There is a function f(t) which is given by:
f(t) = sin(t/T) for 0 ≤ t ≤ 2πT and
f(t) = 0 for 2πT ≤ t
This function repeats periodically outside the interval [0,T] with period T (assuming that 2πT
a) what are the restrictions that would be expected for the Fourier coefficient a_j. Which Fourier coefficient is expected to be the largest?
b) Calculate the Fourier expansion , thus verifying the prediction .
Answers
a) The largest Fourier coefficient is a_1.
b) The final answer is:f(t) = (2/π) [sin(t/T) - (1/3) sin(3t/T) + (1/5) sin(5t/T) - ...]
a) Restrictions for Fourier coefficient a_j
The Fourier coefficients for odd functions are odd and for even functions, the Fourier coefficients are even. This function is odd, so a_0 is equal to zero. This is due to the function being odd about the origin. Hence, only odd coefficients exist.
For the given function f(t), f(t) is continuous, and hence a_0 is equal to 0. So, the restrictions on the Fourier coefficient a_j are:
For j even, a(j) = 0, For j odd, a(j) = (2/T)
= ∫[0,T] sin(t/T) sin(jπt/T) dt = (2/T)
= ∫[0,T] sin(t/T) sin(jt) dt.
The largest Fourier coefficient is the one with the highest value of j. Hence, for this function, the largest Fourier coefficient is a_1.
b) Calculating the Fourier expansion using the Fourier series
We know that the Fourier coefficients for odd functions are odd, and for even functions, the Fourier coefficients are even. This function is odd, so a_0 is equal to zero. Thus, the Fourier expansion of the given function is:
f(t) = Σ[1,∞] a_j sin(jt/T), where a_j = (2/T)
= ∫[0, T] sin(t/T) sin(jt) dt
= (2/T) ∫[0, T] sin(t/T) sin(jπt/T) dt,
since j is odd.
Now, let us evaluate the integral using integration by parts by assuming u = sin(t/T) and v' = sin(jπt/T).
Then we get the following: du = (1/T) cos(t/T) dt
dv' = (jπ/T) cos(jπt/T) dt
Integrating by parts, we have: a(j) = [2/T]
(uv)|_[0,T] - [2/T]
∫[0,T] u' v dt = [(2/T) (cos(Tjπ) - 1) sin(T/T) + jπ(2/T) ∫[0,T] cos(t/T) cos(jπt/T) dt]/jπ
Using the trigonometric identity, cos(A) cos(B) = 0.5 (cos(A-B) + cos(A+B)), we have:
a(j) = [(2/T) (cos(Tjπ) - 1) sin(T/T) + jπ(2/T) ∫[0, T] cos((jπ-Tπ)t/T)/2 + cos((jπ+Tπ)t/T)/2 dt]/jπ
= [(2/T) (cos(Tjπ) - 1) sin(T/T) + (2/T) sin(jπ)/2 + (2/T) sin(jπ)/2]/jπ,
since the integral is zero (because cos((jπ-Tπ)t/T) and cos((jπ+Tπ)t/T) are periodic with period 2T).
Thus, we get the following expression for a(j): a(j) = [(2/T) (cos(Tjπ) - 1) sin(T/T)]/jπ.
So, the Fourier series expansion of the given function is f(t) = Σ[1,∞] [(2/T) (cos(Tjπ) - 1) sin(T/T)] sin(jt/T) / jπ.
Hence, the final answer is:f(t) = (2/π) [sin(t/T) - (1/3) sin(3t/T) + (1/5) sin(5t/T) - ...]
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Related Questions
Help please. Is this a function?
Answers
Answer:
yes, this is a function
I WILL GIVE 50 POINTS EACH TO WHOEVER SOLVES THIS!!!
