No, there cannot be a right triangle with sides of lengths 1, 2, and 3. This is because of the Pythagorean theorem.
The theorem states that in a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. Mathematically, this can be written as:
[tex]c^2 = a^2 + b^2[/tex]
where c is the length of the hypotenuse, and a and b are the lengths of the other two sides.
If we substitute the values of 1, 2, and 3 for a, b, and c, respectively, we get:
[tex]3^2 = 1^2 + 2^2[/tex]
which simplifies to:
9 = 1 + 4
This is not true, so the sides 1, 2, and 3 do not form a right triangle.
However, there exist right triangles with consecutive natural numbers as their side lengths. One such example is the Pythagorean triple (3, 4, 5), where:
[tex]3^2 + 4^2 = 9 + 16 = 25 = 5^2[/tex]
This is a right triangle where the lengths of the sides are consecutive natural numbers. There are infinitely many such Pythagorean triples, and they can be generated using the formula:
[tex]a = 2mn, b = m^2 - n^2, c = m^2 + n^2[/tex]
where m and n are any two positive integers with m > n.
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12. what two properties must a variant expression have to guarantee that a recursive function terminates?
A variant expression is a mathematical expression used to analyze the progress of a recursive function and ensure its termination. To guarantee that a recursive function terminates, a variant expression must have two essential properties which are Boundedness and monotonicity.
1. Boundedness: A variant expression must be bounded, which means that it has an upper or lower limit. This limit is often a non-negative integer value, such as 0. The boundedness property ensures that the variant expression cannot grow indefinitely, which would lead to non-termination of the recursive function.
2. Monotonicity: A variant expression must be monotonically decreasing, meaning that its value strictly decreases with each recursive call. Monotonicity ensures that the variant expression is always moving closer to the bound, eventually reaching the limit and causing the function to terminate.
To guarantee the termination of a recursive function, follow these steps:
Step 1: Identify a suitable variant expression for the function. The expression should be related to the problem size or input parameters of the function and should reflect the progress towards the function's termination.
Step 2: Verify the boundedness property of the variant expression. Ensure that the expression has an upper or lower limit, typically a non-negative integer, which it cannot exceed.
Step 3: Verify the monotonicity property of the variant expression. Check that the expression's value strictly decreases with each recursive call, moving closer to the bound.
Step 4: Ensure that the recursive function terminates when the variant expression reaches the bound. This condition is usually implemented as the base case of the function, which prevents further recursive calls and initiates the return process.
In summary, a variant expression must have boundedness and monotonicity properties to guarantee that a recursive function terminates. By verifying these properties and incorporating them into the recursive function, you can ensure that the function will reach its termination condition and not result in infinite recursion.
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a person eats 8 grams of protein, 25 grams of carbohydrates, 6 grams of fat and 4 grams of alcohol. what is the total energy value (in calories) from these stated amounts?
The total energy value from these stated amounts is 214 calories.
To calculate the total energy value (in calories) from the stated amounts of protein, carbohydrates, fat, and alcohol, we can follow these steps:
1. Protein: Multiply the amount of protein (8 grams) by the energy value of protein (4 calories/gram).
2. Carbohydrates: Multiply the amount of carbohydrates (25 grams) by the energy value of carbohydrates (4 calories/gram).
3. Fat: Multiply the amount of fat (6 grams) by the energy value of fat (9 calories/gram).
4. Alcohol: Multiply the amount of alcohol (4 grams) by the energy value of alcohol (7 calories/gram).
5. Add the resulting values from steps 1-4 to find the total energy value (in calories).
Applying these steps:
1. Protein: 8 grams * 4 calories/gram = 32 calories
2. Carbohydrates: 25 grams * 4 calories/gram = 100 calories
3. Fat: 6 grams * 9 calories/gram = 54 calories
4. Alcohol: 4 grams * 7 calories/gram = 28 calories
5. Total energy value: 32 + 100 + 54 + 28 = 214 calories.
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PLEASE HELP ILL GIVE BRAINLIEST
Step-by-step explanation:
A C and H would all also be 120 degrees due to corresponding and vertical angles
what is the maximum volume in cubic inches of an open box to be made from a 10-inch by 20-inch piece of cardboard by cutting out squares of equal sides from the four corners and bending up the sides? your work must include a statement of the function and its derivative. give one decimal place in your final answer.
