No yes no @ 90, 9, 0,-90,-9 @ 25, 11, -8, -7, -15 @ 4, 2, 0, -2, -4, -42 MIDDLE SCHOOL MATH WITH PIZZAZZ! BOOK E Em56 O Creative Publications 4-R - 34-M

Answer:

Volume = 1346.275 [tex]ft^{3}[/tex]

Step-by-step explanation:

The radius of the pipe is 3.5 ft and height is 35 ft.

The volume of cylinder is found by the formula,

V = (π)([tex]R^{2})(h)[/tex]

(think about the formula as area of individual circles multiplied by the height of cylinder)

taking value of π as 3.14,

Where, the r = radius and h is height of cylinder.

inserting the above values,

[tex]V = (3.14)(3.5^{2})(35)[/tex]

V = 1346.275 [tex]ft^{3}[/tex]

Answer:32years old

Step-by-step explanation:

R= 2c+5

R+C= 101

2c+5+c=101

3c+5=101

3c=96

c=32

Answer:

32 years

Step-by-step explanation:

Let the age of Carlos be represented by x

Then the age of Raul can be represented as the sum of twice the age of Carlos and 5.

In other words, Raul's age = 2x + 5

Sum of Carlos and Raul's age is 101

That is, \[x + 2x + 5 = 101\]

Or, \[3x + 5 = 101\]

Or, \[3x = 96\]

Or, \[x = 32\]

Hence, the age of Carlos is 32 years.

Age of Raul on the other hand is 2*32 + 5 = 69 years

Sum of their ages is 32 + 69 = 101 years.

Answer: length = 7 feet's

Width = 6 feets

Step-by-step explanation:

Perimeter of a rectangle is the distance round the rectangle.

Perimeter of rectangle is expressed as

2(length + width)

The perimeter of the given rectangle is 26 feet. Therefore

2(L + W) = 26

Dividing by 2,

L+W = 13

The area of a rectangle is expressed as length × width

The given area is 42 square feet. Therefore,

L×W = 42

Substituting L = 13 - W into LW = 42, it becomes

W(13-W) = 42

13W - W^2 = 42

W^2 -13W + 42 = 0

W^2 -7W - 6W + 42 = 0

W(W - 7) - 6(W - 7) = 0

(W-6)(w-7)=0

W-6 = 0 or W-7=0

W=6 or W = 7

L= 13-6 or L= 13-7

L= 7 or L = 6

Final answer:

Using the formulas for the perimeter and the area of a rectangle, we set up two equations, solved them simultaneously, factored a quadratic equation, and found that the dimensions of the rectangle are 7 feet in length and 6 feet in width.

Explanation:

To find the dimensions of a rectangle given the perimeter and the area, we can set up two equations based on these two pieces of information. The perimeter (P) of a rectangle is given by the formula P = 2l + 2w, where l is the length and w is the width. The area (A) of a rectangle is given by the formula A = lw. For the problem given, we have:

Let's denote the length as l and the width as w. We can now write:

Solving these two equations simultaneously will give us the length and width of the rectangle. Firstly, we can rearrange the perimeter equation to express one variable in terms of the other:

l = (26 - 2w) / 2

Substitute this expression for l into the area equation:

((26 - 2w) / 2)w = 42

Multiplying both sides of the equation by 2 to clear the fraction:

(26 - 2w)w = 84

Distribute w across:

26w - 2w² = 84

Rearrange the quadratic equation:

2w² - 26w + 84 = 0

Divide by 2 to simplify:

w² - 13w + 42 = 0

Factor the quadratic equation:

(w - 6)(w - 7) = 0

Therefore, w could be either 6 or 7. Since the width must be less than the length for a rectangle (based on the problem context where length is typically the longer dimension), we can deduce that:

Width (w) = 6 feet

Then substitute the width into the perimeter equation to find the length:

2l + 2(6) = 26

2l + 12 = 26

2l = 14

l = 14 / 2

Length (l) = 7 feet

Hence, the dimensions of the rectangle are 7 feet by 6 feet.

