Let's use the same bag of 12 balls: S are red, 3 are brown, 2 are green and 2 are white. If you draw a red ball and put it aside: a) How many balls are left in the bag? b) How many of the balls in the bag are red? c) How many are brown? Green? White? d) Draw a second ball from the bag. What is the probability that it is green? (Hint: How many of the balls in the bag are green? How many balls are in the bag?) Now put both balls back in the bag and draw two balls without replacing them. e) What is the probability that the first ball is red and the second ball is green (without replacement)? What is the probability that the first ball is red and the second ball is also red (without replacement)? f) What is the probability that the first ball is green and the second ball is also green (without replacement)? g) What is the probability that the first ball is brown and the second ball is white (without replacement)? Would this be different than the probability that the first ball is white and the second ball is brown (without replacement)? Explain. (Feel free to use calculations in your explanation.) h) 27

Answer:

586 people stated that they were in opposition to the war.

Step-by-step explanation:

Percentage problems can be solved as a simple rule of three problem:

In a rule of three problem, the first step is identifying the measures and how they are related, if their relationship is direct of inverse.

When the relationship between the measures is direct, as the value of one measure increases, the value of the other measure is going to increase too. In this case, the rule of three is a cross multiplication.

When the relationship between the measures is inverse, as the value of one measure increases, the value of the other measure will decrease. In this case, the rule of three is a line multiplication.

A percentage problem is an example where the relationship between the measures is direct.

The problem states that  69% of the said they opposed the current war. 69% is 69 of 100. How much it is of 850.

So

69 - 100

x - 850

100x = 69*850

[tex]x = \frac{58650}{100}[/tex]

x = 586.5

586 people stated that they were in opposition to the war.

Answer:

1) 15

2) 4.5

Step-by-step explanation:

1 ) 10, 12, 16, 17,20

Average = [tex]\frac{\text{Sum of all observations}}{\text{Total no. of observations}}[/tex]

Average = [tex]\frac{10+12+16+17+20}{5}[/tex]

Average = [tex]15[/tex]

2) 0, 3,6,9

Average = [tex]\frac{\text{Sum of all observations}}{\text{Total no. of observations}}[/tex]

Average = [tex]\frac{0+3+6+9}{4}[/tex]

Average = [tex]4.5[/tex]

Answer:

500 cubic feet equals 14158.4 liters or 3740.25 gallons.

Step-by-step explanation:

We are asked to convert 500 cubic feet to liters then to gallons.

We know that one cubic feet equals 28.3168 liters.

[tex]\text{500 cubic feet to liters}=\text{500 cubic feet}\times \frac{\text{28.3168 liters}}{\text{cubic feet}}[/tex]

[tex]\text{500 cubic feet to liters}=500\times \text{28.3168 liters}[/tex]

[tex]\text{500 cubic feet to liters}=\text{14158.4 liters}[/tex]

We know one liter equals 0.264172 gallons.

[tex]\text{14158.4 liters to gallons}=\text{14158.4 liters}\times\frac{0.264172\text{ gallons}}{\text{liter}}[/tex]

[tex]\text{14158.4 liters to gallons}=14158.4 \times 0.264172\text{ gallons}[/tex]

[tex]\text{14158.4 liters to gallons}=3740.2528448\text{ gallons}[/tex]

[tex]\text{14158.4 liters to gallons}\approx 3740.25\text{ gallons}[/tex]

Therefore, 500 cubic feet equals 14158.4 liters or 3740.25 gallons.

Answer:

147

Step-by-step explanation:

Given:

Progesterone =  3.8 g

Glycerin = 7 mL

2% methylcellulose solution 50 mL

Cherry syrup ad = 90 mL

Now,

The total volume of the solution = 7 + 50 + 90 = 147 mL

Also,

2% methylcellulose solution 50 mL is concluded as:

the volume of  methylcellulose in the solution is 2% of the total volume of the solution

thus,

volume of methylcellulose = 0.02 × 50 mL = 1 mL

Therefore,

Ratio strength of methylcellulose in the finished product

=[tex]\frac{\textup{volume of methylcellulose}}{\textup{ total volume of the solution}}[/tex]

or

= [tex]\frac{\etxtup{1}}{\textup{ 147}}[/tex]

