Answer:
$8036.784
Step-by-step explanation:
It is easy
The area of carpet is 13.4×24=321.6sq meter
Now, for 1 sq. meter the cost is $24.99
So for 321.6sq.meters, 321.6×24.99
=$8036.784
It will cost amount of $8036.784 to carpet a rectangular room.
What is the area of the rectangle?The area of a rectangle is defined as the product of the length and width.
The area of a rectangle = L × W
Where W is the width of the rectangle and L is the length of the rectangle
To carpet a rectangular room that measures 13.4 meters by 24 meters, it would area:
13.4 x 24 = 321.6 square meters
Since one square meter, the cost is $24.99
So for 321.6 square meters, 321.6×24.99
= $8036.784
Thus, it will cost amount of $8036.784 to carpet a rectangular room.
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someone help me pls it maths
Answer:
a.) 7/9 × 5 2/5
= 7/9 × 27/5
= = 189/25
= 21/5 or 4 1/5
b.) 1 1/3 × 4 1/6
= 4/3 × 25/6
= 100/18
= 50/9 or 5 5/9
Hope this helps
Answer:
A) 21/5. B) 175/24
Step-by-step explanation:
Turn the mixed numbers into improper fractions then just multiply across to get the answers
Hope this helps!
Which point on the y-axis lies on the line that passes through point g and is parallel to line df?.
The point on the y-axis that lies on the line passing through point g and is parallel to line df is (0, 9).
To find the point on the y-axis that lies on the line passing through point g and is parallel to line df, we first need to determine the slope of line df. Since the line is parallel to the new line passing through point g, the slope will be the same.
Once we have the slope, we can use point-slope form to find the equation of the new line. Then, we can set x=0 (since we want to find the point on the y-axis) and solve for y to find the y-coordinate of the point.
So, let's begin.
First, let's find the slope of line df. We can use the formula:
slope = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are any two points on the line. We can use the points d and f, which are (5, 3) and (10, 8), respectively.
slope = (8 - 3) / (10 - 5) = 1
So the slope of line df is 1.
Now, using point-slope form, we can find the equation of the new line passing through point g (which is (-2, 7)) and having a slope of 1. The formula for point-slope form is:
y - y1 = m(x - x1)
where m is the slope and (x1, y1) is any point on the line (in this case, point g). Substituting in our values, we get:
y - 7 = 1(x - (-2))
y - 7 = x + 2
y = x + 9
So the equation of the new line is y = x + 9.
To find the point on the y-axis that lies on this line, we set x=0:
y = 0 + 9
y = 9
So the point on the y-axis that lies on the line passing through point g and is parallel to line df is (0, 9).
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the surface area of earth is about 196.9 million square miles. The land area is about 57.5 million square miles, and the rest is water. What is the probability that a meteorite that reaches the surface of earth will hit land
Answer: 29.2%
Step-by-step explanation:
The total surface area is 196.9 million square miles and out of this, land occupies 57.5 million square miles.
The probability that a meteorite would hit land is the percentage of and out of the entire planet area:
= Land area/ Total earth surface area
= 57.5/ 196.9
= 29.2%
for f(x)=x2+1, which function below has a graph that is a translation up 3 units of the graph
The function that has a graph that is a translation up 3 units of the graph is g(x) = (x - 3)^2 + 1
How to determine the equation of the function after the translation transformation?The equation of the function before transformation or translation is given as:
f(x) = x^2 + 1
When the given function f(x) = x^2 + 1 is translated up by 3 units, the rule of the function translation is
g(x) = f(x - 3)
Next, we substitute the expression x - 3 for x in the function f(x) = x^2 + 1
So, we have:
f(x - 3) = (x - 3)^2 + 1
Next, we substitute the equation f(x - 3) = (x - 3)^2 + 1 in the function equation g(x) = f(x - 3)
So, we have:
g(x) = (x - 3)^2 + 1
Hence, the function that has a graph that is a translation up 3 units of the graph is g(x) = (x - 3)^2 + 1
So, the complete parameters are:
f(x) = x^2 + 1
g(x) = (x - 3)^2 + 1
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you go to the pet store and see 9 cats and 12 dogs.if this ratio stays the same and there are 108 cats in the store,how many dogs would be in the store?
Answer:
144
Step-by-step explanation:
Because if you multiply 9*12=108 and 12*12=144
TRAVEL A certain airline requires that carry-on luggage is designed to fit within a rectangular prism with a length of 24 inches, a height of 12 inches, and width of 8 inches. What is the maximum possible volume of a piece of carryon luggage?
