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In the figure below, AD=32 cm, CD=24 cm, and points C and D are on the perpendicular bisector of segment AB. What is the perimeter of △ABC?
Answers
Answer:
The perimeter of △ABC is 144 cm.
------------------------------
Since points C and D are on the perpendicular bisector of AB, we have:
CD⊥AB;AD = BD;AC = BC.Find the length of AC using Pythagorean:
[tex]AC=\sqrt{AD^2+CD^2}=\sqrt{32^2+24^2} = \sqrt{1600}=40\ cm[/tex]Find the perimeter of ΔABC:
P = AB + BC + AC = 32*2 + 40 + 40 = 64 + 80 = 144 cmRelated Questions
Jamila approximates the areas of circles
using the equation A = 3r², and records areas
of circles with different radius lengths in a table.
Is the relation a linear function? Explain.
Answers
triangle abc and triangle def are smillar
which proportion must be true ?
Answers
The proportion EF/DF = BC/AC is true. Hence, option A is the accurate answer to this question.
We are given that the two triangles ΔABC and ΔDEF are similar.
According to the definition of a similar triangle, we know that the ratio of the corresponding sides is proportional and the corresponding angles are always congruent.
Since ΔABC and ΔDEF are similar, this implies that -
∠A corresponds to ∠D,
∠B corresponds to ∠E, and
∠C corresponds to ∠F.
Thus we can say -
EF/DF = BC/AC
AB/DF = DE/AC
AB/EF = DE/BC
Only option A satisfies the condition. Hence, the proportion EF/DF = BC/AC is true.
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The complete question is -
The triangle abc and triangle def are similar. Which proportion must be true?
A charity needs to report its typical donations received. The following is a list of the donations from one week. A histogram is provided to display the data.
11, 18, 35, 39, 45, 45, 46, 47, 49, 49, 50, 50, 50, 50, 56, 57, 58, 59
A graph titled Donations to Charity in Dollars. The x-axis is labeled 10 to 19, 20 to 29, 30 to 39, 40 to 49, and 50 to 59. The y-axis is labeled Frequency. There is a shaded bar up to 2 above 10 to 19, up to 2 above 30 to 39, up to 6 above 40 to 49, and up to 8 above 50 to 59. There is no shaded bar above 20 to 29.
Which measure of center should the charity use to accurately represent the data? Explain your answer.
The median of 45.2 is the most accurate to use to show that they need more money.
The median of 49 is the most accurate to use, since the data is skewed.
The mean of 49 is the most accurate to use to show that they have plenty of money.
The mean of 45.2 is the most accurate to use, since the data is skewed.
Answers
The best measure of center for this data is the median, and its value is 49.
What is the median in statistics?
In statistics, the median is a measure of central tendency that represents the value separating the higher half of a dataset from the lower half.
To find the median of a dataset, the values must first be arranged in order of magnitude. If the dataset has an odd number of values, the median is the middle value. If the dataset has an even number of values, the median is the average of the two middle values.
To determine the best measure of center for this data, we need to consider the shape of the distribution. Looking at the histogram, we can see that the data is skewed to the right, with a long tail of higher values.
In this case, the median is the best measure of center, because it is not affected by the extreme values in the dataset. The median is the middle value of the dataset when it is arranged in order, and in this case, the middle value is between 49 and 50. Since there are an even number of data points, we take the average of the two middle values, which gives a median of 49.
We are given that;
The list of donation is as follows
11, 18, 35, 39, 45, 45, 46, 47, 49, 49, 50, 50, 50, 50, 56, 57, 58, 59
Now,
Mean is the middle term of the data
Here,
Middle term of data = 49
Therefore, the mean of the given list will be 49.
Therefore, the best measure of center for this data is the median, and its value is 49.
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Answer: The median of 49 is the most accurate to use, since the data is skewed.
