More than 450 students went on a field trip. Ten buses were filled and 5 more students traveled in a car. How many students were on each bus?

Answer:

Part a) [tex]6x+4y=48[/tex]

Part b) The graph in the attached figure

Part c) (6,3) and (4,6)

Step-by-step explanation:

Part a) Write a linear equation to represent the number of items you can buy if she spends $48

Let

x ----> number of package of cupcakes you can buy

y ---> number of package of cookies you can buy

we know that

The number of package of cupcakes you can buy multiplied by it cost ($6 for a package) plus the number of package of cookies you can buy multiplied by it cost ($4 for a package) must be equal to $48

so

The linear equation that represent this problem is

[tex]6x+4y=48[/tex]

Part b) Graph the equation

To graph the line we need two points

Find the intercepts

Find the x-intercept (value of x when the value of y is equal to zero)

For y=0

[tex]6x+4(0)=48[/tex] ----> [tex]x=8[/tex]

the x-intercept is the point (8,0)

Find the y-intercept (value of y when the value of x is equal to zero)

For x=0

[tex]6(0)+4y=48[/tex] ----> [tex]y=12[/tex]

the y-intercept is the point (0,12)

Plot the intercepts and join the points to graph the line

see the attached figure

Part c) State two possible solutions in the context of the problem

1) First possible solution

the ordered pair (6,3)

That means

You can buy 6 package of cupcakes and 3 package of cookies

Verify in the linear equation

[tex]6(6)+4(3)=48[/tex]

[tex]48=48[/tex] ---> is true

therefore

The ordered pair is a solution of the linear equation

2) Second possible solution

the ordered pair (4,6)

That means

You can buy 4 package of cupcakes and 6 package of cookies

Verify in the linear equation

[tex]6(4)+4(6)=48[/tex]

[tex]48=48[/tex] ---> is true

therefore

The ordered pair is a solution of the linear equation

Answer:

convert 17 meters into feet and then write it down to teh nearest 10thof a foot

Step-by-step explanation:

Answer:

a = 6 units

[tex]c= 6\sqrt{2}\ units[/tex]

Step-by-step explanation:

Given:

Let Labelled the diagram first

Δ ABC right angle at ∠ C = 90°

∠ B = 45 °

AB = c

BC = a

AC = 6

To Find:

a =?

c =?

Solution:

In Δ ABC

∠ A + ∠ B + ∠ C = 180°.....{Angle Sum Property of a Triangle}

∴ ∠ A + 45 + 90 = 180°

∴ ∠ A = 180 - 135

∴ ∠ A = 45°

Now ∠ A = ∠ B = 45°  in Δ ABC

∴ Δ ABC is an Isosceles Triangle.

∴ Two sides are equal of an  Isosceles Triangle.

∴ AC = BC = a = 6 units

Now for c we use Pythagoras theorem

[tex](\textrm{Hypotenuse})^{2} = (\textrm{Shorter leg})^{2}+(\textrm{Longer leg})^{2}[/tex]

Substituting the given values we get

c² = a² + 6²

c² = 6² + 6²

c² = 36 + 36

c² = 72

∴ c = ±√72

as c cannot be negative

∴ [tex]c = 6\sqrt{2}\ units\\[/tex]

a = 6 units

[tex]c= 6\sqrt{2}\ units[/tex]

Step-by-step explanation:

denominator separately

[tex]277 \sqrt{3} [/tex]

the below 160 Number 2) Same thing with the first one Multiply the Fraquetions, then multiply the numerator and denominator separately so

[tex] \frac{590}{160} \sqrt{17} [/tex]

then change to

[tex]590 \sqrt{17} [/tex]

the below that is a 5 and above it on the top right corner out 160

Answer:

Step-by-step explanation:

The product is 0 if one of the factors is 0.

[tex]x(x - 2)(x - 1) = 0\iff x=0\ \vee\ x-2=0\ \vee\ x-1=0\\\\(1)\ x=0\\\\(2)\ x-2=0\qquad\text{add 2 to both sides}\\.\qquad x=2\\\\(3)\ x-1=0\qquad\text{add 1 to both sides}\\.\qquad x=1[/tex]

Answer:

.8

Step-by-step explanation:

because 24/30

The probability that a randomly selected house with 2 bathrooms

has 3.