______in^3
If a certain airline requires that carry-on luggage. the maximum possible volume of a piece of carryon luggage is: 2,304 cubic inches.
What is the maximum possible volume?Maximum possible volume describes the biggest amount of space that a 3 dimensional object may occupy .
Using this formula to determine the maximum possible volume of a piece of carryon luggage
V = length x height x width
Where:
Length = 24 in
Height = 12 in
Width = 8 in
Let plug in the formula
V = 24 in x 12 in x 8 in
V = 2,304 cubic inches
Therefore the maximum possible volume is 2,304 cubic inches.
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Which of the following is a right triangle?
A)
a triangle with sides measuring 8, 12, and 13
B)
a triangle with sides measuring 6, 8, and 12
C)
a triangle with sides measuring 1, 2, and 3
D)
a triangle with sides measuring 10, 24, and 26
Answer:
D): a triangle with sides measuring 10, 24, and 26
I hope I helped you! If my answer is correct, could you please mark me brainliest? Thanks!
(iii) a source emits sound of wavelengths 2.54 m and 2.72 m in air. (a) how many beats per second will be heard? (assume t
The number of beats per second that will be heard when a source emits sound waves of wavelengths 2.54 m and 2.72 m in the air at a temperature of \(20^ \circ C\) is approximately 0.18 beats per second.
When two sound waves of slightly different frequencies or wavelengths interfere with each other, they produce a phenomenon known as beats. The number of beats per second can be calculated using the formula:
Beats per second = \(|f_1 - f_2|\),
where \(f_1\) and \(f_2\) are the frequencies of the two sound waves. In this case, since we are given the wavelengths, we can calculate the frequencies using the formula:
Frequency = Speed of sound / Wavelength,
where the speed of sound in air at 20 degrees Celsius is approximately 343 m/s.
For the wavelength 2.54 m, the frequency is 343 m/s / 2.54 m = 135.04 Hz.
For the wavelength 2.72 m, the frequency is 343 m/s / 2.72 m = 126.10 Hz.
Substituting these frequencies into the beats per second formula:
Beats per second = |135.04 Hz - 126.10 Hz| = 8.94 Hz.
Therefore, the number of beats per second that will be heard is approximately 8.94 beats per second or simply 0.18 beats per second when rounded to two decimal places.
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Nevertheless, it appears that the question is not fully formed; the appropriate request should be:
A source emits the sound of wavelengths 2.54 m and 2.72 m in the air. How many beats per second will be heard? (Assume \(T = 20^\circ C\) .)Can anyone help?
For what values of K does the equation (2k+1)x^2 +2x= 10x - 6 have two real and equal roots?
For the equation (2k + 1)x² + 2x = 10x - 6 to have two real and equal roots, the value of k = 5/6.
Since the equation is (2k + 1)x² + 2x = 10x - 6, we collect subtract 10x from both sides and add 6 to both sides.
So, we have (2k + 1)x² + 2x - 10x + 6 = 10x - 6 - 10x + 6
(2k + 1)x² - 8x + 6 = 0
For the equation, (2k + 1)x² + 2x = 10x - 6 to have two real and equal roots, this new equation (2k + 1)x² - 8x + 6 = 0 must also have two real and equal roots.
For the equation to have two real and equal roots, its discriminant, D = 0.
D = b² - 4ac where b = -8, a = 2k + 1 and c = 6.
So, D = b² - 4ac
D = (-8)² - 4 × (2k + 1) × 6 = 0
64 - 24(2k + 1) = 0
Dividing through by 8, we have
8 - 3(2k + 1) = 0
Expanding the bracket, we have
8 - 6k - 3 = 0
Collecting like terms, we have
-6k + 5 = 0
Subtracting 5 from both sides, we have
-6k = -5
Dividing through by -6, we have
k = -5/-6
k = 5/6
So, for the equation (2k + 1)x² + 2x = 10x - 6 to have two real and equal roots, the value of k = 5/6.
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It takes 1/4 of an hour to mow 1/5 of his yard. How much of the yard can he mow per hour
Answer:
4/5
Step-by-step explanation:
if it take 1/4 of an hour to mow 1/5 of his yard.