Step-by-step explanation:
Graph the solution to the following system of inequalities.
y>-3x-5
y≥ 3x-2
Answers
Answer:
Solution to system of inequalities is y≥ 3x-2
Graph attached
Step-by-step explanation:
We have the following two inequalities:
y>-3x-5
y≥ 3x-2
To get the solution and graph
1. Graph each inequality separately
2. Choose a test point to determine which side of the line needs to be shaded
3. The solution will be the area where the shadings from each inequality overlap one another
The graph is attached. The heavily shaded region is the set of all points which satisfy all inequalities
We see that any point that satisfies y≥ 3x-2 also satisfies y>-3x-5
In other words, the solution set for y≥ 3x-2 is a subset of the solution set for y>-3x-5
Find the area of each part of the figure and the whole figure
Answers
1) The area of the first figure is 99 square units, 2) Total area = Area of rectangle + Area of triangle = 30 + 6 = 36 square units, 3) Total area = Area of rectangle + Area of triangle = 13.5 + 10.5 = 24 square units.
i) The first figure is a trapezium with height 9 units, bottom base 9 units and top base 13 units.
To find the area of the trapezium, we can use the formula:
Area of trapezium = h/2[a + b]
where h is the height, a and b are the lengths of the parallel sides.
Substituting the given values, we get:
Area = 9/2[13 + 9] = (9 x 22)/2 = 99 square units
Therefore, the area of the first figure is 99 square units.
ii) The second figure is composed of a rectangle with sides 6 and 5 units and a right-angled triangle with base 3 and height 4 units.
To find the area of the rectangle, we can use the formula:
Area of rectangle = length x breadth
Substituting the given values, we get:
Area of rectangle = 6 x 5 = 30 square units
To find the area of the right-angled triangle, we can use the formula:
Area of right-angled triangle = 1/2 x base x height
Substituting the given values, we get:
Area of right-angled triangle = 1/2 x 3 x 4 = 6 square units
Therefore, the total area of the second figure is:
Total area = Area of rectangle + Area of triangle
= 30 + 6 = 36 square units
iii) The third figure is made up of a rectangle with length 9 and breadth 1.5 units and a right-angled triangle with base 6 (9-3) and height 3.5 (5-1.5) units.
To find the area of the rectangle, we can use the formula:
Area of rectangle = length x breadth
Substituting the given values, we get:
Area of rectangle = 9 x 1.5 = 13.5 square units
To find the area of the right-angled triangle, we can use the formula:
Area of right-angled triangle = 1/2 x base x height
Substituting the given values, we get:
Area of right-angled triangle = 1/2 x 6 x 3.5 = 10.5 square units
Therefore, the total area of the third figure is:
Total area = Area of rectangle + Area of triangle
= 13.5 + 10.5 = 24 square units
Thus, the areas of all three figures have been determined.
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Complete question is in the image provided.
how many ways can rudy choose 6 pizza toppings from a menu of 14 toppings if each topping can only be chosen once?
Answers
Rudy has 3003 different ways to choose 6 toppings from a menu of 14 toppings if each topping can only be chosen once.
The number of ways to choose 6 toppings from a menu of 14 toppings where each topping can only be chosen once is 14!/(14-6)! = 14!/(8!) = 3003. This is because for the first topping, Rudy has 14 options, for the second topping he has 13 options, for the third topping he has 12 options, and so on. To calculate the total number of combinations, we use the formula n!/(n-r)! where n is the number of items and r is the number of items to choose. This formula gives us the number of ways to choose r items from a set of n items without regard to order, which is the number of combinations.
The formula for combinations is given by n!/(n-r)! where n is the number of items and r is the number of items to choose. Here, n = 14 (the number of toppings on the menu) and r = 6 (the number of toppings Rudy wants to choose). So, the number of combinations is 14!/(14-6)! = 14!/(8!).
The factorial (!) is the product of all positive integers up to that number. For example, 4! = 4 x 3 x 2 x 1 = 24.
So, 14! = 14 x 13 x 12 x ... x 2 x 1 = 87178291199, and 8! = 8 x 7 x 6 x ... x 2 x 1 = 40320.
Now, to calculate the number of combinations, we divide 14! by 8!: 14!/(8!) = 87178291199/40320 = 3003.
