The Pythagorean Theorem - Quiz - Level H
Question 6
What is the length of the diagonal of the rectangle?

The half-life of Potassium-42 can be calculated using the following formula:

t1/2 = (ln 2) / λ

where t1/2 is the half-life, ln is the natural logarithm, and λ is the decay rate.

Substituting the values given, we get:

t1/2 = (ln 2) / 0.055

t1/2 ≈ 12.6 hours

Therefore, the original quantity of Potassium-42 will be halved in approximately 12.6 hours.

It will take approximately 13.51 hours for the original quantity of potassium 42 to be halved with a decay rate of 5.5% per hour.

To find the number of hours it takes for potassium 42 to be halved with a decay rate of 5.5% per hour, we can use the formula:

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19 is the range and 14 is the IQR. So, "The range is 19, and the IQR is 14" is the appropriate response.

The range in statistics is the distinction between the greatest and lowest values within a set of data. It is a gauge of variation that reveals the degree to which values within a dataset are dispersed.

You simply take the difference between the two values to determine the range. The range is given by: When the dataset, for instance, has the values 2, 3, 5, 8, et 10, the lowest value is 2, and the highest is 10.

Range = Highest value – Lowest value = 10 – 2 = 8 .The range in this instance is hence 8.

given

The range is the discrepancy between the data set's maximum and minimum values.

The box plot shows that the range is between 40 and 59, with 40 being the smallest value and 59 being the maximum.

Maximum value - Minimum value equals 59 - 40, which equals 19.

You may determine the values of Q1 and Q3 to be roughly 45 and 59, respectively, using the box plot.

IQR = Q3 - Q1 = 59 - 45 = 14

When the IQR is 14 and the range is 19, the answer is:

19 is the range and 14 is the IQR. So, "The range is 19, and the IQR is 14" is the appropriate response.

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Answer:

Step-by-step explanation:

The number of toy cars to the number of trucks is 5/2,the number of toy trucks is 2/7 of the total number of toy cars and toy trucks and  the number of toy cars is 5/7 of the total number of toy cars and toy trucks.

Let's represent the number of toy cars as 'c' and the number of toy trucks as 't'.

From the problem, we know that:

t = 2/5c

Express the number of toy cars as a fraction of the number of toy trucks:

c = 5/2t

c/t = (5/2t)/t = 5/2

Therefore, the number of toy cars is 5/2 times the number of toy trucks.

The total number of toy cars and toy trucks is:

c + t

Substituting t = 2/5c, we get:

c + (2/5)c = 7/5c

So the ratio of the number of toy trucks to the total number of toy cars and toy trucks is:

t/(c + t) = (2/5c) / (7/5c) = 2/7

Therefore, the number of toy trucks is 2/7 of the total number of toy cars and toy trucks.

The fraction of the total number of toy cars and toy trucks that is the number of toy cars is:

c / (c + t)

Substituting t = 2/5c, we get:

c / (c + 2/5c) = c / (7/5c) = 5/7

Therefore, the number of toy cars is 5/7 of the total number of toy cars and toy trucks.

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Ally will have 91.148 meters of tape in all if Ally needs 90 meters of tape for a project.

What is Algebraic expression ?

An algebraic expression is a mathematical phrase that can include numbers, variables, and operators (such as addition, subtraction, multiplication, and division), as well as grouping symbols like parentheses.

Ally needs a total of 90 meters of tape for her project. She bought three rolls of tape, one with 50 meters, another with 22.86 meters, and the last one with 18.288 meters. To find the total amount of tape she has, she needs to add the length of all three rolls together.

Adding the length of the first two rolls, we get:

50 meters + 22.86 meters = 72.86 meters

Then adding the length of the third roll to the sum above, we get:

72.86 meters + 18.288 meters = 91.148 meters

Therefore, Ally will have 91.148 meters of tape in all.

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Equation for the total number of dessert packages needed for a math club meeting with cookies and brownie bites; 5 packages of brownie bites and 15 packages of cookies needed to serve one dessert per person for a total of 75 attendees.

