Given: RS and TV bisect each other at point X. TR and SV are drawn Prove: TR || SV

Answer:

300pi

Step-by-step explanation:

Final answer:

The rate of change of the surface area of the sphere at that instant is 942.48 meters squared per minute.

Explanation:

To find the rate of change of the surface area S(t)S(t)S, (, t, )of the sphere at that instant, we need to differentiate the surface area formula with respect to time and then substitute the given values.

The formula for the surface area of a sphere is [tex]S = 4\pi r^2.[/tex]

Taking the derivative with respect to time, we have dS/dt = 8πr(dr/dt).

Given that dr/dt = 7.5 meters per minute and r = 5 meters, we can substitute these values into the derivative formula to find the rate of change of the surface area at that instant.

= dS/dt = 8π(5)(7.5)

= 300π

= 942.48 meters squared per minute.

Answer:

The sample space for one spin of this game is: S = {00, 0, 1, 2, 3,…, 40}

Step-by-step explanation:

Consider the provided information.

In probability theory, the set of all possible outcomes or outcomes of that experiment is the sample space of an experiment or random trial. Using set notation, a sample space is usually denoted and the possible ordered outcomes are identified as elements in the set.

Here the possible number of elements in the set are 00, 0, 1, 2,...,40

The sample space is anything the ball can land on.

Thus, the sample space for one spin of this game is: S = {00, 0, 1, 2, 3,…, 40}

Step-by-step explanation:

We need to find um of the first 100 terms of

               [tex]\frac{1}{n}-\frac{1}{n+1}[/tex]

That is

           [tex]\texttt{Sum = }\frac{1}{1}-\frac{1}{1+1}+\frac{1}{2}-\frac{1}{2+1}+\frac{1}{3}-\frac{1}{3+1}.....+\frac{1}{100}-\frac{1}{100+1}\\\\\texttt{Sum = }\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}.....+\frac{1}{100}-\frac{1}{101}\\\\\texttt{Sum = }\frac{1}{1}-\frac{1}{101}\\\\\texttt{Sum = }\frac{101-1}{101\times 1}\\\\\texttt{Sum = }\frac{100}{101}[/tex]

Option E is the correct answer.

Final answer:

The sum of the first 100 terms of the sequence an = 1/n - 1/(n+1) is 100/101 because it's a telescoping series where almost all terms cancel each other out except the very first and the very last term.

Explanation:

The student's question involves finding the sum of the first 100 terms of the sequence an = 1/n - 1/(n+1). To find this sum, we can notice that many terms will cancel each other out when we add up the sequence. This is because the sequence is telescoping. Let's illustrate this with the first few terms:

a1 = 1 - 1/2

a2 = 1/2 - 1/3

a3 = 1/3 - 1/4

...

a99 = 1/99 - 1/100

a100 = 1/100 - 1/101

When we add all these up, notice that every negative term cancels out with the positive term that precedes it, except for the very first term, which is 1, and the very last negative term, which is -1/101. Hence, the sum is 1 - 1/101 which simplifies to 100/101. Therefore, the correct answer is (e) 100/101.

Answer:

8,855

Step-by-step explanation:

The way to solve this problem is by using Combinations.

In Combinations, we can form different collections of k elements from a total of n elements where the order of them does not matter and any member of them is not repeated.

Combinations is expressed mathematically as:

[tex]\\nC_k = \frac{n!}{(n-k)!k!} [/tex] [1]

Where n is the total elements, k is the number of elements selected from n, and n! is n factorial, or, for instance, 3! is 3*2*1 = 6; 4! is 4*3*2*1 = 24.

This formula tells us how to form groups of k members from a total of n elements. These groups of k members have no repeated elements, that is, in the context of this question, no flavor is repeated in any group.

Likewise, different orders of the same members do not matter, or, in other words, if we have two groups of four members flavors (vanilla, chocolate, strawberry, lemon) and (chocolate, vanilla, lemon, strawberry), they are considered the same group since order does not matter in Combinations.

In this way, to determine the number of four dip sundaes (k) from 23 flavors (n) that an ice cream store sells, we need to apply the formula [1], as follows:

[tex]\\23C_4 = \frac{23!}{(23-4)!4!} [/tex]

[tex]\\23C_4 = \frac{23!}{19!4!} [/tex]

[tex]\\23C_4 = \frac{23*22*21*20*19!}{19!4!} [/tex], since 19!/19! = 1.