Therefore, Rudy has 3003 different ways to choose 6 toppings from a menu of 14 toppings if each topping can only be chosen once.
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5. Lucy rode her bike around the block 4 times for a total of 1 mile yesterday. Today she wants to ride her bike 3_ 4 of a mile. How many times will she need to ride her bike around the block
Answers
Answer:
3
Step-by-step explanation:
1 loop around the block is 1/4 of a mile. multiply both sides by three. 3 loops is 3/4 of a mile.
PLEASE HELP ASAP DUE TODAY
Answers
Answer: 1) (0, -3) & 2) (0,0)
Step-by-step explanation:
2. Suppose the New York Yankees play a double-header (two games). Using W for a win and L
for a loss, list all the possible outcomes for the double-header. If we are only interested in the
total number of games the Yankees win, what are the possible events for the two games?
Answers
If we are only interested in the total number of games the Yankees win, the possible events for the two games are: 0 wins: LL , 1 win: LW, WL and 2 wins: WW
What is event in probability ?
In probability, an event is a subset of the sample space of a random experiment. In other words, an event is a collection of possible outcomes of the experiment.
For example, consider flipping a fair coin. The sample space of this experiment is {heads, tails}. An event could be defined as "getting heads", which is a subset of the sample space that includes only one outcome.
There are four possible outcomes for the double-header, which can be represented using W for a win and L for a loss:
WW (Yankees win both games)
WL (Yankees win the first game, but lose the second game)
LW (Yankees lose the first game, but win the second game)
LL (Yankees lose both games)
Therefore , If we are only interested in the total number of games the Yankees win, the possible events for the two games are: 0 wins: LL , 1 win: LW, WL and 2 wins: WW
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Question in picture! Thanks
Answers
Answer:
Step-by-step explanation:
Which situation most accurately matches the graph?
Answers
The situation that matches the information in the graph is a boy rides a bicycle on flat ground and goes quickly down a large hill (option A).
What does the graph show?The graph shows how the movement of a body changes over time. In general, it can be observed the movement is first linear as in moving on a flat surface, then the body goes down, then it goes up, and finally, the movement is linear again.
Based on this, it can be concluded that the option that describes the graph is a boy rides a bicycle on a flat group, then he goes down a large hill and finally he drives on flat ground again (option A).
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Cassie rolls a fair number cube with 6 faces labeled 1 through 6. She rolls the number cube 300 times. Which result is most likely?
Answers
Answer:
she mostly likely found 1 result .
Please help!!
Sandra is looking to buy decorative doughnuts for a party. She is considering two local bakers. She looks up two bakers online to find how much each one charges. Here are the advertisements she saw online. Ms. Spellings Donut Shop Try out this week's special! All decorative donuts are on sale for $1.62 each with
an $8 boxing fee! Phil'd Up Doughnuts Boxing fee included in price! Number of Decorative Doughnuts and price 2 $9 4 $13 6 $17 8 $21 $25 12 $29 a) What is the rate of change and the boxing fee for each doughnut shop? b) If the total cost was Sandra's biggest concern, who should she buy from? Justify you answer mathematically.
Answers
Answer:
Step-by-step explanation:
Dont cheat on your homework
For Phil'd Up Doughnuts, the rate of change is $1.25 per doughnut (calculated as $29/12). The boxing fee is included in the price, so there is no separate fee.
b) If the total cost is Sandra's biggest concern, she should buy from Phil'd Up Doughnuts. This is because the rate of change for Phil'd Up Doughnuts is lower than the rate of change for Ms. Spellings Donut Shop. As the number of doughnuts increases, the cost per doughnut at Phil'd Up Doughnuts will stay the same, whereas the cost per doughnut at Ms. Spellings Donut Shop will increase due to the fixed boxing fee.