We can use the following equation to calculate the total number of packages of each dessert that the club treasurer must purchase:

Total packages of brownie bites (b) + Total packages of cookies (c) = Total number of attendees (75)

We know that brownie bites come in packages of 15, so the total number of packages of brownie bites the treasurer needs to purchase would be:

b = ceil(75/15) = 5

where "ceil" refers to the ceiling function, which rounds to the nearest integer. In this case, we need at least 5 packages of brownie bites to have one dessert per person.

To find the number of packages of cookies needed, we can substitute the value of b into the equation:

c + 12b = 75

c + 12(5) = 75

c + 60 = 75

c = 15

However, this value is not reasonable since 15 packages of cookies would provide 180 cookies, which is more than 2 cookies per person. Therefore, the treasurer would need to adjust the number of packages of cookies based on the actual number of cookies in each package to ensure that they have exactly one dessert per person.

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A player would need to make a goal in approximately 18.85 minutes or faster to be sponsored by the puck company.

To determine how fast a player needs to make a goal to be sponsored by the puck company, we'll need to find the

time cutoff for the top 10% quickest goals.

Step 1: Determine the z-score for the 90th percentile (since we want the top 10%).

We can use a standard normal distribution table (or a calculator) to find that the z-score for the 90th percentile is

approximately 1.28.

Step 2: Use the z-score formula to find the time cutoff.

The formula is: Z = (X - μ) / σ

Where Z is the z-score, X is the time cutoff we're looking for, μ is the mean (12.56 minutes), and σ is the standard

deviation (4.91 minutes).

Step 3: Solve for X.

Rearrange the formula to solve for X: X = Z × σ + μ

Plug in the values: X = 1.28 × 4.91 + 12.56

X ≈ 6.29 + 12.56

X ≈ 18.85

So, a player would need to make a goal in approximately 18.85 minutes or faster to be sponsored by the puck company.

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Using the line plot graph, we can conclude that the 5th distance that is 5/4 is the most common distance from home to the grocery store.

A line plot is a type of graph that shows the frequency of each value together with the data using symbols above a frequency number-line. It is used to organise the information simply and is very easy to comprehend.

In the given graph we can see that the distance values are given.

So, in the first instance the number of times the distance is used = 2.

The 2nd distance has been used = 3.

The 3rd distance has been used = 2.

The 4th distance has been used = 4.

The 5th distance has been used = 5.

The 6th distance has been used = 2.

The 7th distance has been used = 2.

The 8th distance has been used = 2.

Hence, we can conclude that the 5th distance that is 5/4 is the most common distance from home to the grocery store.

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An equation which is the inverse of y = 2x² - 8 include the following: B. [tex]y= \pm \sqrt{\frac{x+8}{2} }[/tex]

In Mathematics, an inverse function simply refers to a type of function that is obtained by reversing the mathematical operation in a given function (f(x)).

In order to determine the inverse of any function, you should swap both the input value (x-value) and output value (y-value) as follows;

f(x) = y = 2x² - 8

x = 2y² - 8

2y² = x + 8

By dividing both sides of the equation by 2, we have the following:

y² = (x + 8)/2

By taking the square root of both sides of the equation, we have the following:

y = √[(x + 8)/2]

[tex]y= \pm \sqrt{\frac{x+8}{2} }[/tex]

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For the given statements the correct answer is B |6| > |−8| and C |−8| < |3|.

What is statement?

In mathematics, a statement is a declarative sentence that is either true or false, but not both. Statements are often expressed using variables, symbols, or mathematical notation, and they can describe mathematical objects, relationships, properties, or operations.

For the first question, the correct response is:

B |6| > |−8|

This is true because the absolute value of a number is always positive, and it represents the distance of that number from zero on the number line. So, the absolute value of 6 is greater than the absolute value of -8, because 6 is further away from zero than -8 on the number line.