Answer:

x = 6

∠B = 126

Step-by-step explanation:

∠A = ∠B through alternate interior angles

∠A = ∠B

8x + 78 = 2x + 114

8x - 2x = 114 - 78

6x = 36

x = 36 ÷ 6

x = 6

∠B = 2x + 114

2(6) + 114

12 + 114

= 126

d1 || d2 => ∡A = ∡B

8x + 78° = 2x + 114°

8x - 2x = 114° - 78°

6x = 36°

x = 36° : 6

x = 6°

∡B = 2x + 114°

∡B = 2×6° + 114°

∡B = 12° + 114°

∡B = 126°

Answer: The value of the card after you buy your 8th coffee will be $61.3

Step-by-step explanation:

The worth of the gift card for the coffee shop is $90. Each day you use the card to get a coffee for $4.10. This means that the worth of the gift card is reducing by $4.10 each day. This rate is in arithmetic progression.

The formula for the nth term of an arithmetic sequence, Tn is expressed as

Tn = a + (n-1)d

Where a is the first term

d is the common difference

n is the number of days

From the information given,

a = $90

d = - $4.1

The explicit formula representing the amount of money available will be

Tn = 90 - 4.1(n - 1)

The value of the card after you buy your 8th coffee will be

T8 = 90 - 4.1(8 - 1) = T8 = 90 - 4.1×7

T8 = 90 - 28.7

T8 = $61.3

Only hole of function [tex]f(x) = \frac{x^{2}+7x+10 }{x^{2}+9x+20 }[/tex] is at x=(-4)

Step-by-step explanation:

Given the function is [tex]f(x) = \frac{x^{2}+7x+10 }{x^{2}+9x+20 }[/tex]

In order to find holes of any function, you should find when function is becoming undefined or say " infinity"

Given function is polynomial function.

It will become undefined become denominator become zero

[tex]x^{2}+9x+20=0[/tex]

Solving for x value when denominator become zero

[tex]x^{2}+9x+20=0\\x^{2}+5x+4x+20=0\\x(x+5)+4(x+5)=0\\(x+4)(x+5)=0[/tex]

we get possible holes at x=(-4) and x=(-5)

Check whether you can eliminate any holes

Now, Solving for x value when numerator become zero

[tex]x^{2}+7x+10=0\\x^{2}+5x+2x+10=0\\(x+5)(x+2)=0[/tex]

x=(-5) and x=(-2)

x=(-5) is common is both numerator and denominator.

So that, we can eliminate it.

[tex]f(x) = \frac{(x+5)(x+2)}{(x+5)(x+4)}[/tex]

[tex]f(x) = \frac{(x+2)}{(x+4)}[/tex]

Therefore, Only hole of function [tex]f(x) = \frac{x^{2}+7x+10 }{x^{2}+9x+20 }[/tex] is at x=(-4)

Answer:

Answer is  12-19 = -7

Step-by-step explanation:

12- 19 you can break 19  into 12 +7

then

it is easy to find  12-12 = 0

Now subtracts  0 - 7 =- 7

then

break 7 into 3 + 4

then

0 - 3 = -3

-3 - 4 = -7

So ,

12 - 19 = -7

Answer:

The answer to your question is 57.3 kg

Step-by-step explanation:

a)

   [tex]X stones \frac{14 pounds}{1 stone} \frac{1 kilogram}{2.2 pounds}[/tex]

   = [tex]\frac{14}{2.2}[/tex]

  = 6.36 X stones

b)    Weight = 6.36 (x)

x = 9 stones

      Weight = 6.36 (9)                

                    = 57.3 kg

Answer:

  2/π ≈ 0.637 m/s

Step-by-step explanation:

The rate of change of area with respect to time is ...

  A = πr²

  dA/dt = 2πr·dr/dt

Filling in given values in the above equations, we can find r and dr/dt.

  25π = πr²   ⇒   r = 5

  20 = 2π·5·dr/dt

  dr/dt = 20/(10π) = 2/π . . . . meters per second

The radius is increasing at the rate of 2/π ≈ 0.637 meters per second.