First you have to find out what is 1/4 of an hour which in order to find you have to multiply 1/4 by 60 because it has 60 mins in an hour. You now have 15 mins and we already know there are 60 mins so we divide 60 by 15 which give you 4 then you multiply 1/5 by 4 which gives you 4/5 or 0.8.
What is the equation of the
horizontal line that goes through
the point (-1, 4)?
Answer:
y = 4
Step-by-step explanation:
an horizontal line has always the same y (in this case is 4)
Which of the following set of matrix can we add?
Answer: Choice C) Matrix X and matrix W
Explanation:
Matrix addition is only possible if the sizes match up.
Matrix X and matrix W both have
2 rows3 columnsSo this means we can add the two matrices.
Let T: P₂ → P4 be the transformation that maps a polynomial p(t) into the polynomial p(t)- t²p(t) a. Find the image of p(t)=6+t-t². b. Show that T is a linear transformation. c. Find the matrix for T relative to the bases (1, t, t2) and (1, t, 12, 1³, 14). a. The image of p(t)=6+t-1² is 6-t+51²-13-14
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T: P₂ → P4, is the transformation that maps a polynomial p(t) into the polynomial p(t)- t²p(t). Let’s find out the image of p(t) = 6 + t - t² and show that T is a linear transformation and find the matrix for T relative to the bases (1, t, t²) and (1, t, 12, 1³, 14).
Step by step answer:
a) The image of p(t) = 6 + t - t² is;
T(p(t)) = p(t) - t² p(t)T(p(t))
= (6 + t - t²) - t²(6 + t - t²)T(p(t))
= 6 - t + 5t² - 13t + 14T(p(t))
= 20 - t + 5t²
Therefore, the image of p(t) = 6 + t - t² is 20 - t + 5t².
b)To show T as a linear transformation, we need to prove that;
(i)T(u + v) = T(u) + T(v)
(ii)T(cu) = cT(u)
Let u(t) and v(t) be two polynomials and c be any scalar.
(i)T(u(t) + v(t))
= T(u(t)) + T(v(t))
= [u(t) + v(t)] - t²[u(t) + v(t)]
= [u(t) - t²u(t)] + [v(t) - t²v(t)]
= T(u(t)) + T(v(t))
(ii)T(cu(t)) = cT (u(t))= c[u(t) - t²u(t)] = cT(u(t))
Therefore, T is a linear transformation.
c)The standard matrix for T, [T], is determined by its action on the basis vectors;
(i)T(1) = 1 - t²(1) = 1 - t²
(ii)T(t) = t - t²t = t - t³
(iii)T(t²) = t² - t²t² = t² - t⁴
(iv)T(1) = 1 - t²(1) = 1 - t²
(v)T(14) = 14 - t²14 = 14 - 14t²
Therefore, the standard matrix for T is;\($$[T] = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & -1 & 1 \\ 0 & -13 & 0 \\ 0 & 0 & -14 \end{bmatrix}$$\)Hence, the solution of the given problem is as follows;(a) The image of p(t) = 6 + t - t² is 20 - t + 5t².(b) T is a linear transformation because it satisfies both the conditions of linearity.(c) The standard matrix for T relative to the bases (1, t, t²) and (1, t, 12, 1³, 14) is;\($$[T] = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & -1 & 1 \\ 0 & -13 & 0 \\ 0 & 0 & -14 \end{bmatrix}$$\)
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City cabs charges a $2.25 pickup fee and $1.25 per mile traveled diego's fare for a cross-town cab ride is $18.50 how far did he travel in the cab?
Using proportions, it is found that Diego traveled 15 miles in the cross-town cab ride.
What is a proportion?A proportion is a fraction of a total amount, and the measures are related using a rule of three.
In this problem, for the miles traveled, Diego paid 18.50 - 2.25 = $16.25. Considering the fee of $1.25 per mile, the distance he traveled is given by:
d = 16.25/1.25 = 15.
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The following table gives hourly compensation costs for production workers in manufacturing in four countries in the Americas. Hourly compensation costs in U.S. dollars for production workers in manuf
To complete Exercise 1 in Excel, follow these steps:
(a) Open a new workbook and rename Sheet1 as Exercise 1:
- Open Microsoft Excel.
- Click on Sheet1 at the bottom of the workbook and rename it to "Exercise 1" by right-clicking on the tab and selecting "Rename."
- Save the workbook as "W2Handout_LastNameFirstName" by clicking on File > Save As and entering the desired name.
(b) Add rows at the bottom for average, maximum, and minimum:
- Go to the bottom of the table in Sheet1.
