To find the area of the region that lies inside the first curve (r = 11 cos(θ)) and outside the second curve (r = 5 + cos(θ)), we need to set up an integral. The curves intersect at two points, so we need to split the region into two parts.
The first part is when θ goes from 0 to π, and the second part is when θ goes from π to 2π. For the first part, we have:
∫[0,π] ½ [(11 cos(θ))² - (5 + cos(θ))²] dθ
For the second part, we have:
∫[π,2π] ½ [(11 cos(θ))² - (5 + cos(θ))²] dθ
Evaluating these integrals, we get:
∫[0,π] ½ [(11 cos(θ))² - (5 + cos(θ))²] dθ = 139.04 units²
and
∫[π,2π] ½ [(11 cos(θ))² - (5 + cos(θ))²] dθ = 139.04 units²
Adding these two areas together, we get a total area of:
278.08 units²
To find the area of the region that lies inside the first curve (r = 11cos(θ)) and outside the second curve (r = 5 + cos(θ)), follow these steps:
1. Identify the points of intersection: Set r = 11cos(θ) = 5 + cos(θ). Solve for θ to get θ = 0 and θ = π.
2. Convert the polar equations to Cartesian coordinates:
- First curve: x = 11cos^2(θ), y = 11sin(θ)cos(θ)
- Second curve: x = (5 + cos(θ))cos(θ), y = (5 + cos(θ))sin(θ)
3. Set up the integral for the area of the region:
Area = 1/2 * ∫[0 to π] (11cos(θ))^2 - (5 + cos(θ))^2 dθ
4. Evaluate the integral:
Area = 1/2 * [∫(121cos^2(θ) - 10cos^3(θ) - cos^4(θ) - 25 - 10cos(θ) - cos^2(θ)) dθ] from 0 to π
5. Calculate the result:
Area ≈ 20.91 square units (after evaluating the integral)
So, the area of the region that lies inside the first curve and outside the second curve is approximately 20.91 square units.
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a box contains 3 red balls, 5 white balls, and 10 green balls. if a ball is chosen at random, what is the probability that it is either white or green?
Step-by-step explanation:
solution:
given,
no. of red balls [n(R)] = 3
no. of white balls [n(W)] = 5
no. of green balls [n(G)] = 10
no. of sample events [n(S)] = 3+5+10 = 18
no. of white or green ball [n(WUG)] = 5+10 = 15
no. of favourable events [n(E)] = 15
probability of favourable events [P(E)] = ?
We know,
P(E) = n(E) / n(S)
= 15/18
= 5/6
Therefore if a ball is chosen at random, the probability that it is either white or green is 5/6.
Let b1 , b2 ,. . . , bt denote a partition of the sample space ω. Pt(a) prove that pr[a] = k=1 pr[a | bk ] pr[bk ]. (b) deduce that pr[a] ≤ max1≤k≤t pr[a | bk ]
The solution is, 4xy is an integer, bₖ₊₁ is divisible by 4.
By mathematical induction, it is proven that bₙ is divisible by 4 for every integer n ≥ 1.
To prove that the sequence b₁, b₂, b₃, ... defined by b₁ = 4, b₂ = 12, and bₖ = bₖ₋₂ bₖ₋₁ for each integer k ≥ 3 is divisible by 4 for every integer n ≥ 1, we can use mathematical induction.
Base case:
For n = 1, b₁ = 4, which is divisible by 4.
For n = 2, b₂ = 12, which is also divisible by 4.
Inductive step:
Assume that bₖ and bₖ₋₁ are divisible by 4 for some integer k ≥ 3. We want to prove that bₖ₊₁ is also divisible by 4. We have:
bₖ₊₁ = bₖ₋₁ bₖ
Since we assumed bₖ and bₖ₋₁ are divisible by 4, there exist integers x and y such that:
bₖ = 4x and bₖ₋₁ = 4y
Then, we can rewrite bₖ₊₁ as:
bₖ₊₁ = (4y)(4x) = 4(4xy)
Since 4xy is an integer, bₖ₊₁ is divisible by 4.
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A voltage V across a resistance R generates a current I =
V/R. If a constant voltage of 22 volts is put across a resistance
that is increasing at a rate of 0.2 ohms per second when the resistance is 5 ohms, at what rate is the current changing?
