases B (larger base) and b (smaller base) and height H:

[tex]Area = \frac{(B+b)\,H}{2}[/tex]

Therefore, for our case we have:

[tex]Area = \frac{(B+b)\,H}{2}\\34 \,ft^2 = \frac{(28\,ft+6\,ft)\,H}{2}\\34 \,ft^2 = \frac{(34 \,ft)\,H}{2}[/tex]

So, now we can solve for the height H:

[tex]34 \,ft^2 = \frac{(34 \,ft)\,H}{2}\\2\,*\,34 \,ft^2 =34\,ft\,* H\\H=\frac{2\,*\,34 \,ft^2}{34\,ft}\\ H=2\,ft[/tex]

Answer:

the least common denominator is 4xy

Answer:

We can use the sample about 42 days.

Step-by-step explanation:

Decay Equation:

[tex]\frac{dN}{dt}\propto -N[/tex]

[tex]\Rightarrow \frac{dN}{dt} =-\lambda N[/tex]

[tex]\Rightarrow \frac{dN}{N} =-\lambda dt[/tex]

Integrating both sides

[tex]\int \frac{dN}{N} =\int\lambda dt[/tex]

[tex]\Rightarrow ln|N|=-\lambda t+c[/tex]

When t=0, N=[tex]N_0[/tex] = initial amount

[tex]\Rightarrow ln|N_0|=-\lambda .0+c[/tex]

[tex]\Rightarrow c= ln|N_0|[/tex]

[tex]\therefore ln|N|=-\lambda t+ln|N_0|[/tex]

[tex]\Rightarrow ln|N|-ln|N_0|=-\lambda t[/tex]

[tex]\Rightarrow ln|\frac{N}{N_0}|=-\lambda t[/tex].......(1)

                            [tex]\frac{N}{N_0}=e^{-\lambda t}[/tex].........(2)

Logarithm:

130 days is the half-life of the given radioactive element.

For half life,

[tex]N=\frac12 N_0[/tex],  [tex]t=t_\frac12=130[/tex] days.

we plug all values in equation (1)

[tex]ln|\frac{\frac12N_0}{N_0}|=-\lambda \times 130[/tex]

[tex]\rightarrow ln|\frac{\frac12}{1}|=-\lambda \times 130[/tex]

[tex]\rightarrow ln|1|-ln|2|-ln|1|=-\lambda \times 130[/tex]

[tex]\rightarrow -ln|2|=-\lambda \times 130[/tex]

[tex]\rightarrow \lambda= \frac{-ln|2|}{-130}[/tex]

[tex]\rightarrow \lambda= \frac{ln|2|}{130}[/tex]

We need to find the time when the sample remains 80% of its original.

[tex]N=\frac{80}{100}N_0[/tex]

[tex]\therefore ln|{\frac{\frac {80}{100}N_0}{N_0}|=-\frac{ln2}{130}t[/tex]

[tex]\Rightarrow ln|{{\frac {80}{100}|=-\frac{ln2}{130}t[/tex]

[tex]\Rightarrow ln|{{ {80}|-ln|{100}|=-\frac{ln2}{130}t[/tex]

[tex]\Rightarrow t=\frac{ln|80|-ln|100|}{-\frac{ln|2|}{130}}[/tex]

[tex]\Rightarrow t=\frac{(ln|80|-ln|100|)\times 130}{-{ln|2|}}[/tex]

[tex]\Rightarrow t\approx 42[/tex]

We can use the sample about 42 days.

Final answer:

The radioactive sample can be used for approximately 390 days.

Explanation:

The half-life of a radioactive element is the time it takes for half of the radioactive nuclei to decay. In this case, the half-life is 130 days. The sample is considered no longer useful when 80% of the nuclei have decayed. To determine how many days the sample can be used, we need to find the number of half-lives it takes for 80% of the nuclei to decay.