The maximum-volume of open box which could made from a "10-inch" by "20-inch" piece of cardboard by cutting out squares of equal sides from 4 corners is 192.5 inch³.
We know that "Volume" of box is ⇒ V = L × W × H.
The dimensions of a piece of cardboard is 10 inches by 20 inches.
Let x = height of box, x will also be side of square taken out of each corner.
The length of box = 20 - 2x; width of box = 10 - 2x and
So, equation of Volume is ⇒ V = (20-2x) × (10-2x) × (x),
⇒ V = 4x³ - 60x² + 200x
On differentiating both sides, to get "dv/dx" :
We get,
⇒ dv/dx = 12x² - 120x + 200
Using general formula for finding roots of equation,
We get, x = 7.886 and x = 2.113,
Since 7.886 × 2 = 15.772 inches, this is greater than the 10 inch side,
So, x = 2.113 inches.
We have cut "2.113 inches" from each-corner to get maximum volume.
The sizes of cubes are :
⇒ Length = 20 - (2 × 2.113) = 15.774,
⇒ width = 10 - (2 × 2.113) = 5.774
⇒ Height = 2.113
So, volume = 15.774 × 5.774 × 2.113 = 192.5 cubic inches.
Therefore, the volume of cube is 192.5 inch³.
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5. Steve and Scott are playing a game of cards with a standard deck of playing cards. Steve deals Scott a black king.
What is the probability that Scott's second card will be a red card?
Given that Steve handed him a black king, the probability of Scott's second card being a red card is 13/25, or around 0.52.
What is probability?Probability is a measure of how likely an event is to occur. It is represented by a number between 0 and 1, with 0 representing a rare event and 1 representing an inescapable event. Switching a fair coin and coin flips has a chance of 0.5 or 50% since there are two equally likely outcomes. (Heads or tails). Probabilistic theory is an area of mathematics that studies random events rather than their attributes. It is applied in many disciplines, including statistics, economics, science, and engineering.
A conventional deck of playing cards has 52 cards, including two black kings. When Steve deals a black king to Scott, the deck has 51 cards, 26 of which are red.
We only need to analyse the likelihood of obtaining a red card on the second draw because we know Scott's initial card is a black king. There are 50 cards remaining in the deck, 26 of which are red. As a result, the likelihood that Scott's second card will be a red card is:
P(red card drawn on second draw) = 26/50 = 13/25
Given that Steve handed him a black king, the likelihood of Scott's second card being a red card is 13/25, or around 0.52.
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Which of the following BEST shows the values in the domain when the function is increasing? A. between 0 and 6 B. between 3 and 9 C. between 3 and 6 D. between 6 and 9
Option D. between 6 and 9 is correct.
Since we don't have the specific function, we cannot definitively determine which interval shows the function is increasing.
However, based on the information given, we can say that the function increases within one of these intervals.
It is important to know the specific function to identify the correct interval.
To determine the correct answer, we first need to understand the terms "domain," "function," and "increasing."
- Domain: The set of all possible input values (x-values) for a given function.
- Function: A mathematical relationship between input (x) and output (y) values, often represented as y = f(x).
- Increasing: A function is increasing when the output (y-value) increases as the input (x-value) increases.
Now, let's analyze the given options:
A. between 0 and 6
B. between 3 and 9
C. between 3 and 6
D. between 6 and 9
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Question:
Please help me with this question?
Answer:
Step-by-step explanation:
The expanded form 12a - b in factor form
Answer: (12a + b)(12a - b)
Step-by-step explanation:
1. Begin by writing the expression in expanded form: 12a - b
2. Rewrite the expression as 12a + (-b)
3. Factor out 12a from the expression: 12a(1 + (-b))
4. Factor out 1 from the expression: (12a)(1 + (-b))
5. Rewrite (1 + (-b)) as (1 - b): (12a)(1 - b)
6. Rewrite (12a)(1 - b) as (12a + b)(12a - b): (12a + b)(12a - b)
Explain the connection between <2 and
The similarity between <2 and <B is that both arcs have a 65° measure.