Hence, the answer according to the question is 147

Answer:

Hence the current weighted average of student = 87.60

Step-by-step explanation:

Grades obtained by student are

Q1= 100

Q2= 93

IW1= 82

IW2= 83

H1= 80

the weighted average = sum of all the grades/ number of subjects

[tex]= \frac{100+93+82+83+80}{5}[/tex]= 87.60

Hence the current weighted average of student = 87.60

1.5x + 1.3x = - 8.4

2.8x = -8.4

x= - 8.4/2.8

x = 3

On simplification of liner equation 1.5x + 1.3x = -8, we get x = -3.

To solve the linear equation 1.5x + 1.3x = -8.4, we need to start by combining like terms.

Both terms on the left side of the equation have the variable x, so we can add them together.

1.5x + 1.3x = 2.8x.

2.8x = -8.4.

To find the value of x, we need to isolate the variable by dividing both sides of the equation by 2.8.

Divide both sides by 2.8:

x = -8.4 / 2.8.

x = -3.

Answer:

Using the rule of 72, the doubling time is 9.35 years.

The exact answer is that the doubling time is 8.89 years.

Step-by-step explanation:

By the rule of 72, we have that the doubling time D is given by:

[tex]D = \frac{72}{Interest Rate}[/tex]

The interest rate is in %.

In our exercise, the interest rate is 7.7%. So, by the rule of 72:

[tex]D = \frac{72}{7.7} = 9.35[/tex].

Exact answer:

The exact answer is going to be found using the compound interest formula(since the rule of 72 is a simplification of this formula).

The compound interest formula is given by:

[tex]A = P(1 + \frac{r}{n})^{nt}[/tex]

Where A is the amount of money, P is the principal(the initial sum of money), r is the interest rate(as a decimal value), n is the number of times that interest is compounded per unit t and t is the time the money is invested or borrowed for.

So, for this exercise, we have:

We want to find the doubling time, that is, the time in which the amount is double the initial amount, double the principal.

[tex]A = 2P[/tex]

[tex]r = 0.077[/tex]

There are 52 weeks in a year, so [tex]n = 52[/tex]

[tex]A = P(1 + \frac{r}{n})^{nt}[/tex]

[tex]2P = P(1 + \frac{0.077}{52})^{52t}[/tex]

[tex]2 = (1.0015)^{52t}[/tex]

Now, we apply the following log propriety:

[tex]\log_{a} a^{n} = n[/tex]

So:

[tex]\log_{1.0015}(1.0015)^{52t} = \log_{1.0015} 2[/tex]

[tex]52t = 462.44[/tex]

[tex]t = \frac{462.44}{52}[/tex]

[tex]t = 8.89[/tex]

The exact answer is that the doubling time is 8.89 years.

Answer:

2x + y = 2.

Step-by-step explanation:

First  find the slope of the required line by writing the line 2x + y = -5 in slope intercept form:

2x + y = -5

y = -2x - 5

- so the slope is -2.

Therefore the required equation is

y = -2x + 2    (where  2 is the y-intercept).

Converting to standard form:

y = -2x + 2

2x + y = 2.

The equation of the line parallel to 2x + y = -5 with a y-intercept of 2 is 2x + y = 2.

To find the equation of a line parallel to the given line, we must first realize that parallel lines have the same slope. The given equation is 2x + y = -5, which can be rearranged into y = -2x - 5, showing us that the slope of the given line is -2. Therefore, the slope of the line we want to find is also -2.

With a slope of -2 and a y-intercept of 2 (since the line is said to intersect with the y-axis at y=2), the slope-intercept form of the line is y = -2x + 2. However, the question requires the answer in standard form. The standard form is Ax + By = C, where A, B and C are integers and A > 0. To convert our slope-intercept equation to standard form, we will add 2x to both sides, obtaining the final equation as 2x + y = 2.

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Answer:

the probability that the user is fraudulent is 0.00299133

Step-by-step explanation:

Let be the events be:

G: The user generates calls from two or more areas.

NG: The user does NOT generate calls from two or more areas.