We have given that,

The two-way table shows the number of houses on the market in the Castillos’ price range.

A 6-column table has 4 rows.

The first column has entries 1 bathroom, 2 bathrooms and 3 bathrooms, in total.

The second column is labeled 1 bedroom with entries 67, 0, 0, 67.

The third column is labeled 2 bedrooms with entries 21, 6, 18, 45.

We are only considering the houses that have 2 bathrooms.

 There are a total of 30 of these houses.

Out of these 30, 24 have 3 bedrooms.  

The probability of an event can only be between 0 and 1 and can also be written as a percentage.

This makes the probability 24/30, which is 0.8.

Therefore, the probability that a randomly selected house with 2 bathrooms has 3.

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Answer:

The answer is B

Answer:

The answer is 24/2=36/3=66/5.5=108/9=180/15.

Step-by-step explanation:

Given:

24/2=_/3=_/5.5=108/_=_/15.

Now, we need to solve this by putting [tex]x[/tex] in the place of _ and then continue:

[tex]\frac{24}{2} =\frac{x}{3}[/tex]

By cross multiplication we get:

[tex]72=2x[/tex]

By dividing with 2 we get:

[tex]36=x[/tex]

Now, we will continue like this process:

[tex]\frac{36}{3}=\frac{x}{5.5}[/tex]

[tex]198=3x[/tex]

[tex]x=66[/tex]

And, then again:

[tex]\frac{66}{5.5}=\frac{108}{x}[/tex]

[tex]66x=594[/tex]

[tex]x=9[/tex]

And, last:

[tex]\frac{108}{9}=\frac{x}{15}[/tex]

[tex]9x=1620[/tex]

[tex]x=180[/tex]

Therefore, the answer is 24/2=36/3=66/5.5=108/9=180/15.

Answer:

Sample size n = 96.04

Explanation:

Given the interval is 5.6 hours to 6.4 hours

Let x be the midpoint, then x = (5.6+6.4)/2 = 6

Let E be the radius, then E = (6.4-5.6)/2 = 0.8/2 = 0.4

σ = 2 hours; as standard deviation is given and α = 0.05

Therefore, the sample size is:

[tex]n=\left(\frac{1.96 * 2}{0.4}\right)^{2}=96.04[/tex]

The student's question pertains to a 95% confidence interval estimate of the mean time required for caffeine to leave the body after consumption, which was found to be 5.6 to 6.4 hours in adults. This suggests that if the study were conducted multiple times, the mean time for caffeine to leave the body would lie within this range in 95% of those studies. The influence of variances in individual's metabolism, body mass, and general health can impact the range of the confidence interval.

The subject of this discussion revolves around statistics and more specifically around the concept of confidence intervals. In the provided context, researchers conducted a study to estimate the mean time it takes for an adult body to process and remove caffeine. They found, with a 95% confidence interval, that this time lies between 5.6 hours and 6.4 hours.

This means that if the researchers were to conduct this study multiple times, in 95% of those studies, the mean (average) time for caffeine to leave the body would lie within that given range (5.6 to 6.4 hours). This is a common method in statistics and is used as a way to give an estimate about where the actual (population) value might lie in the context of sampling error.

However, it's important to note that variance in the sample population can impact the confidence interval. Individual differences such as metabolism rate, body mass, and overall health can influence how quickly a person processes caffeine, which may result in a wider confidence interval.

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Answer:

The probability of picking the first boyis 14/26 and the probability of picking another boy is 13/25. The combined probability is thus 14/26 x 13/25 = 7/25

Step-by-step explanation:

The probability the teacher picks 2 boys in a row is 7/13.

Probability can be defined as the ratio of the number of favourable outcomes to the total number of outcomes of an event.

We know that, probability of an event = Number of favorable outcomes/Total number of outcomes.