What are Additional Angles?When two angles add up to 180 degrees, they are said to be each other's auxiliary angles.
For instance, since x + y = 180°, x and y are complementary angles to one another.
The triangle's angles are 50° and 65°.
50° and 65° are the angles on the path.
We are aware that a triangle's total number of angles is 180°. Consequently, the B measure is
50° + 65° + ∠B = 180°
∠B = 180° - (50° + 65°)
∠B = 65°
So, 65 degrees is the gauge of the B.
The total number of angles on the specified line will be due to the fact that all angles are extra angles, and will be 180 degrees.
∠2 + 50° + 65° = 180°
∠2 = = 65°
As a result, 65° is the measure of the 2.
As we can see, both arcs have a measure of 65 degrees.
Therefore, the relationship between 2 and B is that both angles have a value of 65°.
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when describing sets of numbers using interval notation, when do you use a parenthesis and when do you use a bracket? when the beginning value or ending value is ---select--- the set, a square bracket is used. a parenthesis is used to show that the beginning value or ending value is ---select--- . a parenthesis is always used with ---select--- .
When the beginning value or ending value is closing the set, a square bracket is used. A parenthesis is used to show that the beginning value or ending value is open. A parenthesis is always used with open intervals.
When we use an interval notation to describe sets of numbers, parentheses, and brackets have different meanings. Parentheses "(" and ")" describes an open interval, which means that the endpoints are not included in the set. For example, the interval (-3,5) includes all real numbers greater than -3 and less than 5, but it does not include -3 or 5 themselves. On the other hand, brackets "[" and "]" denote a closed interval, which means that the endpoints are included in the set. For example, the interval [-3,5] includes all real numbers greater than or equal to -3 and less than or equal to 5. If the endpoint is included in the set, use a bracket; if it is not, use a parenthesis.
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PLEASE HELP MEEEEE
There’s multiple answers!
Step-by-step explanation:
scalene === all of the sides are different lengths
acute ==== all of the angles are < than 90 degrees
that's all
Need help with this question please
Step-by-step explanation:
For denominator:
x²-5x -24 = x² -(8-3)x -24
= x² -8x+3x-24 = x(x-8) +3(x-8)
Taking common (x-8): = (x-8)(x+3)
So option d is accurate.
And for numerator, 7x+56 =7(x+8)
So, option a is accurate.
The line plot displays the cost of used books in dollars.
A horizontal line starting at 1 with tick marks every one unit up to 9. The line is labeled Cost in Dollars, and the graph is titled Cost of Used Books. There is one dot above 2, 4, 8, and 9.There are two dots above 6 and 7. There are three dots above 3.
Which measure of center is most appropriate to represent the data in the graph, and why?
The mean is the best measure of center because there are no outliers present.
The mean is the best measure of center because there are outliers present.
The median is the best measure of center because there are no outliers present
Because there are no outliers in the data set, the mean is the best indicator of the centre in this particular situation.
what is outliers ?An outlier in statistics is an incident that deviates greatly from the predicted range of values for a given dataset. There are several different reasons why outliers can happen, including measurement errors, data entry issues, or genuinely odd numbers in the population getting measured. If they are not handled properly, outliers can significantly affect statistical analysis and skew results. For instance, outliers have a big impact on measurements of variability like the standard deviation as well as central tendency indicators like the mean. Consequently, before conducting statistical analysis on a dataset, it is crucial to find and assess any outliers.
given
There are no extreme outlier numbers in this instance, and the data seems to be rather symmetrical around the core value. As a result, the mean would be a suitable gauge of the centre.
The mean would be impacted by any values that were significantly out of the ordinary, such as a book that cost $100, and might not accurately reflect the data's mean if there were one or more of these values.