For a sample size of n=12, the degrees of freedom (df) would be 11, and the critical t value for a two-tailed 95% confidence interval is typically around 2.201, as determined by a t-distribution table or statistical software.

To find the critical t* value corresponding to a 95% confidence level for a sample size of n=12, you must take into account that the degrees of freedom (df) are equal to n - 1, which is 11 in this case. You use a t-distribution, not the z-distribution, because the population standard deviation is unknown. The critical t* value can be found using a t-distribution table or statistical software.

With the degrees of freedom at 11 for a 95% confidence interval (two-sided), the critical t value is typically around 2.201. However, one must look up the exact value using statistical software or a t-distribution table, as the value may vary slightly based on different sources or rounding.

Thus, to answer the student's question: For a two-tailed 95 percent confidence interval and 11 degrees of freedom, the corresponding critical value t* will typically be around 2.201.

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Answer:

270 more pounds of aluminium cans collected by Jessica than Min

Step-by-step explanation:

334 + 124 = 458 (Total pounds of aluminium cans collected by min)

414 + 314 = 728 (Total pounds of aluminium cans collected by Jessica)

728 - 458 = 270 (More pounds Jessica collected than Min)

Final answer:

Jessica collected 270 pounds more of aluminum cans than Min did over the two days. Min collected 458 pounds in total, and Jessica collected 728 pounds in total.

Explanation:

The student's question is asking us to compare the total amount of aluminum cans collected by two individuals, Min and Jessica, over a two-day period and determine how much more Jessica collected compared to Min. To solve this, we need to add the amounts collected by each individual for both days and then subtract Min's total from Jessica's total.

First, we find Min's total collection by adding her collections from Monday and Tuesday: 334 pounds + 124 pounds = 458 pounds.

Next, we find Jessica's total collection by adding her collections from Monday and Tuesday: 414 pounds + 314 pounds = 728 pounds.

Finally, we calculate how much more Jessica collected than Min by subtracting Min's total from Jessica's total: 728 pounds - 458 pounds = 270 pounds. Since the question asks for the answer as a mixed number in simplest form and 270 is already a whole number, our answer is simply 270 pounds. There is no need to convert to a mixed number.

Answer:

You can use A and B systems

Step-by-step explanation:

Lets call z the total speed of the boat. If the boat goes against the current, then, the current will drop the boat natural speed, and therefore z is obtained from substracting y from the rate of the boat on still water, x. Thus, z = x-y. If The boat goes in favor of the current, then the current will raise the speed, and we obtain z by adding y to x. If we want to calculate x and y, we know that:

On the first trip, z = x-y, and it took 6 hours to finish the 120 mile trip, therefore 6z = 120, or equivalently, 6(x-y)=120

On the return trip, z = x+y, and it took 5 hours to finish the trip, so we have 5z = 5(x+y) = 120.

Thus, in order so solve the problem, we can use system A and B.

Note that system A is equivalent to the equation x-y = 20, obtained by dididing everything by 6. If we divide by 5 the second equation we obtain that x+y = 24. We have

By summing this equations it follows that 2x = 44, therefore x = 22. Since x-y = 20, we obtain that y = 2.

Answer:

The correct answer is 6(x - y) = 120 and 5(x + y) = 120

Answer:

  C)  $2,500

Step-by-step explanation:

Madison's profit is 25% of the $2000 paid, so is ...

  0.25 × $2000 = $500

That means the lot was sold for ...

  $2000 + 500 = $2500

_____

Profit can also be expressed as a percentage of the selling price. If that were the case here, Madison's initial cost would be 0.75 of the $2666.67 selling price, and the 25% profit would be $666.67. Since that is not among the answer choices, we presume our assumption is correct that Madison's profit is measured as a percentage of cost.

Answer:

c

Step-by-step explanation:

your welcome

Answer:

A. No, the student is not right. The central limit theorem says nothing about the histogram of the sample values. It deals only with the distribution of the sample means.