L: The user is legitimate.

F: The user is fraudulent.

The probabilities established in the statement are:

[tex]P (G | L) = 0.01//P (G | F) = 0.30//P (F) = 0.0001//P (L) = 0.9999//[/tex]

With these values, the probability that a user is fraudulent, if it has originated calls from two or more areas is:

[tex]P (F|G) = \frac{P(F\bigcap G)}{P(G)} = \frac{P(F)P(G|F)}{P(G)} = \frac{P(F)P(G|F)}{P(F)P(G|F)+P(L)P(G|L)}[/tex]

[tex]\frac{(0.0001)(0.30)}{(0.0001)(0.30)+(0.9999)(0.01)} = 0.00299133[/tex]

Answer:

when graphing f(x) between x=-3 & x=3, the result is a function that comes from x=-infinite and positive y, crosses the x-axis at (-2.646,0), continues to decrease until (-1.871,-12.25) and then increases until (0,0).

This function is symetrical by the y axis, therefore, after reaching (0,0), f(x) decreases until (1.871,-12.25), starts to increase until it crosses the x axis at (2-646,0) and continues to increase until x=+infinite

Step-by-step explanation:

This funcion appears as a large W, with it's points on (-1.871,-12.25) , (0,0) & (1.871,-12.25)

Answer:

[tex]0\le P\le 1[/tex]

Step-by-step explanation:

The probability of an event is a number describing the chance that the event will happen.

Definition: The probability of an evant is

[tex]P=\dfrac{\text{Number of favorable outcomes}}{\text{Number of all possible outcomes}}[/tex]

1. An event that is certain to happen (Number of favorable outcomes = Number of all possible outcomes) has a probability of 1.

2. An event that cannot possibly happen (Number of favorable outcomes = 0) has a probability of 0.

3. If there is a chance that an event will happen, then its probability is between 0 and 1.

Thus,

[tex]0\le P\le 1[/tex]

The probability of an event occurring is always between 0 and 1, inclusive. Therefore, the correct inequality is 0≤P≤1.

The probability of an event (P) in a standard probability model is always defined between 0 and 1. Here, 0 represents the impossibility of the event, whereas 1 represents the certainty of the event. Therefore, the inequality that best describes the probability P is 0≤P≤1. Any other range for probability does not fit into the standard probability model used in mathematics.

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Answer:

The greatest common divisor of two integers a and b (not both 0) is the largest integer that divides both a and b.

Step-by-step explanation:

Think for example of the numbers a=5, and b= -10. The greatest common divisor of 5 and -10, is the largest integer that divides both 5 and -10. We can find it by inspection (although there are more advanced methods to find it). We can list all integers that divide both 5 and -10.

-5 divides 5, and it also divides -10

-1 divides 5, and it also divides -10

1 divides 5, and it also divides -10

5 divides 5, and it also dividies -10

The LARGEST of them all is then 5, so 5 is the greatest common divisor of 5 and -10. The usual way to write it is

[tex]gcd(5,-10)=(5,-10)=5[/tex]

The greatest common divisor (GCD) of two integers a and b is the largest integer that divides both a and b without leaving a remainder. The GCD, denoted as (a, b), can be calculated using the Euclidean algorithm.

The greatest common divisor (GCD) of two non-zero integers a and b, denoted as (a, b), is the largest positive integer that divides both a and b without leaving a remainder. A formal definition could be formulated as follows: The greatest common divisor of a and b is a number d such that:

d is a divisor of a (i.e., a mod d = 0).

d is a divisor of b (i.e., b mod d = 0).

For any other integer e that divides both a and b, e \\leq d.

Moreover, the GCD can be calculated using algorithms such as the Euclidean algorithm, which is based on the principle that the gcd of a and b is the same as the gcd of b and a mod b, assuming a > b and b \\neq 0.

Answer:

The sum of two rational numbers is always a rational number.

Step-by-step explanation:

Rational Number is the number of the form [tex]\frac{p}{q}[/tex], q≠0 and p and q are integers.