Write each of the following expressions as a powers of 2: 4^−6⋅ 4^4⋅ (2^3⋅ 2^−4)^−1
Answers
Answer:
2^-3Step-by-step explanation:
Solving in steps
Using exponent rules: a^b·a^c = a^(b+c) and (a^b)^c = a^(bc)
4^−6 ⋅ 4^4⋅ (2^3⋅ 2^−4)^−1 = 4^(-6 + 4) ⋅ (2^(3 - 4))^-1 =4^-2 ⋅ (2^(-1))^-1 =(2^2)^-2 ⋅ 2^((-1) ⋅ (-1)) =2^-4 ⋅ 2^1 = 2^(-4 + 1) =2^-3Answer:
2 ⁻³Step-by-step explanation:
the expression 4^−6⋅ 4^4⋅ (2^3⋅ 2^−4)^−1 can be solved using exponential rule:
= 4⁻⁶ * 4⁴ * (2³ * 2⁻⁴)⁻¹
= 4⁽⁻⁶ ⁺ ⁴⁾ * (2 ⁽³ ⁻ ⁴⁾ ) ⁻¹
= 4⁻² * (2⁻¹)⁻¹
= (2²) ⁻² * 2 ⁽⁻¹ ˣ ⁻¹⁾
= 2⁻⁴ * 2¹
= 2 ⁽⁻⁴ ⁺ ¹⁾
= 2 ⁻³
Draw a sketch of y = x2 - x - 3for values of x in the domain -3 <=x<= 3. Write down the coordinates of the turning point in your solution. Hence, from your sketch, find approximate solutions to:x2 – X – 3 = 0.
Answers
The sketch of the function y = \(x^{2}\) - x - 3 for -3 <= x <= 3 reveals a parabolic curve that opens upwards. The turning point of the parabola, also known as the vertex, can be identified as (-0.5, -3.25).
To sketch the graph of y = \(x^{2}\) - x - 3, we consider the given domain of -3 <= x <= 3. The function represents a parabola that opens upwards. By calculating the coordinates of the turning point, we can locate the vertex of the parabola.
To find the x-coordinate of the turning point, we use the formula x = -b/2a, where a and b are the coefficients of the quadratic equation. In this case, a = 1 and b = -1. Substituting these values, we have x = -(-1)/2(1) = -0.5.
To find the y-coordinate of the turning point, we substitute the x-coordinate (-0.5) into the equation y = \(x^{2}\) - x - 3. Evaluating this expression, we get y = \(-0.5^{2}\) - (-0.5) - 3 = -3.25.
Therefore, the turning point of the parabola is approximately (-0.5, -3.25).
From the sketch, we can estimate the approximate solutions to the equation \(x^{2}\)- x - 3 = 0 by identifying the x-values where the graph intersects the x-axis. These solutions are approximately x ≈ -2.5 and x ≈ 1.5.
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Rushing to get this done.
Answers
Answer:
180m²
Step-by-step explanation:
To get the height we first have to use Pythagoras theorem.
a²+b²=c²
a² + 9² = 41²
a = 40 which is the height
now we have to find the area of this triangle.
1/2(b)(h)
1/2(9)(40)
area = 180m²
Which ordered pair represents the solution to the system of linear equations shown above? *
(0, 2)
(5,0)
O (2,3)
O (3, 2)
Answers
Answer:
(2,3)
Step-by-step explanation:
-x +5 =1/2x +2
-x - 1/2x = 2-5
-3/2x = - 3
-3/2x ÷ - 3/2 = - 3 ÷ -3/2
x= 2
Substitute x=2 in any equation
y= -(2) + 5
y= 3
Points are (2,3)
in their research study of measuring the correlation between two variables, students of ace college found a nearly perfect positive correlation between the variables. what coefficient of correlation did they arrive at?
Answers
The students of Ace College found a nearly perfect positive correlation between two variables in their research study. The nearly perfect positive correlation suggests that the two variables are closely related and move in sync with each other.
In their research study, the students of Ace College discovered a nearly perfect positive correlation between the two variables they were investigating. The coefficient of correlation they arrived at is known as the Pearson correlation coefficient, which measures the strength and direction of the linear relationship between two variables.
The Pearson correlation coefficient ranges from -1 to +1, where -1 represents a perfect negative correlation, +1 represents a perfect positive correlation, and 0 represents no correlation. Since the students found a nearly perfect positive correlation, the coefficient of correlation would be close to +1.
This indicates a strong and direct relationship between the variables, meaning that as one variable increases, the other variable also tends to increase consistently. The nearly perfect positive correlation suggests that the two variables are closely related and move in sync with each other.
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Helpppppppppppppppp?
Answers
start with 18 multiplied by 16 which is 288
then i believe other side length next to the 6 might be 2
so that would mean you do 6 multiplied by 12 and you subtract that fthe 288
so im pretty sure the answer is 276, but im not entirely sure
pls help will give brainliest!
Derek just bought a bag of his favorite candy, Starbursts. In the bag there were 3 Strawberry, 5 Lemon, 4 Orange, and 3 Cherry.