- Insert three rows by right-clicking on the row number above where you want to add the rows and selecting "Insert."
- In the newly inserted rows, label the first column as "Average," the second column as "Maximum," and the third column as "Minimum."
- Use appropriate Excel functions to calculate the average, maximum, and minimum values for each column.
For example, in cell B22, enter the formula "=AVERAGE(B2:B21)" to calculate the average for the first country (United States).
Repeat the same for other countries and columns.
(c) Use Auto Fill to show the predicted wages for 2006, 2007, 2008, 2009, and 2010 for each country, assuming a linear trend:
- Select the cell B23 (corresponding to the year 2001 for the first country, United States).
- Enter the formula "B2+($B$21-$B$2)/5" in cell B23 to calculate the predicted wage for 2006.
- Select cell B23 and drag the fill handle (a small square in the bottom right corner of the cell) across to cell G23 to autofill the formula for the remaining years.
- Format all the data cells with two decimal places by selecting the range B23:G25, right-clicking, choosing "Format Cells," selecting "Number" tab, and setting the desired decimal places.
Your Excel worksheet should now be set up as described in Exercise 1, with the predicted wages filled for 2006 to 2010.
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Answer this - 3 - 3 (+2)
Answer:
-3-6= -9
.......................
Answer:
if you mean -3-3 +2 then it is -4
if you mean -3-3(2) then it is -9
Step-by-step explanation:
For a recent paint job, Josh mixed red and white paint to make two different shades of pink. When the job was done, Josh ended up with leftover paint: 5 gallons of dark pink paint (80% red) and 4 gallons of light pink paint (30% red). Josh wants to make a medium pink color (50% red) to paint his daughter's bedroom. He will need 3 gallons to completely cover the walls. How much of each of the leftover paints should Josh mix to achieve his desired color?
? gallons of dark pink paint
? gallons of light pink paint
Josh should mix 1.2 gallons of dark pink paint and 1.8 gallons of light pink paint to achieve the desired medium pink color.
To find out how much of each leftover paint Josh should mix to achieve a medium pink color (50% red), we can set up a system of equations based on the percentages of red in the paints.
Let's assume that Josh needs x gallons of dark pink paint and y gallons of light pink paint to achieve the desired color.
The total amount of paint needed is 3 gallons, so we have the equation:
x + y = 3
The percentage of red in the dark pink paint is 80%, which means 80% of x gallons is red. Similarly, the percentage of red in the light pink paint is 30%, which means 30% of y gallons is red. Since Josh wants a 50% red mixture, we have the equation:
(80/100)x + (30/100)y = (50/100)(x + y)
Simplifying this equation, we get:
0.8x + 0.3y = 0.5(x + y)
Now, we can solve this system of equations to find the values of x and y.
Let's multiply both sides of the first equation by 0.3 to eliminate decimals:
0.3x + 0.3y = 0.3(3)
0.3x + 0.3y = 0.9
Now we can subtract the second equation from this equation:
(0.3x + 0.3y) - (0.8x + 0.3y) = 0.9 - 0.5(x + y)
-0.5x = 0.9 - 0.5x - 0.5y
Simplifying further, we have:
-0.5x = 0.9 - 0.5x - 0.5y
Now, rearrange the equation to isolate y:
0.5x - 0.5y = 0.9 - 0.5x
Next, divide through by -0.5:
x - y = -1.8 + x
Canceling out the x terms, we get:
-y = -1.8
Finally, solve for y:
y = 1.8
Substitute this value of y back into the first equation to solve for x:
x + 1.8 = 3
x = 3 - 1.8
x = 1.2
Therefore, Josh should mix 1.2 gallons of dark pink paint and 1.8 gallons of light pink paint to achieve the desired medium pink color.
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URGENT A fair die is rolled 6000 times. Assume the trials are independent and the probability of success = 1/6 on each trial. What's the approximate probability of getting at least 1050 fours in your 6000 rolls of the die?
Answer:
Step-by-step explanation:
This problem looks like a binomial probability question. in order to be for sure we must check that it satisfies all the conditions in BINS.
B: Binomial, Yes there are two options. (rolling a 4 vs. anything else)
I. Independent, The problems states trials are independent.
N. Number of Trials, Yes we have a given number of trials (6000)
S. Success (probability of success), Yes we have a given probability (1/6)/
Depending on the teacher and technology available there are two ways to solve this.