We can verify this by calculating the total cost for each shop for a given number of doughnuts. For example, if Sandra wants to buy 6 decorative doughnuts, the total cost at Ms. Spellings Donut Shop would be $1.62 * 6 + $8 = $12.72. The total cost at Phil'd Up Doughnuts would be $17.
a sample from a refuse deposit near the strait of magellan had 60% of the carbon-14 found in a contemporary living sample. how old was the sample?
Answers
The sample is approximately 50,000 years old, as the amount of Carbon-14 in it is lower than that of a contemporary sample.
1) Calculate the ratio of Carbon-14 in the sample compared to a contemporary sample.
Ratio = (60/100) = 0.6
2) Calculate the number of half-lives since the sample was alive.
Number of half-lives = ln(0.6) / ln(0.5) = 0.69897
3) Calculate the age of the sample.
Age of sample = (0.69897 x 5730) = 4020 years
60% of the Carbon-14 identified in a sample taken from a waste site close to the Strait of Magellan was present in the sample from the trash deposit. We must first determine the sample's carbon-14 ratio in relation to a more recent sample in order to determine the sample's age. This ratio is equivalent to the sample's 60% carbon-14 content, or 0.6, when divided by 100. The number of half-lives since the sample has been alive may then be determined by taking the natural log of the ratio and dividing it by the natural log of 0.5, which is 0.69897. By dividing the quantity of half-lives by the 5730-year half-life of carbon-14, we may finally determine the sample's age.
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Leon is standing 67 feet from a telephone pole. As he looks up, a red-tailed hawk lands on the top of the
pole. Leon's angle of sight up to the bird is 24° and his eyes are 5.1 feet above the ground. How fall is th
pole?
Answers
The pole is 34.93 feet tall. The solution has been obtained by using trigonometry.
What is trigonometry?
Trigonometry is the branch of mathematics that focuses on the study of the sides, angles, and connections of the right-angle triangle.
We are given that Leon is standing 67 feet from a telephone pole. Leon's angle of sight up to the bird is 24° and his eyes are 5.1 feet above the ground.
From this, a figure is obtained which is attached below.
Using trigonometry,
We know that tan theta = opposite side/ adjacent side
So,
⇒tan θ = x/67
Here, θ = 24°
So,
⇒tan 24° = x/67
⇒x = 67 tan 24°
⇒x = 67 * 0.4452
⇒x ≈ 29.83
The height of the pole = 29.83 + 5.1 = 34.93 feet
Hence, the pole is 34.93 feet tall.
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Keisha got a prepaid debit card with $30 on it. For her first purchase with the card, she bought some bulk ribbon at a craft store. The price of the ribbon was 16 cents per yard. If after that purchase there was $23.92 left on the card, how many yards of ribbon did Keisha buy?
Answers
Answer:
38 yards of ribbon
Step-by-step explanation:
23.92 +.16x = 30
30-23.92 = 6.08
.16x =6.08
6.08/.16 = x
x =38 yards
Can somebody please look at the picture and help me pleaseeeeeeee
Answers
The equation 2 and 3 would not present linear functions.
What is linear function?
A linear function is one in which the maximum power of the variables is one.
A linear function is a function that has the form f(x) = mx + b, where m and b are constants. In other words, a linear function is a straight line when plotted on a graph.
Out of the given relationships, the following do not represent a linear function:
2) The ant population for a colony that begins with 5 ants and doubles in number every two weeks.
This relationship is not linear because the rate of change (the number of ants per unit time) is not constant. The number of ants increases exponentially, which results in a curve rather than a straight line when plotted on a graph.
3) A history student's grade on a test that begins at 100 but decreases by 4 points for each question answered incorrectly.
This relationship is not linear because the rate of change (the decrease in points per incorrect question) is not constant. The decrease in points results in a curve rather than a straight line when plotted on a graph.
The other two relationships do represent linear functions.
Therefore, The equation 2 and 3 would not present linear functions.
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After their soccer game, Mark and his friends eat at their favorite restaurant, the Dancing Meatball. When the bill comes, they divide the cost of a meatball pizza evenly among 5 people. Each person pays $3.25. Which equation can you use to find the cost c of the pizza?