Given that, there are 12 girls and 14 boys in math class.

Here, total number of outcomes = 12+14

= 26

Number of favorable outcomes = 14

Now, probability = 14/26

= 7/13

Therefore, the probability the teacher picks 2 boys in a row is 7/13.

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Answer:

$52080

Step-by-step explanation:

46, 500 + (.12×46,500) =52080

Answer:

0.6808

Step-by-step explanation:

First, find the standard deviation of the sample.

s = σ / √n

s = 3 / √49

s = 0.429

Next, find the z-score.

z = (x − μ) / s

z = (28.8 − 29) / 0.429

z = -0.467

Use a calculator or z-score table to find the probability.

Using a table:

P(x > -0.47) = 1 − 0.3192 = 0.6808

Using a calculator:

P(x > -0.467) = 0.6796

The probability that the average mileage of the fleet is greater than 28.8 mpg is approximately 0.680.

Step 1

In order to determine the likelihood that a fleet of 49 automobiles will get more than 28.8 mpg on average, we must apply the Central Limit Theorem, which states that the sample mean's sampling distribution will be roughly normally distributed.

Given:

- Mean [tex](\(\mu\))[/tex]= 29 mpg

- Standard deviation [tex](\(\sigma\))[/tex] = 3 mpg

- Sample size [tex](\(n\))[/tex] = 49 cars

- Sample mean [tex](\(\bar{x}\))[/tex]= 28.8 mpg

Step 2

First, we find the standard error of the mean (SEM):

[tex]\[\text{SEM} = \frac{\sigma}{\sqrt{n}} = \frac{3}{\sqrt{49}} = \frac{3}{7} \approx 0.4286\][/tex]

Next, we convert the sample mean to a z-score to find the probability:

[tex]\[z = \frac{\bar{x} - \mu}{\text{SEM}} = \frac{28.8 - 29}{0.4286} \approx \frac{-0.2}{0.4286} \approx -0.466\][/tex]

We can use a calculator or the conventional normal distribution table to find the probability using the z-score.

With a z-score of -0.466, one may calculate the cumulative probability to be roughly 0.3204. This is the likelihood that the mileage will be on average less than 28.8 mpg.

We deduct this figure from 1 to get the likelihood that the average mileage is higher than 28.8 mpg:

Step 3

[tex]\[P(\bar{x} > 28.8) = 1 - P(\bar{x} < 28.8) = 1 - 0.3204 = 0.6796\][/tex]

Therefore, the probability that the average mileage of the fleet is greater than 28.8 mpg is approximately 0.680 (rounded to three decimal places).

Answer:

[tex]\large\boxed{s(x)\cdot t(x)=2x^3+11x^2+8x-16}[/tex]

Step-by-step explanation:

[tex]s(x)=2x^2+3x-4,\ t(x)=x+4\\\\s(x)\cdot t(x)\\\\\text{use the distributive property:}\ a(b+c)=ab+ac\\\\s(x)\cdot t(x)=(2x^2+3x-4)(x+4)\\\\=(2x^2+3x-4)(x)+(2x^2+3x-4)(4)\\\\=(2x^2)(x)+(3x)(x)+(-4)(x)+(2x^2)(4)+(3x)(4)+(-4)(4)\\\\=2x^3+3x^2-4x+8x^2+12x-16\\\\\text{combine like terms}\\\\=2x^3+(3x^2+8x^2)+(-4x+12x)-16\\\\=2x^3+11x^2+8x-16[/tex]

Final answer:

The product of the functions [tex]s(x) = 2x^2 + 3x - 4[/tex] and t(x) = x + 4 is obtained by multiplying each term of s(x) with each term of t(x), resulting in [tex]2x^3 + 11x^2 + 8x - 16[/tex].