[tex]\\23C_4 = \frac{23*22*21*20}{4*3*2*1} [/tex]

[tex]\\23C_4 = 8,855 [/tex]

To find the number of 4-dip sundaes possible with 23 flavors of ice cream, we use combinations. The formula for combinations is C(n, r) = n! / (r!(n-r)!). Applying this formula, we find that there are 8855 possible 4-dip sundaes.

To determine the number of 4-dip sundaes possible with 23 flavors of ice cream, we can use combinations.

A combination is used when the order does not matter, and no repetitions are allowed.

In this case, we use the formula for combinations of r items selected from a set of n items without replacement: C(n, r) = n! / (r!(n-r)!)

So, the number of 4-dip sundaes possible is C(23, 4) = 23! / (4!(23-4)!) = 23! / (4!19!)

Calculating this using a calculator, we find that there are 8855 possible 4-dip sundaes with 23 flavors of ice cream.

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Answer:

Step-by-step explanation:

Bill received $12 to feed a neighbor's cat for 3 days. His pay rate, x would be 12/3 = 4

He is paid $4 for feeding the cat per day.

To earn $40, the number of days that bill would have to work would be 40/4 = 10 days.

The neighbor's family is going on

vacation for 3 weeks next summer. There are 7 days in a week. Converting 3 weeks to days, it becomes 7 × 3 = 21 days.

The total amount of money that Bill will earn in 21 days would be

21 × 4 = $84

Since Bill wants to earn enough money to buy a CD player that costs $89, $84 won't be enough. He still needs $5 more and that would be from 2 more days.

Answer:

[tex]\frac{7}{32}[/tex] inch.

Step-by-step explanation:

We have been given that the depth of the new tire is 9/32 inch after two month use 1/16 inch worn off. We are asked to find the depth of the tire remaining tire thread.

To find the depth of remaining tire thread, we will subtract worn off value from initial depth as:

[tex]\text{Depth of remaining tire thread}=\frac{9}{32}-\frac{1}{16}[/tex]

Let us make a common denominator.

[tex]\text{Depth of remaining tire thread}=\frac{9}{32}-\frac{1*2}{16*2}[/tex]

[tex]\text{Depth of remaining tire thread}=\frac{9}{32}-\frac{2}{32}[/tex]

Combine numerators:

[tex]\text{Depth of remaining tire thread}=\frac{9-2}{32}[/tex]

[tex]\text{Depth of remaining tire thread}=\frac{7}{32}[/tex]

Therefore, the depth of the remaining tire thread would be [tex]\frac{7}{32}[/tex] inch.

The remaining depth of the tire is 7/32 inches.

To determine the remaining depth of the tire tread, you need to subtract the depth worn off from the initial depth of the tire.

Step-by-Step Solution:

This approach ensures you correctly determine the remaining depth of the tire tread.

Answer:

Rina will need 6 tickets.

Explanation:

Rina needs only 1 ticket to ride the ferris wheel once, and 1 ticket to ride the bumper cars once. If she wants to ride the ferris wheel 5 times, then she'll need 5 tickets since 1 x 5 = 5. If she wants to ride the bumper cars only once, she'll only need 1 ticket since 1 x 1 = 1.

Add the answers together, and you get 6 tickets since 5 + 1 = 6.

Hope this helps! :)

Answer:

Base of triangle is 9 inches and altitude of triangle is 14 inches.

Step-by-step explanation:

Given:

Area of Triangle = 63 sq. in.

Let base of the triangle be 'b'.

Let altitude of triangle be 'a'.

Now according to question;

altitude is 5 inches longer than the base.

hence equation can be framed as;

[tex]a=b+5[/tex]

Now we know that Area of triangle is half times base and altitude.

Hence we get;

[tex]\frac{1}{2} \times b \times a =\textrm{Area of Triangle}[/tex]

Substituting the values we get;

[tex]\frac{1}{2} \times b \times (b+5) =63\\\\b(b+5)=63\times2\\\\b^2+5b=126\\\\b^2+5b-126=0[/tex]

Now finding the roots for given equation we get;

[tex]b^2+14b-9b-126=0\\\\b(b+14)-9(b+14)=0\\\\(b+14)(b-9)=0[/tex]

Hence there are 2 values of b[tex]b-9 = 0\\b=9\\\\b+14=0\\b=-14[/tex]

Since base of triangle cannot be negative hence we can say [tex]b=9\ inches[/tex]

So Base of triangle = 9 inches.