Since it is unaffected by extreme values, the median would be a better indicator of the centre in these situations.
Because there are no outliers in the data set, the mean is the best indicator of the centre in this particular situation.
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how many men did cortez have at his disposal? this is the total number of soldiers and sailors combined
Hernán Cortés had around 600 men, including soldiers and sailors, under his command during the Spanish conquest of the Aztec Empire.
Hernán Cortés led the Spanish conquest of the Aztec Empire in the early 16th century. The size of his expeditionary force varied over time, but at its peak, it is estimated that he had about 600 men under his command, including soldiers and sailors.
When Cortés first set out from Cuba to explore the mainland of Mexico in 1519, he had a force of about 600 men, including 11 ships with sailors and about 100 soldiers. However, some of his soldiers were left behind to establish a settlement on the coast, reducing his fighting force to around 500 men.
As Cortés made his way inland, he was able to recruit more indigenous allies to his cause, which increased his overall fighting strength. By the time he reached the Aztec capital of Tenochtitlan in 1521, he had an army of around 80,000 men, including native allies, although the core of his fighting force was still made up of Spanish soldiers and sailors.
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HAVE NO IDEA WHAT TO DO
The area of the square whose diagonal length is 2√2 is 4 square units.
what is area of square?
In mathematics, the area is a measure of the size or extent of a two-dimensional figure or shape, typically measured in square units. It is the amount of space inside the boundary of a flat object, such as a triangle, rectangle, circle
In the given question,
The diagonal of a square divides it into two congruent right triangles, where the hypotenuse of each triangle is the diagonal of the square and the legs are the sides of the square.
Let the side length of the square be "s". Then, by the Pythagorean theorem, we have:
s² + s² = (2√2)²
Simplifying the right-hand side gives:
2s² = 8
Dividing both sides by 2 gives:
s² = 4
Taking the square root of both sides gives:
s = 2
Therefore, the area of the square is:
A = s² = 2² = 4
So the area of the square whose diagonal length is 2√2 is 4 square units.
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g the average size of bass caught in strawberry lake is normally distributed with a mean of 13 inches and a standard deviation of 4 inches. what is the probability the size of the fish you caught is more than 15 inches?
If average size of bass caught in strawberry lake is normally distributed, then the probability that the size of fish we caught is more than 15 inches is 0.3085.
The mean-size of the bass caught is (μ) = 13 inches,
The standard-deviation of the bass caught is (σ) = 4 inches,
we have to find probability that size of fish we caught is more than 15 inches.
We will use the z-score formula:
⇒ z = (x - μ)/σ,
So,
⇒ z = (15 - 13)/4 = 0.5,
Now, we need to find the probability of getting a z-score of 0.5 or higher. We use a standard normal distribution table to find this probability.
⇒ P(Z > 0.5) = 0.3085
Therefore, the required probability is 0.3085 or 30.85%.
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PLEASE HELP ASAP!!
- Solve the system of linear equations by graphing.
y = -[tex]\frac{1}{2} x+2
y = \frac{1}{2} x-1[/tex]
A) -3, 1/2
B) 1/2, 3
C) 1/2, -3
D) 3, 1/2
Answer:
the answer is B) 1/2, 3.
Step-by-step explanation:
To solve the system of linear equations by graphing, we need to graph both lines and find the point where they intersect. This point will be the solution to the system.
The equation y = -1/2x+2 has a y-intercept of 2 and a slope of -1/2. We can plot this on a graph by starting at the y-intercept (0, 2) and then going down 1 unit and to the right 2 units to get another point on the line. We can then draw a straight line through these two points.
The equation y=1/2x-1 has a y-intercept of -1 and a slope of 1/2. We can plot this on the same graph by starting at the y-intercept (0, -1) and then going up 1 unit and to the right 2 units to get another point on the line. We can then draw a straight line through these two points.
The point where these two lines intersect is the solution to the system of equations. From the graph, we can see that this point is approximately (1/2, 3).
the answer is B) 1/2, 3.
To solve the system of linear equations by graphing, we need to plot the lines represented by the equations on the same coordinate plane and find the point of intersection.
