Answer:
[tex]\frac{49}{45}[/tex] OR [tex]1\frac{4}{45}[/tex]
Step-by-step explanation:
The quadrants go in counter-clockwise position.
O True
O False
Answer:true
Step-by-step explanation:
3. What is the solution of the equation −4(n+5)=−32 - 4 ( n + 5 ) = - 32 ? −13 - 13 −12 - 12 3 3 13
The solution of the given equation -4(n+5)=-32 is n=3.
What is equations ?
An equation is a mathematical statement that shows that two expressions are equal. It contains an equals sign (=) between two expressions. The expressions on both sides of the equals sign have the same value. Equations are used to solve problems in mathematics and science.
To solve the equation, we need to isolate the variable n.
-4(n+5) = -32
Distribute -4 on the left side of the equation:
-4n - 20 = -32
Add 20 to both sides of the equation:
-4n = -12
Divide both sides by -4:
n = 3
Therefore, the solution of the given equation -4(n+5)=-32 is n=3.
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ALGEBRA 1 HW!! I WILL GIVE BRAINLYEST FOR ANSWERS! PLEASE HELP DUE SOON
The analysis of the number of runners, can be presented as follows;
a. The sequence is a geometric sequence
b. r(3) = 72, which indicates that the number of runners after the third week is 72
c. r(6)/r(5) = (2/3)
d. The recursive rule for r(w) is that the number of runners in a week is (2/3) × The number of passengers in the previous week
What is a sequence?A sequence is an ordered list of items.
The number of passengers currently in the running club = 243
The amount by which the number reduces each week = 2/3 the number in the previous week.
a. Therefore, the number of passengers each week can be presented as follows;
First week = 243
Second week = (2/3) × 243 = 162
Third week = (2/3)³ × 243
Therefore, the number of runners in the nth week is; an= (2/3)ⁿ × 243, which is a geometric sequence
b. The number of runners in the wth week, r(w) = 2/3)^w × 243,
Therefore; r(3) = (2/3)³ × 243 = 72
The meaning of the value is that the number of runners in the third week, r(3) = 72 runners
c. r(6))/(r(5)) can be obtained as follows;
r(6) = (2/3)^6 × 243
r(5) = (2/3)^5 × 243
r(6)/r(5) = ((2/3)^6 × 243)/( (2/3)^5 × 243) = (2/3)
d. The recursive formula can be obtained as follows;
The number of passengers in a specified week = (2/3) × The number of passengers in the previous week
r(w + 1) = (2/3)^w × 243
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8. (02.02 LC)
Which of the following rational numbers is equal to 3.1? (5 points)
22/9
24/9
26/9
28/9
Answer: 28/9 (choice D)
Explanation:
You don't need a calculator for this.
Each fraction involves 9 in the denominator. The numerators are the only thing that change. Let x be the numerator.
x/9 = 3.1 can be estimated to x/9 = 3 and that solves to x = 27
The numerator is around 27.
If x = 26, then 26/9 is slightly smaller than 3If x = 27, then 27/9 is exactly 3If x = 28, then 28/9 is slightly larger than 3Choices A,B,C are ruled out since the numerators are smaller than 28. The only thing left is choice D.
You can use a calculator or long division to confirm that 28/9 = 3.1111... where the '1's go on forever.
The probability that a person with certain symptoms has hepatitis is 0.8. The blood test used to confirm this diagnosis gives positive results for 94% of people with the disease and 5% of those without the disease. What is the probability that an individual who has the symptoms and who reacts positively to the test actually has hepatitis?
The likelihood that someone with symptoms and a positive test has hepatitis is therefore 0.994, or almost 99.4%.
