Answer:
Step-by-step explanation:
One way for Paul to display his data is by using a scatter plot. A scatter plot is a graph that uses ordered pairs to represent data points. In this case, each data point would represent a friend's score and the number of minutes they spent studying science over the weekend.
To create a scatter plot, Paul would plot each data point as an ordered pair on a coordinate plane. The x-coordinate would represent the number of minutes spent studying, and the y-coordinate would represent the friend's score on the science test. Paul could then use a title and labels for the x and y axes to make the graph clear and informative.
A scatter plot is a good choice for this data because it allows Paul to easily see any relationship between the amount of time spent studying and the scores on the science test. He can also use the graph to identify any outliers or patterns in the data.
Mary, Jane, Tom, Andy saved for 6 weeks like this:
-
M: 2, 4, 8, 16,
J: 10, 12, 14, 16,
T: 7, 13, 19, 25,
A:
3,6,9..
Work out how much each person saved so that you can put their names in
order of how much they saved, from smallest to largest amount.
Enter your code as a four-lettered "word"
Write 7.725666118 as a percentage
please show the method too.
Answer: 773%
Step-by-step explanation: 7.725666118 x 100 = 772.5666118
772.5666118 = 773 (rounded).
So, 7.725666118 as a percentage is 773%
For example, 50% is written in decimal form as 0.5
And, 0.5 x 100 = 50.
Multiplying by 100 is the same thing as moving the decimal two places to the right.
A salesman's commission varies directly as his sales.If the commission is $10 for $100 in sales, find the commission for $150 in sales.
a. $12 b. $20 c. $15 d. $25
The commission for $150 in sales is $15. This corresponds to option (c) in the given answer choices.
We are told that the salesman's commission varies directly with his sales. This means that if the sales increase by a certain percentage, then the commission will also increase by the same percentage. We can set up a proportion to solve for the commission:
Commission/Sales = Commission Rate
We are given that the commission rate is $10 for every $100 in sales. We can use this information to set up a proportion:
Commission/150 = 10/100
To solve for the commission, we can cross-multiply and simplify:
Commission = (150 x 10)/100 = $15
In general, if we let x represent the sales amount, and c represent the commission, then we can use the formula:
c = (x/100) * 10
where 10 represents the commission rate of $10 per $100 in sales. We can use this formula to calculate the commission for any given sales amount.
Therefore, Correct option is C.
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Click an item in the list or group of pictures at the bottom of the problem and, holding the button down, drag it into the correct position in the answer box. Release your mouse button when the item is place. If you change your mind, drag the item to the trashcan. Click the trashcan to clear all your answers.
Factor completely and then place the factors in the proper location on the grid.
81a2 + 36a + 4
When we factor the expression, we are going to obtain the result;
9a(9a + 4) + 4
How do you factor an expression?Factoring an expression involves breaking it down into simpler parts that can be multiplied together to obtain the original expression. The specific method used to factor an expression depends on the type of expression.
It's important to note that factoring can sometimes be a challenging process, and not all expressions can be factored using real numbers.
We can carry out this factorization by the use of the nesting method, Hence;
81a^2 + 36a + 4
(81a^2 + 36a ) + 4
9a(9a + 4) + 4
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A group of medical professionals predicts that the number of yearly occurrences of a disease will decrease from the year 2008 to the year 2016. They constructed a spreadsheet with formulas to display their predictions. How many occurrences of the disease do they predict in the year 2016?
One way to predict occurrences of a disease for a future year based on previous years' data is to use exponential modeling.
How can we predict occurrences of disease for year mathematically based on previous years data?Specifically, the formula for exponential growth or decay can be used to project the number of cases for a future time period based on the growth or decline rate observed in the past.
For example, let's say that we have data on the number of cases of a certain disease for the past 10 years. We can use this data to fit an exponential model that describes the growth or decline rate of the disease over time.
Once we have this model, we can use it to predict the number of cases for a future year. For instance, if the model predicts that the disease will grow at a rate of 5% per year, we can use this rate to estimate the number of cases for the next year by multiplying the current number of cases by 1.05. By doing this, we can forecast the occurrence of the disease for the next year based on the previous years' data.
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Find the volume of the solid formed by rotating the space bounded by y = 2x², y=0, and x = 2 around the line y = 8
The volume of the solid is (256π/15) - (128π/3) cubic units, which is formed by rotating the space bounded by y = 2x², y=0, and x = 2 around the line y = 8.