A voltage V across a resistance R generates a current I =V/R. If a constant voltage of 22 volts is put across a resistance that is increasing at a rate of 0.2 ohms per second when the resistance is 5 ohms, The current is changing at a rate of -0.176 amperes per second.
Given the formula I = V/R, where V is the voltage, R is the resistance, and I is the current, we can find the rate at which the current is changing.
With a constant voltage of 22 volts and a resistance increasing at a rate of 0.2 ohms per second when the resistance is 5 ohms, we can use the derivative of the current formula with respect to time.
Let I be the current, V be the voltage (22 volts), R be the resistance (5 ohms), and dR/dt be the rate of change of resistance (0.2 ohms/second). We need to find dI/dt, the rate of change of current.
We have the equation I = V/R. Differentiating both sides with respect to time, we get: dI/dt = -V * (dR/dt) / R^2 Now, plug in the given values: dI/dt = -22 * (0.2) / (5)^2 dI/dt = -4.4 / 25 dI/dt = -0.176 A/s.
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The communications department hired Marilyn Curtis at an annual salary of $154,950. Advertising costs were $2,247.50. Interviewing expenses were: Marilyn Curtis, $1,647.43; Tom Hart, $996.94. The agency fee is 20% of the first year's salary. What was the total cost of hiring Marilyn?
Answer:
The total cost of hiring Marilyn Curtis would be $189,978.13----------------------
First, we need to calculate the agency fee, which is 20% of Marilyn's annual salary:
Agency fee = 20% of $154950 = 0.2 * $154950 = $30990Next, we add up all the expenses:
Salary = $154950 Advertising costs = $2247.50 Interviewing expenses = $1647.43 + $996.94 = $2644.37 Agency fee = $30990Total cost is the sum of the four:
$154950 + $2247.50 + $2644.37 + $30990 = $189,978.13The system of differential equations dx/dt = Ax depends on a real-valued constant a. Use the eigenvalues to determine the stability properties of the equilibrium at the origin for all values of a.
The stability properties of the equilibrium at the origin for all values of a in the system of differential equations dx/dt = Ax depend on the eigenvalues of matrix A.
To determine the stability properties, first find the eigenvalues of matrix A by solving the characteristic equation, det(A - λI) = 0, where λ represents the eigenvalues and I is the identity matrix. Once you obtain the eigenvalues, analyze their real parts:
1. If all real parts are negative, the equilibrium is asymptotically stable.
2. If any real part is positive, the equilibrium is unstable.
3. If all real parts are non-positive, and there are no repeated eigenvalues with zero real parts, the equilibrium is stable.
By examining the eigenvalues, you can determine the stability properties for all values of the real-valued constant a.
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Johanna is a crime scene investigator and tapes off two squares at a crime scene. The first square is 3 feet larger than the second, and the total area of the two squares is modeled by the equation (y + 3)² + y² = 369 where y is the length in feet of the side of the smaller square
The length of one side of the smaller square is 12 feet, and the length of one side of the larger square is 3 feet longer, or 15 feet.
Let's start by using the formula for the area of a square, which is A = s² where A is the area and s is the length of one side of the square.
Let y be the length of one side of the smaller square in feet. Then, the length of one side of the larger square is 3 feet longer than y, so it is (y + 3) feet.
The total area of the two squares is given by the equation:
(y + 3)² + y² = 369
Expanding the left side of the equation gives:
y² + 6y + 9 + y² = 369
Simplifying the equation by combining like terms gives:
2y² + 6y - 360 = 0
Dividing both sides of the equation by 2 gives:
y² + 3y - 180 = 0
Now we can solve for y using the quadratic formula:
y = (-b ± √(b² - 4ac)) / 2a
In this case, a = 1, b = 3, and c = -180. Substituting these values into the formula gives:
y = (-3 ± √(3² - 4(1)(-180))) / 2(1)
Simplifying under the square root:
y = (-3 ± √(729)) / 2
y = (-3 ± 27) / 2
We discard the negative solution as it does not make sense in the context of the problem.
y = (24) / 2
y = 12
Therefore, the length of one side of the smaller square is 12 feet, and the length of one side of the larger square is 3 feet longer, or 15 feet.