What is the probability that an individual who has the symptoms and who reacts positively to the test actually has hepatitis?We need to apply Bayes' theorem to determine the likelihood that a person with symptoms and a positive test actually has hepatitis. Let H stand for the occurrence that the person has hepatitis and T stand for the occurrence that the test is positive. Next, we have:
P(T | H) * P(H) / P(T | H) = P(H | T) (T)
where P(H) is the prior probability of the individual having hepatitis (which is 0.8), P(H) is the likelihood of the individual testing positive, and P(T) is the probability of testing positive given that the individual has hepatitis.
We must apply the law of total probability to determine P(T):
P(T) is equal to P(T | H)* P(H) plus P(T | not H)* P. (not H)
P(T | not H) is the complement of P(H), and P(not H) is the probability of testing positive if the person does not have hepatitis (0.05). (i.e., the probability of not having hepatitis, which is 0.2).
Now that the values have been inputted, we can calculate:
P(T) = (0.94 * 0.8) + (0.05 * 0.2) = 0.752 P(H | T) = (0.94 * 0.8) / 0.752 = 0.994
Although there is a strong likelihood that this is the case, it's crucial to remember that no medical test is error-free, and false positives and false negatives can still happen.
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What are the coordinates of each point after the quadrilateral RSTU is rotated 90 degrees about the origin?
Answer:
Step-by-step explanation:
90 degrees is the answer because there is no movement
WILL GIVE 5 STARS
Write the first four terms of the sequence defined by the explicit rule f(n) = n2 - 5. Show all steps and reasoning.
Sequence having rule f(n) = n² - 5, the first four terms will be -4, -1, 4, and 11.
What is a sequence?A sequence in mathematics is a group of integers that adhere to a specific pattern or rule. A sequence of numbers might be infinite or finite, and they are arranged in a certain order.
Every number in the sequence is referred to as a term, and its location within the sequence is referred to as its index. As an illustration, the first term in the series 1, 3, 5, 7, 9, is 1, the second term is 3, the third term is 5, and so on.
Now,
The explicit rule for the sequence is given by f(n) = n² - 5, which means that to find the nth term of the sequence, we need to plug in the value of n into this formula.
To find the first four terms of the sequence, we can simply substitute n = 1, 2, 3, and 4, respectively, into the formula and evaluate the expression.
The first term is f(1) = 1² - 5 = -4.
The second term is f(2) = 2² - 5 = -1.
The third term is f(3) = 3² - 5 = 4.
The fourth term is f(4) = 4² - 5 = 11.
Therefore,
the first four terms of the sequence defined by the explicit rule f(n) = n² - 5 are -4, -1, 4, and 11.
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A researcher watches popular television shows and records every time someone interrupts a speaker. She wants to understand the frequency of interruption on television.
Answer:
the frequency of someone interru
Step-by-step explanation:
The correct answer is (d) noise. Unwanted sound that is regarded loud, unpleasant, or disruptive to hearing is called noise.
What is noise?Even if the label "unwanted" is arbitrary, some sounds are frequently regarded as noise. A sound pressure level, expressed in decibels, is used to measure or anticipate noise (dB).
here, we have,
The measurements and noise meter used depend on the application; different noise metrics can account for magnitude, frequency, and time in different ways.
and, we have,
Another illustration is workplace noise, where the effect of noise on hearing loss is of utmost importance.
To offer a degree of exposure that can be associated to the possibility of hearing loss in employees exposed to the sound during the course of exposure, both the intensity and duration of sound are combined.
Hence, The correct answer is (d) noise. Unwanted sound that is regarded loud, unpleasant, or disruptive to hearing is called noise.
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Complete question:
ivan is trying to understand the concern of an angry customer, but several employees keep interrupting the conversion. these interruptions are an example of
Use the graph to answer the question. Graph of polygon ABCD with vertices at 1 comma negative 1, 3 comma negative 5, 7 comma negative 5, 5 comma negative 1. A second polygon A prime B prime C prime D prime with vertices at 1 comma negative 6, 3 comma negative 10, 7 comma negative 10, 5 comma negative 6. Determine the translation used to create the image. 5 units down 5 units up 1 unit down 1 unit up
The image was created by translating 5 units down, 5 units up, 1 unit down, and 1 unit up. ABCD and A'B'C'D' are two polygons on the graph.