Step-by-step explanation:

No, the student is not right. The central limit theorem says nothing about the histogram of the sample values. It deals only with the distribution of the sample means. The central limit theorem says that if we take a large sample (i.e., a sample of size n > 30) of any distribution with finite mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], then, the sample average is approximately normally distributed  with mean [tex]\mu[/tex] and variance [tex]\sigma^2/n[/tex].

Yes. The student is exactly what the theorem says.

True. According to the central limit theorem, as the sample size increases, the sampling distribution of the sample means becomes more Normal.

An example illustrates how from a population with a uniform distribution, as samples are drawn and the means are calculated, the distribution of these means approximates a normal distribution as the sample size increases.

The central limit theorem ensures that regardless of the population's distribution, with sufficiently large samples, the distribution of sample means tends towards a normal distribution.

Answer:

[tex]\frac{dx}{dt}=\frac{-8}{3}\frac{ft}{s}[/tex]

Step-by-step explanation:

Be [tex]\frac{dy}{dt}=2\frac{ft}{s}[/tex] to find [tex]\frac{dx}{dt}=?, x=6ft[/tex]

[tex]x^{2} +y^{2}=10^{2};6^{2}+y^{2}=100;y^{2}=100-36=64;y=\sqrt{64}=+-8[/tex]→ y=8 for being the positive distance, deriving from t, [tex]x^{2} +y^{2}=100[/tex]→[tex]2x\frac{dx}{dt}+2y\frac{dy}{dt}=0[/tex]→[tex]2x\frac{dx}{dt}=-2y\frac{dy}{dt};\frac{dx}{dt}=-\frac{2y}{2x}\frac{dy}{dt}; \frac{dx}{dt}=-\frac{y}{x}\frac{dy}{dt}[/tex], if x=6 and y=8

[tex]\frac{dx}{dt}=\frac{-8}{6}2[/tex]→[tex]\frac{dx}{dt}=\frac{-8}{3}\frac{ft}{s}[/tex]

we must find the rate of change in radians over seconds, being the speed 8/3 ft / s = 2.66 ft / s the variation in degrees is determined when traveling 6 ft

Answer:

12.2 inches

Step-by-step explanation:

Write and solve a proportion.

Arc length / circumference = central angle / 360°

x / (2π × 7) = 100° / 360°

x = 12.2

The arc length is 12.2 inches

Answer: 6 tubes

Step-by-step explanation:

Create a simple relationship.

4 days = 3 tubes

8 days = ? tubes

Well, if there is twice as many days, twice as many tubes will be used.

4 days x 2 = 8 days

3 tubes x 2 = 6 tubes

Answer:

$19153.26

Step-by-step explanation:

Here is the complete question: Sally has just finished her thirty-fifth year with her company and is getting ready to retire. During her thirty-five years, Sallys average annual salary was 45,603 How much can Sally expect to receive from Social Security annually if she were to retire today? (Assume she will receive 42% of her average annual salary.)

Given: Sally´s average salary while working is $45603.

           Sally will receive 42% of her average annual salary as social security.

Now, finding annual income of sally after retirement.

Sally´s income from social security after retirement= [tex]\frac{42}{100} \times 45603= \$ 19153.26[/tex]

∴ Sally receive $ 19153.26 annually from social security.

Pilar will pay 30 cents less from the original amount by using reusable bags.

Step-by-step explanation:

Bags used by Pilar = b = 6

Price decrease per bag = 5 cents

Let x be the total amount paid by Pilar.

Decrease will lessen the amount paid by Pilar, therefore, according to statement;

P(x) = x - 5b

As she used 6 bags, therefore, putting b=6

[tex]P(x)=x-5(6)\\P(x)=x-30[/tex]

Pilar will pay 30 cents less from the original amount by using reusable bags.

