a) To find the volume of the region A bounded by the sphere x^2 + y^2 + z^2 = 12 and the paraboloid z = x^2 + y^2, we can use cylindrical coordinates.
In cylindrical coordinates, the equations of the surfaces become:Sphere: ρ^2 + z^2 = 12Paraboloid: z = ρ^2The region A is bounded by the sphere and the paraboloid, so we need to integrate over the range of ρ, φ, and z that satisfies both equations. The limits for ρ are 0 to √(12 - z^2), the limits for φ are 0 to 2π, and the limits for z are 0 to 4. So the triple integral for the volume of region A in cylindrical coordinates is:∫∫∫ ρ dρ dφ dz, where the limits of integration are ρ: 0 to √(12 - z^2), φ: 0 to 2π, and z: 0 to 4.b) To find the volume of the region B in the first octant bounded by the surfaces z = x^2 and x^2 + y^2 + z = 1, we can again use cylindrical coordinates. In cylindrical coordinates, the equations of the surfaces become:z = ρ^2 (since we are in the first octant where x and y are non-negative)z = 1 - ρ^2The limits for ρ are 0 to 1, and the limits for φ are 0 to π/2. So the triple integral for the volume of region B in cylindrical coordinates is:∫∫∫ ρ dρ dφ dz, where the limits of integration are ρ: 0 to 1, φ: 0 to π/2, and z: ρ^2 to 1 - ρ^2.c) To find the volume of the region C inside both spheres x^2 + y^2 + (z - 2)^2 = 16 and x^2 + y^2 + 2 = 16, we can once again use cylindrical coordinates. In cylindrical coordinates, the equations of the surfaces become:Sphere 1: ρ^2 + (z - 2)^2 = 16Sphere 2: ρ^2 = 12The region C is bounded by both spheres, so we need to integrate over the range of ρ, φ, and z that satisfies both equations. The limits for ρ are 0 to 2√3, the limits for φ are 0 to 2π, and the limits for z are 2 - √(16 - ρ^2) to 2 + √(16 - ρ^2). So the triple integral for the volume of region C in cylindrical coordinates is:∫∫∫ ρ dρ dφ dz, where the limits of integration are ρ: 0 to 2√3, φ: 0 to 2π, and z: 2 - √(16 - ρ^2) to 2 + √(16 - ρ^2).
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What are the coordinates of the point on the directed line segment from (2, -6) to
(6, 2) that partitions the segment into a ratio of 3 to 5?
If the cos= and tan 0 = 1/32,
what is sin 0?
13
The calculated value of sin of the angle is 1/96
Calculating the value of sin of the angleFrom the question, we have the following parameters that can be used in our computation:
cos(θ) = 1/3
tan(θ) = 1/32
The value of sin of the angle is calculated as
sin(θ) = cos(θ) * tan(θ)
substitute the known values in the above equation, so, we have the following representation
sin(θ) = 1/3 * 1/32
Evaluate the products
so, we have the following representation
sin(θ) = 1/96
Hence, the value of sin of the angle is 1/96
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Shop
Hunter assumed he would only get 64
problems correct on his test, but he
actually got 78 correct. What was his
percent error?
Hint: Percent error =
Prediction - Actual
Actual
x 100
Round to the nearest percent.
[? ]%
Enter
Hunter assumed he would only get 64 problems correct on his test, but he actually got 78 correct, So his percent error is 18%.
To calculate Hunter's percent error, we'll use the given formula:
Percent error = ((Prediction - Actual) / Actual) x 100
Prediction = 64 (the number of problems Hunter assumed he would get correct)
Actual = 78 (the number of problems he actually got correct)
Now, plug in the values:
Percent error = ((64 - 78) / 78) x 100
Percent error = (-14 / 78) x 100
Percent error ≈ -17.95%
Since percent error is typically expressed as a positive value, we can round to the nearest percent and report it as:
Percent error ≈ 18%
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What is the area of a triangle with vertices at (5,0), (-3,1) and (2,6)?
6 units squared
54 units squared
22. 5 units squared
36. 5 units squared
Answer:
To find the area of a triangle given its vertices, we can use the formula:
Area = 1/2 * |(x1*(y2-y3) + x2*(y3-y1) + x3*(y1-y2))|
where (x1,y1), (x2,y2), and (x3,y3) are the coordinates of the three vertices.