Answers
c = 5 x 3.25
In this equation, "5" represents the number of people who are sharing the cost of the pizza, and "3.25" represents the amount each person is paying. So, multiplying these two numbers together gives the total cost of the pizza, "c".
a researcher records the following data: 4, 4, 4, 4, and 3. how would you describe the variability of these data? group of answer choices it is negative because 3 is less than the other scores in the distribution. it is very small (close to 0) because scores are approximately the same. it is very large (much greater than 0) because 3 is an outlier in the data. it is equal to zero because scores are approximately the same.
Answers
The variability of the data is very small (close to 0) because all the scores are approximately the same.
The variability of these data can be calculated using the variance formula, which is the average squared difference from the mean. In this case, the mean is 3.8 (the sum of 4, 4, 4, 4, and 3 divided by 5). Therefore, the variance is 0.08 (the sum of the squared differences [4-3.8, 4-3.8, 4-3.8, 4-3.8, 3-3.8] divided by 5). This indicates that the variability of the data is very small (close to 0) because all the scores are approximately the same.
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A plumber earned AED 1,075 when he worked 3. 5 hours. He charged the customer an extra AED 200
for materials. What is his hourly rate?
Answers
A plumber worked 3. 5 hours for AED 1,075 in total. He added AED 200 for supplies to the customer's bill hourly rate of the plumber is AED 364.29.
How much is charged, paid, or earned for each hour worked?
Instead of paying for the things the advisers try to sell you, you pay a set or hourly charge for their time.
We must first deduct the price of the materials from the plumber's overall profits in order to get his hourly rate:
Income equals AED 1,075 plus AED 200, or AED 1,275.
After that, we may split his total income by the quantity of hours he put in:
Hourly fee is calculated as follows: AED 1,275 / 3.5 hours of work.
Hourly rate is 364.29 AED.
The plumber's hourly charge is therefore AED 364.29.
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Bob tardo 55 minutos en limpiar el garaje. Cuantos segundos tardo bob?un minutos tiene 60 segundos
Answers
Bob tardó 55 minutos en limpiar el garaje, por lo que debemos convertir ese tiempo a segundos para saber cuánto tiempo tardó exactamente.
Sabemos que 1 minuto tiene 60 segundos, por lo que podemos utilizar esta información para convertir los 55 minutos a segundos. La fórmula para realizar esta conversión es:
Segundos = Minutos * 60
Aplicando esta fórmula a los 55 minutos que Bob tardó en limpiar el garaje, obtenemos:
Segundos = 55 * 60
Segundos = 3300
Por lo tanto, Bob tardó 3300 segundos, o aproximadamente 55 minutos, en limpiar el garaje.
Es importante destacar que la conversión de tiempo de minutos a segundos es una operación común en matemáticas y cálculo, y es útil para resolver problemas en los que es necesario trabajar con diferentes unidades de tiempo. Además, entender la relación entre diferentes unidades de tiempo es esencial para realizar cálculos precisos y resolver problemas de manera efectiva.
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There are many ways to save for retirement: putting money away into a treasure chest, purchasing gold nuggets, investing
in real estate, or financial assets. The method that is right for you depends on your income as well as your personal
preferences, such as risk tolerance. You can compare different methods and approximate your savings at retirement to
help you determine the option that best suits your personal needs and preferences. It is a good idea to approximate the
Income you will need at retirement before you start investing. This is where retirement planning becomes important to
consider. How much money do you think you would need to retire by age 67? Which investment options are the most
profitable when saving for retirement?
In this project, you will explore how to construct a retirement plan and consider the various investment options that are
financially beneficial as it pertains to retirement
OBJECTIVES
Construct a retirement plan given a yearly salary.
Compare investment options to determine the most financially beneficial option for a given situation.
VOCABULARY
Inflation rate
the rate of change in prices of a standard basket of consumer products over time
Answers
Most profitable investment option for retirement depends on your individual circumstances, such as your risk tolerance, time horizon, and investment goals. It's essential to consult with a financial advisor who can help you create a personalized retirement plan that meets your specific needs and goals.