80% is the same as 0.8, which means 20% (or 0.2) of the nuclei remain. Each half-life reduces the amount of nuclei by half, so we can set up an equation:

0.2 = (1/2)^n

where n is the number of half-lives.

To solve for n, we can take the logarithm of both sides:

n = log_base(1/2)(0.2)

Using a calculator, we find n ≈ 2.737.

Since we can't have a fraction of a half-life, we round up to the nearest whole number, which gives us n = 3.

Now, we can find the number of days by multiplying the half-life by the number of half-lives:

Number of days = 130 days * 3 = 390 days.

Therefore, the sample can be used for approximately 390 days.

Answer:

10.5 hours

Step-by-step explanation:

Given:

This week Andres will practice with his band for

[tex]1\frac{1}{2} \ means\ \frac{3}{2} \ hours[/tex] on Monday

[tex]1\frac{3}{4} \ means\ \frac{7}{4} \ hours[/tex]  on Tuesday

2 hours Wednesday.

Next week Andres will practice with his band for the same number of hours on Monday, Tuesday, Wednesday.

Question asked:

What was the total number of hours Andres will practice with his band over these 6 days?

Solution:

As she practice for two weeks, three days of each week:

On two Monday, she will practice = [tex]\frac{3}{2} + \frac{3}{2} =\frac{3+3}{2} =\frac{6}{2} =3\ hours[/tex]

On two Tuesday, she will practice = [tex]\frac{7}{4} +\frac{7}{4}=\frac{7+7}{4} =\frac{14}{4} =\frac{7}{2} \ hours[/tex]

On two Wednesday, she will practice = [tex]2+2=4 \ hours[/tex]

Total hours, she will practice over 6 days  [tex]=3+\frac{7}{2} +4[/tex]

                                                                      [tex]=7+\frac{7}{2} \\ \\ =\frac{2\times7+7}{2} \\ \\ =\frac{21}{2} \\ \\ =10.5\ hours[/tex]

Thus, Andres will practice 10.5 hours with his band over these 6 days.

The total number of hours andres will practice with his band over these 6 days is 10 1/2 hours

Given:

Day 1 = 1 1/2 hours

Day 2 = 1 3/4 hours

Day 3 = 2 hours

The total number of hours for six days = 2(1 1/2) + 2(1 3/4) + 2(2)

= 2(3/2) + 2(7/4) + 4

= 6/2 + 14/4 + 4

= 3 + 14/4 + 4

= 7 + 14/4

= (28+14) / 4

= 42 / 4

= 10 2/4

= 10 1/2 hours

Therefore, the total number of hours andres will practice with his band over these 6 days is 10 1/2 hours

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Answer:

[tex]F(t) = 18000(0.6666)^{t}[/tex]

Step-by-step explanation:

The fish population after t years can be modeled by the following equation:

[tex]F(t) = F(0)(1-r)^{t}[/tex]

In which F(0) is the initial population and r is the constant rate of decay.

Year one the fish population was 18000

This means that [tex]F(0) = 18000[/tex]

In year three the fish population was 8000 fish.

Two years later, so [tex]F(2) = 8000[/tex]

[tex]F(t) = F(0)(1-r)^{t}[/tex]

[tex]8000 = 18000(1-r)^{2}[/tex]

[tex](1-r)^{2} = 0.4444[/tex]

[tex]\sqrt{(1-r)^{2}} = \sqrt{0.4444}[/tex]

[tex]1 - r = 0.6666[/tex]

[tex]r = 0.3334[/tex]

So

[tex]F(t) = 18000(0.6666)^{t}[/tex]

Answer:

150

Step-by-step explanation:

1 box is 5 x 5 which=25 x 6 boxes=150

150 squared is your answer

Answer: 150

Step-by-step explanation:

The side of one cube is 5. 5 * 5 = 25 So 25 is the surface area of one cube side. A cube a 6 sides, and 25 * 6 = 150. Therefore, 150 is the surface area of the cube.