Further, when we add or subtract two rational number it is always a rational number. Example:

[tex]\dfrac{4}{64} +\dfrac{25}{4} = \dfrac{4+25\times 16}{64} \\= \dfrac{4+400}{64} = \dfrac{404}{64} =\dfrac{101}{16}[/tex]

which is also a rational number.

Thus, the sum of two rational numbers is always a rational number.

Answer:  0.8531

Step-by-step explanation:

Let x be the random variable that represents the readings on scientific thermometers .

Given : The readings on scientific thermometers are normally distributed,

Population mean : [tex]\mu=0^{\circ}\ C[/tex]

Standard deviation : [tex]\sigma=1^{\circ}\ C[/tex]

Z-score : [tex]z=\dfrac{x-\mu}{\sigma}[/tex]

Now, the z-value corresponding to -1.05 :  [tex]z=\dfrac{-1.05.-0}{1}=-1.05[/tex]

P-value = [tex]P(x>-1.05)=P(z>-1.05)=1-P(z\leq-1.05)[/tex]

[tex]=1-0.1468591=0.8531409\approx0.8531\text{ (Rounded to four decimal places)}[/tex]

Hence, the probability of the reading greater than -1.05 in degrees Celsius.= 0.8531

Answer:

a) $765.13 b) $277,601.23

Step-by-step explanation:

a) The problem is an example of an ordinary annuity (deposits at the end of the period).

The future value of this type of annuity is:

[tex]FV=A*\frac{(1+i)^{n} -1}{i}[/tex]

Clearing the annual deposit A

[tex]A=FV*\frac{i}{(1+i)^{n} -1}[/tex]

[tex]A=360,000*\frac{0.11}{(1.11)^{38}-1 } =360,000*0,002125351=765.13[/tex]

The deposit needed to have $360,000 in 38 years is $765.13

b) We can use the same formula to compute the FV of a known deposit:

[tex]FV=A*\frac{(1+i)^{n} -1}{i}[/tex]

[tex]FV=590*\frac{(1.11)^{38} -1}{0.11}=590*470,5105644=277,601.23[/tex]

With annual deposits of $590 you will have at 38 years an ammount of $277,601.23

Answer:

Step-by-step explanation:

1. Number of boys in the group = 25

Number of girls in the group = 18

Total children = 25 + 18 = 43

Number of ways to arrange the children in a way = 43!

2. If we consider all the boys as an individual then number of ways children can be arranged = 19!

Number of ways boys can sit next to each other = 25!

So the number of ways can be arranged = 19!×25!

3. Number of ways boys can sit next to each other = 25!

Number of ways girls can sit next to each other = 19!

Then number of ways to arrange the children in a row with all boys next to each other and all the girls next to each other will be = 2 × 18! × 25!

4. 1. To choose a chess team if anyone can be chosen

= [tex]^{43}C_{6}[/tex]

= 6096454

4. 2. Exactly 2 girls must be chosen then number of ways

= [tex]^{18}C_{2}\times ^{25}C_{4}=1935450[/tex]

4. 3. At least two boys must be chosen

= [tex]^{25}C_{2}\times ^{18}C_{4}+^{25}C_{3}\times ^{18}C_{3}+^{25}C_{4}\times ^{18}C_{2}+^{25}C_{5}\times ^{18}C_{1}+^{25}C_{6}[/tex]

= 5863690

Answer:

3222212.11113

Step-by-step explanation:

First, you should take care of the fractional separator (the dot) so we split the problem in two parts: one for the integer and other for the fractional part.

Since 4 is a power of 2, we can just take two digits from the orignal number and asign it to its corresponding number in base 4:

[tex]\left[\begin{array}{cc}Binary&Base 4\\00&0\\01&1\\10&2\\11&3\end{array}\right][/tex]

Start with the fractional part from the fractional point to the right:

[tex]\left[\begin{array}{ccccc}01&01&01&01&11\\1&1&1&1&3\end{array}\right][/tex]

Then do the same to the integer part starting from the fractional point to the left.

[tex]\left[\begin{array}{ccccccc}11&10&10&10&10&01&10\\3&2&2&2&2&1&2\end{array}\right][/tex]

By joining them together, we obtain the response.