Keywords: subtraction, function

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C

Step-by-step explanation:

The volume of a cylinder = πr²h

= 33 cubic inches

The volume of a cone = ¹/3 πr²h

If the cylinder and come share the same radius and height then ‘πr²h’ part of the formulas is the same for both;

It means the difference in proportionality is ¹/3 (because even π is the same across board). The volume of the cone is therefore;

¹/3 (33)

= 11

= 11 cubic inches

Learn More:

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The conjunction is false because the number -343/9 is not an integer but is a rational number.

The number [tex]-\frac{343}{9} is an integer, and a rational number.
To determine the truth value of this conjunction, we need to understand that an integer is a whole number like -1, 0, or 1, and a rational number is a ratio of integers like 2/1 or 3/4.
Since the number [tex]-\frac{343}{9} is not an integer and also a rational number (since it can be expressed as -343/9), the conjunction is false.

Answer:

The equation is [tex]\frac{x}{y}= 8[/tex]

Step-by-step explanation:

Let the time for which i bike is = x hours

Let the distance traveled in x hours = y miles

Ratio of distance and time will be = x : y

I travel 11.2 miles in 1.4 hours then the ratio of time and the distance traveled will be 1.4 : 11.2 Or 14 : 112

then the ratio of distance and time will be same in both the cases

So equation will be

[tex]\frac{x}{y} =\frac{14}{112}\\\\\frac{x}{y} =8[/tex]

Hence, equation representing the proportional relationship will be  [tex]\frac{x}{y}= 8[/tex]

Answer:

[tex]\displaystyle 1\frac{119}{250}\:liter[/tex]

[tex]\displaystyle 7,5\:sleps[/tex]

[tex]\displaystyle 37,3\:sleps [/tex]

Step-by-step explanation:

[tex]\displaystyle 1\frac{119}{250} = \frac{1476}{1000}[/tex]

[tex]\displaystyle 1\frac{1}{13} \times 7 = 7\frac{7}{13} ≈ 7,538461538 ≈ 7,5[/tex]

[tex]\displaystyle \frac{41}{1\frac{1}{10}} = 37\frac{3}{11} ≈ 37,3[/tex]

I am joyous to assist you anytime.

Answer:

a. To build systems that can mimic human intelligence.

Step-by-step explanation:

Artificial intelligence or machine intelligence is a part of computer development where computer systems are made to be able to perform tasks that usually require human intelligence like - speech recognition, language translation etc.

Artificial intelligence seeks to build a system that can mimic human intelligence.

Hence, the correct answer is option A.

Artificial intelligence seeks to build systems that can mimic human intelligence. Then the correct option is A.

The ability of an electronic machine that can perform tasks commonly associated with intelligent beings.

Artificial intelligence is a part of computer development where computer systems are made to be able to perform tasks that usually require human intelligence like speech recognition, language translation, etc.

Since all the condition that is fulfilled by option A.

Thus, the correct option is A.

More about the Artificial intelligence link is given below.

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The weight rises at the same constant rate of 2ft/s that the worker walks away, regardless of how far the worker has travelled. When the worker walks away, the length of the rope attached to the weight decreases and thus raises the weight. Therefore, at both distances, 10ft and 30ft, the weight will be rising at 2ft/s.

This is a physics problem involving related rates under the concept of kinematics. When the worker walks, the total length of the rope (80 ft) remains the same, so as the worker's part of the rope increases, the part attached to the weight decreases causing an upward motion. The rates at which the worker walks away and the weight retracts up are directly related.

When the worker has walked 10 feet, the worker's part of the rope has become 50 ft (original 40ft + 10ft walked), thus the weight's part of the rope is 30ft (80ft total - 50ft). Because the worker is walking at a constant rate of 2ft/s, this means that the weight is also rising at that same constant rate of 2ft/s.

Then, when the worker has walked 30 feet, the worker's part of the rope has become 70ft, thus the weight's part of the rope is 10ft. As previously explained, because the worker has a constant rate of walking away, the weight also has a constant rate of 2ft/s in the upward direction. Regardless of the worker's position, it does not impact the rate of the weight's ascent.

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The weight rises at a constant rate of 1 ft/s regardless of whether the worker has walked 10 feet or 30 feet.