The first is very easy!
With a ti-84 calculator we can press [2nd] then [vars] and scroll all the way down until we find binomcdf. By clicking it, we get a screen where we can input number of trials, probability, and x-value. The final product should look something like this: binomcdf(6000, 1/6,1050).
The calculator will then return a value of 0.9592 (rounded to thousandths) which is the answer.
The second way uses the formula
\(_{n}C_{x}p^x(1-p)^{n-x}\)
n = number of trials
x = number of successes
p = probability of success
\(_{n}C_{x}\) = the combination formula which can be found in your calculator under [math] then PROB, written as nCr. It can also be shown as a formula
\(_{n}C_{r}=\frac{n!}{r!(n-r)!}\)
By plugging in you should get the same answer as the first way, 0.9592.
Hope this helps!
What is the solution of the system of equations X plus 3Y equals seven, 3X plus 2Y equals 083, to be -2, 3C2, negative3D -3, 28 -2 -3
Answer:
B
Step-by-step explanation:
x + 3y = 7 → (1)
3x + 2y = 0 → (2)
Multiplying (1) by - 3 and adding to (2) will eliminate x
- 3x - 9y = - 21 → (3)
Add (2) and (3) term by term to eliminate x
0 - 7y = - 21
- 7y = - 21 ( divide both sides by - 7 )
y = 3
Substitute y = 3 into either of the 2 equations and solve for x
Substituting into (1)
x + 3(3) = 7
x + 9 = 7 ( subtract 9 from both sides )
x = - 2
solution is (- 2, 3 ) → B
Electric utility poles in the form of right cylinders are made out of wood that costs $25.37 per cubic foot. Calculate the cost of a utility pole with a diameter of 1.5 ft and a height of 45 ft. Round your answer to the nearest cent.
The cost of the utility pole is approximately $593.96.
What is volume?Volume is a measure of the amount of space occupied by a three-dimensional object. It is usually measured in cubic units such as cubic meters (m³), cubic centimeters (cm³), or cubic feet (ft³). The volume of an object can be found by measuring its length, width, and height and using a formula that relates these dimensions to the amount of space the object occupies. In geometry, the most common shapes for which volume is calculated include cubes, spheres, cylinders, pyramids, and cones.
The volume of a right cylinder is given by the formula V = πr²h, where r is the radius of the base and h is the height.
In this case, the diameter of the utility pole is 1.5 ft, so the radius is 0.75 ft. The height is 45 ft. Therefore, the volume of the utility pole is:
V = π(0.75 ft)²(45 ft) = 23.3826... cubic feet
Rounding to the nearest hundredth, we get:
V ≈ 23.38 cubic feet
The cost of wood per cubic foot is $25.37. Therefore, the cost of the utility pole is:
$25.37/cubic foot × 23.38 cubic feet ≈ $593.96
So the cost of the utility pole is approximately $593.96.
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There are 44,000 adults living in Oak city in examining attitudes according to the news of research group as random sample of Oak city adults what is your main source of news the results are shown below based on the sample predict the number of adults in Oak city whose main source of the news is television on your answer to the nearest whole number do not run by any intermittent calculations
The predicted number of adults in Oak City whose main source of news is newspapers or the radio is 85
To predict the number of adults in Oak City whose main source of news is newspapers or the radio, we need to add the number of adults who answered "Newspapers" to the number of adults who answered "Radio".
According to the sample data
Number of adults who answered "Newspapers": 64
Number of adults who answered "Radio": 21
Therefore, the predicted number of adults in Oak City whose main source of news is newspapers or the radio we have to use the addition
64 + 21 = 85
Rounding this answer to the nearest whole number, we get
85 ≈ 85
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The given question is incomplete, the complete question is:
There are 44,000 adults living in Oak City. In examining attitudes toward the news, a research group asked a random sample of Oak City adults "What is your main source of news?" The results are shown below. Main Source of News Newspapers Number of Adults 64 Internet 80 Television 126 Radio 21 Other 40 Based on the sample, predict the number of adults in Oak City whose main source of news is newspapers or the radio. Round your answer to the nearest whole number.
If an object moves along an x axis from x = 5 m to x = 3 m, what is the sign of its displacement?
The sign of displacement is negative.
Displacement: It is defined as the change in the position of an object. It is a vector quantity, means it has both direction and magnitude. It is the shortest distance between two position of an object.
It is the difference between final and the initial position of an object.