The first polygon, ABCD, has vertices at (1,-1), (3,-5), (7,-5), and (8,-5). (5,-1). The vertices of the second polygon, A'B'C'D', are located at (1,-6), (3,-10), (7,-10), and (8,-10). (5,-6).
What exactly are vertices?Vertices are points in a geometric figure that define its shape. A vertex is defined mathematically as the intersection of two or more lines, edges, or curves.
We can compare the coordinates of the two polygons to determine the translation that was used to create the image. The first coordinate of the two polygons is (1,-1) and the second coordinate is (1,-1). (1,-6). The y-coordinate difference is 5 units, so the translation is 5 units down. The two polygons' second coordinate is (3,-5) and (3,-10). The y-coordinate difference is 5 units, so the translation is 5 units down.
The second polygon's third coordinate is (7,-5) and (7,-10). The y-coordinate difference is 5 units, so the translation is 5 units down. (5,-1) is the fourth coordinate of both polygons (5,-6). The y-coordinate difference is 5 units, so the translation is 5 units down.
The image was translated 5 units down, 5 units up, 1 unit down, and 1 unit up.
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If � p and � q vary inversely and � p is 18 when � q is 28, determine � q when � p is equal to 72.
Answer:
q = 7Step-by-step explanation:
p and q vary inversely:
p = k/qIf p = 18, q = 28, find k:
18 = k/9k = 18*28 = 504The equation is:
p = 504/qWhen p = 72, q is:
18 = 504/qq = 504/72q = 7There are 270 students at Colfax Middle School, where the ratio of boys to girls is 5 : 4. There are 180 students at Winthrop Middle School, where the ratio of boys to girls is 4: 5. The two schools hold a dance and all students from both schools attend. What fraction of the students at the dance are girls?
We have the following response after answering the given question: equation Hence, there are 22/45 female students at the dance.
What is equation?In a mathematical equation, the equals sign (=), which connects two claims and denotes equality, is utilised. In algebra, an equation is a mathematical statement that proves the equality of two mathematical expressions. For instance, in the equation 3x + 5 = 14, the equal sign separates the numbers by a space. Mathematical expressions can be used to describe the relationship between the two sentences on either side of a letter. The logo and the particular piece of software frequently correspond. like, for instance, 2x - 4 = 2.
Now, let's determine how many boys and females attend each school:
Middle School in Colfax
Boys: (5/9) * 270 = 150
Girls: (4/9) * 270 = 120
School in Winthrop:
Boys: (4/9) * 180 = 80
Girls: (5/9) * 180 = 100
Now, we can determine how many males and girls attended the dance overall:
Boys: 150 + 80 = 230
Girls: 120 + 100 = 220
Thus, the proportion of female students attending the dance is:
220 / (230 + 220) = 220 / 450 = 44/90 = 22/45
Hence, there are 22/45 female students at the dance.
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Two landmarks are 65 miles apart. The distance between them on a map is 2 feet 5 inches. What is the distance between two other landmarks on the same map if the actual distance between them is 52 miles? Round your answer to the nearest inch.
The distance between the two other landmarks on the same map is approximately 23 inches.
What is a scale factor?
A scale factor is a measure of the proportional relationship between two similar figures. It is the ratio of any two corresponding lengths in the two figures.
We can set up a proportion to solve this problem:
Distance on map / Actual distance = Scale factor
Let x be the distance between the two other landmarks on the map. We can set up the following proportion:
2 feet 5 inches / 65 miles = x / 52 miles
To solve for x, we first convert 2 feet 5 inches to inches:
2 feet = 24 inches
5 inches = 5 inches
2 feet 5 inches = 24 inches + 5 inches = 29 inches
Substituting the values into the proportion, we get:
29 inches / 65 miles = x / 52 miles
To solve for x, we cross-multiply:
29 inches × 52 miles = 65 miles × x
Simplifying, we get:
x = 29 inches × 52 miles / 65 miles
x = 23.2 inches
Rounding to the nearest inch, we get x ≈ 23 inches.