He reaches in randomly and selects a starburst, eats it, then reaches in and selects another. What is the probability that both the selected Starbursts are Cherry?
Show or explain your work.
Answers
Answer:
To find the probability that both selected Starbursts are Cherry, we need to determine the total number of Starbursts in the bag and the number of Cherry Starbursts.
The total number of Starbursts in the bag is 3 + 5 + 4 + 3 = 15.
The number of Cherry Starbursts in the bag is 3.
When Derek selects the first Starburst, there is a 3/15 probability that it will be Cherry. After eating the first Starburst, there are 14 remaining Starbursts in the bag, with 2 of them being Cherry.
Therefore, the probability of selecting a second Cherry Starburst is 2/14.
To find the probability of both events happening (selecting a Cherry Starburst twice in a row), we multiply the probabilities:
(3/15) * (2/14) = 6/210 = 1/35.
Therefore, the probability that both selected Starbursts are Cherry is 1/35.
6-27: On graph to the right, graph each line in the system shown belowWhat is the point of intersection ? Does more one exist? How do you know for sure? y = - x + 2; y = 3x + 6
Answers
The point of intersection for the two lines \(y = -x + 2\) and \(y = 3x + 6\) is \((2,4).\)
To find the point of intersection, you can set the two equations equal to each other and solve for x and y.
\(y = -x + 2\)
\(y = 3x + 6\)
\(-x + 2 = 3x + 6\)
\(-4x = 4\)
\(x = -1\)
then substitute \(x = -1\) in any of the equation
\(y = -x + 2\)
\(y = -(-1) + 2\)
\(y = 1 + 2\)
\(y = 3\)
So the point of intersection is \((-1,3)\).
It is possible that there may be more than one point of intersection, but in this case, there is only one point of intersection.
To be sure that there is only one point of intersection, you could graph the two lines and visually check if they intersect at only one point.
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A different toy store marks down all
of their toys by 25% in January.
How much does a $50 remote
controlled car cost during January's
sale?
Answers
س 17 / 2.5 درجة For reduce the answer to decimals with 4 digits, we use
Answers
To reduce the answer to decimals with four digits, we use both the `format function` and the `round function` in Python. In the given expression س 17 / 2.5 درجة, we need to divide 17 by 2.5 and then reduce it to decimals with four digits. The steps below are used in Python to reduce the answer to decimals with four digits. Step 1: We start by evaluating the expression using Python. For this, we use the division operator `/`. س 17 / 2.5 درجة evaluates to 6.8, which is the division of 17 by 2.5. To assign this value to a variable, we can write the following code: `result = 17 / 2.5`Step 2: We then use the `format function` to reduce the number of decimal places in the result to four. The `format function` is used as follows:'result = "{:.4f}".format(result)`The `:.4f` part in the above code specifies that we want to format the result with four decimal places. Step 3: Finally, we use the `round function` to ensure that the last digit in the result is rounded up if it is greater than or equal to five. The `round function` is used as follows:'result = round(float(result), 4)`The second argument of the `round function` specifies that we want to round the result to four decimal places. The `float` function is used to convert the result from a string to a float data type. Therefore, the answer to the given question "س 17 / 2.5 درجة To reduce the answer to decimals with 4 digits, we use" "Both the format function and the round function".
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Mario plays basketball on a town league team. The
table below gives Mario's scoring statistics for last
season. How many points did Mario score playing
basketball last season?
Answers
Answer:
Step-by-step explanation:
Total points Mario scored = total 1-point free throw + total 2-point field goal + total 3-point field goal.
In 80 attempts 1-point free throw, Mario had 75% success rate. That is, Mario had 80 x 75/100 success
80 x 75/100 = 60
Thus, Mario scored 1 x 60 (since for each success, 1 points is added) points = 60
In 60 attempts 2-point field goal, Mario had 90% success rate. That is, Mario had 60 x 90/100 success
60 x 90/100 = 54
Thus, Mario scored 2 x 54 points (since for each success, 2 points are added) = 108
In 60 attempts 3-point field goal, Mario had 25% success rate. That is, Mario had 60 x 25/100 success
60 x 25/100 = 15
Thus, Mario scored 3 x 15 points (since for each success, 3 points are added) = 45
Therefore, the total points Mario scored = 60 + 108 + 45
= 213
Natasha has a job putting letters in envelopes to be mailed. in 1 hour she put 42 letters in envelopes. which equation best represents y, the total number of letters natasha puts in envelops in x hours if she continues
Answers
Answer:
36 pages .394kb
Step-by-step explanation:
that's it well done
You can use multiplication along with converting statements to symbols.