Using this formula, we can calculate the area of the triangle with vertices at (5,0), (-3,1), and (2,6) as follows:
Area = 1/2 * |(5*(1-6) + (-3)(6-0) + 2(0-1))|
= 1/2 * |-25 - 18 - 2|
= 1/2 * |-45|
= 22.5
Therefore, the area of the triangle is 22.5 units squared. So the answer is option C: 22.5 units squared.
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Find The Area Of This Shape.
Answer:
34.65 mi²
Step-by-step explanation:
Area of parallelogramam = b · h
b = 6.3 mi
h = 5.5 mi
Let's solve
6.3 · 5.5 = 34.65 mi²
So, the area of the shape is 34.65 mi²
Llus
Find x.
Round to the nearest tenth:
31°
х
y
400 ft
x = [? ]ft
Pls help me
Using the sine function and the given information, we can find that length of y is approximately 203.3 feet.
We know that exterior angle of a triangle is equal to the sum of its opposite interior angles. So, we can find angle BAC as follows
BAC + ABC = 180 degrees
As sum of interior angles of a triangle
BAC + 90 = 180
BAC = 90 degrees
Now, we can use the sine ratio to find y
sin(31) = y/400
y = 400 * sin(31)
y ≈ 203.3 ft
Therefore, y is approximately 203.3 ft when rounded to the nearest tenth.
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--The given question is incomplete, the complete question is given
" Find y. Round to the nearest tenth:
In triangle ABC, 31° is the outer angle at A which is in between a line extended from A parallel to BC.
X is CA
у is AB
400 ft is BC
Angle B is right angle
y = [ ? ]ft Enter"--
Colin surveyed 12 teachers at his school to determine how much each person budgets for lunch. He recorded his results in the table. What does the relationship between the mean and median reveal about the shape of the data? the mean is less than the median, so the data is skewed left. The mean is more than the median, so the data is skewed right. The mean is equal to the median, so the data is symmetrical. The mean is equal to the median, so the data is linear.
The mean is equal to the median, so the data is symmetrical.
Given that :
Colin surveyed 12 teachers at his school to determine how much each person budgets for lunch.
The data is :
10 5 8 10 12 6 8 10 15 6 12 18
Mean is the average of these numbers.
Mean = (10 + 5 + 8 + 10 + 12 + 6 + 8 + 10 + 15 + 6 + 12 + 18) / 12
= 10
Now median is the element in the middle arranged in an order.
Arrange the data in ascending order.
5 6 6 8 8 10 10 10 12 12 15 18
The middle element is the average of the 6th and 7th elements.
Median = (10 10) / 2 = 10
So mean and the median is equal. So the data is symmetrical.
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The OLS estimator is derived by: Group of answer choices connecting the Yi corresponding to the lowest Xi observation with the Yi corresponding to the highest Xi making sure that the standard error of the regression equals the standard error of the slope estimator. Minimizing the sum of absolute residuals. Minimizing the sum of squared residuals
Minimizing the sum of squared residuals.
How is the OLS estimator derived?The OLS (Ordinary Least Squares) estimator is derived by minimizing the sum of squared residuals. This method aims to find the line of best fit that minimizes the vertical distance between the observed data points (Yi) and the predicted values on the regression line. The residuals represent the differences between the observed values and the predicted values.
By minimizing the sum of squared residuals, the OLS estimator ensures that the line fits the data as closely as possible. This approach is based on the principle of least squares, which seeks to find the parameters that minimize the overall discrepancy between the observed data and the predicted values.
Connecting the Yi corresponding to the lowest Xi observation with the Yi corresponding to the highest Xi or ensuring that the standard error of the regression equals the standard error of the slope estimator are not the steps involved in deriving the OLS estimator. The OLS method specifically focuses on minimizing the sum of squared residuals.
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Use the given sample data to find each of the listed values. In your answer, include both numerical answers and an explanation, in complete sentences, for the steps necessary to find the data values. Complete your work in the space provided.
