Find the area of the region bounded by the given curves.
y=x^2, y=8x−x^2
Answers
The area of the region bounded by the curves y = x^2 and y = 8x - x^2 is approximately 16.667 square units. We need to calculate the definite integral of the difference between the two functions over their common interval of intersection.
To find the intersection points of the curves, we set the two equations equal to each other and solve for x:
x^2 = 8x - x^2
2x^2 - 8x = 0
2x(x - 4) = 0
This equation gives us two solutions: x = 0 and x = 4. These are the x-values at which the two curves intersect.
To calculate the area between the curves, we integrate the difference between the upper curve (8x - x^2) and the lower curve (x^2) over the interval [0, 4]. The integral represents the sum of infinitely small areas between the curves.
The integral to calculate the area is given by:
∫[0,4] (8x - x^2 - x^2) dx
Simplifying, we have:
∫[0,4] (8x - 2x^2) dx
Integrating, we get:
[4x^2 - (2/3)x^3] from 0 to 4
Evaluating the integral at the upper and lower limits, we have:
[4(4)^2 - (2/3)(4)^3] - [4(0)^2 - (2/3)(0)^3]
Simplifying further, we get:
[64 - (128/3)] - [0 - 0]
Which equals:
[192/3 - 128/3] = 64/3 ≈ 21.333
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Related Questions
At a school recess, there needs to be a ratio of 2 adults for every 24 children on the playground. The double number line represents the number of adults and children on the playground at recess. How many adults are needed if there are 72 children?.
Answers
A ratio of 2:24 kids (when simplified) is 1 parent for every 12 kids.
Now it’s a matter of simple multiplication. In order to get 72 kids, what times 12 equals 72?
The answer is 6.
The triangle on the right is a scaled copy of the triangle on the left. Identify the scale factor. Express your answer as a whole number or fraction in simplest form.
First triangle
15
15
second triangle
45/2
45/2
Answers
Answer: 3
Step-by-step explanation:
The scale factor is the ratio of corresponding side lengths of the two triangles.
In the given triangles, the length of the corresponding sides of the triangles are:
For the first triangle:
Side A = 15 units
Side B = 15 units
For the second triangle:
Side A = 45/2 units
Side B = 45/2 units
To find the scale factor, we can divide the length of one side of the second triangle by the corresponding length of the first triangle.
For example, we can use Side A:
Scale factor = (length of Side A of the second triangle) / (length of Side A of the first triangle)
Scale factor = (45/2) / 15
Scale factor = (45/2) * (1/15)
Scale factor = 3/2
Since the two triangles are scaled copies of each other, the scale factor should be the same for all corresponding sides.
Thus, the scale factor is 3 (in simplest form).
GIVING BRAINLIEST! The graph shows an image of a dilation about the origin with a scale factor of 1/2.
Answers
Answer:
(a) (-8, -12)
Step-by-step explanation:
The scale factor multiplies each coordinate value.
A' = 1/2A
2A' = A . . . . . . multiply by 2
2(-4, -6) = A = (-8, -12) . . . . read A' from the graph
The coordinates of A are (-8, -12).
Answer:
It is (-12) and (-8)
Step-by-step explanation:
Because A and point B is 1\2away from each other
the boxplot shows the fuel economy ratings for 67 subcompact cars with the same model year. some summary statistics are also provided. the extreme outlier is an electric car whose electricity usage is equivalent to 112 miles per gallon. if that electric car is removed from the data set, how will the standard deviation be affected? the iqr?
Answers
The IQR measures the spread of the middle 50% of the data.
Without knowing the actual values of the fuel economy ratings or the summary statistics, we cannot provide an exact answer. However, we can make some general observations about the effects of removing an extreme outlier on the standard deviation and the interquartile range (IQR).
First, the standard deviation measures the spread of the data around the mean. Removing an extreme outlier that is far from the mean could potentially decrease the standard deviation, as the remaining values may be more tightly clustered around the mean. However, the effect on the standard deviation will depend on the exact values of the data points and how much they vary from the mean.