For the second question, the correct response is:

C |−8| < |3|

This is true because the absolute value of -8 is greater than the absolute value of 3. Absolute values tell us the distance from zero on the number line, and the distance between -8 and 0 is greater than the distance between 3 and 0.

Therefore for the given statements the correct answer is B |6| > |−8| and C |−8| < |3|.

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We can use polynomial long division to divide (4x^4 - 4x^2 - x - 3) by (2x^2 - 3):

         2x^2 + 0x - 1

    ---------------------

2x^2 - 3 | 4x^4 + 0x^3 - 4x^2 - x - 3

         4x^4 - 6x^2

         -----------

                 2x^2 - x

                 2x^2 - 3

                 -------

                      x - 3  

Therefore,

(4x^4 - 4x^2 - x - 3)/(2x^2 - 3) = 2x^2 + 0x - 1 with a remainder of (x - 3)/(2x^2 - 3).

So the final answer is

2x^2 - 1 + (x - 3)/(2x^2 - 3)

Answer:

1) 100m^2 3) 96cm^2 4) 36 inches

Step-by-step explanation:

1) 10*10=100[tex]m^{2}[/tex]

2)

3) 12*8=96[tex]cm^{2}[/tex]

4)9*4=36 inches

Step-by-step explanation:

When the diver hits the water at Height = 0 , the diver is no longer in the air .

0  -16t^2 + 10t + 26      Use Quadratic Formula  

                                       a = -16        b = 10        c= 26

   to find the positive value of t  = 1.625 s

The value of y is 10. we canget this answer by using  fact  ratios of the corresponding sides are equal. Using this fact, we can set up the proportion

what is corresponding sides?

In geometry, corresponding sides are sides in two or more figures that are in the same relative position or location. For example, if you have two triangles ABC and DEF, and AB is corresponding to DE, BC is corresponding to EF, and AC is corresponding to DF, then AB and DE are corresponding sides, BC and EF are corresponding sides, and AC and DF are corresponding sides.

In the given question,

To solve for y, we need to use the fact that line wt and line ye are parallel to each other. This means that the ratios of the corresponding sides are equal. Using this fact, we can set up the following proportion:

wt/ty = we/yc

Substituting the given values, we get:

(8 + x + 6)/16 = 10/y

Simplifying and solving for y, we get:

y = (160)/(14 + x)

Now we need to use the fact that the sum of the lengths of two sides of a triangle is always greater than the length of the third side. Using this fact for triangle ATY, we get:

AT + TY > AY

Substituting the given values, we get:

x + 6 + 16 > y

Simplifying and substituting y with its expression, we get:

x + 22 > (160)/(14 + x)

Multiplying both sides by (14 + x), we get:

x² + 22x + 14x + 352 > 160

Simplifying and rearranging, we get:

x² + 36x - 192 > 0

Now we can solve this quadratic inequality to get the range of possible values for x:

(x - 4)(x + 48) > 0

The solutions are x < -48 or x > 4. Since x represents a length, we can ignore the negative solution. Therefore, we have:

x > 4

Substituting this value into the expression we derived for y, we get:

y = (160)/(14 + x) = (160)/(14 + 4) = 10

Therefore, the value of y is 10.

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The principal should then survey the selected students to gather their opinions on the new dress code.

To answer your question, the principal should choose a sample of students that is both representative and random. This will ensure that the opinions gathered are an accurate reflection of the entire high school population.

How the principal can achieve this:
Determine the sample size:

The principal should decide on a suitable sample size based on the total number of students in the high school.

A larger sample size will yield more accurate results, but it may not be feasible to survey too many students.

To ensure representativeness, the principal should divide the high school population into different strata, such as grade levels or other relevant demographic categories.

This will help ensure that the sample includes a balanced mix of students from different backgrounds and groups.
Select a random sample:

Within each stratum, the principal should choose a random sample of students.

The principal should then survey the selected students to gather their opinions on the new dress code.
By following these steps, the principal can select an appropriate sample of students that will provide an accurate representation of the high school's overall opinion on the new dress code.