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Find the surface area of the prism.
so the trapezoidal prism has four rectangles and two trapezoids
[tex]\stackrel{ \textit{two trapezoids} }{2\left( \cfrac{4(6+12)}{2} \right)}~~ + ~~\stackrel{ \textit{left and right} }{2(5)(3)}~~ + ~~\stackrel{ front }{(6)(3)}~~ + ~~\stackrel{ back }{(12)(3)} \\\\\\ 72+30+18+36\implies \text{\LARGE 156}~in^2[/tex]
A relational database uses row-oriented storage to store an entire row within one _____. a. page b. block c. sector d. Table
A relational database uses row-oriented storage to store an entire row within one:
(b) block.
In row-oriented storage, the entire row of a table is stored in one block of storage. A block is a unit of storage used by the file system or operating system to manage data on a disk or other storage device. Blocks are typically a fixed size and contain a set number of bytes. When data is written to the disk, it is written in blocks.
In contrast, column-oriented storage stores data by column rather than by row. This can be more efficient for queries that only need to access certain columns, as only the required columns need to be read from disk, rather than the entire row. However, it can be less efficient for queries that need to access all columns of a table.
Thus, the correct option is :
(b) block
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if a is a 4×4 matrix with characteristic polynomial λ4+λ3+λ2+λ, then a is not invertible.
The matrix a is not invertible.
Is the 4x4 matrix with characteristic polynomial invertible?We can use the fact that a matrix is invertible if and only if its determinant is non-zero.
Let λ be an eigenvalue of a with corresponding eigenvector x. Then we have:
a x = λ x
Multiplying both sides by a, we get:
[tex]a^2[/tex]x = λ a x
Substituting a x = λ x, we get:
[tex]a^2\\[/tex] x = λ² x
Similarly, we can show that:
[tex]a^3[/tex] x = λ³ x and
[tex]a^4[/tex] x = λ⁴ x
Substituting these results into the characteristic polynomial, we get:
det(a - λI) = (λ⁴+λ³+λ²+λ) = λ(λ³+λ²+λ+1)
Since the characteristic polynomial has degree 4, we know that a must have 4 eigenvalues (counting multiplicity).
Suppose a were invertible. Then all of its eigenvalues would be non-zero, and so we would have:
det(a - λI) = (λ - λ1)(λ - λ2)(λ - λ3)(λ - λ4)
where λ1, λ2, λ3, λ4 are the eigenvalues of a.
But we just saw that the characteristic polynomial has a factor of λ, so at least one of the eigenvalues must be zero. This is a contradiction, so our assumption that a is invertible must be false.
Therefore, a is not invertible.
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En una fiesta los niños se comieron la cuarta parte del pastel. El total de niños que comieron pastel fueron 38. ¿Cuántos niños más pudieron comer pastel hasta que se acabara?
114 more children can eat a cake if 38 children finish a quarter of the cake at a party.
Fraction refers to a part of a whole. A quarter refers to the one-fourth of an object. It can be represented as [tex]\frac{1}{4}[/tex].
Given in the question,
Number of children that eat a quarter of cake = 38
Number of children that eat whole cake = 38 * 4
= 152
Cake left = Whole cake - a quarter of the cake
= 1 - [tex]\frac{1}{4}[/tex] = [tex]\frac{3}{4}[/tex]
Number of children that can eat three-fourths of cake = [tex]\frac{3}{4}[/tex] * 152
= 114 children
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The question is given in Spanish and the question in English is:
At a party, the children ate a quarter of the cake. The total number of children who ate cake was 38. How many more children could eat cake until it ran out?
Sara made a withdrawal of $80. Tom deposited $47. 50.
Which transaction involved more money?
Tom's deposit of $47.50 involved more money than Sara's withdrawal of $80.
This is because the amount involved in a transaction is determined by the magnitude of the transaction, which is the absolute value of the transaction. In other words, the amount involved in a transaction is determined by the size of the number, regardless of whether it is positive or negative.
Tom's deposit of $47.50 is a larger number than Sara's withdrawal of $80 when considering the absolute value of the transactions. Hence, Tom's deposit involved more money.
The direction of the transaction (whether it is a deposit or withdrawal) does not necessarily indicate the amount involved. The magnitude of the transaction is what determines the amount involved.
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a store will give 3 baseball cards to each of 5 kids. there are a total of 40 cards and they are all different. the store will give 3 cards to each kid and keep the remaining 25 cards. how many ways are there to distribute the cards?
(40-15)! / 40! this the number of ways that cards can be distributed to each child.