To find the volume of the solid formed by rotating the space bounded by y = 2x², y=0, and x = 2 around the line y = 8, we can use the disk method.
First, we need to find the bounds of integration. Since x = 2 is the right boundary, we can integrate from 0 to 2.
The distance between the line y=8 and the curve y=2x² is 8 - 2x². Therefore, the radius of the disk at a given x-value is 8 - 2x².
The area of each disk is given by π(radius)². Therefore, the volume of the solid is given by the integral of π(radius)² dx from 0 to 2;
V = ∫₀² π(8 - 2x²)² dx
This integral can be evaluated by expanding the square and using the power rule of integration. After simplification, we get;
V = (256π/15) - (128π/3)
Therefore, the volume of the solid is (256π/15) - (128π/3) cubic units.
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What is the y-intercept of the line with the equation y = one-half x minus 3? a. -3 c. -6 b. 6 d. One-half
Pre-Caclus 75 points + [tex]Brainliest[/tex]
Question shown in picture.
Options:
[tex]\frac{x - 28}{x-46}[/tex]
[tex]\frac{x - 13}{x-28}[/tex]
[tex]\frac{x - 7}{x-13}[/tex]
[tex]\frac{x - 13}{x-46}[/tex]
Expectations:
Correct
Explantion
Reasonable
Must not:
Incorrect
No explanation
Spam
Gibberish
Question:
Answer:
Factoring the polynomials and then simplifying:
[tex] \frac{ {x}^{2} - 20x + 91}{ {x}^{2} - 35x + 196} \times \frac{ {x}^{2} - 74x + 1288}{ {x}^{2} - 92x + 2116 } [/tex]
[tex] \frac{(x - 7)(x - 13)}{(x - 7)(x - 28)} \times \frac{(x - 28)(x - 46)}{(x - 46)(x - 46)} [/tex]
[tex] \frac{x - 13}{x - 46} [/tex]
Answer:
x−46
x−13
Step-by-step explanation:
Explanation in picture :)
Solve the system of linear equations using any method
Answer:
(-1,-1)
Step-by-step explanation:
Rearange the equation
5y=6x+1 + 4y=-6x-10
9y = -9
y = -1
The only option with y as -1 is the second one
You can plug in -1 as x into any of the equations
5y=6x+1
-5 = -6+1
Hope this helps
trigonometry, please help.
The value of x is given as follows:
x = 9.40 cm.
What are the trigonometric ratios?The three trigonometric ratios are the sine, the cosine and the tangent, and they are defined as follows:
Sine of angle = length of opposite side to the angle divided by the length of the hypotenuse.Cosine of angle = length of adjacent side to the angle divided by the length of the hypotenuse.Tangent of angle = length of opposite side to the angle divided by the length of the adjacent side to the angle.The parameters for this problem are given as follows:
Hypotenuse of 10 cm.Side length of x cm adjacent to the angle of 20º.Applying the cosine, the value of x is obtained as follows:
cos(20º) = x/10
x = 10 x cosine of 20 degrees
x = 9.40 cm.
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A teacher has a prize box that has 6 fidgets,2 rubix cubes,and 2 pop its-as well as a bag of candy that has 4 lollipops and 3 fieces of chocolate. Esther wins the weekly prize in class and can pick one toy from the prize box and one candy from the candy bag. What is the probability she will pick a fidget and a piece of chocolate?
The probability of Esther picking a fidget and a piece of chocolate is 3/14.
How to obtain the probabilityTo obtain the probability, we first need to list the items and then determine the chances of them being picked. Fisrt, there are a total of 12 items in the prize box and 7 items in the candy bag.
The probability of Esther picking a fidget from the prize box is 6/12 or 1/2, and the probability of her picking a piece of chocolate from the candy bag is 3/7. Since these events are independent, we can multiply their probabilities to find the probability of both events occurring:
P(fidget and chocolate) = P(fidget) x P(chocolate)
P(fidget and chocolate) = (1/2) x (3/7)
P(fidget and chocolate) = 3/14
Therefore, the probability of Esther picking a fidget and a piece of chocolate is 3/14.
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● • 4√13 ft
.
143 ft
• 14.33 ft
Which list shows these diagonal lengths in order from greatest to least?