The equation for the given condition is given by \(y = 42x\)
where y represents the total number of letters Natasha puts in envelops in x hours.
How to form symbolic expressions from the description?You need to know how to convert the statement to symbolic expression.
Suppose if it is said that a thing is increased by 4,then we write it as
that thing + 4
Suppose if it is written that 'x' objects are sold on p price per object. Since 1 object is sold at p price, then x objects will be sold at:
\(p + p + ... + p \: \: \: (x \: \: times) = p \times x\)
Such conversion can help us to form symbolic expressions from the given description.
Since for the given case, in 1 hour, Natasha put 42 letters in envelopes, then in x hour she will put:
\(42 + 42 + ... + 42 \: \: (x \: times) = 42 \times x\)
Since in x hours, she puts y envelopes but envelopes put in x hours came out as 42 times x, thus, both the values are equal, or
\(y = 42 \times x = 42x \text{\:\:(written in short)}\)
The equation for the given condition is given by \(y = 42x\)
where y represents the total number of letters Natasha puts in envelops in x hours.
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Which quadrant is the following point in?
Point A (1,3)
Answers
Answer:
Its first quadrant bro.
I don’t know how to solve this
Answers
The solution of the given system of equations include the following:
C. a = -3 and b = 6.
How to solve the given system of equations?In order to solve the given system of equations, we would apply the substitution method. For instance, we have the following system of equations:
2a + 3b = 12 .......equation 1.
a = 1/2(b) - 6 .......equation 2.
By using the substitution method to substitute equation 2 into equation 1, we have the following:
2(b/2 - 6) + 3b = 12
b - 12 + 3b = 12
4b = 12 + 12
4b = 24
b = 24/4 = 6.
For the value of a, we have:
a = 1/2(b) - 6
a = 1/2(6) - 6
a = 3 - 6
a = -3.
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suppose that f(0)=−3 and f′(x)≤8 for all values of x. use the mean value theorem to determine how large f(4) can possibly be. answer: f(4)≤
Answers
The largest value that f(4) can possibly be is 29.
The mean value theorem states that for a function f(x) that is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), there exists a number c in the open interval (a, b) such that:
f(b) - f(a) = f'(c)(b - a)
In this case, we are given that f(0) = -3 and that f'(x) ≤ 8 for all values of x. To determine how large f(4) can possibly be, we can use the mean value theorem with a = 0 and b = 4:
f(4) - f(0) = f'(c)(4 - 0)
Substituting the given values:
f(4) - (-3) = f'(c)(4)
f(4) + 3 = 4f'(c)
Since f'(x) ≤ 8 for all values of x, we can say that f'(c) ≤ 8. Therefore:
f(4) + 3 ≤ 4f'(c) ≤ 4(8) = 32
Therefore, we have:
f(4) ≤ 32 - 3 = 29
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Jesse Deposits $3,500 in an account at Green Market Group that earns 5% simple interest. Find the amount of interest earned after 15 years.
Answers
The amount of interest earned after 15 years at rate of 5% on $3,500 amount is $6125.
What is simple interest?Simple interest is a method to calculate the amount of interest which is applied on the principal amount, rate for a given time intereval.
SI = P × R × T/100
The principal amount is given as $3,500, rate is 5% and time period is 15 years.
SI = P × R × T/100
SI = $3,500 × 5 × 15 /100
SI = $2625
Since, the total value of the amount of interest = $2625 + $3,500
= $6125
Therefore, the amount of interest earned after 15 years at rate of 5% on $3,500 amount is $6125.
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Answer:
$2,625
Step-by-step explanation:
I=prt
3,500(.05)(15)
2,625+3,500=Account balance ($6,125).
20% of $80 and add it to the original amount!
Answers
Answer:
96
Step-by-step explanation:
20 percent of 80 is 16, and added to 80 would equal 96
it says to find the rule on this question but I need help finding the rule.