62 52 52 52 64 69 69 76 54 58 61 67 73
Range _____
Q1 _____
Q3 _____
IQR _____
Using the given sample data, the data values are:
Range 24
Q1 52
Q3 69
IQR 17
1. Arrange the data in ascending order:
52, 52, 52, 54, 58, 61, 62, 64, 67, 69, 69, 73, 76
2. Calculate the range (highest value - lowest value):
Range = 76 - 52 = 24
3. Find the first quartile (Q1) - the median of the first half of the data:
Since there are 13 data points, we'll take the median of the first 6 numbers: (52 + 52) / 2 = 52
Q1 = 52
4. Find the third quartile (Q3) - the median of the second half of the data:
We'll take the median of the last 6 numbers: (69 + 69) / 2 = 69
Q3 = 69
5. Calculate the interquartile range (IQR) by subtracting Q1 from Q3:
IQR = Q3 - Q1 = 69 - 52 = 17
So, the requested values are:
Range = 24
Q1 = 52
Q3 = 69
IQR = 17
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O is the centre of the given circle. if OX⊥PQ, OY⊥RS and PQ=RS, write down the relation between OX and OY.
Since OX is perpendicular to PQ, and OY is perpendicular to RS, we know that OX and OY are both radii of the circle. Therefore, we can write:
OX = OY
This is because all radii of a circle are equal in length. Alternatively, we could also say that OX and OY are both the distance from the center O to the respective lines PQ and RS. Since PQ=RS, OX and OY are equal in length.
What is the circle about?In a circle, the center is the point from which all points on the circumference are equidistant. This means that any line segment from the center to a point on the circle is a radius of the circle.
In this problem, we have two lines PQ and RS, both of which are tangent to the circle at points P and R respectively. We also have two lines OX and OY, each of which is perpendicular to one of the tangent lines.
Because the tangent lines are perpendicular to their respective radii (PQ is perpendicular to OX, and RS is perpendicular to OY), we can conclude that OX and OY are both radii of the circle, and therefore, they have the same length.
Note that both are still angles at 90 degrees.
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A woman applies a 10 Newton force and uses 100 joules of energy to push a cart of groceries. How much work did she perform
The woman performed 100 joules of work to push the cart of groceries over a distance of 10 meters.
To find the work performed by the woman, we will use the work-energy theorem which states that work (W) is equal to the change in energy. In this case, the woman applies a 10 Newton force and uses 100 joules of energy. The formula for work is:
W = F × d × cos(θ)
Where W is work, F is force, d is the distance over which the force is applied, and θ is the angle between the force and the direction of motion. Since the woman used 100 joules of energy, we can rewrite the equation as:
100 J = 10 N × d × cos(θ)
We don't have information about the angle θ, but if we assume that she applied the force horizontally, which is in the same direction as the motion, the angle θ would be 0 degrees, and the cosine of 0 is 1. Therefore, the equation becomes:
100 J = 10 N × d
To find the distance (d), we can now solve for d:
d = 100 J / 10 N
d = 10 meters
So, 100 joules of work was performed by the women to push the cart of groceries over a distance of 10 meters.
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A school’s art teacher designs a circular flower bed inside a rectangular sandbox. The sandbox is 6 feet wide and 10 feet long. How many square feet will there be for sand after the flower bed is installed? Use 3.14 for pi. ROUND the answer to the NEAREST SQUARE FOOT.
A. 22 square feet
B. 28 square feet
C. 32 square feet
D. 41 square feet
Pairs of twins are numbered 1, 1, 2, 2, and so on seated around a circle so that the least number of gaps between two twins always equals the assigned number. Find two different twin circles with 5 pairs of twins and explain why there is no twin circle with 3 pairs of twins
Therefore , the solution of the given problem of circle comes out to be since the numbers 1, 2, and 3 cannot be dispersed evenly around the circle with the necessary amount of gaps.
What is circle?Each element of the aeroplanes creates a circle when viewed from this new angle and at a distance. (center). Its structure is composed of surfaces and undulating regions that contrast with one another. Additionally, it rotates equally within the centre in all directions. Every ultimate extent of a circular or restricted double sphere is the same as the sphere's "center."
Here,
One potential twin circle that we can create is as follows:
=> 1 2 1 3 2 5 4 5 4 3
Here, the first twin pair 1 is divided into three pairs, with the second twin pair 2 being divided into two pairs, the third twin pair being divided into three pairs, and so on. This satisfies the criteria of the puzzle, and we can verify that every twin pair appears precisely twice.