To distribute the 3 baseball cards to each of the 5 kids from a total of 40 different cards, you'll first need to determine the total number of cards being given out and the number of combinations for each child.
Since each kid gets 3 cards, there will be a total of 3 * 5 = 15 cards given out, leaving 25 cards for the store.
Now, let's calculate the ways to distribute the cards to each kid. For the first kid, there are 40 cards to choose from, so there are 40 choose 3 (denoted as C(40,3)) ways to select the cards. Similarly, for the second kid, there are 37 remaining cards to choose from, so there are C(37,3) ways. Following the same logic, we have C(34,3) ways for the third kid, C(31,3) ways for the fourth kid, and C(28,3) ways for the fifth kid.
To determine the total number of ways to distribute the cards, you'll need to multiply the combinations for each kid together: C(40,3) * C(37,3) * C(34,3) * C(31,3) * C(28,3). This will give you the total number of ways to distribute the 15 cards among the 5 kids while keeping the remaining 25 cards in the store.
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Write the equation of the line in fully simplified slope-intercept form.
The equation of the line in fully simplified slope-intercept form is y = -7/8x - 7
Writing the equation of the line in fully simplified slope-intercept form.from the question, we have the following parameters that can be used in our computation:
The linear graph
Where we have the points
(0, -7) and (-8, 0)
A linear equation is represented as
y = mx + c
Where
c = y when x = 0
So, we have
y = mx - 7
Next, we have
0 = -8m - 7
Evaluate
m = -7/8
So, we have
y = -7/8x - 7
Hence, the equation of the line in fully simplified slope-intercept form is y = -7/8x - 7
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please solve the problemIf y = sin(log(x² + 2x + 1) prove that (x + 1)?) + (x+1)y, - - 4y |
We have proved that (x + 1)dy/dx + (x + 1)y - 4y = 0, which means that the expression is true.
To solve this problem, we need to use some algebraic manipulations and the properties of the derivative of sin(x) with respect to x.
First, let's simplify the expression inside the sine function:
log(x² + 2x + 1) = log((x + 1)²) = 2log(x + 1)
Substituting this into the original equation, we get:
y = sin(2log(x + 1))
Now, let's take the derivative of both sides of this equation with respect to x:
dy/dx = d/dx(sin(2log(x + 1)))
dy/dx = cos(2log(x + 1)) * d/dx(2log(x + 1))
dy/dx = cos(2log(x + 1)) * 2/(x + 1)
Now, let's simplify the expression we're trying to prove:
(x + 1)dy/dx + (x + 1)y - 4y
= (x + 1)cos(2log(x + 1)) * 2/(x + 1) * sin(2log(x + 1)) + (x + 1)sin(2log(x + 1)) - 4sin(2log(x + 1))
= 2(x + 1)cos(2log(x + 1))sin(2log(x + 1)) + (x + 1)sin(2log(x + 1)) - 4sin(2log(x + 1))
= (2x + 2)sin(2log(x + 1)) - 2sin(2log(x + 1)) - 4sin(2log(x + 1))
= 0
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What is the nearest degree to ABC
Answer:
Your answer is <ABC = 156°
Step-by-step explanation:
I am not sure how to explain this, but you look at the values between 0°-180°. Starting from A right counterclockwise from 0 to C. You can see the degree.
Find T,N, and κ for the space curve r(t)=t^7/7 i+t^5/5 j , t > 0.
The values of T, N and k for the space curve can be seen in the given image.
What is a Vector Space?A vector space in mathematics represents a set of vectors that meet particular requirements.
A vector, detected as an array of values, presents the direction and magnitude of any object. As per this definition, within a vector space, these configured objects can be added together and even oscillated by a definitive number referred to as scalar multiplication.
Closure under addition or scalar multiplication, commutativity, associativity, zero-vector existence and inverses are some critical features necessary for classifying this ensemble of vectors as a vector space.
Significantly utilized in geometry, linear algebra, functional analysis, not forgetting physics, engineering, and computer science being areas where their application is regular.
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Fish management researchers estimate that approximately 60% of spawning Atlantic salmon are female (Sutterby and Greenhaigh, 2005). Suppose that a group of students takes a random sample of 10 salmon from a river where salmon are spawning. The number of females in the sample follows a binomial distribution with = 10 and p=0.60, where nequals the number of trials and p equals the probability of success on any given trial. Which histogram shows the distribution of the number of females in samples of size 10 when the population proportion of females is 0.00
Based on the given information, we can infer that you're looking for the histogram that shows the distribution of the number of females in samples of size 10 when the population proportion of females is 0.60.