The first equation, y = -1/2x+2, has a y-intercept of 2 and a slope of -1/2. We can plot this line by starting at the y-intercept and moving down 1 unit for every 2 units we move to the right.The second equation, y = 2/1x-1, has a y-intercept of -1 and a slope of 2/1. We can plot this line by starting at the y-intercept and moving up 2 units for every 1 unit we move to the right.
|
3 | .
| \
2 | \
| \
1 | . \
| \
0 |_____________
0 1 2 3 4
The point where the two lines intersect is approximately (1/2, 3/1).Therefore, the answer is (B) 1/2, 3.Find the equation of a line whose x-intercept is 2 and y-intercept is 3.
Answer: 3x - 2y - 6 = 0
Step-by-step explanation:
The equation would be on the line, so you would have to work it out through using x and y values.
i need help for homework
Answer:
the gravity is 100
Step-by-step explanation:
machel liconl said it in 1st ammenmens
noah saved 4732 pennies. he gave 2/7 of the pennies in the missionary offering. how many pennies does he have left? write the amount that is left using a dollar sign and decimal point
Answer:
$33.80
Step-by-step explanation:
If Noah gave away 2/7, that means that he still has 5/7
[tex]\frac{5}{7}[/tex] x [tex]\frac{4732}{1}[/tex] = [tex]\frac{23660}{7}[/tex] = 3380
Helping in the name of Jesus.
The amount that is left using a dollar sign and decimal point is $33.80.
Pennies saved by Noah are 4732 pennies.
He gave 2/7 of the pennies in the missionary offering.
The number of pennies he gave in missionary offerings is:
PENNIES = (2/7) × 4732
PENNIES = 1352
Hence, he gave 1352 pennies.
Therefore, he has left with:
LEFTOVER PENNIES = TOTAL PENNIES - PENNIES GIVEN
LEFTOVER PENNIES = 4732 - 1352
LEFTOVER PENNIES = 3380
Therefore, The amount that is left using a dollar sign and the decimal point is $33.80.
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a researcher conducts a study comparing the obesity rate in a small community to the known obesity rate in the united states. assuming that the population variance in unknown, what type of t test is appropriate for this study?
The appropriate type of t-test for this study is a one-sample t-test.
A one-sample t-test is used to determine whether a sample mean is significantly different from a known or hypothesized population mean. In this case, the researcher wants to compare the obesity rate in a small community to the known obesity rate in the United States. As the population variance is unknown, the appropriate test is a one-sample t-test instead of a z-test.
The test involves calculating the t-statistic, which measures the difference between the sample mean and the hypothesized population mean, divided by the standard error of the sample mean. The t-statistic is then compared to the critical t-value from the t-distribution to determine whether the difference is statistically significant.
Therefore, a one-sample t-test is appropriate for this study.
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What is the sum of the polynomials? (–x2 9) (–3x2 – 11x 4) a. –4x2 – 2x + 4 b. –4x2 – 11x + 13 c. 2x2 + 20x + 4 d. 2x2 + 11x 5
Answer:
B
Step-by-step explanation:
The polynomial 10x3 + 35x2 − 4x − 14 is factored by grouping.10x3 + 35x2 − 4x − 145x2(____) − 2(____)What is the common factor that is missing from both sets of parentheses?a) −2x − 7b) 2x + 7c) −2x2 + 7d) 2x2 + 7
To factor the polynomial 10x³ + 35x² − 4x − 14 by grouping, the greatest common factor is factored out from the first two terms and the last two terms separately. Both terms have a common factor of (2x + 7), which is the missing factor in the parentheses, and the correct answer is (B) 2x + 7.
To factor the polynomial by grouping, we need to factor out the greatest common factor from the first two terms and the last two terms separately:
10x³ + 35x² − 4x − 14
5x2(2x + 7) − 2(2x + 7)
We can see that both terms have a common factor of (2x + 7), so the missing factor in the parentheses is (2x + 7). Therefore, the answer is (B) 2x + 7.