Answers
y = 2x - 6
n = 22
Explanations:By careful observation of the data in the table, the table has a constant rate of change, that is, a constant slope, hence it represents a staight line
The slope is calculated using the formula:
\(m\text{ = }\frac{y_2-y_1}{x_2-x_1}\)In the table:
x₁ = 8, y₁ = 10, x₂ = 9, y₂ = 12
\(\begin{gathered} m\text{ = }\frac{12-10}{9-8} \\ m\text{ = }\frac{2}{1} \\ m\text{ =2 } \end{gathered}\)Since we have confirmed that the table shows a linear equation, the equation of a line shown below can be used to form the relationship
\(y-y_1=m(x-x_1)\)\(\begin{gathered} y\text{ - 10 = 2(x - 8)} \\ y\text{ - 10 = 2x - 16} \\ y\text{ = 2x - 16 + 10} \\ y\text{ = 2x - 6} \end{gathered}\)The rule is therefore y = 2x - 6
To find the variable n, substitute y = n, and x = 14 into the equation y = 2x - 6
n = 2(14) - 6
n = 28 - 6
n = 22
a laboratory worker finds that 3% of his blood samples test positive for the hiv virus. in a random sample of 70 blood tests, what is the mean number that test positive for the hiv virus? (round your answer to 1 decimal place)
Answers
In linear equation ,The mean number that the test positive for the HIV virus is 2.1 .
What in mathematics is a linear equation?
A linear equation is a first-order (linear) term plus a constant in the algebraic form y=mx+b, where m is the slope and b is the y-intercept. Sometimes, the aforementioned is referred to as a "linear equation of two variables," where x and y are the variables.Let x be the number of blood samples test positive for the HIV virus in a sample of 70 blood test.
Here X ~ Binom( n = 70, p = 0.03)
Using binomial distribution property , the mean of X is
μ = ηp
= 70 * 0.03 = 2.1
The mean number that the test positive for the HIV virus is 2.1 .
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Identify the national road that follows the coastline on the eastern side of the south Africa?
Answers
The name of the national road that follows the eastern coastline of South Africa is the N2.
Which road follows South Africa's eastern coastline?The N2 is a national road in South Africa that starts at Cape Town and goes towards the eastern part of South Africa.
It passes though Port Elizabeth, East London, Durban, and Richard's Bay which are all cities on the eastern side of South Africa and are along the coastline.
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In this diagram, ZEDH is a right angle, and ZEDF LFDG.
E
V
32°
D
H
What is the measure of ZEDF?
d
A
26°
B
28°
C
29°
D
32°
Answers
Answer: If you measure the circumference and the diameter of a circle and divide the circumference by the diameter, you get the number Pi, which is about 3.14.
Pi is often shown by the Greek letter π .
The ancient Greeks used the symbol π to indicate the ratio between the circumference and diameter of a circle. If your calculator does not have the symbol π , you can use the rounded off number 3.14 for π .
A = 5m × 5m × 3.14 78.5m2 (Note: the symbol means about the same as).
In the diagram above, the area of the small yellow square is a quarter of the area of the big square.
If the area of the shaded square is A = 5m × 5m = 25m2
then the area of the big square is A = 4 × 25m2 =100m2
You can see that the area of the circle is less than that of the square since the circle fits inside it. You can also find the area of the circle by multiplying the area of the small square by Pi:
A = 3.14 × 25m2 =78.5m2
Which of the following are rational numbers?
5.48
78.541
97
42.111...
Answers
Answer:
5.48
78.541
97
42.111...
(all of above)
Step-by-step explanation:
because a rational number is a number that can be written as a fraction
a decimal number that ends can be written as a fraction (ex: 5/4 = 1.25)
a decimal number that has a repeating number after the decimal point can also be written as a fraction (ex: 10/3 = 3.33...)
any
help is much appreciated!!
Problem. 8: Use the given transformation z = 7u+v, y=u+7u to set up the integral. dA, where R is the triangular region with vertices (0,0), (7, 1), and (1,7). 0 0 1-u R ? du du
Answers
The integral dA, where R is the triangular region with vertices (0,0), (7, 1), and (1,7), can be set up using the given transformation z = 7u+v and y = u+7u.
To set up the integral, we need to express the differential area element dA in terms of the variables u and v. Since the transformation relates z and y to u and v, we can express dA as a product of the absolute value of the determinant of the Jacobian matrix.
The Jacobian matrix J of the transformation is given by:
J = |∂z/∂u ∂z/∂v|
|∂y/∂u ∂y/∂v|
Taking the partial derivatives, we have ∂z/∂u = 7 and ∂z/∂v = 1, and ∂y/∂u = 1 and ∂y/∂v = 7.