To create this histogram, we'll use the binomial distribution with n=10 trials and p=0.60 success probability.
Step 1: Identify the range of possible outcomes
The range of possible outcomes for the number of females in a sample of 10 salmon can be from 0 to 10.
Step 2: Calculate the probabilities for each outcome
Using the binomial distribution formula, calculate the probability of each outcome from 0 to 10 females. The formula is:
P(x) = C(n, x) * p^x * (1-p)^(n-x)
Step 3: Create the histogram
Plot the probabilities for each outcome (number of females) on the x-axis and their respective probabilities on the y-axis. This will give you a histogram that represents the distribution of the number of females in samples of size 10 when the population proportion of females is 0.60.
Note: As you didn't provide any specific histograms for comparison, we can't tell you which one matches this description. However, the explanation above should help you understand how to create and interpret the correct histogram.
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Which ordered pair is a solution to the equation? y=x32−2
The solution is, : OPTION D: NEITHER, ordered pair is a solution to the equation.
Here, we have,
The given equation is: 7x - 2y = - 5
To find a solution to this, we substitute the options and compare LHS and RHS.
OPTION A: (1, 5)
LHS = 7(1) - 2(5) = 7 - 10 = -3
RHS = - 5
LHS RHS.
So, this option is eliminated.
OPTION B: (-1, 1)
LHS = 7(-1) - 2(1) = -7 - 2 = - 9
RHS = - 5
Again, LHS ≠ RHS.
So, this Option is eliminated as well.
OPTION C: It says both A and B. Clearly, this is eliminated as well.
This is a two variable equation. So, we need a minimum of two equations to determine the solution. Since, only one equation is given here, we use the help of options.
Therefore, the answer is: OPTION D: NEITHER, ordered pair is a solution to the equation.
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5. Given the velocity in meters/second for v(t) = 8 - 20,1st s 6 a.) find the displacement of the particle over the given time interval; b.) find the distance traveled by the particle over the given time interval.
The displacement of the particle over the given time interval is -310 meters and the distance traveled by the particle over the given time interval is 310 meters.
First, let's clarify the given information:
v(t) = 8 - 20t (velocity function in meters/second)
Time interval: [1, 6]
Now, let's address each part of the question:
a) Find the displacement of the particle over the given time interval:
To find the displacement, we need to integrate the velocity function v(t) to get the position function s(t) and then evaluate the difference in position at the endpoints of the time interval.
1. Integrate v(t): ∫(8 - 20t) dt = 8t - 10t^2 + C (position function s(t))
2. To find the displacement, evaluate s(t) at the endpoints of the interval and find the difference:
Displacement = s(6) - s(1)
= (8(6) - 10(6)^2) - (8(1) - 10(1)^2)
= (48 - 360) - (8 - 10)
= (-312) - (-2)
= -310 meters
b) Find the distance traveled by the particle over the given time interval:
To find the distance traveled, we need to find the absolute value of the integral of the velocity function over the given interval.
1. Since we already have the position function s(t), we can find the distance by evaluating the absolute value of the difference in position:
Distance = |s(6) - s(1)|
= |-310|
= 310 meters
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Consider the function: f(x) = x³ – 3x²+6 (a) Find the critical value(s) (b) On what intervals is the function increasing and decreasing? (c) Find the location (x-values) of the local min and max. Please clearly label your answers. (d) On what intervals is the function concave up and concave down?
The function: f(x) = x³ – 3x²+6: (a) The critical values of f(x) are x=1 and x=2. (b) The function is increasing on the intervals (-∞, 1) and (2, ∞), (c) The location of the local minimum is at x=1, and the local maximum is at x=2, (d) The function is concave up on the interval (2, ∞) and concave down on the interval (-∞, 2).
(a) To find the critical values of f(x), we take the derivative of f(x) and set it equal to zero: f'(x) = 3x² - 6x = 3x(x-2). Setting f'(x) = 0, we get x=0 and x=2 as the critical values. However, x=0 is not in the domain of the function, so we discard it.