F 14.33, 14,4√13
G 14.33, 4√13, 14
H 14, 14.33, 4√/13
J 4√13, 14, 14.33
The correct order from greatest to least is option G. 14.33, 4√13, 14
How did we get the value?To convert a root number to a whole number, you simply need to evaluate the root expression by performing the indicated operation. The result of the evaluation will be a whole number.
Here are the steps to convert a root number to a whole number:
Determine the type of root: square root (√), cube root (∛), etc.
Write the root expression using the appropriate symbol and the radicand (the number inside the root symbol).
Evaluate the expression by performing the indicated operation. For example, if it is a square root, multiply the number inside the root symbol by itself. If it is a cube root, multiply the number inside the root symbol by itself three times.
The result of the evaluation will be a whole number.
To compare the diagonal lengths, we need to convert them to the same units. We can convert 4√13 ft to approximately 9.17 ft by using a calculator. Now we have:
14.33 ft
0.143 ft
9.17 ft
So, the order from greatest to least is:
G. 14.33, 4√13, 14
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Assume that
log 4 ~ 0.6021, log 5 ~ 0.6990, and log 6 – 0.7782
Use the properties of logarithms to evaluate
the expression. Do not use your calculator.
(Round your answer to 4 decimal places. If your
answer includes a negative, enter it like: -2)
log 16 = [a]
Answer:
Step-by-step explanation:
?
Roller Coaster Crew
Ray and Kelsey have summer internships at an engineering firm. As part of their internship, they get to assist in the planning of a brand new roller coaster. For this assignment, you help Ray and Kelsey as they tackle the math behind some simple curves in the coaster's track.
Part A
The first part of Ray and Kelsey's roller coaster is a curved pattern that can be represented by a polynomial function.
Ray and Kelsey are working to graph a third-degree polynomial function that represents the first pattern in the coaster plan. Ray says the third-degree polynomial has four intercepts. Kelsey argues the function can have as many as three zeros only. Is there a way for the both of them to be correct? Explain your answer.
Kelsey has a list of possible functions. Pick one of the g(x) functions below and then describe to Kelsey the key features of g(x), including the end behavior, y-intercept, and zeros.
g(x) = (x + 2)(x − 1)(x − 2)
g(x) = (x + 3)(x + 2)(x − 3)
g(x) = (x + 2)(x − 2)(x − 3)
g(x) = (x + 5)(x + 2)(x − 5)
g(x) = (x + 7)(x + 1)(x − 1)
Create a graph of the polynomial function you selected from Question 2.
Part B
The second part of the new coaster is a parabola.
Ray needs help creating the second part of the coaster. Create a unique parabola in the pattern f(x) = (x − a)(x − b). Describe the direction of the parabola and determine the y-intercept and zeros.
Create a graph of the polynomial function you created in Question 4.
Part C
Now that the curve pieces are determined, use those pieces as sections of a complete coaster. By hand or by using a drawing program, sketch a design of Ray and Kelsey's coaster that includes the shape of the g(x) and f(x) functions that you chose in the Parts A and B. You do not have to include the coordinate plane. You may arrange the functions in any order you choose, but label each section of the graph with the corresponding function for your instructor to view.
Part D
Create an ad campaign to promote Ray and Kelsey's roller coaster. It can be a 15-second advertisement for television or radio, an interview for a magazine or news report, or a song, poem, or slideshow presentation for a company. These are just examples; you are not limited to how you prepare your advertisement, so be creative. Make sure to include a script of what each of you will say if you are preparing an interview or a report. The purpose of this ad is to get everyone excited about the roller coaster.
Find the volume of a cone that has a height of 9 inches and a diameter of 12 inches. Use 3. 14 for pi and round to the nearest tenth
The volume of the cone with height of 9 inches and a diameter of 12 inches is approximately 339.3 cubic inches.
To find the volume of a cone, we need to use the formula V = (1/3)πr^2h, where r is the radius of the base of the cone and h is the height of the cone.
Since the diameter of the cone is given as 12 inches, we can find the radius as half of the diameter, which is 6 inches. The height of the cone is given as 9 inches.
Now, we can substitute the values of r and h in the formula to get the volume as follows
V = (1/3)πr^2h
= (1/3)×3.14×6^2×9
= 339.12 cubic inches (rounded to the nearest tenth)
Therefore, the volume of the given cone is approximately 339.1 cubic inches.