When the numbers are reversed, a second potential twin circle results:
=> 3 4 5 4 5 2 3 1 2 1
As a result, there is no such twin circle since the numbers 1, 2, and 3 cannot be dispersed evenly around the circle with the necessary amount of gaps.
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Stephanie throws a paper airplane off the roof of her school. Her classmates measure the plane’s height, in feet, after each second. The table shows some of her data.
T= 0, 1, 2, 3, 4
h(t)= 200, 208, 212, 212, 208
Find the equation for h(t), and use it to determine how long it will take the plane to hit the ground. Round to the nearest hundredth
The paper airplane will not hit the ground within the given time frame.
To find the equation for h( t), we need to find the rate of change or the pitch of the function. We can do this by chancing the difference in height and time for any two points on the line. Let's choose the first two points,( 0,200) and( 1,208).
pitch = ( change in height)/( change in time)
pitch = ( 208- 200)/( 1- 0) = 8
So, the equation for h( t) can be written in pitch- intercept form as h( t) = 8t b
To find the value of b, we can use any of the given points.
Let's use( 0,200) 200 = 8( 0) b b = 200
So, the equation for h( t) is h( t) = 8t 200
To find how long it'll take the aeroplane to hit the ground, we need to find when h( t) = 0,
since that would mean the height is 0 bases or the ground position. 0 = 8t 200 working for t,
we get t = -25
Since time cannot be negative the paper airplane will not hit the ground within the given time frame.
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A 10-foot ladder is leaning against a wall. If we pull the ladder away from the wall at a rate of 8 ft/s how fast is the top of the ladder moving down the wall when the bottom of the ladder is 8ft from the wall? (Enter an exact answer.) Provide your answer below: The ladder is moving down the wall at a rate of__ feet per second
The top of the ladder is moving down the wall at a rate of 32/3 feet per second when the bottom of the ladder is 8 feet from the wall.
Let's denote the distance between the bottom of the ladder and the wall as x, and the distance between the top of the ladder and the ground as y. Since the ladder is leaning against the wall, we have a right triangle formed by the ladder, the wall, and the ground.
We know that the ladder has a length of 10 feet, so by the Pythagorean theorem, we have:
x^2 + y^2 = 10^2
Differentiating both sides with respect to time t, we get:
2x(dx/dt) + 2y(dy/dt) = 0
We want to find the rate of change of y (the speed at which the top of the ladder is moving down the wall), when x = 8 ft and dx/dt = 8 ft/s. We can substitute these values into the equation above and solve for dy/dt:
2(8)(8) + 2y(dy/dt) = 0
Simplifying this equation, we get:
dy/dt = -64/y
Now we need to find the value of y when x = 8 ft. We can use the Pythagorean theorem again:
x^2 + y^2 = 10^2
8^2 + y^2 = 100
y^2 = 100 - 64
y = sqrt(36) = 6 ft
Substituting this value of y into our equation for dy/dt, we get:
dy/dt = -64/6 = -32/3 ft/s
Therefore, the top of the ladder is moving down the wall at a rate of 32/3 feet per second when the bottom of the ladder is 8 feet from the wall.
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What is the sum?
√24 + √81
A 53/3
B 63
C 2√3+3
D 2√3+9
Answer:
2√6 + 9
Step-by-step explanation:
√24 + √81
= √4√6 + 9
= 2√6 + 9
PLEASE HELP WITH 4 AND 5
1. The area of the shaded region is
2. percentage of the shaded part is 89.6%
What is area of shape?The area of a shape is the space occupied by the boundary of a plane figures like circles, rectangles, and triangles.
The area of the shaded part = area of the rectangle - area of unshaded part
Area of rectangle = 7× 11 = 77 unit²
area of rectangle = 1/2 bh
= 1/2 × 4 × 4
= 1/2 × 8
= 4 unit²
area of second triangle = 4 units²
area of unshaded part = 4+4 = 8 units²
area of shaded part = 77-8 = 69units²
2. percentage of the rectangle shaded = 69/77 × 100
= 6900/77 = 89.6%
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company A charges $50 per month plus $0.05 per minute. Company B has no monthly charge but charges $0.15 per minute. When is the charge for the companies the same?