Second, the IQR measures the spread of the middle 50% of the data. Removing an extreme outlier that is far from the bulk of the data may not have a significant effect on the IQR, as the remaining.
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school sold 2 adult tickets and 10 child tickets for a total of $56. The school took in $52 on the second
The school that Elisa goes to is selling tickets to a fall musical. On the first day of ticket sales the
second day by selling 4 adult tickets and 5 child tickets. What is the price each of one adult
ticket and one child ticket?
Answers
Answer:
x+y=4+8=12
$12 for one of each
or if you mean indivually (the wording confused me)
$4 for child, $8 for adult
Step-by-step explanation:
x=adult
y=child
2x+10y=56
4x+5y=52
iin order to be able to subtract values and find x and y im going to multiply 4x+5y=52 by two and get 8x+10y=104
8x+10y=104
-2x+10y=56
6x=48 divide each side by 6
x=8
plug 8 back into an equation to find cost of y
2(8)+10y=56
-16 -16
10y=40
y=4
Suppose an investment account is opened with an initial deposit of $11,000
earning 6.2% interest compounded monthly.
a) How much will the account be worth after 20 years?
b) How much more would the account be worth if compounded continuously?
Answers
a) The account will be worth $39,277.54 after 20 years.
b) If compounded continuously $2,434.90 more the account would be worthy.
a) To find the future value of the account after 20 years, we can use the formula:
FV = \(P(1 + r/n)^{(nt)\)
Where FV is the future value, P is the principal (initial deposit), r is the annual interest rate as a decimal, n is the number of times the interest is compounded per year, and t is the number of years.
Plugging in the given values, we get:
FV = 11,000(1 + 0.062/12)²⁴⁰
FV = $39,277.54
b) If the account is compounded continuously, then we use the formula:
FV = \(Pe^{(rt)\)
Where e is the mathematical constant approximately equal to 2.71828.
Plugging in the given values, we get:
FV = 11,000\(e^{(0.062*20)\)
FV = $41,712.44
Therefore, if the account is compounded continuously, it will be worth $41,712.44 after 20 years. The difference between the two values is $2,434.90, which is the amount the account would earn in interest with continuous compounding over 20 years.
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find a point on the hyperboloid x2 4y2 − z2 = 1 where the tangent plane is parallel to the plane x 4y − z = 0. (x, y, z) = (smaller x-value) (x, y, z) = (larger x-value)
Answers
To find the points on the hyperboloid x^2 + 4y^2 - z^2 = 1 where the tangent plane is parallel to the plane x + 4y - z = 0, we need to compare the gradients (normal vectors) of both planes.
First, find the gradient of the given plane x + 4y - z = 0 by taking the coefficients of x, y, and z. The gradient (normal vector) is (1, 4, -1).
Now, we need to find the gradient of the tangent plane to the hyperboloid. To do this, we'll calculate the gradient of the hyperboloid equation and set it equal to the gradient of the given plane:
∇(x^2 + 4y^2 - z^2) = (2x, 8y, -2z)
Since the gradients are parallel, we have:
2x = 1 (the x component)
8y = 4 (the y component)
-2z = -1 (the z component)
Solving these equations, we get two sets of points (x, y, z) due to the symmetry of the hyperboloid:
(x, y, z) = (1/2, 1/2, 1/2) (smaller x-value)
(x, y, z) = (-1/2, -1/2, -1/2) (larger x-value)
These are the points on the hyperboloid where the tangent planes are parallel to the given plane x + 4y - z = 0.
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In a certain Algebra 2 class of 26 students, 10 of them play basketball and 5 of them
play baseball. There are 3 students who play both sports. What is the probability that
a
student chosen randomly from the class plays basketball or baseball?