Altitude = 5 + base  = 5 + 9 = 14 in.

Hence Base of triangle is 9 inches and altitude of triangle is 14 inches.

Answer:

"minimum value = 0; maximum value = 8"

Step-by-step explanation:

This is the absolute value function, which returns a positive value for any numbers (positive or negative).

For example,

| -9 | = 9

| 9 | = 9

| 0 | = 0

Now, the domain is from -8 to 7 and we want to find max and min value that we can get from this function.

If we look closely, putting 7 into x won't give us max value as putting -8 would do, because:

|7| = 7

|-8| = 8

So, putting -8 would give us max value of 8 for the function.

Now, we can't get any min values that are negative, because the function doesn't return any negative values. So the lowest value would definitely be 0!

|0| = 0

and

ex:  |-2| = 2 (bigger),  |-5| = 5 (even bigger).

So,

Min Value = 0

Max Value = 8

Answer:

minimum value = 0; maximum value = 8

Step-by-step explanation:

The function [tex]f(x)[/tex] is an absolute value function, which means that for negative values in it's domain it gives positive values of  [tex]f(x)[/tex], and therefore it's minimum value is 0.

In the given domain interval the maximum value of the function is 8 because [tex]f(-8)=8[/tex].

Answer:

  1 cm : 800 km   or   1/80,000,000

Step-by-step explanation:

A model or map scale is often expressed as ...

  (1 unit of A on the model) : (N units of B in the real world)

We're given the relative measurements as ...

  15 cm : 12,000 km

Dividing by 15 gives the unit ratio as above:

  1 cm : 800 km

__

A scale can also be expressed as a unitless fraction. To find that, we need to convert the units of both parts of this ratio to the same unit.

  0.01 m : 800,000 m

Multiplying by 100, we get ...

  1 m : 80,000,000 m

Since the units are the same, they aren't needed, and we can write the scale factor as ...

  1 : 80,000,000   or   1/80,000,000

Answer:

No; Because g'(0) ≠ g'(1), i.e. 0≠2, then this function is not differentiable for g:[0,1]→R

Step-by-step explanation:

Assuming:  the function is [tex]f(x)=x^{2}[/tex] in [0,1]

And rewriting it for the sake of clarity:

Does there exist a differentiable function g : [0, 1] →R such that g'(x) = f(x) for all g(x)=x² ∈ [0, 1]? Justify your answer

1) A function is considered to be differentiable if, and only if  both derivatives (right and left ones) do exist and have the same value. In this case, for the Domain [0,1]:

[tex]g'(0)=g'(1)[/tex]

2) Examining it, the Domain for this set is smaller than the Real Set, since it is [0,1]

The limit to the left

[tex]g(x)=x^{2}\\g'(x)=2x\\ g'(0)=2(0) \Rightarrow g'(0)=0[/tex]

[tex]g(x)=x^{2}\\g'(x)=2x\\ g'(1)=2(1) \Rightarrow g'(1)=2[/tex]

g'(x)=f(x) then g'(0)=f(0) and g'(1)=f(1)

3) Since g'(0) ≠ g'(1), i.e. 0≠2, then this function is not differentiable for g:[0,1]→R

Because this is the same as to calculate the limit from the left and right side, of g(x).

[tex]f'(c)=\lim_{x\rightarrow c}\left [\frac{f(b)-f(a)}{b-a} \right ]\\\\g'(0)=\lim_{x\rightarrow 0}\left [\frac{g(b)-g(a)}{b-a} \right ]\\\\g'(1)=\lim_{x\rightarrow 1}\left [\frac{g(b)-g(a)}{b-a} \right ][/tex]

This is what the Bilateral Theorem says:

[tex]\lim_{x\rightarrow c^{-}}f(x)=L\Leftrightarrow \lim_{x\rightarrow c^{+}}f(x)=L\:and\:\lim_{x\rightarrow c^{-}}f(x)=L[/tex]

Answer:

he rank from least to great based on their axis of symmetry:

0, 1, -3 ⇒ g(x), h(x), f(x)

So, option C is correct.

Step-by-step explanation:

A quadratic equation is given by:

[tex]ax^2+bx+c =0[/tex]

Here, a, b and c are termed as coefficients and x being the variable.