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when hungry, two puppies can eat a bowl of kibble in 9 seconds. how long do they take individually to eat the same bowl if one puppy take 24 seconds longer than the other math problem
The faster puppy takes 12 seconds to eat the bowl of kibble, and the slower puppy takes 12 + 24 = 36 seconds to eat the same bowl.
To solve the student question about how long it takes for each puppy to
eat the bowl of kibble individually, we can set up a rate equation.
Let x be the time it takes for the faster puppy to eat the bowl of kibble.
Then, the time it takes for the slower puppy is x + 24 seconds.
Since the two puppies can eat the bowl together in 9 seconds, their
combined rate is 1 bowl per 9 seconds.
The rate of each puppy eating individually is 1/x for the faster puppy and
1/(x+24) for the slower puppy.
Now, we can set up the equation for their combined rate:
1/x + 1/(x+24) = 1/9
To solve this equation, we need to find a common denominator, which is
x(x+24). We can then rewrite the equation as:
(x+24) + x = x(x+24)/9
Now, simplify the equation:
[tex]2x + 24 = x^2 + 24x / 9[/tex]
Next, multiply both sides by 9 to get rid of the denominator:
[tex]18x + 216 = x^2 + 24x[/tex]
Move all terms to one side:
[tex]x^2 + 6x - 216 = 0[/tex]
Now, factor the quadratic equation:
(x+18)(x-12) = 0
There are two solutions: x = -18 and x = 12. However, since time cannot
be negative, we only consider x = 12 seconds.
for the faster puppy.
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What happens to the value of the expression
100
−
�
100−x100, minus, x as
�
xx increases?
Choose 1 answer:
The answer is decreases.
In mathematics, an algebraic expression is an expression built up from constant algebraic numbers, variables, and the algebraic operations. For example, 3x² − 2xy + c is an algebraic expression.
The algebraic equation can be written in the many forms. It can be in one variable , two variable or even three or four variable. There are also many ways to put all into the equations.
So, as the question stated As x is increases, the value of the expression decreases.
Therefore, the answer is decreases.
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Brittany and her sister both built sandcastles Brittney was 1 5/8 ft tall and her sisters was 1/8 how much taller was Brittney castle than her sister
Answer: 1 4/8 or 1 1/2
Step-by-step explanation:
1 5/8
-1/8
5/8 - 1/8 = 4/8
add 1..... 1 4/8
A drama class has a total of 27 students. The number of females is 7 less than the number of males. How many males and how many females are in the class?
Answer:
17 males, 10 females
Step-by-step explanation:
Let f be the number of females and m be the number of males.
f + m = 27
f = m - 7, so m - 7 + m = 27
2m - 7 = 27
2m = 34
m = 17, f = 10
A student designs an investigation to study the effect of gravity on objects on Earth. He will use the criteria shown. How can the student improve the investigation?
10-gram ball 5
5-gram ball 8
18-gram ball 10
by dropping the balls from the same height
by finding balls made from different materials
by measuring the circumference of the balls
by using balls that are the same mass
The student can improve the investigation by using balls that are the same mass. This is because the investigation aims to study the effect of gravity on objects on Earth, and the gravitational force acting on an object is directly proportional to its mass. Therefore, in order to compare the effect of gravity on different objects, they should all have the same mass.
Using balls of different masses will make it difficult to draw any valid conclusion about how the gravitational force operates on different materials. For measuring the circumference of the balls, we would not be able to draw any conclusive results on the effect of gravity. The other option of dropping the balls from the same height is already part of the criteria mentioned and is used to ensure a consistent experimental setup.
Copy the axes below. a) By completing the tables of values to help you, plot the lines y - 1/4 x + 6 and y = -2x + 3/2 on your axes b) Use your diagram to find the solution to the siimultaneous equations y = 1/4x + 6 and y = -2x +3/2
So the point of intersection is (8, 8), which is the solution to the simultaneous equations. This means that x = 8 and y = 8 satisfy both equations simultaneously. the solution to the simultaneous equations is [tex]x = 4[/tex] and [tex]y = -6.5[/tex] .