Therefore, the distance between the two other landmarks on the same map is approximately 23 inches.
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Solve the system of inequalities by graphing.
y < 6
y > x + 3
The solution to the inequalities, y < 6 and y > x + 3 is the common intersecting region.
What are inequalities and it's types?Inequality is a relation that compares two numbers or other mathematical expressions in an unequal way.
The symbol a < b indicates that a is smaller than b.
When a > b is used, it indicates that a is bigger than b.
a is less than or equal to b when a notation like a ≤ b.
a is bigger or equal value of an is indicated by the notation a ≥ b.
Given, y < 6.
Now, To determine wether it inclues the origin or not we'll set, y = 0.
So, 0 < 6 is true so y < 6 includes the origin.
Now, y > x + 3.
0 > 0 + 4 is not true therefore, y > x + 3 doest not inclue the origin.
The solution to these now can be determine by the common intersecting region.
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a highway has at least one rest stop every 25 miles. Write solve and graph an inequality to represent the number of rest stops in the first 200 miles of the highway.
In a linear equation, The number of rest stops in the first 200 miles would be 8 or more.
What in mathematics is a linear equation?
An algebraic equation with simply a constant and a first-order (linear) term, such as y=mx+b, where m is the slope and b is the y-intercept, is known as a linear equation.
Sometimes, the aforementioned is referred to as a "linear equation of two variables," where x and y are the variables. Equations with variables of power 1 are referred to as linear equations. One example with only one variable is where ax+b = 0, where a and b are real values and x is the variable.
25x > 200
25 25
x > 8
The number of rest stops in the first 200 miles would be 8 or more.
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what is the answer i’m lost
Note that where the conditions above subsist the height determined by the other colleague is not the same so the answer is No. To solve this problem, you need to be conversant with the Pythagorean Theorem, and Angle Bisector Theorems.
What does the Pythagorean Theorem say?
The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides.
The formula for the Pythagorean Theorem is:
c² = a² + b²
where "c" represents the length of the hypotenuse and "a" and "b" represent the lengths of the other two sides (the legs) of a right triangle.
With this, we can find the c for the lower Right angle triangle on the left and B for the lower right angle triangle on the right.
Since C² = A² + B²
B = √( A²- C²)
B = √( 9.5²- 6²)
B [tex]\approx[/tex] 7.36546
To determine c on the right angle to the left,
c² = a² + b²
c = √(a² + b²)
c = √(5.5² + 7.2²)
c = 9.060353194
c [tex]\approx[/tex] 9.06
With the postulate of the Angle Bisector Theorem, we can begin to find the height of the two upper right angle triangles.
The Angle Bisector Theorem states that in a triangle, an angle bisector of a vertex of the triangle divides the opposite side into two segments that are proportional to the lengths of the other two sides.
In other words, if a line segment bisects an angle of a triangle and intersects the opposite side or the extension of the opposite side, then the ratio of the lengths of the segments on the opposite side is equal to the ratio of the lengths of the other two sides of the triangle.
As seen from the image, the right angles where each lady is holding is bisected. This means that we have for each Triangle Two angles and one side.
Right Angle on the Left:
90°
45°; and
7.2ft
Given the above, Since the angle opposite the height is 45 degrees, we know that the ratio of the height to the base is 1:1, or h/7.2 = 1/1.
Therefore, the height of the right triangle is:
h = 7.2 × 1
h = 7.2 ft
So the height of the right triangle is 7.2 feet.
Right Angle on the Right :
90°
45°; and
7.36546ft
Given the above, we can use the trigonometric ratios of the angles in the right triangle.