Axis of symmetry can be obtained using the formula

[tex]x = \frac{-b}{2a}[/tex]

Identification of a, b and c in f(x), g(x) and h(x) can be obtained as follows:

[tex]f(x) = x^2 + 6x - 1[/tex]

⇒ a = 1, b = 6 and c = -1

[tex]g(x) = -x^2 + 2[/tex]

⇒ a = -1, b = 0 and c = 2

[tex]h(x) = 2^2 - 4x + 3[/tex]

⇒ a = 2, b = -4 and c = 3

So, axis of symmetry in [tex]f(x) = x^2 + 6x - 1[/tex] will be:

[tex]x = \frac{-b}{2a}[/tex]

x = -6/2(1) = -3

and axis of symmetry in [tex]g(x) = -x^2 + 2[/tex] will be:

[tex]x = \frac{-b}{2a}[/tex]

x = -(0)/2(-1) = 0

and axis of symmetry in [tex]h(x) = 2^2 - 4x + 3[/tex] will be:

[tex]x = \frac{-b}{2a}[/tex]

x = -(-4)/2(2) = 1

So, the rank from least to great based on their axis of symmetry:

0, 1, -3 ⇒ g(x), h(x), f(x)

So, option C is correct.

Keywords: axis of symmetry, functions

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Answer:

  12

Step-by-step explanation:

The difference of the two coordinates is ...

  23 -(-10) = 33

The desired coordinate is 2/3 of that length from J, so is ...

  J + (2/3)·33 = J +22 = -10 +22 = 12

The desired coordinate is 12.

Answer:

Y=4x^5-12x^4+6x and y=5x^3-2x

Answer:

Y = 4x5 - 12x4 + 6x

Y = 5x3 - 2x

Step-by-step explanation:

Answer:

[tex]P(\bar X >80)=P(Z>2.143)=1-P(z<2.143)=1-0.984=0.016[/tex]

Step-by-step explanation:

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".

Let X the random variable that represent the Student scores on exams given by a certain instructor, we know that X have the following distribution:

[tex]X \sim N(\mu=74, \sigma=14)[/tex]

The sampling distribution for the sample mean is given by:

[tex]\bar X \sim N(\mu,\frac{\sigma}{\sqrt{n}})[/tex]

The deduction is explained below we have this:

[tex]E(\bar X)= E(\sum_{i=1}^{n}\frac{x_i}{n})= \sum_{i=1}^n \frac{E(x_i)}{n}= \frac{n\mu}{n}=\mu[/tex]

[tex]Var(\bar X)=Var(\sum_{i=1}^{n}\frac{x_i}{n})= \frac{1}{n^2}\sum_{i=1}^n Var(x_i)[/tex]

Since the variance for each individual observation is [tex]Var(x_i)=\sigma^2 [/tex] then:

[tex]Var(\bar X)=\frac{n \sigma^2}{n^2}=\frac{\sigma}{n}[/tex]

And then for this special case:

[tex]\bar X \sim N(74,\frac{14}{\sqrt{25}}=2.8)[/tex]

We are interested on this probability:

[tex]P(\bar X >80)[/tex]

And we have already found the probability distribution for the sample mean on part a. So on this case we can use the z score formula given by:

[tex]z=\frac{\bar X -\mu}{\frac{\sigma}{\sqrt{n}}}[/tex]

Applying this we have the following result:

[tex]P(\bar X >80)=P(Z>\frac{80-74}{\frac{14}{\sqrt{25}}})=P(Z>2.143)[/tex]

And using the normal standard distribution, Excel or a calculator we find this:

[tex]P(Z>2.143)=1-P(z<2.143)=1-0.984=0.016[/tex]

Final answer:

Using the Central Limit Theorem and the z-score formula, we calculate that the approximate probability that the average test score in the class of size 25 exceeds 80 is approximately 1.62%.

Explanation:

To approximate the probability that the average test score in the class of size 25 exceeds 80, we can use the Central Limit Theorem which tells us that the sampling distribution of the sample mean will be approximately normally distributed if the sample size is large enough (typically n ≥ 30 is considered sufficient, but we can still use this for a sample of 25 when the population distribution is not overly skewed).

The formula for the z-score of a sample mean is:

z = (X - μ) / (σ / √n)

where X is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size. Given the population mean μ = 74, population standard deviation σ = 14, and sample size n = 25, we can calculate the z-score for a sample mean of 80.