The charge for the companies is the same at 500 minutes
When is the charge for the companies the same?From the question, we have the following parameters that can be used in our computation:
Company A charges $50 per month plus $0.05 per minute. Company B has no monthly charge but charges $0.15 per minute.This means that
A(x) = 50 + 0.05x
B(x) = 0.15x
When the charge for the companies are the same, we have
0.15x = 50 + 0.05x
So, we have
0.1x = 50
Divide
x = 500
Hence, the number of minutes is 500
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What value of k makes the equation true?
k – 3(k + 5) – 0. 5 = 3(0. 25k + 4)
For -10 as the value of k the equation k – 3(k + 5) – 0. 5 = 3(0. 25k + 4) is true.
Given equation is k – 3(k + 5) – 0. 5 = 3(0. 25k + 4)
To solve the equation, firstly we have to solve or open the brackets using the distributive property that is a(x + y) = ax +ay
Thus the equation becomes, k - 3k - 15 - 0.5 = 0.75k + 12
Then we have to take all the terms with k on one side and other on the other or segregate the like terms, the equation now is
k - 3k - 0.75k = 12 + 15 + 0.5
Simplify the equation by adding or subtracting the like terms.
-2.75k = 27.5
Then we divide the coefficient of k by the other side and get our answer
k = 27.5 ÷ (-2.75)
k = -10
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An phone playlist has songs of various genres
according to the table below. What is the probability
that, of three random songs, the first two are country
and the third is R&B? Once a song is played it will not
be repeated until all other songs are played.
Genre :Number of Songs
Rock: 12
Country :15
R&B :10
Classical :3
The probability of selecting two country songs followed by an R&B song is 0.0202.
What is the probability?The probability of selecting two country songs followed by an R&B song is determined below as follows:
The total number of songs in the playlist = 40
The probability of the first song being country = 15/40.
The probability of the second song also being country = 14/39
The probability of the third song being R&B =s 10/38
Therefore, the probability of selecting two country songs followed by an R&B song is:
Probability(country, country, R&B) = (15/40) x (14/39) x (10/38)
Probability(country, country, R&B) = 0.0202
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A basic pattern of 1 blue bead and 1 green bead is used to make a bracelet that is 37cm long. The bracelet is made by repeating the basic pattern 10 times. The length of a blue bead is Bcm. The length of a green bead is 1. 2cm. Complete the question to represent the length of the bracelet
Answer: Therefore, the length of a blue bead is 2.5 cm, and the length of a green bead is 1.2 cm. And the length of the bracelet is:
10 × (2.5 + 1.2) = 37 cm.
Step-by-step explanation:
To represent the length of the bracelet, we need to determine the length of each repetition of the basic pattern and then multiply it by the number of times the pattern is repeated.
The length of each repetition of the basic pattern is the sum of the length of one blue bead and one green bead, which is:
B + 1.2 cm
Since the basic pattern is repeated 10 times, the total length of the bracelet is:
10 × (B + 1.2) cm
And we know that the total length of the bracelet is 37 cm, so we can set up an equation:
10 × (B + 1.2) = 37
Simplifying the equation, we can divide both sides by 10:
B + 1.2 = 3.7
Subtracting 1.2 from both sides, we get:
B = 2.5
Therefore, the length of a blue bead is 2.5 cm, and the length of a green bead is 1.2 cm. And the length of the bracelet is:
10 × (2.5 + 1.2) = 37 cm.
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Triangle ABC has vertices A(-1,1), B(1,3) and C(4,1). The image of ABC after the transformation matrix T=
The coordinates of transforming image of the vertices of the triangle ABC are A' (1, -1) ,B' (3, 1) , and C' (1, 4).
In triangle ABC,
Coordinates of the vertices of triangle ABC are,
A(-1,1), B(1,3) and C(4,1)
The transformation T y=x reflects the points across the line y=x.
The image of each point, we simply swap the x and y coordinates of each point.
So, applying the transformation T y=x to the vertices of triangle ABC, we get,
A' = (-1, 1) → (1, -1)
B' = (1, 3) → (3, 1)
C' = (4, 1) → (1, 4)
This implies,
The image of triangle ABC under the transformation T y=x is triangle A'B'C', where,
A' is located at (1, -1)
B' is located at (3, 1)
C' is located at (1, 4)
Therefore, in triangle ABC labeling the coordinates of the vertices of A'B'C' after transformation are as follows,
A' (1, -1)
B' (3, 1)
C' (1, 4)
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The above question is incomplete, the complete question is:
Triangle ABC has vertices A(-1,1), B(1,3) and C(4,1). The image of ABC after the transformation T y=x is A’ B’ C’. State and label the coordinates of A’ B’ C’.