Answer:

a) yes

b) no

Step-by-step explanation:

[tex](\mathbb{Z}, *)[/tex] is a gruop if satisfies the following conditions:

1. If a and b are two elements in [tex]\mathbb{Z}[/tex], then the product a*b is also in [tex]\mathbb{Z}[/tex].

2. The defined multiplication is associative, i.e., for all a,b,c in [tex]\mathbb{Z}[/tex], (a*b)*c=a*(b*c).

3. There is an identity element e such that e*a=a*e=a for every element a in [tex]\mathbb{Z}[/tex].

4. There must be an inverse of each element. Therefore, for each element a of [tex]\mathbb{Z}[/tex], the set contains an element b=a^(-1) such that a*a^(-1)=a^(-1)*a=e.

Let's see if the conditions are satisfied:

a)

1. if x and y are integers then x+y-1=a*y is an integer

2. If x,y and z are integers then

   (x*y)*z= (x+y-1)*z= (x+y-1) + z - 1= x +y+z-2,

   x*(y*z)= x*(y+z-1)= x + (y+z-1) -1 = x+ y + z -2

  Then (x*y)*z=x*(y*z), i.e, * is associative.

3. Let e=1 and b an integer. Observe that

    1*b=1+b-1=b and b*1= b + 1 -1= b.

  Then e is an identity element.

4. a and integer and b= 2- a. Observe that

    b*a= 2-a+a-1= 1 and a*b= a+2-a-1=1,

the b= a^(-1) is the inverse of a.

We conclude that [tex](\mathbb{Z}, *)[/tex] is a group.

b)

1. If x,y and z are integers then

   (x*y)*z= (x-y+xy)*z= (x-y+xy) - z + (x-y+xy)z= x -y-z+xy+xz-yz+xyz

   x*(y*z)= x*(y-z+yz)= x - (y-z+yz) +x(y-z+yz) = x-y +z + xy -xz -yz+xyz

  Then (x*y)*z≠x*(y*z), i.e, * isn't associative.

We conclude that [tex](\mathbb{Z}, *)[/tex] isn't a group.

To determine the maximum length of the shipping carton given a maximum combined length and girth of 108 inches, we calculate the girth as 52 inches using the provided width and height. Subtracting this from 108 inches gives us a maximum allowable length of 56 inches.

The problem is a geometry optimization problem where we need to maximize the length of a shipping carton given a constraint on its combined length and girth. Girth is the perimeter around the width and height sides of the package. According to the U.S. postal regulations, the maximum combined length and girth is 108 inches.

First, let's calculate the girth using the given dimensions of the carton. The width is 14 inches and the height is 12 inches. The girth is twice the width plus twice the height (since girth is the perimeter of the cross-section):

girth = 2  × width + 2 × height

girth = 2  × 14in + 2 × 12in

girth = 28in + 24in

girth = 52in

Now, to find the maximum length, we can subtract the girth from the maximum allowed combined length and girth:

max length = max combined length and girth - girth

max length = 108in - 52in

max length = 56in

Therefore, the maximum length that the carton can be is 56 inches.

Answer:

a) [tex]A \cap B = \{\phi\}[/tex]

b)[tex]\{0, 2, 3, 4, 5,6, 7, 8\}[/tex]

c)[tex]\{0, 2, 3, 4, 5,6, 7, 8\}[/tex]

Step-by-step explanation:

We are given the following information:

The universal set is : U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}

A = {0, 2, 4, 6, 8}

B = {1, 3, 5, 7, 9}

C = {2, 3, 5, 7}

a) [tex]A \cap B = \{\phi\}[/tex]

b)[tex] C/B = C \cap \overline{B} = \{2, 3, 5, 7\} \cap \{0, 2, 4, 6, 8\} = \{2\}[/tex]  

c)

[tex]A \cup (B \cap C) = \{0, 2, 4, 6, 8\} \cup (\{1, 3, 5, 7, 9\} \cap \{2, 3, 5, 7\})\\=\{0, 2, 4, 6, 8\} \cup \{3, 5, 7\}\\= \{0, 2, 3, 4, 5,6, 7, 8\}[/tex]

Answer:

So, the times the ball will be 48 feet above the ground are t = 0 and t = 2.