Why are retirement planning crucial for generating money?Retirement planning is crucial to ensure that you have enough money to live comfortably during your golden years. The amount of money you will need in retirement depends on several factors, including your lifestyle, health, and expected lifespan.
However, a general rule of thumb is that you should aim to have saved 10-12 times your current annual income by the time you retire. Assuming a retirement age of 67, if you earn $50,000 per year, you would need to have saved between $500,000 and $600,000 by the time you retire. This amount may vary depending on your specific circumstances, such as your expected retirement lifestyle, expected lifespan, and anticipated expenses.
Which investment options are the most profitable when saving for retirement?When it comes to investment options for retirement, there are several options that can be financially beneficial. Some of the most profitable investment options for retirement include:
401(k) plans: A 401(k) plan is an employer-sponsored retirement plan that allows employees to contribute pre-tax dollars towards retirement savings. Traditional IRAs: A traditional IRA is an individual retirement account that allows you to contribute pre-tax dollars towards retirement savings. Roth IRAs: A Roth IRA is an individual retirement account that allows you to contribute after-tax dollars towards retirement savings. Mutual funds: Mutual funds are a type of investment that pools money from multiple investors to purchase a diversified portfolio of stocks, bonds, or other securities.Read more about retirement planning
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Solve for x ? Question in picture
Answers
Answer:
22
Step-by-step explanation:
x+35=56
56-35=22
x=22
a population of 100 frogs increases at an annual rate of 22%. after how many years will the population of the frogs double?
Answers
The population of the frogs will be double after 4 years when the current number of frogs were 100 and the annual rate of increment is 22%.
In the given question,
the current population of frogs are: 100
Annual rate of increase in population of frogs = 22%
Population required = double the current population = 2 x 100
= 200
Using the compound interest formula for amount, we can calculate the number of years in which the population of the frogs will be double:
Amount = principal ((1 + r%)^n) ,
here we will consider the amount to be population required, principal will be the current population, r is the annual rate of increment, and n is the time in years.
200 = 100 ((1 + 0.22) ^ n)
⇒ 200/100 = (1.22)^n
⇒ 2 = (1.22) ^ n
We plug digits in place of n and check for the closest result as 2.
We observe that (1.22)^4 = 2.2153 ≈ 2
Hence, we can say that the population of the frogs will be double after 4 years.
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State how the triangles are congruent using SSS, SAS, ASA, AAS, or
HL. If they are not congruent, type NOT.
Answers
Answer:
AAS
Step-by-step explanation:
There are two pairs of congruent angles, and the pair of congruent sides is not included in between the angles.
Maxine is pre-heating the oven to bake a pie. The temperature of the oven increases at a constant rate.
How long will it take the oven to reach a temperature of at least 375
degrees?
Responses
Answers
It will take approximately 12.96 minutes for the oven to reach a temperature of at least 375 degrees.
What is Approximation?
Approximation is the act of estimating or finding a value that is close to the true or exact value, but not necessarily identical to it. In other words, an approximation is a rough calculation or an educated guess that is used when the exact value is not known or is too difficult to calculate. Approximations are commonly used in mathematics, science, engineering, and many other fields where exact values are often difficult to obtain or not required. The level of accuracy of an approximation depends on the method used and the degree of precision needed for the particular situation.
Let's use the information given to find the rate of temperature increase per minute.
Between the first two data points, the temperature increased by 3 degrees in 1 minute. Between the next two data points, the temperature increased by (112.5 - 3) = 109.5 degrees in (5 - 1) = 4 minutes. So the rate of temperature increase is (109.5/4) = 27.375 degrees per minute.
To find out how long it will take the oven to reach 375 degrees, we can use the formula:
time = (final temperature - initial temperature) / rate of temperature increase
In this case, the initial temperature is 3 degrees and the final temperature is 375 degrees. Substituting these values and the rate of temperature increase we calculated earlier, we get:
time = (375 - 3) / 27.375 = 12.96 minutes (rounded to two decimal places)
So it will take approximately 12.96 minutes for the oven to reach a temperature of at least 375 degrees.