Using these values:

z = (80 - 74) / (14 / √25) = (6) / (14 / 5) = 6 / 2.8 = 2.14

Now, we need to find the probability corresponding to a z-score of 2.14. We check the standard normal distribution table or use a calculator with normal distribution functions to find that the area to the left of z = 2.14 is approximately 0.9838. The probability that the average is above 80 is the area to the right of 2.14, so we subtract this value from 1.

Probability = 1 - 0.9838 = 0.0162

The approximate probability that the average test score in the class of size 25 exceeds 80 is approximately 0.0162, or 1.62%.

Answer:16

Step-by-step explanation:

Answer:

C. 0.006

Step-by-step explanation:

Here we have to calculate the probability of two events happen at once, so the probability is the product of the probability of having a 2 and the probability of having a 10.

There are four 2 cards out of 52 in the poker game, so the probability of having a 2 is:

[tex]P(2)=\frac{4}{52}=0.077[/tex]

Now the probability of having a 10 is 4 out of 51 because we substracted the card labeled as 2.

[tex]P(10)=\frac{4}{51}=0.079[/tex]

so the probability is:

[tex]P(P(2)andP(10))=0.077*0.079=0.006[/tex]

Answer:

Step-by-step explanation:

Let x represent the cost of an adult ticket and

Let y represent the cost of a child ticket.

Your school is sponsoring a pancake dinner to raise money for a field trip. You estimate that 200 adults and 250 children will attend.

The equation that can be used to find what ticket prices to set in order to raise $3800 would be

200x + 250y = 3800

The equation that can be used to find what ticket prices to set in order to raise $3800 is [tex]3800=200x+250y[/tex].

An equation is formed when two equal expressions are equated together with the help of an equal sign '='.

As it is given that the cost of an adult ticket is x while the number of adult tickets sold was 200. Similarly, the cost of a child ticket is y while the number of child tickets sold will be 250. And the total money that is needed to be raised is $3800, therefore, the equation can be written as,

Total amount= Total amount of Adult Tickets + Total amount of Child Ticket

[tex]\$3,800 = (\$x \times 200)+(\$y \times 250)\\\\3800=200x+250y[/tex]

Hence, the equation that can be used to find what ticket prices to set in order to raise $3800 is [tex]3800=200x+250y[/tex].

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Answer:

16 Weeks.

Step-by-step explanation:

1050 - 150 = 900.

900 divided by 55 = 16.3.

Round down, because 3 is less than 5.

Therefore, Johnny can spend $55 each week for 16 weeks and have at least $150 left in his account.

Hope this helps.

Answer:

The probability that the amount dispensed per box will have to be increased is 0.0062.

Step-by-step explanation:

Consider the provided information.

Sample of 16 boxes is selected at random.

If the mean for 1 hour is 1 pound and the standard deviation is 0.1

1 Pound = 16 ounces , then 0.1 Pound = 16/10 = 1.6 ounces

Thus: μ = 16 ounces and σ = 1.6 ounces.

Compute the test statistic [tex]z=\frac{\bar x-\mu}{\frac{\sigma}{\sqrt{n}}}[/tex]

[tex]z=\frac{15-16}{\frac{1.6}{\sqrt{16}}}[/tex]

[tex]z=\frac{-1}{\frac{1.6}{4}}[/tex]

[tex]z=\frac{-1}{0.4}[/tex]

[tex]z=-2.5[/tex]

By using the table.

P value = P(Z<-250) = 0.0062

Thus, the probability that the amount dispensed per box will have to be increased is 0.0062.

A clothing store is selling a shirt for a discounted price of $43.61 . The original price of the shirt was approximately $49.01.

To find the original price of the shirt, we can use the formula: Original Price = Discounted Price / (1 - Discount Rate). In this case, the discounted price is $43.61 and the discount rate is 11%, or 0.11. Plugging these values into the formula, we get: Original Price = 43.61 / (1 - 0.11) = 43.61 / 0.89 ≈ 49.01. Therefore, the original price of the shirt was approximately $49.01.

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Answer: 5/7

Step-by-step explanation:please see attachment for explanation.

Answer: the lowest commission is $163.96

Step-by-step explanation:

Let x represent the lowest monthly commission that a salesman earned.

Let y represent the highest monthly commission that a salesman earned.