Finding the Rate of the Weight Rising

Distance from the connection point to the ceiling: 40 ft

Worker’s walking speed: 2 ft/s

As the worker walks away, the rope is extended, and the weight rises. We need to determine how fast the weight is rising after the worker has walked different distances. Here’s how to do it step-by-step:

Let x be the distance the worker has walked away from the initial point. Therefore, the total length of the rope now serves both the height the weight has risen (h) and the horizontal distance the worker has walked (x).

80 ft = 2h + x

0 = 2(dh/dt) + dx/dt

2(dh/dt) + 2 = 0

2(dh/dt) = -2

dh/dt = -1 ft/s

a) For 10 feet: Plugging into the length equation:

2h + 10 = 80

2h = 70

⇒ h = 35 ft

b) For 30 feet: Plugging into the length equation:

2h + 30 = 80

2h = 50

∴ h = 25 ft

The weight rises at a rate of 1 ft/s when the worker has walked either 10 feet or 30 feet because the rate change of height (dh/dt) is constant.

To determine the atomic mass of naturally occurring gallium, convert the percentages of each isotope to decimals, multiply each decimal by its atomic mass, and add the products. The average atomic mass of gallium is 69.72 u.

The atomic mass of naturally occurring gallium can be determined using the following numerical setup:

Convert the percentages of the isotopes (60.11% and 39.89%) to decimals (0.6011 and 0.3989).

Multiply the decimal for each isotope by its atomic mass.

Add the products from step 2 to get the average atomic mass.

Round the final result to the appropriate number of significant figures.

In this case, the numerical setup would be:

(0.6011 x 68.93 u) + (0.3989 x 70.92 u) = 69.72 u

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Final answer:

To find the atomic mass of naturally occurring gallium, you perform a weighted average, multiplying the abundance of each isotope by its atomic mass, then summing the results. Using the given abundances and atomic masses, the atomic mass of gallium is approximately 69.75 u.

Explanation:

To calculate the atomic mass of naturally occurring gallium, which is a mixture of isotopes, you would set up a weighted average based on the isotopic composition and the atomic masses of the individual isotopes. You have Ga-69 with an atomic mass of 68.93 u, comprising 60.11% of gallium, and Ga-71 with an atomic mass of 70.92 u, making up 39.89% of gallium.

The numerical setup is as follows:

Atomic mass of gallium = (% abundance of Ga-69 × atomic mass of Ga-69) + (% abundance of Ga-71 × atomic mass of Ga-71)

Converting the percentage to a decimal fraction (60.11% = 0.6011 and 39.89% = 0.3989), the equation becomes:

Atomic mass of gallium = (0.6011 × 68.93 u) + (0.3989 × 70.92 u)

We then perform the multiplication and add the results:

Atomic mass of gallium = (0.6011 × 68.93 u) + (0.3989 × 70.92 u) = 41.433863 u + 28.320708 u = 69.754571 u

Thus, the atomic mass of naturally occurring gallium can be approximated to 69.75 u.

Answer:

Step-by-step explanation:

The given equation is of the form ...

  ax² +bx +c = 0

where a=2, b=-1, c=-8.

The quadratic formula gives the solution to the above equation as ...

  [tex]x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}[/tex]

So, for your equation, the solution is ...

  [tex]m=\dfrac{-(-1)\pm\sqrt{(-1)^2-4(2)(-8)}}{2(2)}=\dfrac{1\pm\sqrt{65}}{4}[/tex]

Comparing this to the form ...

  [tex]m=\dfrac{N\pm\sqrt{D}}{M}[/tex]

we see ...