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The expression for the probability of getting at least one six in n throws of fair dice is 1 - (5/6)ⁿ.

The complement rule states that the probability of an event occurring is equal to 1 minus the probability of the event not occurring.

In this problem, we use the complement rule to find the probability of getting at least one six in n throws of a fair dice by subtracting the probability of not getting any sixes in n throws from 1.

Here we have

A fair dice is thrown n times.

The probability of getting at least one six in n throws of a fair dice is equal to 1 minus the probability of not getting any sixes in n throws.

The probability of not getting a six on any one throw  = 5/6

[ Since there are 5 possible outcomes that are not six and 6 possible outcomes in total ]

Therefore,

The probability of not getting any sixes in n throws is (5/6)ⁿ.

So, the probability of getting at least one six in n throws is:

P(at least one six) = 1 - P(no sixes) = 1 - (5/6)ⁿ

Therefore,

The expression for the probability of getting at least one six in n throws of fair dice is 1 - (5/6)ⁿ.

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Answer:

6 degrees Celsius

Step-by-step explanation:

The temperature at 2am was 3.7 degrees C.

Since the temp dropped 1.15 degrees each hour, at 1am the temp was 3.7+1.15 = 4.85 degrees

So at midnight, 12am, the temp was 4.85+1.15 = 6 degrees Celsius

Answer:

x=21

Step-by-step explanation:

Answer:

Sure, I can help you with that geometry problem! To find the center of a circle given two points on its circumference, we need to find the perpendicular bisector of the line segment connecting those two points. Let's first find the midpoint of line segment PQ. Midpoint M = ( (1+7)/2 , (-2+1)/2 ) M = (4, -1/2) The slope of the line PQ is: Slope of PQ = (1-(-2)) / (7-1) Slope of PQ = 3/2 The slope of the perpendicular bisector of PQ is the negative reciprocal of the slope of PQ. Slope of perpendicular bisector = -2/3 Now we can find the equation of the perpendicular bisector of PQ in point-slope form using the midpoint M:

Answer:

where is the pic

Step-by-step explanation:

Using area formula, we can find that the diagonal length is not 10ft. So, Jordan should not buy the pen.

A square's area is a measurement of the volume or surface that it takes up. It is equivalent to the two sides' combined lengths. Since the product of a square's two sides determines its size, the area is expressed in square units.

Here in the question,

Jordan wants an area that is of 35 ft² and 10ft diagonal length.

Now in the figure,

Dimensions of the pen are:

Length, l = 6ft.

Width, w = 6ft.

Height, h = 6ft.

So, it's a square.

Area that the puppy pen will provide = l × w

= 6 × 6

=36ft².

Diagonal of the pen is: (As per Pythagoras theorem)

√ (6² + 6²)

= √ (36+36)

= √ 72

= 8.48

The pen does provide 35ft² of area as it has 36ft², but it does not have a diagonal length of 10ft.

Therefore, Jordan should not buy the puppy pen.

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The complete question is:

Jordan wants a collapsible puppy pen that gives his puppy at least 35 square feet of area and at Least 10 feet of diagonal length. Should Jordan buy the pen shown? Explain.

We may determine that the diagonal length is not 10 feet using the area formula. Jordan shouldn't buy the pen.

The volume or area that a square occupies is determined by its area. It is equal to the sum of the lengths of the two sides. The area is given in square units since a square's size is determined by the product of its two sides.

Jordan requests a 35ft² square foot area with a 10-foot diagonal.

Now look at the figure.

The pen is the following size:

Length, l = 6ft.

Width, w = 6ft.

Height, h = 6ft.

So, it's a square.

The area that the puppy kennel will offer

= 6 × 6

=36ft².

Diagonal of the pen is:

= √ (6² + 6²)

= √ (36+36)

= √ 72

= 8.48

The pen has 36ft² square feet, so it does offer 35ft² square feet, but it lacks a 10 foot diagonal.

Jordan shouldn't purchase the Puppy pen as a result.