The lowest monthly commission that a salesman earned was only 1/5 more than 1/4 as high as the highest commission he earned. This means that

x = y/4 + 1/5 - - - - - - - - 1

The highest and lowest commissions when added together equal $819. This means that

x + y = 819

x = 819 - y - - - - - - -2

Substituting equation 2 into 1, it becomes

819 - y = y/4 + 1/5

Multiplying through by 20, it becomes

16380 - 20y = 5y + 4

25y = 16380 - 4 = 16376

y = 16376/25 = 655.04

x = 819 - 655.04 = 163.96

Final answer:

Larry studied a total of 5 1/12 hours on Monday and Tuesday. To find this, convert the mixed numbers to improper fractions, find a common denominator, add the fractions together, and simplify to get the final sum.a

Explanation:

To calculate the total amount of time Larry spent studying on Monday and Tuesday, we need to add the hours together:

    1.  Monday: 2 1/4 hours
    2.  Tuesday: 2 5/6 hours

Let's convert these mixed numbers to improper fractions to simplify the addition:

Next, we find a common denominator, which is 12, and rewrite the fractions:

Now that they have the same denominator, we can add them together:

To simplify, divide 61 by 12, which is 5 with a remainder of 1. Thus, the mixed number is 5 1/12. Therefore, the addition sentence to show how many hours Larry spent studying Monday and Tuesday is:

Answer:

[tex]\sigma =\frac{478}{6}=79.667[/tex]

Step-by-step explanation:

The empirical rule, also referred to as "the three-sigma rule or 68-95-99.7 rule, is a statistical rule which states that for a normal distribution, almost all data falls within three standard deviations (denoted by σ) of the mean (denoted by µ)". The empirical rule shows that 68% falls within the first standard deviation (µ ± σ), 95% within the first two standard deviations (µ ± 2σ), and 99.7% within the first three standard deviations (µ ± 3σ).

And on this case since we are within 3 deviations (because we have 99.7% of the data between 568 and 1066hours), the result obtained using the z score agrees with the empirical rule.  

So on this case we can find the standard deviation on this ways:

[tex]\mu -3\sigma = 568[/tex]     (1)

[tex]\mu +3\sigma = 1066[/tex]   (2)

If we subtract conditions (2) and (1) we got:

[tex]1066-588 =\mu +3\sigma -\mu +3\sigma[/tex]

[tex]478= 6\sigma[/tex]

[tex]\sigma =\frac{478}{6}=79.667[/tex]

Answer:

the correct option is D.

Step-by-step explanation:

x+3y>6

y≥2x+4

consider the equation x+3y=6

3y = 6-x

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

this line is in the form of y = mx + c

where m is the slope os the line and c is the y intercept of the line

therefore the line has a y intercept of 2  and slope of-1/3

therefore the line has negative slope with positive intercept.

now consider the line y=2x+4

this line is in the form of y = mx + c

where m is the slope os the line and c is the y intercept of the line

therefore slope = 2 and y intercept = 4

therefore the line has positive slope and positive y intercept.

in option a both line has positive intercept so it cant be an answer.

in option b one line has positive intercept of 2 and another with negative intercept of -4 but we need intercept of both line to be positive so it cant be an answer.

in option c both line has negative intercept of -2 and -4 but we need intercept of both line to be positive so it cant be an answer.

in option d both line has positive intercept of 2 and 4 and also one of the line has negative slope and another line has positive slope so it should be an answer

further to confirm consider x+3y>6

put the point 0,0 in the inequality

0>6 which is wrong so 0,0 cant lie in the region which is true according to the graph.

Answer:

d

Step-by-step explanation:

Answer:

3.8 [tex]in^{2}[/tex]

Step-by-step explanation:

We are given a square of size 6x6 in. So, area of this square is equal to 6 x 6 = 36 [tex]in^{2}[/tex]. Now, the shaded region is [tex]\frac{1}{2}[/tex] x (Area of square - area of semicircles)

The diameter of both semicircles = side of the square = 6in

So, radius (r) = [tex]\frac{1}{2}[/tex] x diameter = [tex]\frac{1}{2}[/tex] x 6 = 3in

And hence, area of semicircle is  = [tex]\frac{1}{2}[/tex] x π[tex]r^{2}[/tex]

= [tex]\frac{1}{2}[/tex] x π[tex]3^{2}[/tex]

Since, there are two semicircles we multiply above by 2, so area of both semicircles = 2 x [tex]\frac{1}{2}[/tex] π[tex]3^{2}[/tex] = 9π

Area of shaded region =  [tex]\frac{1}{2}[/tex] (36 - 9π) = 3.8628 = 3.8 [tex]in^{2}[/tex] to the nearest tenth.