Initial position, \(x_i\) = 5m
Final position, \(x_f\) = 3m
Displacement, \(d = x_f - x_i\) = (3 - 5)m
= -2m
negative sign shows that the direction of the an object is towards negative x-axis.
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If this trapezoid is moved through the translation (x+3,y-2),what will the coordinates of A’ be
9514 1404 393
Answer:
A'(-3, 0)
Step-by-step explanation:
The coordinates of A are (-6, 2). Use these (x, y) values in the translation expression to find the coordinates of A'.
(x, y) ⇒ (x +3, y -2)
(-6, 2) ⇒ (-6 +3, 2 -2) = (-3, 0)
The coordinates of A' are (-3, 0).
The table shows the information about the type and number of sandwiches ordered by 80 customers. If 120 customers order a sandwich, how many would you expect to order a club?
A. 36
B. 50
C.54
D.63
Evaluate the expression for s = 2.
9s- 2(s+ 1)2 =
Answer:
s=\(\frac{40}{21}\)
Step-by-step explanation:
Answer:
6
Step-by-step explanation:
I need the answer to this question √225/16
Answer:
15/4 or 3 3/4 or 3.75
Step-by-step explanation:
We can split it up like 225 (square root) divided by 16 (square root). If you recall squares, you will know 225 square root is 15, and 16 square root is 4. So 15/4 is the answer, or if you want to simplify 3 3/4, and decimal will be 3.75.
Please help me with this question
Answer:
wheres the formula
Noah and Lin are marketing paper cones to hold popcorn to hand out at parent math night. They want the cones to hold 9 cubic inches of popcorn. What are two different possible values for height h and radius r for the cones?
if h=3, then r=
if r=2, then h=
Answer: r = 1.69 inches
h = 2.15 inches
Step-by-step explanation: Volume of a solid is the amount of space contained within a solid.
Volume of a cone is directly proportional to radius and height:
\(V=\frac{1}{3}\pi r^{2}h\)
They want the cones to hold the same volume of 9 cubic inches.
If height is 3:
\(9=\frac{1}{3}\pi r^{2}3\)
\(\pi r^{2}=9\)
\(r^{2}=2.864\)
r = 1.69 inches
When height is 3, radius is 1.69 inches for a cone to have 9 cubic inches of volume.
If radius is 2:
\(9=\frac{1}{3}\pi (2)^{2}h\)
\(h=\frac{9(3)}{4 \pi}\)
h = 2.15 inches
If radius is 2 inches, height of the cone is 2.15 inches.
use a power series to approximate the definite integral, I, to six decimal places. ∫ 0.2 0 (x^4/1+x5) dx
By using power series the approximate value of the given definite integral is 0.000397.
What is power series?A power series is a numerical portrayal of a capability as a boundless amount of terms, where each term is a steady increased by a variable raised to a particular power.
Based on the information provided:
To approximate the definite integral ∫[0.2 to 0] (x^4/(1+x^5)) dx using a power series, we can use the technique of Taylor series expansion.
First, we need to find a power series representation for the integrand \((x^4/(1+x^5))\). We can start by expressing the denominator as a power of \((1+x^5)\) using the binomial theorem:
\((1+x^5)^{-1}= 1 - x^5 + x^{10} - x^{15} + ...\)
Now we can multiply the numerator x^4 with the power series for (1+x^5)^(-1) to get the power series representation for the integrand:
\(x^4/(1+x^5) = x^4(1 - x^5 + x^{10} - x^{15} + ...)\)
The power series can then be integrated term by term within the specified interval, from 0.2 to 0. We can integrate the integrand's power series representation from 0 to 0.2 because power series can be integrated term by term within their convergence interval.