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The complete question is attached below,

The required probability of randomly selected people among 11% of left handed are,

Having some lefties (≥1) among the 5 people is 0.4416.

Having exactly 3 lefties in the group of 5 people is 0.01054.

Percentage of people left handed = 11%

Number of people selected at random = 5

Probability that there are some lefties (≥1) among the 5 people,

Probability of the complement event, which is that there are no lefties in the group.

Probability of selecting a right-handed person is

=1-0.11

=0.89.

Probability that all 5 people are right-handed is,

P(all right-handed)

= (0.89)^5

= 0.5584

Probability of there being some lefties (≥1) is the complement of this,

P(some lefties)

= 1 - P(all right-handed)

= 1 - 0.5584

= 0.4416

Probability of there being some lefties (≥1) among the 5 people is 0.4416.

Probability that there are exactly 3 lefties in the group of 5 people,

Use the binomial distribution formula,

P(X=3) = (⁵C₃) × (0.11)^3 × (1-0.11)^2

where X is the number of lefties in the group.

Using the binomial coefficient formula

ⁿCₐ= n! / (a! * (n-a)!)

P(X=3)

= (5!/(3!*(5-3)!)) × (0.11)^3×(0.89)^2

= 0.01054

Probability of there being exactly 3 lefties in the group of 5 people is 0.01054

Therefore, the probability of having some lefties ( ≥ 1) among the 5 people and  exactly 3 lefties in the group is equal to 0.4416 and 0.01054 respectively.

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The above question is incomplete, the complete question is:

Assume that 11% of people are left-handed. if we select 5 people at random, find the probability of each outcome described below, rounded to four decimal places:

a. There are some lefties ( ≥ 1) among the 5 people.

b. There are exactly 3 lefties in the group.

That we know that Mandy sells 12 tickets, we can use the first equation to find out how many tickets Alonzo sells: a = 3n - 5 = 3(12) - 5 = 31

Alonzo sells 31 tickets.

What is algebra?

Algebra is a branch of mathematics that deals with mathematical operations and symbols used to represent numbers and quantities in equations and formulas. It involves the study of variables, expressions, equations, and functions.

Let's use algebra to solve the problem:

Let's start by defining some variables:

Let n be the number of tickets Mandy sells.

Let a be the number of tickets Alonzo sells.

We know that:

Alonzo sells 5 tickets less than three times what Mandy sells. So we can write:

a = 3n - 5

Together they sell a total of 43 tickets. So we can write:

a + n = 43

We have two equations with two variables, so we can solve for a by substituting the first equation into the second equation:

(3n - 5) + n = 43

4n - 5 = 43

4n = 48

n = 12

Now that we know that Mandy sells 12 tickets, we can use the first equation to find out how many tickets Alonzo sells:

a = 3n - 5 = 3(12) - 5 = 31

Therefore, Alonzo sells 31 tickets.

So the answer is: Alonzo sells 31 tickets.

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The volume of the given pyramid is 86 in³ respectively.

A three-dimensional shape is a pyramid.

A pyramid's flat triangular faces unite at a common point known as the apex and have a polygonal base.

The bases are joined to the peak to create a pyramid.

The lateral face is a triangle face formed by the connection of each edge of the base to the apex.

A three-dimensional figure is a pyramid.

Its base is a flat polygon.

The remaining faces are all triangles and are referred to as lateral faces.

So, the formula for the volume of the pyramid is:

V = lwh/3

Now, insert values as follows:

V = lwh/3

V = 7.4*7*5/3

V = 86.33333

Rounding off: 86 in³

Therefore, the volume of the given pyramid is 86 in³ respectively.

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Answer: A=78

Step-by-step explanation: 936/12=78

936 eggs were produced last year by the sizable dairy farm. 78 dozen eggs are produced by this farm. The answer is option (A).

The study of symbols for math and the procedures for using them to solve equations and comprehend the relationships between variables is the subject of the branch of mathematics known as algebra. When representing quantities and their relationships using letters and symbols, algebra uses graphs, functions, and expression manipulation in addition to equation and inequalities solving.