Step-by-step explanation:

The height h of the ball is modeled by the following equation

[tex]h(t)=-16t^2+32t+48[/tex]

The problem want you to find the times the ball will be 48 feet above the ground.

It is going to be when:

[tex]h(t) = 48[/tex]

[tex]h(t)=-16t^{2}+32t+48[/tex]

[tex]48=-16t^{2}+32t+48[/tex]

[tex]0=-16t^{2}+32t+48 - 48[/tex]

[tex]16t^{2} - 32t = 0[/tex]

We can simplify by 16t. So

[tex]16t(t-2)= 0[/tex]

It means that

16t = 0

t = 0

or

t - 2 = 0

t = 2

So, the times the ball will be 48 feet above the ground are t = 0 and t = 2.

Answer:

The 33rd percentile of the time it takes Finn to get to work on any given day is 31.04 minutes.

There is a 61.92% probability that Finn took more than 40.5 minutes to get to work on the first day or more than 38.5 minutes to get to work on the second day.

Step-by-step explanation:

This can be solved by the the z-score formula:

On a normaly distributed set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a value X is given by:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

Each z-score value has an equivalent p-value, that represents the percentile that the value X is:

The problem states that:

Mean = 35, so [tex]\mu = 35[/tex]

Variance = 81. The standard deviation is the square root of the variance, so [tex]\sigma = \sqrt{81} = 9[/tex].

Find the 33rd percentile of the time it takes Finn to get to work on any given day. Do not include any units in your answer.

Looking at the z-score table, [tex]z = -0.44[/tex] has a pvalue of 0.333. So what is the value of X when [tex]z = -0.44[/tex].

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]-0.44 = \frac{X - 35}{9}[/tex]

[tex]X - 35 = -3.96[/tex]

[tex]X = 31.04[/tex]

The 33rd percentile of the time it takes Finn to get to work on any given day is 31.04 minutes.

Over the next 2 days, find the probability that Finn took more than 40.5 minutes to get to work on the first day or more than 38.5 minutes to get to work on the second day.

[tex]P = P_{1} + P_{2}[/tex]

[tex]P_{1}[/tex] is the probability that Finn took more than 40.5 minutes to get to work on the first day. The first step to solve this problem is finding the z-value of [tex]X = 40.5[/tex].

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]Z = \frac{40.5 - 35}{9}[/tex]

[tex]Z = 0.61[/tex]

[tex]Z = 0.61[/tex] has a pvalue of 0.7291. This means that the probability that it took LESS than 40.5 minutes for Finn to get to work is 72.91%. The probability that it took more than 40.5 minutes if [tex]P_{1} = 100% - 72.91% = 27.09% = 0.2709[/tex]

[tex]P_{2}[/tex] is the probability that Finn took more than 38.5 minutes to get to work on the second day. Sine the probabilities are independent, we can solve it the same way we did for the first day, we find the z-score of

[tex]X = 38.5[/tex]

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]Z = \frac{38.5 - 35}{9}[/tex]

[tex]Z = 0.39[/tex]

[tex]Z = 0.39[/tex] has a pvalue of 0.6517. This means that the probability that it took LESS than 38.5 minutes for Finn to get to work is 65.17%. The probability that it took more than 38 minutes if [tex]P_{1} = 100% - 65.17% = 34.83% = 0.3483[/tex]

So:

[tex]P = P_{1} + P_{2} = 0.2709 + 0.3483 = 0.6192[/tex]

There is a 61.92% probability that Finn took more than 40.5 minutes to get to work on the first day or more than 38.5 minutes to get to work on the second day.

Answer: 376.98192 mL

Step-by-step explanation:

We are going to use this equation.