Explanation:

To find the product of s(x) and t(x), given  [tex]s(x) = 2x^2 + 3x - 4[/tex], and t(x) = x + 4, we use the distributive property to multiply each term in s(x) by each term in t(x). The steps are as follows:

Adding up all these products:

[tex]2x^3 + (8x^2 + 3x^2) + (12x - 4x) - 16[/tex]

Simplifying:

[tex]2x^3 + 11x^2 + 8x - 16[/tex]

Thus, the product s(x) · t(x) is  [tex]2x^3 + 11x^2 + 8x - 16[/tex]

Answer: 3.sas

4.sas

5.sas

6.asa

7.sas

8. not possible

9. hl

Step-by-step explanation:

Answer:

⇒[tex](Base)^2 + ( Perpendicular)^2 = (Hypotenuse)^2[/tex]  is the required relationship.

Step-by-step explanation:

Let us assume, the given right angled triangle is ΔPQR.

Here. PQ = Perpendicular of the triangle.

QR = Base of the triangle.

PR = Hypotenuse of the triangle.

Now, PYTHAGORAS THEOREM states:

In a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides“

⇒[tex](Base)^2 + ( Perpendicular)^2 = (Hypotenuse)^2[/tex]

Hence in ΔPQR:  [tex](QR)^2 + ( PQ)^2 = (PR)^2[/tex]

And the above expression is the required relationship between the sides of a right triangle.

The sides of a right triangle have a specific relationship given by the Pythagorean Theorem, which states a² + b² = c², where a and b refer to the lengths of the sides and c refers to the length of the hypotenuse. Additionally, the sides have relationships to the angles of the triangle expressed through trigonometric functions.

The relationship between the sides of a right triangle is given by the Pythagorean Theorem which was demonstrated by the ancient Greek philosopher, Pythagoras. This theorem states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. In mathematical terms, if the lengths of the sides are a, b, and c (where c represents the length of the hypotenuse), then the relationship can be represented as a² + b² = c².

Moreover, the sides of a right triangle also have specific relationships to the measures of the angles of the triangle, which are expressed through the trigonometric functions sine, cosine, and tangent. For example, for an angle in a right triangle, the sine is the length of the opposite side divided by the length of the hypotenuse, the cosine is the length of the adjacent side divided by the hypotenuse, and the tangent is the length of the opposite side divided by the adjacent side.

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Answer:

21, 24, 29

5

Step-by-step explanation:

The first term is when n = 1.

(1)² + 20 = 21

The second term is when n = 2.

(2)² + 20 = 24

The third term is when n = 3.

(3)² + 20 = 29

To find how many terms are less than 50, set n² + 20 equal to 50 and solve for n:

n² + 20 = 50

n² = 30

n ≈ 5.477

Rounding down, n = 5 is the last term that is less than 50.

Therefore, there are 5 terms in the sequence that are less than 50.

Answer:

Thomas still has 1 3/5 suitcases available for his own belongings.

Step-by-step explanation:

1. Let's review the data given to us for solving the question:

Number of souvenirs bought by Thomas = 6

Space that each souvenir takes of Thomas suitcase = 1/15

Number of Thomas suitcases = 2

2.  How much room is left for his own belongings in his suitcases?

Let's find out how much space the souvenirs take:

Number of souvenirs * Space that each souvenir takes

6 * 1/15 = 6/15 = 2/5 (Dividing by 3 the numerator and the denominator)

The souvenirs take 2/5 of one suitcase.

Now, we can calculate the room that is left for Thomas' belongings.

2 Suitcases - 2/5 for the souvenirs

2 - 2/5 = 10/5 - 2/5 = 8/5 = 1 3/5

Thomas still has 1 3/5 suitcases available for his own belongings.

Answer:

1 9/15

Step-by-step explanation:

Answer:

$60

Step-by-step explanation:

75% divided by 100 x 80 = 60

              OR

$80 divided by 100 x 75 = 60

Answer:

The standard error of distribution for n = 4 is 5 and for n = 25 is 2.