(b) To determine the intervals where the function is increasing and decreasing, we use the first derivative test. On the interval (-∞, 1), f'(x) is negative, so the function is decreasing. On the interval (1, 2), f'(x) is positive, so the function is increasing. On the interval (2, ∞), f'(x) is positive, so the function is increasing.
(c) To find the location of the local minimum and maximum, we use the second derivative test. The second derivative of f(x) is f''(x) = 6x - 6. At x=1, f''(1) is negative, so the function has a local maximum at x=1. At x=2, f''(2) is positive, so the function has a local minimum at x=2.
(d) To determine the intervals where the function is concave up and concave down, we use the second derivative test. The function is concave up on the interval (2, ∞), where f''(x) is positive, and concave down on the interval (-∞, 2), where f''(x) is negative.
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the 3px, 3py, and 3pz orbitals look the same, but they point in different directions. T/F?
True. The 3px, 3py, and 3pz orbitals are similar in shape but differ in their orientation or direction.
The p orbitals are one type of orbital that corresponds to the angular momentum number l = 1. These p orbitals are designated as 3px, 3py, and 3pz to indicate their orientations along the x, y, and z axes, respectively.
The p orbitals have a shape with a node at the nucleus. They consist of two lobes of electron density, one on either side of the nucleus, separated by a region of zero electron density. The lobes are oriented along the designated axes. The 3px orbital points along the x-axis, the 3py orbital points along the y-axis, and the 3pz orbital points along the z-axis. Although they have different orientations, their shapes and sizes are the same.
So, while the 3px, 3py, and 3pz orbitals differ in their orientation in space, they share the same overall shape and size.
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If a ball is thrown into the air with a velocity of 40 ft/s, its height in feet after tt seconds is given by y=40t−16t2y=40t−16t2.
(a) Find the average velocity for the time period beginning with t=2t=2 and
(1) lasting 0.5 seconds:
(2) lasting 0.1 seconds:
(3) lasting 0.05 seconds:
(4) lasting 0.01 seconds:
b) Find the instantaneous velocity when t=2t=2:
(a) The average velocity for the time period beginning with t=2t=2 and
(1) lasting 0.5 seconds: -40 ft/s
(2) lasting 0.1 seconds: -88.4 ft/s
(3) lasting 0.05 seconds: -67.2 ft/s
(4) lasting 0.01 seconds: -64.36 ft/s
(b) The instantaneous velocity when t=2 is -24 ft/s.
(a) To find the average velocity for a given time period, we need to find the displacement during that time period and divide it by the duration of the time period.
(1) For the time period lasting 0.5 seconds, the initial time is t=2 and the final time is t=2.5. The displacement during this time period is:
[tex]y(2.5) - y(2) = (402.5 - 162.5^2) - (402 - 162^2) = -20 ft[/tex]
The duration of the time period is 0.5 seconds. Therefore, the average velocity is:
average velocity = displacement / duration = -20 / 0.5 = -40 ft/s
(2) For the time period lasting 0.1 seconds, the initial time is t=2 and the final time is t=2.1. The displacement during this time period is:
[tex]y(2.1) - y(2) = (402.1 - 162.1^2) - (402 - 162^2) = -8.84 ft[/tex]
The duration of the time period is 0.1 seconds. Therefore, the average velocity is:
average velocity = displacement / duration = -8.84 / 0.1 = -88.4 ft/s
(3) For the time period lasting 0.05 seconds, the initial time is t=2 and the final time is t=2.05. The displacement during this time period is:
[tex]y(2.05) - y(2) = (402.05 - 162.05^2) - (402 - 162^2) = -3.36 ft[/tex]
The duration of the time period is 0.05 seconds. Therefore, the average velocity is:
average velocity = displacement / duration = -3.36 / 0.05 = -67.2 ft/s
(4) For the time period lasting 0.01 seconds, the initial time is t=2 and the final time is t=2.01. The displacement during this time period is:
[tex]y(2.01) - y(2) = (402.01 - 162.01^2) - (402 - 162^2) = -0.6436 ft[/tex]
The duration of the time period is 0.01 seconds. Therefore, the average velocity is:
average velocity = displacement / duration = -0.6436 / 0.01 = -64.36 ft/s
(b) To find the instantaneous velocity when t=2, we need to find the derivative of the position function y(t) and evaluate it at t=2:
[tex]y(t) = 40t - 16t^2[/tex]
y'(t) = 40 - 32t
y'(2) = 40 - 32(2) = -24 ft/s
Therefore, the instantaneous velocity when t=2 is -24 ft/s.