V  = π  *  r²  * h

according to the question we have the value for r and h, if you replace the values into the equation we will get  following product:

note: also keep in mind that value of π is 3.141516

V  = π  *  r²  * h

V  = π  *  (4in)²  * (10in)

V = 502.64256 in³

after we can divide this value in 4 equals parts

then  we get the following equation:

502.64256 in³ / 4 = 125.66064 in³

after that that you can multiply this value by 3 to get the 3 parts of the cylinder vase for example:

125.66064 in³ * 3 = 376.98192 in³

and this result is the volume of water that we have to pour into the vase

Answer:

The solution to this set of linear equations is:

[tex]x=-\frac{1}{3}\\y=-\frac{2}{3}[/tex]

Step-by-step explanation:

This is a system of two equations with two unknown variables x and y, let's call them

Equation 1: [tex]9x-3y=-1[/tex]

Equation 2: [tex]\frac{1}{5}x+\frac{2}{5}y=-\frac{1}{3}[/tex]

The first step is to solve Equation 1 for y, this means to leave the y alone on one side of the equal

[tex]y= 3x+\frac{1}{3}[/tex]

Then with this equation, you can find the value of x by replacing y in Equation 2

[tex]\frac{1}{5}x+\frac{2}{5}(3x+\frac{1}{3})=-\frac{1}{3}[/tex]

Then simplify this equation to find x

[tex]\frac{1}{5}x+\frac{6}{5}x+\frac{2}{15}=-\frac{1}{3}[/tex]

[tex]\frac{1}{5}x+\frac{6}{5}x=-\frac{1}{3}-\frac{2}{15}[/tex]

[tex]\frac{7}{5}x=-\frac{5}{15}-\frac{2}{15}[/tex]

[tex]\frac{7}{5}x=-\frac{7}{15}[/tex]

Now you solve for x

[tex]x=-\frac{1}{3}[/tex]

Now you use this value of x to find y

[tex]y=3(-\frac{1}{3})+\frac{1}{3}\\y=-\frac{2}{3}[/tex]

You can check if this answer is correct by replacing the values of x and y into Equation 1 or 2, in this case, let's take Equation 1:

[tex]9(-\frac{1}{3})-3(-\frac{2}{3})=-1\\-3+2=-1\\-1=-1\\[/tex]

Answer:

You can complete your schedule in 13 different ways.

Step-by-step explanation:

You initially have two general options:

1) You  can take Calculus at 9.30 and Physics at 11.30 OR

2) You can take Physics at 9.30 and Calculus at 11.30.

Let's examine each option:

1) If you take Calculus at 9.30 you'd have 3 options (since there are 3 Calculus sections), and then you'd have 3 options at 11.30 to take Physics. This makes 3 x 3 = 9 options.

2) If you choose to take Physics at 9.30, you'd have 2 options and then you'd have 2 Calculus options at 11.30. This makes 2x2 = 4 options.

Since you can take either option one OR two, we will sum up both results, and therefore you have 9 +4 = 13 different ways to complete your schedule.

Answer:

67,68

Step-by-step explanation:

Consecutive numbers are the numbers that follow each other. They may be arranged from the smaller to larger or larger to smaller ones.

Some examples of consecutive numbers are 42,43,45,46,... or 67,68,69,70,...

Now, let x be the smaller gym locker number.

The number consecutive to x will be x+1. Thus the larger gym locker number is x+1.

It is given in the question that the sum of these two locker number is 135.

⇒(x)+(x+1) = 135

⇒ 2x + 1 = 135

⇒ 2x = 134

⇒ x =67

Thus, the two consecutive locker number are x = 67 and x+1 = 68.

Final answer:

The two consecutively numbered gym lockers that add up to 135 are 67 and 68.

Explanation:

The locker numbers are 67 and 68.

To find the locker numbers, we can set up an equation where x represents the smaller locker number. Since the lockers are consecutively numbered, the larger locker number is x+1. The sum of the two locker numbers is x + (x + 1) = 135. By solving this equation, we find that x = 67, making the locker numbers 67 and 68.