Step-by-step explanation:

1. Let's review the information given to us for solving the question:

Population mean = 72

Standard deviation  =  10

Sample₁ = 4

Sample₂ = 25

2. For finding the standard error of the mean, we use the following formula:

Standard error = Standard deviation / √Size of the sample

Standard error for Sample₁ = 10/√4

Standard error for Sample₁ = 10/2 = 5

Now, let's find the standard error for Sample₂

Standard error for Sample₂ = 10/√25

Standard error for Sample₂ = 10/5 = 2

Answer:

step 2 i believe

Step-by-step explanation:

hope this helps

Answer:

The answer is B

Step-by-step explanation:

Hope this helps :))

Answer:

3

Step-by-step explanation:

Given: [tex]$  (\frac{1}{4} + \frac{5}{4} )^{2} + \frac{3}{4} $[/tex]

This is equivalent to: [tex]$ ( \frac{6}{4}) ^2 + \frac{3}{4} $[/tex]

⇒  [tex]$ (\frac{3}{2})^2 + \frac{3}{4} $[/tex]

⇒ [tex]$ \frac{9}{4} + \frac{3}{4} $[/tex]

⇒ [tex]$\frac{12}{4} = 3$[/tex]

Therefore,  [tex]$(\frac{1}{4} + \frac{5}{4} )^{2} + \frac{3}{4} = 3 $[/tex]

Answer:

(1/4 + 5/4)^2 +3/4 = 12

Step-by-step explanation:

To solve the problem given, we will follow the steps below;

(1/4 + 5/4)^2 +3/4

First, we will find the value of  (1/4 + 5/4)^2

1/4 + 5/4 = 6/4

(1/4 + 5/4)^2   =   (6/4)^2 =  [tex]\frac{36}{16}[/tex]

 [tex]\frac{36}{16}[/tex]  can be reduced to   [tex]\frac{9}{4}[/tex]

This implies that ; (1/4 + 5/4)^2    =   [tex]\frac{9}{4}[/tex]

Then, we can now add  [tex]\frac{9}{4}[/tex]   and [tex]\frac{3}{4}[/tex]  together  

(1/4 + 5/4)^2 +3/4    =   [tex]\frac{9}{4}[/tex]   +  [tex]\frac{3}{4}[/tex]   =  [tex]\frac{12}{3}[/tex]   = 4

Therefore    (1/4 + 5/4)^2 +3/4 = 12

Answer:

The length of the side of the square is 6 feet.

Step-by-step explanation:

Given,

Length of side of square = [tex](24 - 3x)\ feet[/tex]

According to question, x = 6

So we have to substitute x with 6 in the given expression.

Length of side of square = [tex](24-3x)= 24-3\times6=24-18=6\ feet[/tex].

Thus the length of the side of the square is 6 feet.

Final answer:

To find the length of the side of the square when x is 6, substitute 6 for x into the given expression (24 - 3x). Thus, the length of the side is calculated as 6 feet.

Explanation:

The student asked for the length of the side of the square when x = 6. To find this, we substitute x with 6 into the expression representing the side length of the square, which is (24 - 3x) feet.

Replacing x with 6, we get:

The length of the side of the square when x is equal to 6 is therefore 6 feet.

Answer:

The temperature at noon is 6ºF.

Step-by-step explanation:

3:00am: -10ºF

               -12ºF

6:00am: -22ºF

              +8ºF

9:00am: -14ºF

              +20ºF

Noon:     6ºF

So at 6:00 a.m. the temperature is 33 F

12:00 p.m. the temperature increased by 10 F so it is 43 F

3:00 p.m. the temperature increased by another 12 F making it 55 F

At 10:00 p.m. it would decrease by 15 F making it 40 F.

The temperature would need to fall/decrease 7 F to reach the original temperature of 33 F so it would be A.

Hope this helps!

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D. g(x) = f(x) +9 is the right answer

Step-by-step explanation:

The shifting of graphs is mathematically denoted by adding or subtracting a number from the output or input.

When the graph is shifted up, a number is added to the functions's output

For example

For a function

f(x)

Shifting up is represented as g(x) = f(x)+b where b is a number

So,

Given function is:

f(x) = x

Shifting the graph 9 units upwards we will get

g(x) = f(x) + 9

Hence,

D. g(x) = f(x) +9 is the right answer

Keywords: Functions, Graphs

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Answer: D. g(x) = f(x) +9 is the right answer

Step-by-step explanation:

Answer:

Subtract the second equation from the first one.