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Cars lose value the farther they are driven. A random sample of 11 cars for sale was taken. All 11 cars were the same make and model. a line was to fit to the data to model the relationship between how far each car had been driven and its selling price
The equation is y=-(1/4)x+40.
The price of a car that had been driven 56 thousand kilometers is $27,500
We have,
The plot is shows y represents price (thousands of dollars) and x represents kilometers driven (in thousands)
and, the y-intercept of line is (0, 40)
So, the line also pass through (20, 35), then its slope is
= (35 - 40)/(20 - 0)
= -1/4
Now, put into the equation with x = 56, we get
y=-(1/4)(50)+40
y = 27.5
and, The price of a car that had been driven 56 thousand kilometers is
= 27.5 x 1000
= $27,500
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Differentiate the function. 1 y = (7x + 3)* dy 11 dx (Simplify your answer.)
The final answer after differentiating the function is y' = 7(dy/dx) + (7x + 3)* d^2y/dx^2.
To differentiate the function y = (7x + 3)* dy/dx, we need to use the product rule of differentiation. The product rule states that the derivative of the product of two functions is equal to the first function multiplied by the derivative of the second function plus the second function multiplied by the derivative of the first function.
In this case, we have y = (7x + 3)* dy/dx, so we can apply the product rule as follows:
y' = (7x + 3)* d/dx(dy/dx) + dy/dx* d/dx(7x + 3)
The first term can be simplified by using the chain rule, which states that the derivative of a composite function is equal to the derivative of the outer function multiplied by the derivative of the inner function. In this case, the outer function is (7x + 3) and the inner function is dy/dx. So, we get:
d/dx(dy/dx) = d/dy(dy/dx)* dy/dx = d^2y/dx^2
Substituting this back into the equation, we get:
y' = (7x + 3)* d^2y/dx^2 + dy/dx* 7
Simplifying further, we get:
y' = 7(dy/dx) + (7x + 3)* d^2y/dx^2
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The thickness of a conductive coating in micrometers has a density function of 600x-2 for 100 100 μm < x < 120 μm. What is the mean of the coating thickness (in ?m). What is the variance of coating thickness (in m^2 ). If the coating costs $0.50 per ?m of thickness on each part, what is the average cost of the coating per part? (in dollars)
The density function of the conductive coating is given by 600x-2 for 100 μm < x < 120 μm. To find the mean of the coating thickness, we need to calculate the integral of the density function over the given range and divide by the range:
Mean = (1/(120-100)) * ∫100^120 (600x-2) dx
= (1/20) * [300x^2 - 2x] from 100 to 120
= 118.4 μm
To find the variance of the coating thickness, we need to calculate the integral of the squared deviation of the density function from the mean over the given range and divide by the range:
Variance = (1/(120-100)) * ∫100^120 [(x-118.4)^2 * (600x-2)] dx
= (1/20) * [1.2x^4 - 480.8x^3 + 71530.8x^2 - 4405046.4x + 86304223.8] from 100 to 120
= 29.3333 m^2
The average cost of the coating per part is given by multiplying the thickness by the cost per micrometer and taking the mean:
Average cost = $0.50 * 118.4
= $59.20 per part.
Hi! To calculate the mean, variance, and average cost of the conductive coating, we'll use the given density function, 600x-2, and the given range (100 μm < x < 120 μm).
1. Mean (μ) of the coating thickness:
Mean (μ) = ∫(x * f(x) dx) over the interval [100, 120]
2. Variance (σ²) of the coating thickness:
First, we'll need to calculate E(x²) = ∫(x² * f(x) dx) over the interval [100, 120]
Then, Variance (σ²) = E(x²) - (Mean)²
3. Average cost of the coating per part:
Average Cost = Mean (μ) * Cost per μm = Mean (μ) * $0.50
By calculating these values, you will find the mean, variance, and average cost of the conductive coating.
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I need help! +21 points
Kiki has a piece of string that she cuts into smaller pieces. This line plot shows the lengths of the pieces. Raj has a piece of string that is 1/2 as long as Kiki's third-longest piece. (Note: The problem says third-longest piece, not third-longest length.)
How long is Raj's piece of string?