006
Answers
K=basketball, S= baseball
10K
5S
3 K & S
26 total
P(K) = 10/26, P(S) = 5/26 P(K^S) = 3/26
P(K v S) = P(K) + P(S) - P(K^S) = 10/26 + 5/26 - 3/26 = 12/26 = 6/13
Probability of randomly getting a student who plays either basketball or baseball = 6/13
A rectangle with vertices (3,-2),(3,-4), (7, -2), (7, -4) is reflected across the x-axis and then rotated 90° counterclockwise. What other transformations produce the sameimage?
Answers
Answer: A reflection about the y-axis and a clockwise rotation of 90° around the origin.
another example is:
A counterclockwise rotation of 90° around the origin, and then a reflection about the y-axis.
Step-by-step explanation:
Let's do this for a single point because it is essentially the same.
Let's use the point (3, -2)
First, we have a reflection across the x-axis, this only changes the sign of the y-component.
Then the new point will be: (3, -(-2)) = (3, 2)
Notice that this point is on the first quadrant.
Now we do a 90° counterclockwise rotation.
Then we move to the second quadrant, and we change the order of the components in the point (and because we are in the second quadrant, the x-component is negative and the y-component is positive)
Then the new point is (-2, 3)
Now, another transformation that is equivalent to this one is to do first a reflection about the y-axis, so we only change the sign of the y-component
Then the new point is (-3, -2)
And now we are in the third quadrant.
and then we do a rotation of 90° clockwise, which moves our point to the second quadrant, changes the order of the components and leaves the y-component positive and the x-component negative, then the new point is:
(-2, 3)
Same as before.
(notice that we used the same transformations, but applied to different line and direction of rotation)
Another example is if we first start with a counterclockwise rotation of 90° around the origin, the original point (3, -2) is on the fourth quadrant, so this rotation leaves our point in the first quadrant, so we change the order of the components and both of them will have a positive sign.
The new point is: (2, 3)
Now let's do a reflection about the y-axis, which changes the sign of the x-component.
Then the final point is (-2, 3), same as before.
And there are a lot of other transformations that will be equivalent to this ones, these are just two examples.
if
it is estimated that 80% people recieve a call back after an
interview and 20% dont. in a random sample of 100, how many recieve
a call back
Answers
If it is estimated that 80% people receive a call back after an interview and 20% don't in a random sample of 100, then 80 people receive a call back.
To find the number of people who get a call back, follow these steps:
It is given that the total number of people= 100 and 80% of people receive a call back. So, the number of people who get a call back can be found by multiplying the percentage of people who get a call back by the total number of people.So, the number of people who get a call back = 80% of 100= (80/100) × 100 = 80.So, we can estimate that 80 people will receive a call back after the interview.
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Match the vocabulary word to its correct definition.
1. complex conjugate the sum of a real number and an imaginary number: a + bi, where a is a real number and b is the imaginary part
2. complex number the part of a complex that is multiplied by i; so, the imaginary part of the complex number a + bi is b; the imaginary part of a complex number is a real number
3. i the part of a complex that is not multiplied by i. So, the real part of the complex number a + bi is a; the real part of a complex number is a real number
4. imaginary number a number defined with the property that 12 = -1, so that
5. imaginary part of a complex number the inverse of a complex number a + bi is a complex number c + di such that the product of these two numbers equals 1
6. multiplicative inverse any nonzero multiple of i; this is the same as the square root of any negative real number
7. real part of a complex number the conjugate of a complex number has the opposite imaginary part. So, the conjugate of a + bi is a - bi. Likewise, the conjugate of a - bi is a + bi. So, complex conjugates always occur in pairs
Answers
Answer:
1. Complex number.
2. Imaginary part of a complex number.
3. Real part of a complex number.
4. i
5. Multiplicative inverse.
6. Imaginary number.
7. Complex conjugate.
Step-by-step explanation:
1. Complex number: is the sum of a real number and an imaginary number: a + bi, where a is a real number and b is the imaginary part.
