A force of 3i -2j+4k displace an object from a point ( 1, 1, 1) to another (2, 0, 3) the work done by force is
A. 10
B. 12
C. 13
D. 29
E. non of these

To find the maximum likelihood estimator for theta, we need to first find the likelihood function by taking the product of the density function for each observation. Assuming we have n observations, the likelihood function is given by:

L(theta) = (1/theta^2) * Π[i=1 to n] (xi * e^(-xi/theta))

Taking the logarithm of the likelihood function and simplifying it, we get:

ln(L(theta)) = -2ln(theta) + Σ[i=1 to n] ln(xi) - Σ[i=1 to n] (xi/theta)

To find the maximum likelihood estimator for theta, we need to differentiate ln(L(theta)) with respect to theta and set it equal to zero. Solving for theta, we get:

θ = Σ[i=1 to n] xi / n

Therefore, the maximum likelihood estimator for theta is the sample mean of the n observations.

It is important to note that this estimator is unbiased and efficient, meaning that it has the smallest possible variance among all unbiased estimators. This makes it a desirable estimator for practical applications.

In conclusion, the maximum likelihood estimator for theta in the given probability density function is the sample mean of the n observations.

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Sophie needs approximately 14.82 ounces of flour to bake her cake .

To convert grams to ounces, we can use the conversion factor that 1 ounce is approximately equal to 28.35 grams . The mass m in grams (g) is equal to the mass m in ounces (oz) times 28.34952

1 ounces = 28.35 gram

So, to find the number of ounces of flour Sophie needs, we can divide the weight in grams by the conversion factor .

420 g × 1 ounces / 28.35 g

420 g / 28.35 g = 14.82 ounces

Therefore, Sophie needs approximately 14.82 ounces of flour to bake her cake .

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The step necessary for incorporating randomization into a simulation is to use a chance device.

A chance device, such as a random number generator, is necessary to introduce randomness into the simulation. This allows the simulation to generate random outcomes that reflect the uncertainty and variability of real-world situations. By incorporating randomization, the simulation can simulate a range of possible outcomes and estimate the probabilities of different outcomes occurring.

Creating a data table and running the simulation more than once are important steps in a simulation, but they are not specifically related to incorporating randomization. Writing an alternative hypothesis is also not related to incorporating randomization into a simulation, but rather a step in the hypothesis testing process.

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The fact that f(9) = 42.5 tells us that the numerator of the function, x^2, evaluated at x = 9 is equal to 42.5.

In the given function f(x) = x^2 / (4x - 7),

evaluating it at x = 9 yields f(9) = 9^2 / (4(9) - 7) = 81 / 29 ≈ 2.7931.

Since the numerator of the function is x^2, the fact that f(9) = 42.5 indicates that the numerator x^2 evaluated at x = 9 is equal to 42.5.

In this case, it means that 9^2 = 81 is equal to 42.5, which is not true. Therefore, there seems to be an error or inconsistency in the given information or calculation.

The numerator x^2 should evaluate to 81, not 42.5.

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The limit as a definite integral on the given interval is lim n→∞ nΣi=1n exi* Δxi = ∫0⁹ ex dx = e⁹ - 1.

To express the limit as a definite integral on the given interval, use the definition of a Riemann sum:

lim n→∞ Σi=1n f(xi*) Δxi = ∫aᵇ f(x) dx

where f(x) = ex, a = 0, b = 9, and Δx = (b - a)/n = 9/n. Also, xi* = point in the i-th subinterval [xi-1, xi], where xi = a + iΔx.

Substituting the values:

lim n→∞ Σi=1n exi* Δxi = ∫0⁹ ex dx

Integrating:

lim n→∞ Σi=1n exi* Δxi = [ex]0⁹ = e⁹ - 1

Therefore, the limit as a definite integral on the given interval is:

lim n→∞ nΣi=1n exi* Δxi = ∫0⁹ ex dx = e⁹ - 1

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Therefore, the population will exceed 10,000 cells in approximately 43.1 hours.