Enter your answer as a mixed number in simplest form by filling in the boxes.
The length of Raj's string is 54 in long.
How to solveA dot chart or dot plot is a statistical chart consisting of data points plotted on a fairly simple scale, typically using filled in circles. There are two common, yet very different, versions of the dot chart.
Given that, Kiki has a piece of string that she cuts into smaller pieces. This line plot shows the lengths of the pieces. Raj has a piece of string that is 12 as long as Kiki's third-longest piece.
The third-longest piece = 4 1/2 in
The length of Raj's string = 4 1/2 x 12 = 9/2 x 12 = 54
Hence, the length of Raj's string is 54 in long.
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determine whether each relation r on the set of all real numbers is reflexive, symmetric, antisymmetric, or transitive. (i) r = { ( x , y ) ∈ r × r : x y = 0 } .
The relation r is symmetric and transitive, but not reflexive or antisymmetric.
Analyzing the relation r = {(x, y) ∈ ℝ × ℝ : xy = 0} with respect to the following properties:
1. Reflexive: A relation is reflexive if (x, x) ∈ r for all x ∈ ℝ. Since x * x = 0 only when x = 0, the relation is not reflexive.
2. Symmetric: A relation is symmetric if (x, y) ∈ r implies (y, x) ∈ r for all x, y ∈ ℝ. In this case, if xy = 0, then yx = 0, so the relation is symmetric.
3. Antisymmetric: A relation is antisymmetric if (x, y) ∈ r and (y, x) ∈ r imply x = y for all x, y ∈ ℝ. Since r contains non-identical pairs (x, y) with xy = 0 (e.g., (2, 0) and (0, 2)), the relation is not antisymmetric.
4. Transitive: A relation is transitive if (x, y) ∈ r and (y, z) ∈ r imply (x, z) ∈ r for all x, y, z ∈ ℝ. In this case, if xy = 0 and yz = 0, either x = 0 or y = 0, and either y = 0 or z = 0. Therefore, either x = 0 or z = 0, implying xz = 0. So, the relation is transitive.
In summary, the relation r is symmetric and transitive, but not reflexive or antisymmetric.
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Faces
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11. The following shape is made up of 6 cubes. The volume of the shape is 384 cm³. If the
shape is dipped in paint then taken apart, what is the area of the unpainted surfaces?
The area of four surfaces unpainted in the 6 cubes is 64 cm².
We have,
The volume of the shape = 384 cm³
Number of cubes = 6
This means,
Area of one cube.
= 384/6
= 64 cm³
Now,
Area of cube = side³
So,
side³ = 64
side³ = 4³
side = 4
Now,
There are four surfaces unpainted.
so,
One surface is in the shape of a rectangle.
This means,
One surface area = 4 x 4 = 16 cm²
Now,
Area of four surfaces unpainted.
= 4 x 16
= 64 cm²
Thus,
The area of four surfaces unpainted is 64 cm².
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is this 11 yards or 8 yards?
A cylinder with a diameter of 8 yards has a volume of 552.64 yd3. What is the height of the cylinder? Use 3.14 for π.
44 yards
11 yards
8 yards
3 yards
Answer:
11 yards.
Step-by-step explanation:
A cylinder's volume is π r² h
Where π = 3.14
r = 4 (since radius is half of diameter)
v (volume) = 552.64
So in this case we are solving for h, height.
So rewrite:
552.64 = (3.14)(4)^2(h)
So now we solve:
1. Evaluate exponent:
552.64 = (3.14)(16)(h)
2. Multiply
552.64 = 50.24h
3. Divide to get h by itself
552.64 / 50.24 = 50.24 / 50.24 (h)
11 = h
Hence the height of the cylinder is 11 yards.
Also I’ve attached below pictures to help further prove. The 10.99 is rounded up to 11.
20 POINTS!!Which coordinate plane shows the graph of the function displayed in the input/output table??
x y
0 1
1 2
2 3
3 4
The coordinate plane that shows the graph of the function is the first graph in the second attachment
Which coordinate plane shows the graph of the functionFrom the question, we have the following parameters that can be used in our computation:
x y
0 1
1 2
2 3
3 4
From the above, we can see that
The x value is added to 1 to get the y value
This means that
The input value is added to 1 to get the output value
So, we have
y = x + 1
From the list of options, the graph that represent the relation is the second graph (first in the second attachment)
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