Answer:

  QR = 28 inches or 44 inches

Step-by-step explanation:

In right triangle QNP, the length of QN is given by the Pythagorean theorem as ...

  QP² = QN² +PN²

  QN = √(QP² -PN²) = √(1521 -225) = √1296 = 36

In right triangle RNP, the length of RN is similarly found:

  RN = √(RP² -PN²) = √(289 -225) = √64 = 8

So, we have N on line QR with QN = 36 and RN = 8.

If N is between Q and R, then ...

  QR = QN +NR = 36 +8 = 44

If R is between Q and N, then ...

  QR = QN -NR = 36 -8 = 28

The possible lengths of QR are 28 in and 44 in.

Final answer:

To determine QR in ∆PQR, the Pythagorean theorem is used on the two right triangles formed by the altitude PN. Calculating gives QN = 36 inches and RN = 8 inches, hence, QR = QN + RN = 44 inches.

Explanation:

To find the length QR in ∆PQR, where PQ = 39 inches, PR = 17 inches, and the altitude PN = 15 inches, we can use the properties of right triangles. Since PN is the altitude to base QR, it forms two right triangles, ∆PNQ and ∆PNR, within ∆PQR. We can use the Pythagorean theorem to solve for the lengths of QN and RN, and then sum these to find QR.

Firstly, let’s find QN in ∆PNQ:

Secondly, we do the same for RN in ∆PNR:

Therefore, QR = QN + RN = 36 inches + 8 inches = 44 inches.

Answer:

1. 96.08%; 2. x=764.8; 3. 63.31%; 4. 93.57%; 5. 99.74%

Step-by-step explanation:

The essential tool here is the standardized cumulative normal distribution which tell us, no matter the values normally distributed, the percentage of values below this z-score. The z values are also normally distributed and this permit us to calculate any probability related to a population normally distributed or follow a Gaussian Distribution. A z-score value is represented by:

[tex]\\ z=\frac{(x-\mu)}{\sigma}[/tex], and the density function is:

[tex]\\ f(x) = \frac{1}{\sqrt{2\pi}} e^{\frac{-z^{2}}{2} }[/tex]

Where [tex]\\ \mu[/tex] is the mean for the population, and [tex]\\ \sigma [/tex] is the standard deviation for the population too.

Tables for z scores are available in any Statistic book and can also be found on the Internet.

The combined math and verbal scores for students taking a national standardized examination for college admission, is normally distributed with a mean of 500 and a standard deviation of 170. If a college requires a minimum score of 800 for admission, what percentage of student do not satisfy that requirement?

For solve this, we know that [tex]\\ \mu = 500[/tex], and [tex]\\ \sigma = 170[/tex], so

z = [tex]\frac{800-500}{170} = 1.7647[/tex].

For this value of z, and having a Table of the Normal Distribution with two decimals, that is, the cumulative normal distribution for this value of z is F(z) = F(1.76) = 0.9608 or 96.08%. So, what percentage of students does not satisfy that requirement? The answer is 96.08%. In other words, only 3.92% satisfy that requirement.

The combined math and verbal scores for students taking a national standardized examination for college admission, is normally distributed with a mean of 630 and a standard deviation of 200. If a college requires a student to be in the top 25 % of students taking this test, what is the minimum score that such a student can obtain and still qualify for admission at the college?

In this case [tex]\\ \mu = 630[/tex], and [tex]\\ \sigma = 200[/tex].

We are asked here for the percentile 75%. That is, for students having a score above this percentile. So, what is the value for z-score whose percentile is 75%? This value is z = 0.674 in the Standardized Normal Distribution, obtained from any Table of the Normal Distribution.

Well, having this information:

[tex]\\ 0.674 = \frac{x-630}{200}[/tex], then

[tex]\\ 0.674 * 200 = x-630[/tex]

[tex]\\ (0.674 * 200) + 630 = x[/tex]

[tex]\\ x = 764.8 [/tex]

Then, the minimum score that a student can obtain and still qualify for admission at the college is x = 764.8. In other words, any score above it represents the top 25% of all the scores obtained and 'qualify for admission at the college'.

[...] The collagen amount was found to be normally distributed with a mean of 69 and standard deviation of 5.9 grams per milliliter.

In this case [tex]\\ \mu = 69[/tex], and [tex]\\ \sigma = 5.9[/tex].