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skinner produce buys fresh boston lettuce daily. daily demand is normally distributed with a mean of 100 units and standard deviation of 15 units. at the beginning of the day skinner orders 140 units of lettuce. what is the probability that skinner will have at least 20 units left over by the end of the day?
For a normally distribution of daily demand of fresh boston lettuce produce by skinner. the probability that skinner will have at least 20 units left over by the end of the day is equals the one.
Skinner produce buys fresh boston lettuce daily. There is daily demand is normally distributed,
Mean = 100 units
Standard deviations = 15 units
At beginning of the day skinner orders 140 units of lettuce. We have to determine the probability that skinner will have at least 20 units left over by the end of the day, P( X ≥ 20). Using the Z-Score for normal distribution is written as
[tex]z = \frac{ X - \mu}{\sigma} [/tex]
where, z --> z-score
X --> excepted value
--> standard deviations
--> population mean
Subsritute the known values in above formula, [tex]z = \frac{ 20 - 100}{15} [/tex]
= - 5.33
Now, probability value, P ( X < 20)
[tex]= P( \frac{X - \mu}{\sigma} < \frac{ 20 - 100}{15} )[/tex] = P( z < - 5.33 )
= 0.000
So, P( X < 20) = P( z< -5.33) = 0 . Also, required probability is P( X≥ 20) = 1 - 0
= 1
Hence, required value is one.
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Three stages at a music festival are arranged in a triangle. The distances between the centers of each stage are given:
Stage A and Stage B: 100 yards
Stage B and Stage C: 135 yards
Stage C and Stage A: 210 yards
List the interior angle measures formed at the center of each stage from greatest to least.
The interior angle measures formed at the center of each stage at a music festival from greatest to least are as follow,
Stage A = 126.01° > Stage B =31.33° > Stage C = 22.66°
Shape of the arrangement of three stages at a music festival is triangle.
Distance between the centers of each stage are as follow,
Stage A and Stage B = 100 yards
Stage B and Stage C = 135 yards
Stage C and Stage A = 210 yards
For Interior angles,
Use the Law of Cosines, which states that for a triangle with sides a, b, and c and opposite angles A, B, and C,
c² = a² + b² - 2abcos(C)
Label the sides of the triangle formed by the three stages,
Side AB = 100 yards
Side BC = 135 yards
Side CA = 210 yards
Then, find the interior angles at each stage as follows,
At Stage A,
a = 210 yards (CA)
b = 100 yards (AB)
c = 135 yards (BC)
cos(A) = (b² + c² - a²) / (2bc)
⇒cos(A) = (100² + 135² - 210²) / (2 × 100 ×135)
⇒cos(A) = -0.5880
⇒A = arccos(-0.5880)
A = 126.01 degrees
At Stage B,
a = 100 yards (AB)
b = 135 yards (BC)
c = 210 yards (CA)
cos(B) = (a² + c² - b²) / (2ac)
⇒cos(B) = (100² + 210² - 135²) / (2 × 100 ×210)
⇒cos(B) = 0.8542
⇒B = arccos(0.8542)
⇒B = 31.33 degrees
At Stage C,
a = 135 yards (BC)
b = 210 yards (CA)
c = 100 yards (AB)
cos(C) = (a²+ b² - c²) / (2ab)
⇒cos(C) = (135² + 210² - 100²) / (2 × 135 × 210)
⇒cos(C) = 0.9228
⇒C = arccos(0.9228)
⇒ C = 22.66 degrees
Therefore, the interior angle measures formed at the center of each stage from greatest to least are,
Stage A = 126.01 degrees
Stage B =31.33 degrees
Stage C = 22.66 degrees
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What are the zeros of the function f (x) = (2x +6)(x - 4)?
Answer:
x = - 3 , x = 4
Step-by-step explanation:
to find the zeros let f(x) = 0 , that is
(2x + 6)(x - 4) = 0
equate each factor to zero and solve for x
2x + 6 = 0 ( subtract 6 from both sides )
2x = - 6 ( divide both sides by 2 )
x = - 3
x - 4 = 0 ( add 4 to both sides )
x = 4
the zeros are x = - 3 , x = 4
What are 2 questions that would require an interpretation of the table?