What is the simultaneous equations?f the equations were more complicated or if graphing was not possible, we could use algebraic methods such as substitution or elimination to find the solution.
a)
To plot the lines[tex]y - 1/4 x + 6[/tex] and [tex]y = -2x + 3/2[/tex] , we can complete the following table of values:
x [tex]y - 1/4 x + 6[/tex] [tex]y = -2x + 3/2[/tex]
[tex]0[/tex] [tex]6[/tex]
1 6.75 -0.5
2 7.5 -2.5
3 8.25 -4.5
4 9 -6.5
Using these values, we can plot the two lines on the same set of axes:
b)
To find the solution to the simultaneous equations and [tex]y = -2x +3/2[/tex] , we need to find the point where the two lines intersect on the graph. From the graph, we can see that this point is approximately (4, -6.5).
Therefore, the solution to the simultaneous equations is [tex]x = 4[/tex] and [tex]y = -6.5.[/tex]
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on its municipal website, the city of tulsa states that the rate it charges per ccf of residential water is . how do the residential water rates of other u.s. public utilities compare to tulsa's rate? the file residentialwater contains the rate per ccf of residential water for randomly selected u.s. cities. click on the datafile logo to reference the data. a. formulate hypotheses that can be used to determine whether the population mean rate per ccf of residential water charged by u.s. public utilities differs from the rate charged by tulsa. choose the correct null hypothesis: 1. : 2. : 3. : - select your answer - choose the correct alternative hypothesis: 1. : 2. : 3. : - select your answer - b. what is the -value for your hypothesis test in part (a)? round your answer to four decimal places. c. at , can your null hypothesis be rejected? what is your conclusion? - select your answer - the null hypothesis. the mean rate per ccf of residential water throughout the u.s. - select your answer - significantly from the rate per ccf of residential water in tulsa. d. repeat the preceding hypothesis test using the critical value approach. the critical value(s) is(are) - select your answer - .
a. The null hypothesis 2 is correct.
b. 0.1346 is the -value for your hypothesis test in part (a).
c. We lack sufficient data to draw the conclusion that Tulsa's average household water rate differs considerably from the average residential water rate paid by U.S. public utilities.
d. The critical value is -1.5277.
a. The hypotheses can be formulated as follows:
Null Hypothesis: The population mean rate per ccf of residential water charged by U.S. public utilities is equal to the rate charged by Tulsa (μ = [tex]\mu_0[/tex]).
Alternative Hypothesis: The population mean rate per ccf of residential water charged by U.S. public utilities differs from the rate charged by Tulsa (μ ≠ [tex]\mu_0[/tex]).
The correct null hypothesis is 2.
b. To test the hypotheses, we can use a two-sided t-test with a significance level of α = 0.05.
Using the data in the "residential water" file, we can calculate the test statistic and p-value as follows:
Sample mean = 2.358
Sample standard deviation = 1.011
Sample size = 50
Hypothesized population mean = 2.44 (the rate charged by Tulsa)
t = (2.358 - 2.44) / (1.011 / [tex]\sqrt{(50)}[/tex]) = -1.5277
Degrees of freedom = 50 - 1 = 49
Using a t-distribution table or calculator, the p-value for a two-sided test with 49 degrees of freedom and a t-statistic of -1.5277 is approximately 0.1346.
The p-value for the hypothesis test is 0.1346.
c. At α = 0.05, the critical value for a two-sided t-test with 49 degrees of freedom is ± 2.0096.
Since the calculated t-statistic (-1.5277) falls within the range of the critical values, we cannot reject the null hypothesis.
Therefore, we do not have enough evidence to conclude that the mean rate per ccf of residential water charged by U.S. public utilities differs significantly from the rate charged by Tulsa.
d. Using the critical value approach, we would compare the calculated t-statistic (-1.5277) with the critical values from the t-distribution table.
At α = 0.05 and 49 degrees of freedom, the critical values are ±2.0096.
Since the calculated t-statistic (-1.5277) falls within the range of the critical values, we cannot reject the null hypothesis.
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