Solve for a.
5
a = [?]
3
a
evaluate the square root
before entering your
answer.
pythagorean theorem: a2 + b2 = c2
By evaluating square root and using Pythagorean Theorem the value of a is a= 3.33 (rounded to two decimal places)
The given expression is 5/3, which can be simplified as follows:
a = 5/3
To evaluate the square root of a, we can rewrite it in terms of exponents:
a = (5/3)¹/₂
Using a calculator, we get:
a ≈ 1.83
Next, we can use the Pythagorean Theorem to find the value of c, given that a = 3.33 and b = 4.66:
a² + b² = c²
(3.33)² + (4.66)² = c²
11.0889 + 21.7156 = c²
32.8045 = c²
c ≈ 5.72
Therefore, the final answer is a = 3.33 and c ≈ 5.72.
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Two cars start together and travel in the same direction.
One car goes twice as fast as the other. After five
hours, they are 225 kilometers apart.
How fast is each car traveling?
Faster car's speed:
Slower car's speed:
The speed of the faster car is 90 kph and the speed of the slower car is 45 kph.
We are given that two cars are starting together and they travel in the same direction. Let one car be car A and the other car B. Speed of car B is twice the speed of car A. Let r be the rate of speed of car A and 2r be the rate of speed of car B.
We know that these two cars are 225 km apart. We will use the formula distance = speed * time. Let the distance car A travels after 5 hours be 5r. So, the distance traveled by car B after 5 hours will be 5(2r) = 10r.
Since car B is faster, it will have traveled farther after 5 hours. Therefore,
Distance traveled by car B - distance traveled by car A = 225
10r - 5r = 225
5r = 225
r = 45 kph
and
2r = 90 kph
Therefore, car A is traveling at 45 kph and car B is traveling at 90 kph.
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A scale model of a statue has a height of 10 cm. The real statue has a height of 2 m. The model and statue are both made of the same stone which has a density of 2 g/cm(3). The mass of the real statue is 1632 kg. Find the volume of the model in cm(3)
Answer:
102 cm³
Step-by-step explanation:
density = m/v
1 g = 0.001 kg
Volume of real statue = v = m/d = 1632 kg / 0.002 kg/cm³ = 816,000 cm³
scale factor for height: 200 cm : 10 cm = 20 : 1
scale factor for volume: 20³ : 1³ = 8,000 : 1
Volume of model = 816,000 cm³ / 8,000 = 102 cm³
30 POINTS!!!!!! I WILL GIVE BRAINLIEST!!!!!!!!
If the measure of arc QR is 40 degrees, what is the measure of angle PQR?
The measure of angle PQR is 20 degrees.
We have,
The measure of angle PQR can be found by using the property that the measure of an inscribed angle is half the measure of its intercepted arc.
Given that the measure of arc QR is 40 degrees, we can conclude that the measure of angle PQR is half of that, which is:
The measure of angle PQR = 1/2 x Measure of arc QR
= 1/2 x 40 degrees
= 20 degrees
Therefore,
The measure of angle PQR is 20 degrees.
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Question 11 (Multiple Choice Worth 5 points)
(Laws of Exponents with Integer Exponents MC)
Choose the expression that is equivalent to
9
92
Therefore, the equivalent expression to [tex]9^2[/tex] is 81.
How to solve equation?To solve an equation, you need to isolate the variable on one side of the equation, typically the left side, and simplify the other side until you are left with an expression that has only the variable you are trying to solve for.
Here are the general steps to solve an equation:
Start by simplifying both sides of the equation as much as possible. This may involve distributing or combining like terms.
Get all terms with the variable you are solving for on one side of the equation. To do this, you can add or subtract terms from both sides of the equation. For example, if you have the equation 2x + 5 = 9, you can subtract 5 from both sides to get 2x = 4.
Isolate the variable by dividing both sides of the equation by its coefficient. In the above example, you can divide both sides by 2 to get x = 2.