1. What is the relationship between education level and income in this population? 2. How does the popularity of different music genres vary across different age groups?
What do you mean by an interpretation?An interpretation is a way of understanding or explaining something. Interpretation often involves making connections between different elements or pieces of information to create a coherent understanding. It can also involve applying theories, frameworks, or methods to analyze or explain a particular subject. Ultimately, interpretations are subjective and may vary based on individual experiences and perspectives, making it important to consider multiple interpretations when analyzing a particular subject.
Here is two example questions:
1. What is the relationship between education level and income in this population?
Interpretation: To answer this question, you would need to examine a table that includes columns for education level and income, such as a cross-tabulation or a summary statistics table. You could look for patterns or trends in the data, such as higher incomes being associated with higher levels of education. You might also look for any outliers or anomalies in the data that could be affecting the overall relationship between education level and income.
2. How does the popularity of different music genres vary across different age groups?
Interpretation: To answer this question, you would need to examine a table that includes rows for different age groups and columns for different music genres, such as a frequency table or a bar chart. You could look for patterns or trends in the data, such as certain music genres being more popular among younger age groups or older age groups. You might also look for any outliers or anomalies in the data that could be affecting the overall relationship between age and music genre popularity.
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At Bruno’s Video World, the regular price for a DVD is d dollars. How many DVDs can be purchased for x dollars when the DVDs are on sale at 20% off the regular price?
The number of DVDs can be purchased for x dollars = 5x/4d.
The correct answer is an option (e)
Here you have x dollars to spend.
The regular price for a DVD is d dollars.
We need to find the number of DVDs can be purchased for x dollars when the DVDs are on sale at 20% off the regular price.
To find number of DVDs can be purchased for x dollars, we must divide x by the sale price of a CD.
After 20% off, the sale price would be,
(100 - 20)% of d
= 80% of d
Using percentage formula, we can write 80 percent of d as,
80%d
= (80/100) × d
= (4/5)d .
So, we need to divide x by (4/5d) which is equal to 5x/4d.
Therefore, the correct answer is an option (e)
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The complete question is:
At Bruno’s Video World, the regular price for a DVD is d dollars. How many DVDs can be purchased for x dollars when the DVDs are on sale at 20% off the regular price?
a) 4/5x
b) 5/4x
c) 4/5d
d) 4x/5d
e) 5x/4d
a person wants to know the true proportion of customers that write a review. the owner randomly selects a random sample of 560 customers and only 340 of them wrote a review. when calculating a 90% confidence interval, what is the margin of error used:
The margin of error used in calculating the 90% confidence interval for the true proportion of customers who write a review is 0.043.
To calculate the margin of error, we first need to find the sample proportion of customers who wrote a review:
sample proportion[tex]\hat p =\frac{ 340}{560} = 0.607[/tex]
Next, we can use the following formula to calculate the margin of error:
margin of error [tex]= z* \sqrt{(\hat p * (1-\hat p) / n)[/tex]
where z* is the z-score associated with the desired level of confidence (90% in this case), and n is the sample size.
The z-score for a 90% confidence interval is 1.645. Plugging in the values, we get:
margin of error =[tex]1.645 * \sqrt{(0.607 * (1-0.607) / 560)[/tex]
= 0.043
Therefore, the margin of error used in calculating the 90% confidence interval for the true proportion of customers who write a review is 0.043.
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Please help to hard and I’ll give u 20 points
Answer: 2:4
Step-by-step explanation:
we randomly select 100 pell grant recipients from two states. state a is a relatively small state with approximately 4,000 pell grant recipients. state b is a large state with approximately 200,000 pell grant recipients. suppose that the mean and standard deviation in individual pell grants is approximately the same for both states: and . for which state is the sample mean for our 100 pell grant recipients most likely to be within $80 of $2,600? group of answer choices state a because the sample represents a larger segment of this small population. state b because there is less variability in larger populations so estimates from samples are more accurate. equally likely because for both states.
Answer: The answer is state B because there is less variability in larger populations so estimates from samples are more accurate. The standard error of the mean is proportional to the standard deviation of the population and inversely proportional to the square root of the sample size. Since state B has a much larger population, it is more likely that our sample mean will be within $80 of $2,600 for state B than for state A.