Since the angle opposite the height is 45 degrees, we know that the ratio of the height to the base is 1:1, or h/7.36546 = 1/1.
Therefore, the height of the right triangle is:
h = 7.36546 × 1
h = 7.36546 ft
So the height of the right triangle is 7.36546 feet.
So both heights approximated to the nearest tenth:
Since 7.2 ≠ 7.4, the heights are not equal.
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The mean score of the students in Dr. Battle’s math class on the most recent quiz was 78 points with a standard deviation of 6 points. If these scores can be modeled using a normal distribution, which percentage best represents the number of students with quiz scores between 72 and 90 points?
Approximate Percentages of Data in a Normal Distribution:
• 68% of the data lie within 1 standard deviation of the mean.
• 95% of the data lie within 2 standard deviations of the mean.
• 99.7% of the data lie within 3 standard deviations of the mean.
Responses
A.68.0%
B.81.5%
C.95.0%
D.97.5%
The percentage of students with quiz scores between 72 and 90 points would be 81.5%.
What is standard deviations?Standard deviations is a statistical measure of the spread of data points around the mean in a set of data. It is calculated by taking the square root of the variance, which is the average of the squared differences from the mean. Standard deviations are used to measure the uncertainty of a statistic or the variability of a population or sample of observations.
This answer is correct because the data is being modeled using a normal distribution. This means that the quiz scores can be described as a bell-shaped curve, with the mean score of 78 points being the highest peak. The 68% of the data that lies within 1 standard deviation of the mean would be between 72 and 90 points, which is the range specified in the question, so the answer is 81.5%. This is because 68% of the data lies within 1 standard deviation of the mean and 13.5% of the data lies within the second standard deviation of the mean. The sum of these two figures (68% + 13.5%) is 81.5%. Therefore, the percentage of students with quiz scores between 72 and 90 points would be 81.5%.
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I need help with part d!!
The probability that the randomely selected fertilized egg will hatch between 15 days to 17 days as 0.5.
What is Probability?The probability of an event is a number that indicates how likely the event is to occur.
It is expressed as a number in the range from 0 and 1 or it can be expressed using percentage notation, in the range from 0% to 100%
Given is that the probability of a randomely selected fertilized egg will take 21 days to hatch is 0.028.
We can write the probability that the randomely selected fertilized egg will hatch between 15 days to 17 days as -
P{E} = {0.028 + (1 - 0.028)}/2
P{E} = 1/2
P{E} = 0.5
Therefore, the probability that the randomely selected fertilized egg will hatch between 15 days to 17 days as 0.5.
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[tex]-3\sqrt{x} +2\ \textless \ 15[/tex]
Answer:
12
Step-by-step explanation:
Find the cube root of these expression
³√5׳√5׳√5=
We can simplify this expression by multiplying the cube roots together:
³√5 × ³√5 × ³√5 = ³√(5 × 5 × 5)
Simplifying the product inside the cube root, we get:
³√(125)
Therefore, the cube root of 5 × 5 × 5 is equal to:
³√5׳√5׳√5 = ³√125 = 5
How many terms are in this expression? 7p + 6 + r
PLEASE HELP WILL MARK BRAINLIEST!
1) when a certain polynomial is divided by x+3, the quotient is x^2-3x+5 and the remainder is 6. what is the polynomial?
2) when a certain polynomial is divided by x-2, the quotient is x^2+4x-7 and the remainder is -4. what is the polynomial?
Consider the expressions 3x(x − 2) + 2 and 2x2 + 3x − 18.
Part 1
Evaluate each expression for x = 4 and for x = 5. Based on your results, do you know whether the two expressions are equivalent? Complete the explanation.
For x = 4, each expression has a value of
26
. For x = 5, each expression has a value of
47
. These results suggest that the expressions
may be
equivalent, but
do not prove that
the expressions are equivalent.