Check your solution by plugging it back into the original equation and verifying that both sides of the equation are equal.
[tex]a^m/a^n = a^{(m-n)}[/tex]
to simplify the expression.
The expression [tex]9^2[/tex] means 9 multiplied by itself 2 times:
[tex]9^2 = 9 * 9[/tex]
Using the laws of exponents, we can rewrite this expression as:
[tex]9^2 = 9^{(1+1)} (since 2 = 1 + 1)[/tex]
Then, using the rule that [tex]a^{(m+n)} = a^m * a^n[/tex], we have:
[tex]9^2 = 9^1 *9^1[/tex]
Finally, using the fact that. [tex]a^1 = a[/tex] for any value of a, we get:
We can use the rule of exponents that states:
[tex]9^2 = 9 * 9 = 81[/tex]
Therefore, the equivalent expression to [tex]9^2[/tex] is 81
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Marshall is renting a bike for the day. It costs $13 for up to one hour. After one hour, the price increases to $20. After three hours, the price increases again to $50. The maximum time he can rent the bike is 10 hours total
The piecewise function that represents the situation is therefore:
Costs Possible hours
13 0 < x ≤ 1
20 1 < x ≤ 3
50 3 < x ≤ 10
How to find the piecewise function ?We see that up to one hour, the cost to Marshall would be $ 13 so the possible hours at that price is 0 < x ≤ 1 .
Likewise from one hour, the price goes up to $ 20 which means that possible hours become 1 < x ≤ 3 because the price will increase at 3 hours again.
From 3 hours, the price becomes $ 50 and there are a maximum of 10 hours so the hours become 3 < x ≤ 10 .
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3 For y=f(x) = 9x, x= 3, and Ax = 0.03 find a) y for the given x and Ax values, b) dy = f'(x)dx, to) dy for the given x and Ax values.
a) To find y for the given x and Δx values, first calculate x + Δx:
x + Δx = 3 + 0.03 = 3.03
Now, use the function y = f(x) = 9x to find the y values:
y = 9(3) = 27 (for x = 3)
y = 9(3.03) = 27.27 (for x = 3.03)
b) To find dy, we first need to find the derivative of the function (f'(x)). The function is y = f(x) = 9x, and its derivative (using differentiation) is:
f'(x) = 9
c) To find dy for the given x and Δx values, we can now use the formula dy = f'(x)dx:
dy = f'(x)dx = 9(0.03) = 0.27
So, for the given x and Δx values, a) y is 27 and 27.27, b) dy is equal to 9, and c) dy for the given x and Δx values is 0.27.
Alejandro will deposit $1,750 in an account that earns 6. 5% simple interest every year. His sister Anallency will deposit $1,675 in an account that earns 7. 5% interest compounded annually. The deposits will be made on the same day, and no additional money will be deposited or withdrawn from the accounts. How much more money will Anallency have?
Anallency will have $134.46 more money than Alejandro after one year of their respective deposits.
How much more money will Anallency have than Alejandro after one year of their respective deposits?Alejandro and Anallency are depositing money in separate accounts with different interest rates. Alejandro is depositing $1,750 in an account that earns a simple interest rate of 6.5% per year, while Anallency is depositing $1,675 in an account that earns a compounded interest rate of 7.5% per year. After one year of their respective deposits, Anallency will have $134.46 more money than Alejandro.
The difference in interest earned between the two accounts is the reason for the difference in money earned by Alejandro and Anallency. Alejandro's account earns a simple interest rate, which means that he earns 6.5% of his initial deposit every year, regardless of how much interest he has already earned.
On the other hand, Anallency's account earns a compounded interest rate, which means that she earns interest on both her initial deposit and on any interest earned in previous years.
The formula for calculating simple interest is I = Prt, where I is the interest earned, P is the principal (or initial deposit), r is the interest rate, and t is the time in years. Using this formula, we can calculate that Alejandro will earn $113.75 in interest after one year.
The formula for calculating compounded interest is A = P(1 + r/n)^(nt), where A is the amount of money at the end of the time period, P is the principal, r is the interest rate, n is the number of times the interest is compounded per year, and t is the time in years.
Using this formula, we can calculate that Anallency will have $248.21 more in her account than Alejandro after one year.
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