The determinant of the Jacobian matrix is |J| = (∂z/∂u)(∂y/∂v) - (∂z/∂v)(∂y/∂u) = 7(7) - 1(1) = 48.
Therefore, the differential area element dA can be expressed as dA = |J| du dv = 48 du dv.
Now, we need to express the limits of integration in terms of u and v. Since R is a triangular region with vertices (0,0), (7, 1), and (1,7), we can set the limits as follows:
For u, the lower limit is 0 and the upper limit is 1.
For v, the lower limit is 0 and the upper limit is 7u.
Therefore, the integral becomes:
∫∫R dA = ∫[0,1] ∫[0,7u] 48 du dv.
This integral represents the calculation of the area of the triangular region R using the given transformation
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four cubes of volumes $1 \text{ cm}^3$, $8 \text{ cm}^3$, $27 \text{ cm}^3$, and $125 \text{ cm}^3$ are glued together at their faces. what is the number of square centimeters in the smallest possible surface area of the resulting solid figure?
Answers
Answer: 194
Step-by-step explanation:
From the volumes, we deduce that the side lengths of the cubes are 1 cm, 2 cm, 3 cm, and 5 cm. We position the cubes as follows:
[asy]
unitsize(0.5 cm);
draw((0,0)--(5*dir(-30))--(5*dir(-30) + 5*dir(30))--(10*dir(-30))--(5*dir(-30) + 5*dir(-90))--(5*dir(-90))--(0,0));
draw((5*dir(-30))--(5*dir(-30) + 5*dir(-90)));
draw((0,0)--(0,2)--((0,2) + 2*dir(-30))--(2*dir(-30)));
draw((0,2)--((0,2) + 2*dir(30))--((0,2) + 2*dir(30) + 2*dir(-30))--(2*dir(30)));
draw((2*dir(-30))--(2*dir(-30) + dir(30))--(2*dir(-30) + dir(30) + (0,1))--(2*dir(-30) + 2*dir(30) + (0,1))--(2*dir(-30) + 2*dir(30) + (0,2)));
draw((2*dir(-30) + dir(30))--(3*dir(-30) + dir(30))--(3*dir(-30) + dir(30) + (0,1))--(2*dir(-30) + dir(30) + (0,1)));
draw((3*dir(-30) + dir(30) + (0,1))--(3*dir(-30) + 2*dir(30) + (0,1))--(2*dir(-30) + 2*dir(30) + (0,1)));
draw((2*dir(30) + (0,2))--(2*dir(30) + (0,3))--(2*dir(30) + 3*dir(-30) + (0,3))--(2*dir(30) + 3*dir(-30))--(dir(30) + 3*dir(-30)));
draw((2*dir(30) + (0,3))--(5*dir(30) + (0,3))--(5*dir(30) + 3*dir(-30) + (0,3))--(5*dir(30) + 3*dir(-30))--(5*dir(30) + 5*dir(-30)));
draw((3*dir(-30) + 2*dir(30))--(3*dir(-30) + 5*dir(30)));
draw((3*dir(-30) + 2*dir(30) + (0,3))--(3*dir(-30) + 5*dir(30) + (0,3)));
[/asy]
The surface area of a cube with side length $s$ is $6s^2$, so the total surface area of the cubes is $6 \cdot 1^2 + 6 \cdot 2^2 + 6 \cdot 3^2 + 6 \cdot 5^2 = 234$.
Note that every pair of cubes touches, and furthermore, they have maximum contact. (This is why this solid has the smallest possible area.) The area of contact of the 1-cube and the 2-cube is 1 square centimeter, so we must subtract this twice from 234 (because this portion of the area from both the 1-cube and 2-cube is not seen anymore).
Doing this for every pair of cubes, we find that the surface area of this solid is $234 - 2 \cdot 1^2 - 2 \cdot 1^2 - 2 \cdot 1^2 - 2 \cdot 2^2 - 2 \cdot 2^2 - 2 \cdot 3^2 = \boxed{194}$.