What percentage of compounds have an amount of collagen greater than 67 grams per milliliter?

z = [tex]\frac{67-69}{5.9} = -0.3389[/tex]. The z-score tells us the distance from the mean of the population, then this value is below 0.3389 from the mean.

What is the value of the percentile for this z-score? That is, the percentage of data below this z.

We know that the Standard Distribution is symmetrical. Most of the tables give us only positive values for z. But, because of the symmetry of this distribution, z = 0.3389 is the distance of this value from the mean of the population. The F(z) for this value is 0.6331 (actually, the value for z = 0.34 in a Table of the Normal Distribution).

This value is 0.6331-0.5000=0.1331 (13.31%) above the mean. But, because of the symmetry of the Normal Distribution, z = -0.34, the value F(z) = 0.5000-0.1331=0.3669. That is, for z = -0.34, the value for F(z) = 36.69%.

Well, what percentage of compounds have an amount of collagen greater than 67 grams per milliliter?

Those values greater that 67 grams per milliliter is 1 - 0.3669 = 0.6331 or 63.31%.

What percentage of compounds have an amount of collagen less than 78 grams per milliliter?

In this case,

z = [tex]\frac{78-69}{5.9} = 1.5254[/tex].

For this z-score, the value F(z) = 0.9357 or 93.57%. That is, below 78 grams per milliliter, the percentage of compounds that have an amount of collagen is 93.57%.

What exact percentage of compounds formed from the extract of this plant fall within 3 standard deviations of the mean?

We need here to take into account three standard deviations below the mean and three standard deviations above the mean. All the values between these two values are the exact percentage of compounds formed from the extract of this plant.

From the Table:

For z = 3, F(3) = 0.9987.

For z = -3, F(-3) = 1 - 0.9987 = 0.0013.

Then, the exact percentage of compounds formed from the extract of this plant fall within 3 standard deviations of the mean is:

F(3) - F(-3) = 0.9987 - 0.0013 = 0.9974 or 99.74%.

A z-score is used to determine how many standard deviations a value is from the mean. A score of 720 on the SAT is 1.74 standard deviations above the mean, whereas a score of 692.5 is 1.5 standard deviations above the mean. To compare scores from different tests, like the SAT and ACT, you compute the z-scores for each and compare them.

In statistics and probability theory, when comparing values from different normal distributions, one useful tool is the z-score. It informs us of how many standard deviations an element is from the mean of its distribution. A z-score is calculated using the formula Z = (X - μ) / σ, where X is the value in question, μ is the mean, and σ is the standard deviation.

To calculate a z-score for an SAT score of 720 when the mean is 520 and the standard deviation is 115:
Z = (720 - 520) / 115 = 200 / 115 ≈ 1.74.

This z-score of approximately 1.74 implies that the score of 720 is 1.74 standard deviations above the mean SAT score.

To find an SAT score that is 1.5 standard deviations above the mean:
X = μ + 1.5σ = 520 + 1.5 × 115 = 520 + 172.5 = 692.5.

So, a score of approximately 692.5 is 1.5 standard deviations above the mean, indicating a well-above-average performance.

Comparing an SAT math score of 700 and an ACT score of 30 with respect to their respective mean and standard deviation:

Based on their z-scores, the individual with the ACT score performed slightly better relative to others who took the same test than the individual who took the SAT math test.

Answer:

Fourth degree polynomial (aka: quartic)

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

Work Shown:

There isnt much work to show here because we can use the fundamental theorem of algebra. The fundamental theorem of algebra states that the number of roots is directly equal to the degree. So if we have 4 roots, then the degree is 4. This is assuming that there are no complex or imaginary roots.

-------------------

If you want to show more work, then you would effectively expand out the polynomial

(x-m)(x-n)(x-p)(x-q)

where

m = 4, n = 2, p = sqrt(2), q = -sqrt(2)

are the four roots in question

(x-m)(x-n)(x-p)(x-q)

(x-4)(x-2)(x-sqrt(2))(x-(-sqrt(2)))

(x-4)(x-2)(x-sqrt(2))(x+sqrt(2))

(x^2-6x+8)(x^2 - 2)

(x^2-2)(x^2-6x+8)

x^2(x^2-6x+8) - 2(x^2-6x+8)

x^4-6x^3+8x^2 - 2x^2 + 12x - 16

x^4 - 6x^3 + 6x^2 + 12x - 16

We end up with a 4th degree polynomial since the largest exponent is 4.