\(\int\limits^{0.2}_ 0 \,(x^4/(1+x^5)) dx = \int\limits^{0.2}_0} \, (x^4(1 - x^5 + x^{10} - x^{15} + ...)) dx\)
After a certain number of terms, we can now approximate the integral by truncating the power series. To get a good estimate, let's truncate the power series after the x-10 term:
\(\int\limits^{0.2}_0 \, (x^4/(1+x^5)) dx\) ≈ \(\int\limits^{0.2}_0 (x^4(1 - x^5 + x^{10}) dx\)
Now we can integrate the truncated power series term by term within the interval [0 to 0.2]:
\(\int\limits^{0.2}_0 \, x^4(1 - x^5 + x^{10} )dx\)\(= \int\limits^{0.2}_0 \, (x^4 - x^9 + x^{14}) dx\)
We can integrate each term separately:
\(\int\limits^{0.2}_0 \, x^4 dx - \int\limits^{0.2}_0 \, x^9 dx + \int\limits^{0.2}_0 \, x^{14} dx\)
Using the power rule for integration, we can find the antiderivatives of each term:
\((x^5/5) - (x^{10}/10) + (x^{15}/15)\)
Now we can evaluate the antiderivatives at the upper and lower limits of integration and subtract the results:
\([(0.2^5)/5 - (0.2^{10})/10 + (0.2^{15})/15] - [(0^5)/5 - (0^{10})/10 + (0^{15})/15]\)
Plugging in the values and rounding to six decimal places, we get the approximate value of the definite integral:
0.000397 - 0 + 0 ≈ 0.000397
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Using a power series to approximate the given definite integral to six decimal places, we get:
∫ 0.2 0 (x⁴/1+x⁵) dx ≈ -0.000023
What is power series?A power series is a numerical portrayal of a capability as a boundless amount of terms, where each term is a steady increased by a variable raised to a particular power.
We can use a power series to approximate the given definite integral:
∫ 0.2 0 (x⁴/1+x⁵) dx = ∫ 0.2 0 (x⁴)(1 - x⁵ + x¹⁰ - x¹⁵ + ...) dx
The series representation of (1/(1-x)) is 1 + x + x² + x³ + ..., so we can substitute (-x⁵) for x in this series and get:
1 + (-x⁵) + (-x⁵)² + (-x⁵)³ + ... = 1 - x⁵ + x¹⁰ - x¹⁵ + ...
Substituting this series in the original integral, we get:
∫ 0.2 0 (x⁴/1+x⁵) dx = ∫ 0.2 0 (x⁴)(1 - x⁵ + x¹⁰ - x¹⁵ + ...) dx
= ∫ 0.2 0 (x⁴)(1 + (-x⁵) + (-x⁵)² + (-x⁵)³ + ...) dx
= ∫ 0.2 0 (x⁴)∑((-1)^n)\((x^{(5n)}) dx\)
= ∑((-1)ⁿ)∫ 0.2 0 \((x^{(5n+4)}) dx\)
= ∑((-1)ⁿ)\((0.2^{(5n+5)})/(5n+5)\)
We can truncate this series after a few terms to get an approximate value for the integral. Let's use the first six terms:
∫ 0.2 0 (x⁴/1+x⁵) dx ≈ (-0.2⁵)/5 - (0.2¹⁰)/10 + (0.2¹⁵)/15 - (0.2²⁰)/20 + (0.2²⁵)/25 - (0.2³⁰)/30
≈ -0.0000226667
Therefore, using a power series to approximate the given definite integral to six decimal places, we get:
∫ 0.2 0 (x⁴/1+x⁵) dx ≈ -0.000023
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how many significant figures should be retained in the result of the following calculation:
12.00000 x 0.9893 + 13.00335 x 0.0107
To find the average value of the function f(x, y) = 8x + 5y over the given triangle, we need to calculate the double integral of f(x, y) over the region and then divide it by the area of the triangle.
The vertices of the triangle are (0, 0), (2, 0), and (0, 7). We can set up the integral as follows:
∬R f(x, y) dA = ∫₀² ∫₀ᵧ (8x + 5y) dy dx
Integrating with respect to y first, the inner integral becomes:
∫₀ᵧ (8x + 5y) dy = 8xy + (5y²/2) |₀ᵧ = 8xᵧ + (5ᵧ²/2)
Now integrating with respect to x, the outer integral becomes:
∫₀² (8xᵧ + (5ᵧ²/2)) dx = (4x²ᵧ + (5ᵧ²x)/2) |₀² = (8ᵧ + 10ᵧ² + 20ᵧ)
To find the area of the triangle, we can use the formula for the area of a triangle: A = (1/2) * base * height.
The base of the triangle is 2 and the height is 7.
A = (1/2) * 2 * 7 = 7
Finally, to find the average value, we divide the double integral by the area of the triangle:
Average value = (8ᵧ + 10ᵧ² + 20ᵧ) / 7
Simplifying this expression gives:
Average value = (8 + 10ᵧ + 20ᵧ) / 7 = (8 + 10(7) + 20(7)) / 7 = 142/7 = 20 2/7
Therefore, the correct answer is not listed among the options provided.
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