The solution is (1.71, 0.93).

Step-by-step explanation:

2x + 3y = 7

2x = 4y - 5

2x + 3y - 2x = 7 - (4y - 5)   So we eliminate the x term:

3y =  -4y + 12

7y = 12

y = 12/7 = 1.714

Plug this into the first equation:

2x + 3(1.714) = 7

2x = 1.858

x = 0.929.

Answer:

Step-by-step explanation:

The total amount of liquid in the three beakers is 0.6 liters.

Calculating Total Volume of Liquid

To determine the total amount of liquid in 3 beakers, we need to multiply the volume of liquid in each beaker by the number of beakers.

So, the total amount of liquid in the three beakers is 0.6 liters.

Answer:

f(x) = (8/3)^x

Step-by-step explanation:

Since f increasing, the base value must be greater than 1.

8/3 is the only base value greater than 1.

The base (3/8) would be a decreasing graph because it is less than 1.

The base (-3/8) would result in a wavering graphing passing the x-axis many times. (Because whether the result is negative depends on if x is odd or even.)

in (8/3)^(-x), it is the same as (3/8)^x by applying the negative exponent rule a^(-x) = 1/(a^x).

Answer:

Choice C)

x^2 - 2x + 4 - 12/(x+2)

[tex]x^2 - 2x + 4 - \frac{12}{x+2}[/tex]

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

Explanation:

To see how I got that answer, I have provided two attached images below. One of them shows the polynomial long division. The other shows synthetic division. Both are valid options to get to the same answer.

For each method, I used 1x^3 + 0x^2 + 0x - 4 in place of x^3 - 4 so that the proper terms could align.

Answer:

[tex]x=0,x=\pi,x=\frac{\pi}{3},x=\frac{2\pi}{3},x=\frac{4\pi}{3},x=\frac{5\pi}{3}[/tex]

Step-by-step explanation:

This is a trigonometric equation where we need to use the cosine of the double-angle formula

[tex]cos4x=2cos^22x-1[/tex]

Replacing in the equation

[tex]cos4x - cos2x = 0[/tex]

We have

[tex]2cos^22x-1 - cos 2x = 0[/tex]

Rearranging

[tex]2cos^22x - cos 2x-1 = 0[/tex]

A second-degree equation in cos2x. The solutions are:

[tex]cos2x=1,cos2x=-\frac{1}{2}[/tex]

For the first solution

cos2x=1 we find two solutions (so x belongs to [0,2\pi))

[tex]2x=0, 2x=2\pi[/tex]

Which give us

[tex]x=0,x=\pi[/tex]

For the second solution

[tex]cos2x=-\frac{1}{2}[/tex]

We find four more solutions

[tex]2x=\frac{2\pi}{3},2x=\frac{4\pi}{3},2x=\frac{8\pi}{3},2x=\frac{10\pi}{3}[/tex]

Which give us

[tex]x=\frac{\pi}{3},x=\frac{2\pi}{3},x=\frac{4\pi}{3},x=\frac{5\pi}{3}[/tex]

All the solutions lie in the interval [tex][0,2\pi)[/tex]

Summarizing. The six solutions are

[tex]x=0,x=\pi,x=\frac{\pi}{3},x=\frac{2\pi}{3},x=\frac{4\pi}{3},x=\frac{5\pi}{3}[/tex]

16/36=4/9

(16)(9)=(36)(4) cross multiply

144=144 proportional

After cross-multiplying the ratios 16/36 and 4/9 and finding that both products are equal (144), it is determined that these ratios are proportional.

The question asks to classify the pair of ratios 16/36 and 4/9 as either proportional or not proportional. We can determine if two ratios are proportional by cross-multiplying and checking if the products are equal. In this case, for the ratios 16/36 and 4/9, we cross-multiply:

Since the products are equal (144 = 144), this confirms that the ratios 16/36 and 4/9 are indeed proportional.