Answer:

The percentage of men and women favoring a higher legal drinking age is the same

Step-by-step explanation:

A random sample of 1000 women showed that 65% were in favor of increasing the legal drinking age

n = 1000

No. of females were in favor of increasing the legal drinking age = [tex]\frac{65}{100} \times 1000=650[/tex]

y=650

In a random sample of 1000 men, 60% favored increasing the legal drinking age

n = 1000

No. of males were in favor of increasing the legal drinking age = [tex]\frac{60}{100} \times 1000=600[/tex]

y=600

[tex]n_1=1000 , y_1=650\\n_2=1000 , y_2=600[/tex]

We will use Comparing Two Proportions

[tex]\widehat{p_1}=\frac{y_1}{n_1}[/tex]

[tex]\widehat{p_1}=\frac{650}{1000}[/tex]

[tex]\widehat{p_1}=0.65[/tex]

[tex]\widehat{p_2}=\frac{y_2}{n_2}[/tex]

[tex]\widehat{p_2}=\frac{600}{1000}[/tex]

[tex]\widehat{p_2}=0.6[/tex]

Let p_1 and p_2 be the probabilities of men and women favoring a higher legal drinking age is the same.

So, [tex]H_0:p_1=p_2\\H_a:p_1 \neq p_2[/tex]

[tex]\widehat{p}=\frac{y_1+y_2}{n_1+n_2} =\frac{600+650}{1000+1000}=0.625[/tex]

Formula of test statistic :[tex]\frac{\widehat{p_1}-\widehat{p_2}}{\sqrt{\widehat{p}(1-\widehat{p})(\frac{1}{n_1}+\frac{1}{n_2})}}[/tex]

test statistic : [tex]\frac{0.65-0.6}{\sqrt{0.625(1-0.625)(\frac{1}{1000}+\frac{1}{1000})}}[/tex]

test statistic : 2.3094

Refer the z table for p value :

p value : 0.9893

α = 0.05.

p value > α

So, we failed to reject null hypothesis .

So,  the percentage of men and women favoring a higher legal drinking age is the same

Answer:6.042km/h 6km/h approximately

Step-by-step explanation:

First off we have to know the formula relating speed, distance and time which is

Speed = distance/time

Now we are looking for Rachel's running speed

We are to find Rachel's running speed, so let's label is x

We are given that the distance Rachel runs to her bus stop is 2km

We were not given the time she uses to run to the bus stop

So let's label the time Rachel uses to run to her bus stop as y

So from the formula speed = distance/time

We have x = 2/y

Now we are told that the speed the bus uses to get to school is 45km/h faster than her speed used to run

So speed of bus = 45 + x

And the overall time for the whole journey is 25mins, changing this to hours, because the speed details given is in km/h we divide 25 by 60 which will give 0.417

Now if the total time is 0.417 hours, and we labeled the time for Rachel to run to the bus as y, so the time for the time for the bus to get to school will be 0.417 - y

We are also told the bus rides for 4.5km to school

So adding this together to relate the speed, distance and time of the bus with the formula speed = distance/time

We get 45 + x = 4.5/(0.417 - y)

So we have two equations

x = 2/y (1)

45+x = 4.5/(0.417-y) (2)

So putting (1) in (2) we have

45 + (2/y) = 4.5/(0.417-y)

Expanding further

(45y + 2)/y = 4.5(0.417-y)

Cross multiplying

(45y + 2)(0.417 - y) = 4.5y

Opening the brackets

18.765y - 45y2 + 0.834 - 2y = 4.5y

Collecting like terms

-45y2 + 18.765y -2y - 4.5y + 0.834 = 0

-45y2 + 12.265 + 0.834 = 0

Dividing all sides by -45 to make the coefficient of y2 1

y2 - 0.273y - 0.019 = 0

Now we have gotten a quadratic equation, and since it's with decimal numbers we can use either completing the square method of almighty formula

I'm using almighty formula her

For solving

ax2 + bx + c = 0

x = (-b +-root(b2-4ac)/2a

For our own equation, we are finding y

From our our quadratic equation

a = 1, b=-0.273, c = -0.834

you = (-(-0.273)+-root(-0.273-4(1)(-0.019))/2(1)

y = (273+-root(0.151))/2

y = (0.273+0.389)/2 or (0.273-0.389)/2

y = 0.331 or -0.085

So we use the positive answer which is 0.331, because time can't be negative

Then we put y = 0.331 in (1)

x = 2/y

x = 2/0.331

x = 6.042km/h

x = 6km/h approximately

Rachel's running speed was calculated by forming equations based on the given scenario of her running and riding the bus, solving these equations simultaneously provides the answer.