Step-by-step explanation:
Marley owns a music store. The first week a new album was released, he sold 800 copies at his store. Each week after the release, the number of copies sold decreased by 11%
Select all the functions that can be used to find the number of copies sold, f(n), n weeks after the release.
The correct options that shows the the functions that can be used to get the number of copies are:
f(1) = 800 , f(n) = 0.89 * f(n-1) n ≥ 2Option 2 is also correctHow to find the number of copiesfirst week sold 800 copies
∴ at n=1 ⇒⇒⇒ f(1) = 800
Each week the number of copies sold decreased by 11%.
∴ at n = 2 ⇒ f(2) = 800 * (1-.11) = 800 * 0.89 = f(1) * 0.89
at n = 3 ⇒ f(3) = 800 * 0.89 *0.89 = 800 * 0.89² = f(1) * 0.89 * f(2)
at n = 4 ⇒⇒⇒ f(4) = 800 * 0.89³ = f(1) * 0.89 * f(3)
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Please help me solve my add math question
A result of 2 = 7 - = 7 - (-5) = 12 and = -5 is as follows as after combining similar terms , (2-μ)a + (u+1).
what is vectors ?A vector seems to be an object in mathematics that also has scale (or length) and motion. Arrows can be used to graphically depict vectors, with the arrow's trajectory denoting the vector's magnitude and its length denoting its direction. In order to construct a new vector with the a different magnitude but the same direction, vectors can be multiplied by a scalar (a real number) or joined together to make a resultant vector (or the opposite direction if the scalar is negative).
given
First, we can make the given equation simpler:
(2-μ)a + (u+1)
b = 7a - (2+2)b
2a - a + ub + + + b = 7a - 2b - 2b
combining similar terms
(2-μ)a + (u+1)
b = 7a - 4b
We currently have two equations:
2 - μ = 7 (1)
u + 1 = -4 (2) (2)
Equation (2) provides us with:
u = -5
Equation (1) is amended as follows:
2 - μ = 7
μ = -5
A result of 2 = 7 - = 7 - (-5) = 12 and = -5 is as follows as after combining similar terms , (2-μ)a + (u+1).
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The complete question is:-
The vectors a and b are not parallel and (2-μ)a + (u + 1)b = 7a-(2+2)b where λ and μ are scalars. Find the values of 2 and μ.
write an equation that describes the line: a line has a slope of 0 and through the point (3,-4)
Answer:
Step-by-step explanation:
slope (m) =0
point =(x, y)
=(3, -4)
equation:
(y-y1) =m(x-x1)
{y-(-4)} =0
y+4 =0
y =-4
Explanation:
as we know, the equation to find the slope of a line is (y-y1) =m(x-x1). here, it is given that the slope of the line (m) is 0 and it passes through points 3 and -4. therefore, we can conclude that equation (y) would be 0 by calculating.
50 POINTS ASAP Triangle NMO is drawn with vertices N(−5, 2), M(−2, 1), O(−3 , 3). Determine the image vertices of N′M′O′ if the preimage is reflected over x = −2.
N′(5, −2), M′(2, 1), O′(3, 3)
N′(−2, 2), M′(1, 1), O′(0, 3)
N′(1, 2), M′(−2, 1), O′(−1, 3)
N′(−5, −2), M′(−2, −1), O′(−3, −3)
Answer:
To reflect a point over a vertical line x = c, where c is a constant, we can use the formula (2c - x, y).
Given the line of reflection x = -2, and the original points N(-5, 2), M(-2, 1), O(-3, 3), we can apply the formula as follows:
For N(-5, 2):
N' = (2(-2) - (-5), 2) = (1, 2)
For M(-2, 1):
M' = (2(-2) - (-2), 1) = (2, 1)
For O(-3, 3):
O' = (2(-2) - (-3), 3) = (1, 3)
So, the correct image vertices of N'M'O' after reflecting over x = -2 are N'(1, 2), M'(2, 1), O'(1, 3), which corresponds to the option:
N′(1, 2), M′(2, 1), O′(1, 3)
Please double-check to make sure if its right ;D
Find the area of the shaded regions:
Please help!!!