Answer:

(a) [tex]\frac{7}{8}[/tex]

(b) [tex]\frac{9}{10}[/tex]

Step-by-step explanation:

Given,

P(A) = 0.5 ⇒ [tex]P(A^c)=1-P(A) = 1 - 0.5 = 0.5[/tex]

P(B) = 0.6 ⇒ [tex]P(B^c)=1-P(B) = 1 - 0.6 = 0.4[/tex]

P(A∩B) = 0.15

∵ [tex]P(A\cap B^c)=P(A) - P(A\cap B) = 0.5 - 0.15 = 0.35[/tex]

Similarly,

[tex]P(B\cap A^c)=P(B) - P(B\cap A) = 0.6 - 0.15 = 0.45[/tex]

Now,

(a) [tex]P(\frac{A}{B^c})=\frac{P(A\cap B^c)}{P(B^c)}=\frac{0.35}{0.4}=\frac{35}{40}=\frac{7}{8}[/tex]

(b) [tex]P(\frac{B}{A^c})=\frac{P(B\cap A^c)}{P(A^c)}=\frac{0.45}{0.5}=\frac{45}{50}=\frac{9}{10}[/tex]

Answer:

amount of pepper required= 7.5 tsp

amount of garlic powder required = 30 tsp

Step-by-step explanation:

Given,

amount of salt used for small batch of the recipe = 2 tsp

amount of pepper used for small batch of the recipe = 1 tsp

amount of garlic powder used for small batch of the recipe = 4 tsp

amount of salt used for the larger batch = 15 tsp

                                                             = 2 x 7.5 tsp

                                                             = amount of salt used for small batch the recipe x 7.5

So,

the amount of pepper needed for the larger batch= 7.5 x amount of pepper used for the small batch of recipe

                                                                      = 7.5 x 1 tsp

                                                                       = 7.5 tsp

the amount of garlic powder needed for the larger batch= 7.5 x amount of garlic powder used for the small batch of recipe

                                                                                          = 7.5 x 4 tsp

                                                                                          = 30 tsp

Final answer:

To adjust the recipe for 15 tsp of salt, you will need 7.5 tsp of pepper and 30 tsp of garlic powder, by applying a scaling factor based on the original recipe proportions.

Explanation:

The question asks how much pepper and garlic powder are needed if a recipe is scaled up to use 15 tsp of salt, from an original recipe that calls for 2 tsp of salt, 1 tsp of pepper, and 4 tsp of garlic powder. To solve this, we first determine the scaling factor for the recipe by dividing the new quantity of salt by the original quantity of salt, which is 15 tsp ÷ 2 tsp = 7.5. Next, we apply this scaling factor to the measurements for pepper and garlic powder.

Pepper needed = 1 tsp (original amount) x 7.5 (scaling factor) = 7.5 tsp of pepper.

Garlic Powder needed = 4 tsp (original amount) x 7.5 (scaling factor) = 30 tsp of garlic powder.

Answer:

The angles between the curves at the points of intersection are:

0º, 1.3º

Step-by-step explanation:

The intersections points are found by setting the equations equal to each other and solving the resulting equation:

[tex]7x^2=7x^3\\x^3-x^2=0\\x^2(x-1)=0\\x=0,x=1[/tex]

The angles of the tangent lines can be found by stating their slopes.

To find the slope we differentiate the equations:

[tex]y'_1=14x,y'_2=21x^2[/tex]

Then we plug the x-coordinates of the intersections:

For x=0 we get the slopes are both 0:

[tex]y'_1=14(0)=0,y'_2=21(0)^2=0[/tex]

So the angles of inclination of the lines are the same their difference is 0. Hence the angle  between the tangent curves is also 0º at the point of intersection at x=0

For x=1 we get the following slopes:

[tex]y'_1=14(1)=14,y'_2=21(1)^2=21[/tex]

The slopes are the tangents of the angles. Therefore, to get the angle between the lines we do:

[tex]arctan(21)-arctan(14)\approx87.2737\º-85.9144\º\approx1.3\º[/tex]

So, 1.3º is the angle between the curves at the second point of intersection at x=1.

Answer:

178 meters in 2 seconds = 890 meters in 10 seconds.

Step-by-step explanation:

Given : 178 meters in 2 seconds.

To find : How many meters in 10 seconds ?

Solution :

Applying unitary method,

In 2 seconds there is 178 meter.

In 1 second there is [tex]\frac{178}{2}[/tex] meter.

In 1 second there is 89 meter.

In 10 seconds there is [tex]10\times 89[/tex] meter.

In 10 seconds there is 890 meter.

Therefore, 178 meters in 2 seconds = 890 meters in 10 seconds.