2. Imaginary part of a complex number: the part of a complex that is multiplied by i; so, the imaginary part of the complex number a + bi is b; the imaginary part of a complex number is a real number.
3. Real part of a complex number: the part of a complex that is not multiplied by i. So, the real part of the complex number a + bi is a; the real part of a complex number is a real number.
4. i: a number defined with the property that 12 = -1.
5. Multiplicative inverse: the inverse of a complex number a + bi is a complex number c + di such that the product of these two numbers equals 1.
6. Imaginary number: any nonzero multiple of i; this is the same as the square root of any negative real number.
7. Complex conjugate: the conjugate of a complex number has the opposite imaginary part. So, the conjugate of a + bi is a - bi. Likewise, the conjugate of a - bi is a + bi. So, complex conjugates always occur in pairs.
Answer:
Complex number: is the sum of a real number and an imaginary number: a + bi, where a is a real number and b is the imaginary part.
2. Imaginary part of a complex number: the part of a complex that is multiplied by i; so, the imaginary part of the complex number a + bi is b; the imaginary part of a complex number is a real number.
3. Real part of a complex number: the part of a complex that is not multiplied by i. So, the real part of the complex number a + bi is a; the real part of a complex number is a real number.
4. i: a number defined with the property that 12 = -1.
5. Multiplicative inverse: the inverse of a complex number a + bi is a complex number c + di such that the product of these two numbers equals 1.
6. Imaginary number: any nonzero multiple of i; this is the same as the square root of any negative real number.
7. Complex conjugate: the conjugate of a complex number has the opposite imaginary part. So, the conjugate of a + bi is a - bi. Likewise, the conjugate of a - bi is a + bi. So, complex conjugates always occur in pairs.
Step-by-step explanation:
The reduced gradient is analogous to the ___________ for linear models.
Answers
The reduced gradient is analogous to the residual for linear models.
In linear regression, the residual represents the difference between the observed values and the predicted values of the dependent variable. Similarly, in optimization, the reduced gradient represents the difference between the current solution and the optimal solution. It is a measure of how far the current solution is from the optimal solution in the direction of the search. By minimizing the reduced gradient, we can move closer to the optimal solution.
The reduced gradient is a widely used optimization technique in non-linear programming that allows for efficient computation of the descent direction at each iteration while accounting for constraints. It involves calculating a partial derivative of the objective function with respect to the variables that are not restricted by the constraints, and then projecting the resulting gradient onto the space defined by the constraints. The resulting vector is called the reduced gradient, and it points in the direction of the steepest descent that is feasible.
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Solve 3x²+7x-13 = 0 Give your solutions correct to 2 decimal places
Answers
Answer:
Step-by-step explanation:
\(3x^2+7x-13=0\\x = \frac{-7 +- \sqrt{49+156} }{6}\\ x = \frac{-7 +- \sqrt{215} }{6}\\ x = \frac{-7}{6}+-\frac{14.66}{6}\\ x = \frac{7.66}{6} = 1.27\\ or\\x = \frac{-21.66}{6} = -3.61\)
I need help cancelling units
Answers
Answer:
60:1
Step-by-step explanation:
60:1 = 366:x
60x = 366
x = 6.1
366 minutes is equivalent to 6.1 hours.
Convert 366 minutes to hours.
Knowing that in 60 minutes = 1 hour.\(\boldsymbol{\sf{Therefore \ \to \ 366 \not{m}*\dfrac{1 \ hr}{60\not{m}}=6.1 \ hr }}\)
We conclude that a time of 366 minutes is equal to 6.1 hours
There are 60 minutes in 1 hour. To convert from minutes to hours, divide the number of minutes by 60. For example, 120 minutes equals 2 hours because 120/60=2.
I need help once again :(
Answers
2 is x = 5
3 is no solution
4 is s=6
5 is infinite solutions
6 x=-1
I don't know if u need the work or just the answers but I hope I helped :)
Simon likes to bake pies. His favorite to make is
apple pies. He used a ratio of 9 apples to 3 ounces of
sugar to make a pie. Which ratio is equivalent to
ounces of sugar to apples?