Numerous branches of science, technology, economics, and other disciplines make considerable use of algebraic concepts and methods. It is a fundamental area of mathematics that is frequently studied as a pre-requisite for more difficult maths and science courses.

Algebraic concepts that are important to know include:

Addition, subtraction, multiplication, and division are operations using numbers and variables.

Using variables to solve equations and inequalities

Algebraic expressions manipulation and simplification

Knowing how to graph both linear and nonlinear functions

Utilising matrices and vectors

studying trigonometric, rational, and polynomial functions

We must divide the total number of eggs by 12 to determine the number of dozen eggs produced.

936/12 = 78

We must divide the total number of eggs by 12 to determine the number of dozen eggs produced.

The answer is A) 78.

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Answer:

Step-by-step explanation:

A = 2πrh

Find radius

Diagonal = 14 so radius 14 : 2 = 7

A = 2πrh

A = 2π×7×2

A = 2π x 14

A = 28π meters  or 87.92 meters

Answer:

The interest rate is already given as 16%. If you want to calculate the total interest earned on the investment, you can use the formula:

Simple Interest = (Principal x Rate x Time)/100

Where:

Principal = $1,000

Rate = 16%

Time = 20 years

Substituting the values, we get:

Simple Interest = (1000 x 16 x 20)/100 = $3,200

Therefore, the total interest earned on the investment is $3,200.

Step-by-step explanation:

C. Angle bisector can be constructed by drawing an angle on tracing paper and then folding the paper so that the rays forming the angle lie upon each other. When you fold the paper, the crease that is formed will be the angle bisector of the original angle.

The following statements are accurate conclusions:

The null hypothesis is: The proportion of California college students who currently drink alcohol is the same as the proportion nationwide (p = 0.60).

The alternative hypothesis is: The proportion of California college students who currently drink alcohol is different from the proportion nationwide (p ≠ 0.60).

p in the hypotheses represents the proportion of college students who currently drink alcohol.

The Z-score is calculated as:

Z = (0.66 - 0.60) / 0.023 = 2.61

The P value can be obtained from a standard normal distribution table or calculator:

P = 0.0045 (for a two-tailed test)

We reject the null hypothesis and come to the conclusion that there is a statistically significant difference between the proportion of drinkers in the population of California college students and the proportion of drinkers in the American adult population because the P value (0.0045) is less than the significance level (which is not stated in the question) (0.66).

For the given question:

The following statements are accurate conclusions:

The result that "There is a 0.06 difference in the proportion of drinkers when we compare the US adult population to the population of CA college students" is untrue since the hypothesis test is not concerned with proportional differences.

The complete question is:-

Normal Distribution Calculator Drag the orange flag left or right to change the Z-score Alternatively, enter a Z-score into the textbox and click enter. Enter 8 -1 Z-score: -1.000 0 Z-score The area to the left of the Z value is 0.1587 The area to the right of the Z value is 0.9413 Question 1 6 pts California College Students Who Drink According to the Centers for Disease Control and Prevention, 60% of all American adults ages 18 to 24 currently drink alcohol. Is the proportion of California college students who currently drink alcohol different from the proportion nationwide? A survey of 450 California college students indicates that 66% currently drink alcohol. The null hypothesis is (Select] The alternative hypothesis is Select] p in the hypotheses represents [Select) The standard error is 0.60(1-0.60) 450 = 0.023 The Z-score is (Select] The P value is Select) The conclusion is Select Question 2 4 pts For the previous example, which statements are accurate conclusions? The proportion of drinkers in the population of CA college students is not 0.60. There is a statistically significant difference between the proportion of drinkers in the American adult population (0.60) and the proportion of drinknes in the population of CA college students A larger proportion of CA college students drink alcohol compared to the adults nationwide When we compare the American adult population to the population of CA college students, there is a 0.06 difference in the proportion of drinkers

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