The areas of the composite figures are listed below:
Case 6: A = 134.126 ft²
Case 7: A = 212.5 cm²
Case 8: A = 957.305 m²
Case 9: A = 316.643 in²
How to determine the area of a composite figure
In this problem we find four cases of composite figure, that is, figures that are the result of adding and subtracting figures, whose area formulas are introduced below:
Semicircle
A = 0.5π · r²
Circle
A = π · r²
Triangle
A = 0.5 · b · h
Rectangle
A = b · h
Where:
A - Arear - Radiusb - Baseh - HeightNow we proceed to determine the area of each composite figure:
Case 6
A = (25 ft)² - π · (12.5 ft)²
A = 134.126 ft²
Case 7
A = (25 cm) · (14 cm) - 0.5 · (25 cm) · (11 cm)
A = 212.5 cm²
Case 8
A = 0.5π · [(21.6 m)² + (9 m)²] + 0.5 · (21.6 m) · (9 m)
A = 957.305 m²
Case 9
A = 0.5 · (20 in) · √[(25 in)² - (20 in)²] + (17 in) · √[(25 in)² - (20 in)²] - 0.5π · 0.25 · [(25 in)² - (20 in)²]
A = 316.643 in²
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Destiny sells newspapers at a
newsstand outside of her grocery
store. She normally charges $2.35 for
a newspaper. Today, Destiny has
lowered the price by $0.50. So far, she
has sold 8 newspapers at the new
price. How much money has Destiny
made so far from selling newspapers
at the new price?
Answer:
$14.80
Step-by-step explanation:
First you need to subtract .50 from 2.35, which is 1.85.
Next you need to multiply 1.85 times 8.
1.85 x 8 = 14.80
Therefore, she has made $14.80 since the new price.
Prove the following identity:
(cos 2x + cos 4x)/(cos 2x - cos 4x) = (cot 3x/tan x)
We have shown that the left-hand side of the equation is equal to the right-hand side of the equation, and the trigonometric identity is proved.
We have,
(cos 2x + cos 4x)/(cos 2x - cos 4x)
We can use the identity cos 2A = 1 - 2sin² A to rewrite cos 4x as:
cos 4x = 1 - 2sin² 2x
Then, substituting this into the expression, we get:
(cos 2x + 1 - 2sin² 2x)/(cos 2x - 1 + 2sin² 2x)
Using the identity sin² A = 1 - cos^2 A, we can rewrite sin² 2x as:
sin² 2x = 1 - cos² 2x
Substituting this into the expression, we get:
(cos 2x + 1 - 2(1 - cos² 2x))/(cos 2x - 1 + 2(1 - cos² 2x))
Simplifying the numerator and denominator separately, we get:
(2cos² 2x + 1)/(2cos² 2x - 1)
Using the identity cot A = 1/tan A, we can rewrite tan 3x as:
tan 3x = sin 3x/cos 3x
We can then use the identity sin 3A = 3sin A - 4sin^3 A and
cos 3A = 4cos³ A - 3cos A to rewrite sin 3x and cos 3x as:
sin 3x = 3sin x - 4sin³ x
cos 3x = 4cos³ x - 3cos x
Substituting these expressions into the right-hand side of the equation, we get:
cot 3x/tan x = (cos x/sin x)/(3sin x - 4sin³ x)/(cos x)
Simplifying by multiplying the numerator and denominator by the reciprocal of the fraction in the denominator, we get:
cot 3x/tan x = (cos x/sin x) * (cos x)/(3sin x - 4sin^3 x)
Multiplying the two fractions together, we get:
cot 3x/tan x = cos^2 x/(sin x(3 - 4sin^2 x))
Using the identity 1 - cos^2 A = sin^2 A, we can rewrite cos^2 x as:
cos^2 x = 1 - sin^2 x
Substituting this into the expression, we get:
cot 3x/tan x = (1 - sin^2 x)/(sin x(3 - 4sin^2 x))
Simplifying the numerator and denominator separately, we get:
cot 3x/tan x = (1/sin x) x (1 + sin x)/(3 - 4sin^2 x)
Using the identity 1 + sin A = 1/cos A, we can rewrite the expression as:
cot 3x/tan x = (1/sin x) x (1/cos x)/(3 - 4sin^2 x)
Finally, using the identity cot A = cos A/sin A, we can rewrite the expression as:
cot 3x/tan x = cos x/(sin x(3 - 4sin^2 x))
Therefore,
We have shown that the left-hand side of the equation is equal to the right-hand side of the equation, and the trigonometric identity is proved.
Learn more about trigonometric identities here:
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