Answers
Answer:
C.
Step-by-step explanation:
Hoped I helped! Im Eve btw Hope you have a great day and consider marking this brainliest
what is 339.12 rounded to the nearest hundreth
Answers
Answer:
339.12 rounded to the nearest hundreth is 339
Because the hundreth place right now is 1 so anything below five keep the same
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Step-by-step explanation:
Answer: 339.10
Step-by-step explanation: The value of 2 is the hundredth and it is rounded up to 0
Xavier bought 3 bags of chips for $0.75 each. he was charged a 7% sales tax fee. how much did he pay altogether
Answers
Answer: $2.40
Step-by-step explanation:
0.75 * 0.07 = 0.0525
round = 0.05
0.75+0.05 = 0.8
0.8*3 = 2.4
$2.40
A 3.5 inch-by-5 inch image is being enlarged by
a scale factor of x. Express the area of the
enlarged image in terms of x.
Answers
side of image = 3.5 inch and 5 Inch
Area = 3.5 × 5 = 17.5 inch²
Now length is Enlarge by x
new length and Breadth are (3.5+x) and (5+x)
Area of new Image =(x+3.5)(x+5)
Area of new Image = x² + 5x + 3.5x + 17.5
Area of new Image = x² + 8.5x + 17.5
i hope it's helpful
If a 40-foot tree casts a 16-foot shadow, find the length of the shadow cast by a 23-foot tree
Answers
The shadow of the 24-foot tree will be 9.6 feet.
In mathematics, ratios are used to determine the relationship between two numbers it indicates how many times is one number to another number.
Given that:-
If a 40-foot tree casts a 16-foot shadow, find the length of the shadow cast by a 24-foot tree.
Now, The shadow of the 24-foot tree will be calculated as follows:-
40/16 = 24/x
x = 24 x 16 / 40
x = 9.6
Therefore, the shadow of the 24-foot tree will be 9.6 feet.
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How many solutions are there to the equation below?
7(x + 2) = 7x-10
O A. Infinitely many
O B. o
O c. 1
Answers
Hi there! :)
Answer:
B. 0 solutions.
Step-by-step explanation:
Given the equation 7(x + 2) = 7x - 10
Simplify:
7(x + 2) = 7x - 10
Distribute the 7 with the terms inside of the parenthesis:
7x + 14 = 7x - 10
Subtract "7x" and 14 from both sides:
7x - 7x = -10 - 14
0 ≠ - 24. The equation has no solutions.
Answer:
\(\boxed{\sf B. \ 0}\)
Step-by-step explanation:
\(\sf 7(x + 2) = 7x-10\)
Expand brackets.
\(\sf 7x+14 = 7x-10\)
Subtract 7x and 14 from both sides.
\(\sf 7x+14 -7x-14= 7x-10-7x-14\)
\(\sf 0=-24\)
There are 0 solutions.
A grinding process has an upper specification of 1.649 inches and a lower specification of 1.507 inches. A sample of parts had a mean of 1.57 inches with a standard deviaiton of 0.026 inches.
Round your answer to four decimal places.
What is the process capability index for this system?
Answers
Answer:
The process capability index for this system is 0.9103
Step-by-step explanation:
Upper specification = 1.649 inches
Lowe Specification = 1.507 inches
Mean = 1.57 inches
Standard Deviation = 0.026 inches
Process Capability Index = PCI
PCI = (Upper specification - lower specification)/6(standard deviation)
So, we get,
PCI = (1.649 - 1.507)/((6)(0.026))
PCI = 0.142/0.156
PCI = 0.9102564
To four decimal places, we get,
PCI = 0.9103
f(x)=x^2. What is g(x)
Answers
Answer:
\(g(x) = \frac14x^2\)
Step-by-step explanation:
Since f(x) and g(x) share the same top at (0,0), their difference is only a scale factor, so g(x) = ax² but we need to find a.