We have a function of the form: p(t) = P0(1.13)^t

The function shows that the population of bacteria grows exponentially over time.

Here, we have to find the time it takes for the population to exceed 10,000 cells given that the initial population is 500 cells. To find this, we need to use the following formula:

p(t) = P0(1.13)^t  ≥  10,000 cells

P0 = 500 cells

Putting the values in the formula, we get:10,000 cells = 500 cells (1.13)^tt = ln(10,000/500) / ln(1.13)t = 43.09 hours.

It will take the population approximately 43.1 hours to exceed 10,000 cells.

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The extreme values of f(x,y) can be expressed in terms of the marginal cumulative distribution functions f_x(x) and f_y(y) using the formulas above.

To express the extreme values of f(x,y) in terms of the marginal cumulative distribution functions f_x(x) and f_y(y), we can use the following formulas:

f(x,y) = (d^2/dx dy) F(x,y)

where F(x,y) is the joint cumulative distribution function of X and Y, and

f_x(x) = d/dx F(x,y)

and

f_y(y) = d/dy F(x,y)

are the marginal cumulative distribution functions of X and Y, respectively.

To find the maximum value of f(x,y), we can differentiate f(x,y) with respect to x and y and set the resulting expressions equal to zero. This will give us the critical points of f(x,y), and we can then evaluate f(x,y) at these points to find the maximum value.

To find the minimum value of f(x,y), we can use a similar approach, but instead of setting the derivatives of f(x,y) equal to zero, we can find the minimum value by evaluating f(x,y) at the corners of the rectangular region defined by the range of X and Y.

Therefore, the extreme values of f(x,y) can be expressed in terms of the marginal cumulative distribution functions f_x(x) and f_y(y) using the formulas above.

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the right answer on this question is 7,9

Thus, list the elements in the set (d ∪ e) ∩ F is {4, 6, 7, 9}.

To find the elements in the set (d ∪ e) ∩ F, we first need to determine what the union of d and e is.

Given that:

d={4,7,9}, e={4,6,7,8} and f={3,5,6,7,9}.

The union of two sets, denoted by the symbol ∪, is the set of all elements that are in either one or both of the sets.

So, in this case, d ∪ e would be the set {4, 6, 7, 8, 9}.

Next, we need to find the intersection of the set {4, 6, 7, 8, 9} and f.

The intersection of two sets, denoted by the symbol ∩, is the set of all elements that are in both sets.

So, the elements in the set (d ∪ e) ∩ F would be the elements that are common to both {4, 6, 7, 8, 9} and {3, 5, 6, 7, 9}. These elements are 4, 6, 7, and 9.

Therefore, the answer to the question is (d ∪ e) ∩ F = {4, 6, 7, 9}.

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According to given fractions, No, this situation cannot be modeled as 1/2 - 1/3.

To model Kara's situation, we need to start with her total allowance. Let's say she started with $X.

On Saturday, she spent half of her allowance, or 1/2X.

After Saturday, she had 1/2X left.

On Sunday, she spent 1/3 of what she had left, or 1/3(1/2X) = 1/6X.

So her total spending can be modeled as 1/2X + 1/6X = 2/3X.

Therefore, the correct model for Kara's situation is 2/3X, not 1/2 - 1/3.
Hi! The situation where Kara spent ½ of her allowance on Saturday and 1/3 of what she had left on Sunday cannot be modeled as ½ - 1/3. Here's why:

1. On Saturday, Kara spent ½ of her allowance. Let's assume her total allowance is A. So, she spent ½A on Saturday.
2. After spending ½A on Saturday, she has (1 - ½)A = ½A left.
3. On Sunday, she spent 1/3 of what she had left, which is 1/3 * ½A = 1/6A.

To model the total amount she spent, you need to add her spending on both days: (½A) + (1/6A) = (4/6)A = 2/3A.

So, the situation is modeled as 2/3A, not ½ - 1/3.