This problem is a classic case of the combined speed-time-distance problem. It involves two segments: Rachel running and then riding the bus. The time taken for these two segments combined is given as 25 minutes. We can denote Rachel's running speed as 'r' and her running time as 't1', and the bus speed as 'r+45' and bus time as 't2'.

From the question, we can formulate the following two equations:

The total time 't1 + t2' is equal to 25/60 hours (converting to the same unit).

Now, we solve these equations together to find the value of 'r', Rachel's running speed.

This shows the importance and application of average speed, time and distance in real-world situations.

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Answer:

a) [tex]\bar x= 56.8[/tex]

b) The 95% confidence interval is given by (55.282;58.318)  

c) [tex]P(X<52) = P(Z<\frac{52-56.8}{\sqrt{6}})=P(Z<-1.96) = 0.025[/tex]

Step-by-step explanation:

1) Notation and definitions  

n=10 represent the sample size  

[tex]\bar X[/tex] represent the sample mean  

[tex]s[/tex] represent the sample standard deviation  

[tex]\sigma^2= 6[/tex]

m represent the margin of error  

Confidence =95% or 0.95

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".  

The margin of error is the range of values below and above the sample statistic in a confidence interval.

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".  

2) Calculate the mean (Point of estimate for [tex]\mu[/tex]) and standard deviation for the sample

On this case we need to find the sample standard deviation with the following formula:

[tex]s=\sqrt{\frac{\sum_{i=1}^15 (x_i -\bar x)^2}{n-1}}[/tex]

And in order to find the sample mean we just need to use this formula:

[tex]\bar x =\frac{\sum_{i=1}^{15} x_i}{n}[/tex]

The sample mean obtained on this case is [tex]\bar x= 56.8[/tex] and the deviation s=2.052. Since we know the population standard deviation [tex]\sigma^2=6[/tex] and [tex]\sigma=2.449[/tex]

3) Calculate the critical value tc  

In order to find the critical value is important to mention that we don't know about the population standard deviation, so on this case we need to use the t distribution. Since our interval is at 95% of confidence, our significance level would be given by [tex]\alpha=1-0.95=0.05[/tex] and [tex]\alpha/2 =0.025[/tex].

We can find the critical values in excel using the following formulas:  

"=NORM.INV(0.025,0,1)" for [tex]t_{\alpha/2}=-1.96[/tex]  

"=NORM.INV(1-0.025,0,1)" for [tex]t_{1-\alpha/2}=1.96[/tex]  

The critical value [tex]zc=\pm 1.96[/tex]  

3) Calculate the margin of error (m)  

The margin of error for the sample mean is given by this formula:  

[tex]m=z_c \frac{\sigma}{\sqrt{n}}[/tex]  

[tex]m=1.96 \frac{2.449}{\sqrt{10}}=1.518[/tex]  

4) Calculate the confidence interval  

The interval for the mean is given by this formula:  

[tex]\bar X \pm z_{c} \frac{\sigma}{\sqrt{n}}[/tex]  

And calculating the limits we got:  

[tex]56.8 - 1.96 \frac{2.449}{\sqrt{10}}=55.282[/tex]  

[tex]56.8 + 1.96 \frac{2.449}{\sqrt{10}}=58.318[/tex]  

The 95% confidence interval is given by (55.282;58.318)  

On the basis of these very limited data, what is the probability that an individual snack pack selected at random is filled with less than 52 grams of candy?

On this case we can use as point of estimate for the mean the result from part a, [tex]\bar x= 56.8[/tex] and the variance is [tex]\sigma^2 =6[/tex], so [tex]\sigma =\sqrt{6}[/tex]. And we are interested on this probability:

[tex]P(X<52) = P(Z<\frac{52-56.8}{\sqrt{6}})=P(Z<-1.96) = 0.025[/tex]