Answer:

94.5%

Step-by-step explanation:

Percent decrease can be represented as

[original price] * percentage = [new price]

When trying to find the percentage, you can manipulate the equation by dividing both sides by the original price to get:

percentage = [new price] / [original price]

In this case, this is represented by 192300/203400, or 94.5%

The median home price in the West fell from $203,400 to $192,300 then the percent decrease is 5.5%

percentage, a relative value indicating hundredth parts of any quantity.

Given that In one month, the median home price in the West fell from $203,400 to $192,300.

We have to find the percent decrease.

Percent decrease =[ (difference in price)/(original price) ] ×(100)

=203,400-192,300/203,400 ×(100)

=11100/203,400 ×(100)

=5.457 %

Hence, the median home price in the West fell from $203,400 to $192,300 then the percent decrease is 5.5%

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After simplifying [tex]\(\frac{{21^x}}{{7^x}}\)[/tex], we find it equivalent to [tex]\(3^x\)[/tex], confirming options a, b, and f as correct.

To simplify the expression [tex]\(\frac{{21^x}}{{7^x}}\)[/tex], we can use the property of exponents which states that [tex]\(a^m/a^n = a^{m-n}\)[/tex]. Applying this property, we get:

[tex]\[\frac{{21^x}}{{7^x}} = \frac{{(7 \cdot 3)^x}}{{7^x}} = \frac{{7^x \cdot 3^x}}{{7^x}} = 3^x\][/tex]

So, the simplified expression is [tex]\(3^x\).[/tex]

Now, let's check each option:

a. [tex]\(\frac{{7^x \cdot 3^x}}{{7^x}} = 3^x\)[/tex]. This expression is equivalent to the simplified form.

b. [tex]\((\frac{{21}}{{7}})^x = 3^x\)[/tex]. This expression is equivalent to the simplified form.

c. [tex]\(3\)[/tex]. This expression is not equivalent to the simplified form.

d. [tex]\((21 - 7)^x = 14^x\)[/tex]. This expression is not equivalent to the simplified form.

e. [tex]\(3^{x-7}\)[/tex]. This expression is not equivalent to the simplified form.

f. [tex]\(3^x\)[/tex]. This expression is equivalent to the simplified form.

Therefore, the expressions equivalent to [tex]\(\frac{{21^x}}{{7^x}}\)[/tex] are options a, b, and f.

The question probable maybe:

Given in the attachment

Final answer:

The given system of equations has one unique solution because the lines represented by the equations have different slopes, hence they intersect at exactly one point.

Explanation:

To answer how many solutions there are for the given system of equations:

6y=12x+36
15y=45x+60 (Assuming the '+60' was meant instead of '60', as '+60' makes it a linear equation)

We first simplify both equations to determine if they are identical, parallel, or intersecting.

For the first equation, we divide by 6:

y = 2x + 6

For the second equation, assuming a typo and it should be '+60', we divide by 15:

y = 3x + 4

Since the slopes (2 and 3) are different, the lines are not parallel and will intersect at exactly one point. Therefore, there is one unique solution to this system.

Answer:

third graph

Step-by-step explanation:

Answer:15

Step-by-step explanation:

I’m on USA test prep

In a triangle, the midsegment is always half the length of the base. Given that the length of AC is 30, the midsegment DE will be half of this which is 15. Thus, the answer is B) 15.

In geometry, a midsegment of a triangle is a line segment that connects the midpoints of two sides of the triangle. For every triangle, the length of the midsegment is always half the length of the base of the triangle.

From your question, the length of AC (which is the base of the triangle) is 30. Therefore, the length of the midsegment DE would be half of 30, which is 15.

So the answer to your question is: B) 15.

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Answer:

50

Step-by-step explanation:

The factors of 150 are: 1, 2, 3, 5, 6, 10, 15, 25, 30, 50, 75, 150

The factors of 250 are: 1, 2, 5, 10, 25, 50, 125, 250

Then the greatest common factor is 50.