Part 2 out of 2
Evaluate each expression for x = 3. Based on your results, do you know whether the two expressions are equivalent? Complete the explanation.
For x = 3, the first expression has a value of
, and the second expression has a value of
. Since these values are
(select)
, the expressions
(select)
equivalent.
hercules counted some animals at the farm. He saw that 2 of them were sheep and 6 were other animals. If this trend continues, how many of the next 12 animals will be sheep?
Assuming the trend continues, the next 12 animals will include 4 sheep.
What is number?Number is a mathematical concept that represents a quantity or amount. It is usually used to identify a specific value or set of values that can be used for calculation and measurement. Numbers are used in many different fields, including mathematics, science, engineering, and economics. They help us to keep track of quantity and count objects, measure distances and time, and even help to solve problems. Numbers are an essential part of our everyday lives and are essential for understanding the world around us.
This is calculated by taking 2 sheep from the initial group of 8 animals and then multiplying that number by the additional 12 animals for a total of 4 sheep.
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the wind speed y (in miles per hour) of a tropical storm is y = 2x + 66, where x is the same number of hours after the storm enters the gulf of mexico "15 by 15 graph first quadrant" A: Complete the table (show work) B: Graph the equation C: When does the storm become a hurricane
Answer:
Step-by-step explanatio
A. To complete the table, we need to plug in different values of "x" into the equation y = 2x + 66 and solve for "y". We can choose any values of "x", but it's usually helpful to choose values that are easy to calculate. Here's the table with "x" values ranging from 0 to 7:
x y
0 66
1 68
2 70
3 72
4 74
5 76
6 78
7 80
B. To graph the equation, we can plot the points from the table on a coordinate plane. We can use the x-axis to represent the number of hours after the storm enters the Gulf of Mexico (x), and the y-axis to represent the wind speed in miles per hour (y). The points should fall on a straight line with a slope of 2 and a y-intercept of 66. We can plot the points and draw the line as follows:
C. The wind speed of a tropical storm becomes a hurricane when it reaches or exceeds 74 miles per hour. Looking at the graph, we can see that the wind speed reaches 74 miles per hour when x = 4. Therefore, the storm becomes a hurricane 4 hours after entering the Gulf of Mexico.
You need 8 feet of copper pipe for a plumbing job. You have two options.
Option 1: Pipe for $1.50 per foot and a rebate offering $2.00 off the total
Option 2: Pipe for $1.75 per foot and a coupon offering $0.25 off each foot
You find a third company that offers pipe at $1.50 less than the total cost of the better deal between Option 1 and Option 2.
What does the third company charge for the 8 feet of copper pipe?
$8.50
$10
$10.50
$12
$12.25
The charge for the third company for the 8 feet of copper pipe as required to be determined is; $8.50.
What is the charge for the 8 feet of copper pipe from the third company?As evident from the task content;
For the first company; Pipe for $1.50 per foot and a rebate offering $2.00 off the total.
Therefore, total charge is; 8(1.50) - 2 = $10.
For the second company; Pipe for $1.75 per foot and a coupon offering $0.25 off each foot.
Therefore, total charge is 8(1.75) - 8(0.25) = $12.
Hence, since the better deal is; $10.
Ultimately, the charge of the third company whose offer is; $1.50 less than the charge of the better deal is; 10 - 1.50 = $8.50.
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Can someone answer this just read the picture? Right answers only please!
The linear model would be a reasonable way to predict the temperature as temperature determines the number of ice cream that would be sold.
What is temperature?Temperature is a unit of hotness and coldness that can be described in terms of any number of arbitrary scales. It also represents the direction wherein heat energy will naturally flow, from either a hotter (i.e., higher temperature) body to a colder body.
Temperature is not the same as the energy of such a thermodynamic system; for instance, an iceberg has a significantly larger total heat energy than a match, despite the fact that a match is burning at a significantly higher temperature. The linear model would be a reasonable way to predict the temperature as temperature determines the number of ice cream that would be sold.