To find it, we use the given point of g, (2,1)
So g(2) = 1
meaning that
a·2² = 1
4a = 1
a = 1/4
I need to find the values of the variable. Then find the angle measures if the polygon
Answers
Answer:
b=20
5b=100
4b=80
Step-by-step explanation:
5b+4b+4b+5b=360
18b=360
b=20
A bike shop X it costs $1.50 per hour plus a $4.00 deposit to rent a bike. At Bike Shop Z it costs $0.50 per hour plus a $9.00 deposit to rent a bike. The total cost, c, of renting a bike for n hours at either bike shop can be represented by an equation. Write and solve a system of equations to find how many hours you have to rent the bike for the cost to be the same.
Answers
Answer:
4+1.5n=c
9+0.5n=c
5 hours
Step-by-step explanation:
(n) represents the number of hours
(c) represents the total cost
Bike shop A) $4+$1.5n=c
Bike shop Z) $9+$0.5n=c
4+1.5n=9+0.5n
n=5 Both shops would charge the same amount at 5 hours.
the smallest possible interval between two different sounding pitches is called a half step. true false
Answers
True, a half step is the shortest distance that can exist between two pitch-sounding frequencies. In Western musical notation, a half step is thought to be the shortest interval, or space between two notes.
The smallest musical interval typically utilized in Western tonal music is a semitone, sometimes known as a half step or a half tone. It is described as the space in a 12-tone scale between two neighbouring notes. For instance, there is a semitone's distance between C and C. Any interval in a 12-note scale that is almost evenly divided into semitones can be described in terms of the proper number of semitones, for example, a whole tone or major second is two semitones wide, a major third is four semitones wide, and a perfect fifth is seven semitones wide.
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One baseball team played 24 games throughout their entire season. If this baseball team
won 75% of those games, then how many games did they win? Round your answer to the
nearest whole number if necessary.
Answers
What shape is produced when slicing a right rectangular prism perpendicular to
the base?
Answers
Answer:
rectangle
Step-by-step explanation:
If the slice is perpendicular to the base, it can intersect at most two lateral faces and the parallel base. Hence the shape produced will be a rectangle.
the national center for health statistics has found that there is a 0.41% chance that an american citizen will die from falling. what is the probability that you will not die from a fall? (round to the nearest hundredth of a percent)
Answers
If there is a 0.41% chance that an American citizen will die from falling, then the probability that you will not die from a fall is 99.59%
The chances of an American citizen will die from falling = 0.41%
The probability is the ratio of number of favorable outcomes to the total number of outcomes
The equation will be
The probability = Number of favorable outcomes / Total number of outcomes
The probability that you will not die from a fall = 1 - The chances of an American citizen will die from falling
Substitute the values in the equation
The probability that you will not die from a fall = 1 - 0.41
= 99.59%
Therefore, the probability that you will not die from a fall is 99.59%
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in the citation 319 n.w. 2d 459, the number 459 represents the
Answers
The number 459 in the citation 319 N.W. 2d 459 represents the page number where the case or legal opinion can be found in the North Western Reporter, Second Series. The North Western Reporter is a series of case reporters that includes court decisions from various states in the northwestern region of the United States.
In legal citations, the number before the "N.W." indicates the volume number of the reporter. So in this case, "319" refers to the specific volume where the case is located. The abbreviation "N.W." stands for North Western Reporter, which is the name of the reporter series. The number "2d" after "N.W." indicates that this is the second series of the North Western Reporter.
To locate the case within the volume, the number "459" refers to the page number where the case begins. This allows readers, such as lawyers, judges, or researchers, to easily find and reference the specific case within the reporter.
In summary, the number "459" in the citation 319 N.W. 2d 459 represents the page number where the case can be found in the North Western Reporter, Second Series.
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