Hope this helped

:)

Answer:

22.9 miles

Step-by-step explanation:

We are given that

Acceleration=Deceleration=a=[tex]4ft/s^2[/tex]

Maximum cruising speed,v=[tex]90mi/h=90\times \frac{5280}{3600}=132ft/s[/tex]

1 hour=3600 s

1 mile=5280 feet

Time,t=15 minutes=[tex]15\times 60=900 s[/tex]

1 min=60 s

Initial speed,u=0

[tex]v=u+at[/tex]

Substitute the values

[tex]132=0+4t[/tex]

[tex]t=\frac{132}{4}=33 s[/tex]

[tex]s=u+\frac{1}{2}at^2=0+\frac{1}{2}(4)(33)^2=2178 ft[/tex]

Distance,d=[tex]speed\times time=vt=132\times 900=118800ft[/tex]

Total distance=s+d=2178+118800=120978ft

Total distance=[tex]\frac{120978}{5280}=22.9miles[/tex]

Hence, the maximum distance traveled by train =22.9 miles

The scarf is $20 without tax.

The scarf is 20% off of 25, 5 is 20% of 25 so all you have to do is 25 - 5 to get your answer.

25 - 5 = 20

The sale price of the scarf , not including the tax is $20.00.

A sale price is the discounted price at which goods or services are being sold.

sale price = original price - discount price

According to the given question,

the original price of the scarf is $25.00

discount price = 20%of original price

⇒ discount price =[tex]\frac{20}{100}[/tex]×[tex]25[/tex]

⇒discount price = $5.00

Therefore,

sale price  of scarf = $25.00-$5.00

Sale price of scarf = $20.00

Hence, the sale price of the scarf, not including tax is $20.00

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There are 3 even numbers out of A total of 8 numbers.

Each spin would have a 3/8 chance of landing in an even number.

Multiply the chance of landing on even by number of spins:

120 x 3/8 = 360/8 = 45

The answer would be 45 times.

Answer:

The amount made in a month with 9 sales people = 334,734.53

Step-by-step explanation:

The amount, Y, made by a sales company in dollars is related to the number of sales people, x, at the company through the relation,

Y = 60,209.47x - 207,150.70

where Y = amount made in dollars

x = number of sales people

when x = 9, what is Y.

Y = 60,209.47x - 207,150.70

Y = (60,209.47×9) - 207,150.70

Y = 541,885.23 - 207,150.70

Y = 334,734.53

Hope this Helps!!!

Answer:

8906.4 feet

Step-by-step explanation:

The worker is walking around the park which is the circumference.

The equation for the circumference is diameter times (pi)

since the diameter is 975, you multiply 975 by pi which gives you 2968.8

Because the worker walks around the park 3 times you multiply the circumference by 3 which is 2968.8 times 3 which equals 8906.4

Therefore the worker walks 8906.4 feet a day

There are 75 birds in each section

First, lets see how many in each enclosure so we have an amount to multiply by 3

450/18 = 25 birds per enclosure.

Now lets see how many birds would be in 3 enclosures i.e. a section

25 * 3 = 75

Answer: The answer is B. 1.0

Step-by-step explanation:

That’s the closest mean

Answer:

A is 15.27 and B is 12.05

Step-by-step explanation:

Answer:

22/100 or 0.22 or 22%

Step-by-step explanation:

Out of the 50 times you drew a marble, you got 11 blue ones.

11/50

We can divide 11/50 and get 0.22, which is the same as 22/100 or 22%.

Hope this helps! ;-)

The required probability is,

[tex]P(B)=\frac{11}{50} [/tex]

The total number f marbles are 50

Now, the formula for finding the experimental probability is,

P(B)=Possible outcomes/ Number of outcomes

[tex]P(B)=\frac{11}{50} [/tex]

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