Therefore, the linear model would be a reasonable way to predict the temperature as temperature determines the number of ice cream that would be sold.
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Multiple choice: The distance between the bases on a softball field is 60 ft
By answering the above question, we may infer that The equation's answer is B) 60 ft.
What is equation?A mathematical equation links two statements and utilises the equals sign (=) to indicate equality. In algebra, an equation is a mathematical assertion that proves the equality of two mathematical expressions. For instance, in the equation 3x + 5 = 14, the equal sign separates the numbers by a gap. A mathematical formula may be used to determine how the two sentences on either side of a letter relate to one another. The logo and the particular piece of software are usually identical. like, for instance, 2x - 4 = 2.
What is the distance between the bases on a softball field
A) 50 ft
B) 60 ft
C) 70 ft
D) 80 ft
The answer is B) 60 ft.
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What is the total price of the 50-inch Panasonic Plasma TV?
The total price of the 50-inch Panasonic Plasma TV is $687.44.
What is tax?In mathematics, the tax calculation is related to the selling price and income of taxpayers. It is a charge imposed by the government on the citizens for the collection of funds for public welfare and expenditure activities. There are two types of taxes: direct tax and indirect tax.
Given that, 50-65 inch TV's $150 rebate and delivery charge is $35.
50 inch Panasonic Plasma TV costs $734.95
Rebate is $150
So, cost of TV is 734.95-150
= 584.95
Cost of TV including delivery charge
= 584.95-35
= 549.95
Here, tax is 25%
So, cost of TV including tax
= 549.95+25% of 549.95
= 549.95+25/100 ×549.95
= 549.95+0.25×549.95
= 687.4375
Therefore, the total price of the 50-inch Panasonic Plasma TV is $687.44.
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pls show work i’m struggling
Answer:
Step-by-step explanation:
The Pythagorean Theorem states that: a^2+b^2=c^2.
Therefore 6^2+8^2= c^2
36+64= 100^2
Then square root that. The square root of 100 is 10.
The length of XY is 6. The length of XZ is 10. The difference is 4 units
Dennis charges $1.75 for gasoline plus $7.50 per hour for mowing lawns. Which inequality represents xx, the number of hours Dennis must work to earn at least $100?
Dennis must work for at least 10.81 hours to earn at least $100.
What is inequality in mathematics?
An inequality in mathematics is a statement that compares two values or expressions using one of the inequality symbols: < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to).
For example, x < 5 is an inequality that says that the value of x is less than 5. Another example is y ≥ 2x + 3, which is an inequality that says that the value of y is greater than or equal to 2 times x plus 3.
Inequalities are often used to describe ranges of values, such as the set of all numbers that are less than or equal to 5 (written as x ≤ 5) or the set of all numbers that are greater than 2 and less than 7 (written as 2 < x < 7). Inequalities can be solved, graphed on a number line, and used to make comparisons between values or expressions.
Let's start by writing an expression for the total amount of money Dennis earns in x hours:
Total earnings = 1.75x + 7.5x
Simplifying this expression, we get:
Total earnings = 9.25x
Now, we want to find the inequality that represents the number of hours Dennis must work to earn at least $100. We can set up the inequality as follows:
9.25x ≥ 100
This inequality states that Dennis must earn at least $100, which is represented by the value on the right-hand side of the inequality. The left-hand side of the inequality represents Dennis's earnings, which we found to be 9.25x. Since we want to find the minimum number of hours Dennis must work to earn at least $100, we need to solve for x:
9.25x ≥ 100
x ≥ 100/9.25
Simplifying the right-hand side of the inequality, we get:
x ≥ 10.81
Therefore, the inequality that represents the number of hours Dennis must work to earn at least $100 is:
x ≥ 10.81
This means that Dennis must work for at least 10.81 hours to earn at least $100.
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