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miguel spent 1/2 hour riding his bike, as shown on the clockface. He spent 1/3 hour cleaning his room. How much more time did miguel spend riding his bike than cleaning his room? Express the answer as a fraction of an hour. Write numbers in the boxes provided.
Answers
Answer: 1/6 of an hour
Step-by-step explanation: multiply the numerator and denominator by 3 of 1/2 and 1/3 to get 3/6 and 2/6. There is 1/6 more
Two similar triangles, ∆ABC and ∆JKL are shown.
What is the length of side JL? Show all work solving for segment JL.
- show the proportion you set up to solve.
- show all work solving for JL
Answers
The solution is, the value of JL = 15.
What is triangle?A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices A, B, and C is denoted \triangle ABC. In Euclidean geometry, any three points, when non-collinear, determine a unique triangle and simultaneously, a unique plane.
here, we have,
Two similar triangles, ∆ABC and ∆JKL are shown.
so, from the diagram we get,
BC/KL = AB/JK = AC/JL
so, 1/3 = AC/JL
or, JL = 3 * 5
or, JL = 15
Hence, The solution is, the value of JL = 15.
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Find all six trigonometric functions of θ if the given point is on the terminal side of θ. Answer exactly. ( − 4 , − 1 )
Answers
The six trigonometric functions of the angle θ for the given point (-4, -1) are sin θ = -1/5, cos θ = -4/5, tan θ = 1/4, cot θ = -4, sec θ = -5/4, and csc θ = -5.
The six trigonometric functions of an angle θ are sine (sin θ), cosine (cos θ), tangent (tan θ), cotangent (cot θ), secant (sec θ), and cosecant (csc θ). These functions can be used to calculate the coordinates of a point on the terminal side of θ when the angle and one coordinate are known.
For example, if you know the angle θ and one coordinate (in this case, -4) of a point on the terminal side of θ, you can use the six trigonometric functions to calculate the remaining coordinate. In this case, the point is (-4, -1). The sine of θ is equal to the y-coordinate (-1) divided by the hypotenuse (distance from the origin). Therefore, sin θ = -1/5. The cosine of θ is equal to the x-coordinate (-4) divided by the hypotenuse. Therefore, cos θ = -4/5. The tangent of θ is equal to the y-coordinate (-1) divided by the x-coordinate (-4). Therefore, tanθ = -1/-4 = 1/4. The cotangent of θ is equal to the x-coordinate (-4) divided by the y-coordinate (-1). Therefore, cot θ = -4/-1 = -4. The secant of θ is equal to the hypotenuse (distance from the origin) divided by the x-coordinate (-4). Therefore, sec θ = 5/-4 = -5/4. The cosecant of θ is equal to the hypotenuse divided by the y-coordinate (-1). Therefore, csc θ = 5/-1 = -5.So, the six trigonometric functions of the angle θ for the given point (-4, -1) are sin θ = -1/5, cos θ = -4/5, tan θ = 1/4, cot θ = -4, sec θ = -5/4, and csc θ = -5.
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unit 9: transformations homework 1: translations
Answers
The translation of a function and coordinates are given below..
What is translation?It is the movement of the shape in the left, right, up, and down directions.
The translated shape will have the same shape and shape.
There is a positive value when translated to the right and up.
There is a negative value when translated to the left and down.
We have,
The following functions:
y = x²
y = 2x
The transformation is:
Translate two units to the left.
So,
y = (x - 2)²
y = 2(x - 2)
Translation 5 units up.
y = x² + 5
y = 2x + 5
The following coordinates:
Translate 2 units to the right.
(1, 4) = (1 + 2, 4) = (3, 4)
(5, 8) = (5 + 2, 8) = (7, 8)
Translation 5 units down.
(1, 4) = (1, 4 - 5) = (1, -1)
(5, 8) = (5, 8 - 5) = (5, 3)
Thus,
The translation of a function and coordinates are given above.
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