A person is standing 1 foot below sea level. Which integer would best represent that elevation on a number line? –2 -1 1 0
Answers
Answer:
-1 feet
Step-by-step explanation:
below means that your answer will be a negative because sea level is at 0 how many feet below 1 so your answer is -1
From previous experience, the owner of an apple orchard knows that the mean weight of Gala apples is 140 grams. There has been more precipitation than usual this year, and the owner believes the weights of the apples will be heavier than usual. The owner takes a random sample of 30 apples and records their weights. The mean weight of the sample is 144 grams with a standard deviation of 13.2 grams. A significance test at an alpha level of produces a P-value of 0.054. What is the correct interpretation of the P-value
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In statistical hypothesis testing, the P-value is a significant factor. It is the probability of obtaining a test statistic at least as extreme as the one calculated from the data, assuming the null hypothesis to be true. If the null hypothesis is false, the P-value is the probability of a type I error. It is the probability of rejecting the null hypothesis when it is true.
To interpret the P-value correctly, a P-value of 0.054 means that if the null hypothesis is correct, there is a 5.4% probability that the sample will produce a test statistic as extreme as, or more extreme than the one that was observed. If the calculated P-value is higher than the significance level, which is usually 0.05 or 0.01, we cannot reject the null hypothesis.
In the given situation, the sample provides insufficient evidence to reject the owner's claim that the mean weight of Gala apples this year is heavier than usual because the calculated P-value is higher than the significance level. Hence, the correct option is that the P-value suggests that there is not sufficient evidence to reject the null hypothesis.
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Which property of multiplication is shown below?
If x = a + bi and y = c + di, x × y = y × x.
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The commutative property of multiplication is as shown x × y = y × x .
The given values of x and y are:
x = a + bi and y = c + di
By the commutative property which states that it is a binary operation in which ab=ba whenever you change the order of the operands.
Therefore,
x = a + bi and y = c + di
Now, x × y = (a + bi) × (c + di)
= ac + adi + bci + bdi²
= ac - bd + i(ad + bc)
Now, y × x = (c+ di) × (a + bi)
= ac + bci + adi + bdi²
= ac - bd + i(ad + bc)
Therefore, x × y = y × x
and when ab = ba then the property is known as commutative property.
Hence commutative property of multiplication is as shown above.
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two cards are selected in a sequence from a standard deck. what is the probability that the second card is a jack given that the first card was a 2. (assume the 2 was not replaced.)
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The probability that the second card is a jack given that the first card was a 2 is 52/51.
To calculate the probability that the second card is a jack given that the first card was a 2, we need to consider the remaining cards in the deck after the first card is drawn.
When the first card is drawn and it is a 2, there are 51 cards remaining in the deck, out of which there are 4 jacks.
The probability of drawing a jack as the second card, given that the first card was a 2, can be calculated using conditional probability:
P(Second card is a jack | First card is a 2) = P(Second card is a jack and First card is a 2) / P(First card is a 2)
Since the first card is already known to be a 2, the probability of the second card being a jack and the first card being a 2 is simply the probability of drawing a jack from the remaining 51 cards, which is 4/51.
The probability of the first card being a 2 is simply the probability of drawing a 2 from the initial deck, which is 4/52.
P(Second card is a jack | First card is a 2) = (4/51) / (4/52)
Simplifying the expression:
P(Second card is a jack | First card is a 2) = (4/51) * (52/4)
P(Second card is a jack | First card is a 2) = 52/51
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ten days after it was launched toward mars in december 1998, the mars cli- mate orbiter spacecraft (mass 629 kg) was 2.87 x 106km from the earth and traveling at 1.20 x 104km/h relative to the earth
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The kinetic energy of the Mars Climate Orbiter spacecraft is approx 3.31 x 10^7 joules.
To determine the kinetic energy of the Mars Climate Orbiter spacecraft, we can use the formula:
Kinetic energy = (1/2) * mass * velocity^2
Given:
Mass of the spacecraft (m) = 629 kg
Velocity of the spacecraft (v) = 1.20 x 10^4 km/h
First, we need to convert the velocity from km/h to m/s:
1 km = 1000 m
1 h = 3600 s
Velocity in m/s = (1.20 x 10^4 km/h) * (1000 m/km) / (3600 s/h) ≈ 333.33 m/s
Now, we can calculate the kinetic energy:
Kinetic energy = (1/2) * (629 kg) * (333.33 m/s)^2
Kinetic energy ≈ 3.31 x 10^7 joules
Therefore, the kinetic energy of the Mars Climate Orbiter spacecraft is approximately 3.31 x 10^7 joules.
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