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The program segment is not correct with respect to the given initial and final assertions.

To verify that the program segment x := 2; z := x * y; if y > 0 then z := z + 1 else z := 0 is correct with respect to the initial assertion y = 3 and the final assertion z = 6, we need to check that the program produces the expected values of x, y, and z at every step.

1. Initial assertion: y = 3

This is given in the problem statement.

2. x := 2

After executing this statement, we have x = 2.

3. z := x * y

After executing this statement, we have z = x * y = 2 * 3 = 6.

4. if y > 0 then z := z + 1 else z := 0

Since y = 3 > 0, this condition is true and we execute the first branch of the if statement. Therefore, we have z := z + 1, which gives z = 6 + 1 = 7.

5. Final assertion: z = 6

This assertion is not satisfied, since we have z = 7 instead of z = 6.

Therefore, the program segment is not correct with respect to the given initial and final assertions.

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The resulting PDA has the same language as the CFG G. It recognizes strings of the form a^n b^n a^m b^m, where n and m are non-negative integers, and can be used to generate such strings by tracing the transitions of the PDA while keeping track of the stack contents.

The context-free grammar G is:

R → XRX | S

S → aTb | bTa

T → XTX | X | ε

X → a | b

To convert this CFG into an equivalent PDA, we can follow the procedure given in Theorem 2.20:

1) Create a PDA with one state and an empty stack.

2) For each production in the grammar, add a corresponding transition to the PDA. For example, for the production R → XRX, add a transition from the initial state to itself that reads X from the input and pushes R onto the stack, then transitions to a new state, reads X from the input, pops R from the stack, and transitions back to the initial state. Similarly, for the production S → aTb, add a transition that reads a from the input and pushes Tb onto the stack, and so on.

3) Add an accepting state and a transition from the initial state to the accepting state that pops the start symbol R from the stack.

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The new arithmetic mean or mean is 96. Information given that the mean of 4 numbers is 90. 120 has been added to the sum.

We need to calculate the new mean.

Step 1:

To find the sum of the four numbers, lets use the formula:

mean = (sum of all the numbers) / (number of numbers)

If the mean of 4 numbers is 90, then the sum of these 4 numbers is:

90 × 4 = 360

Step 2:

Now that we know the sum of the original 4 numbers is 360, we can find the sum of all five numbers by adding 120. So the new sum is:

360 + 120 = 480

Step 3:

In order to find the new mean, we use the formula for mean once again, but this time we use the new sum and the total number of numbers, which is 5.

mean = (sum of all the numbers) / (number of numbers)

mean = 480 / 5 = 96

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Starting with the given equation F sin(θ) = 24/26, we can use trigonometric identities to find expressions for cos(θ), tan(θ), and sec(θ).

First, we square both sides of the equation to get:

F^2 sin^2(θ) = (24/26)^2

Then, we use the identity sin^2(θ) + cos^2(θ) = 1 to solve for cos(θ):

cos^2(θ) = 1 - sin^2(θ)

cos^2(θ) = 1 - (24/26)^2

cos(θ) = ± √(1 - (24/26)^2)

Since 0 ≤ θ ≤ π/2, we know that cos(θ) must be positive, so we take the positive square root:

cos(θ) = √(1 - (24/26)^2)

Next, we can use the fact that tan(θ) = sin(θ)/cos(θ) to find an expression for tan(θ):

tan(θ) = sin(θ)/cos(θ)

tan(θ) = (F sin(θ))/cos(θ)

tan(θ) = (F sin(θ))/√(1 - (24/26)^2)

Finally, we can use the fact that sec(θ) = 1/cos(θ) to find an expression for sec(θ):

sec(θ) = 1/cos(θ)

sec(θ) = 1/√(1 - (24/26)^2)

So, in summary, we have:

cos(θ) = √(1 - (24/26)^2)

tan(θ) = (F sin(θ))/√(1 - (24/26)^2)

sec(θ) = 1/√(1 - (24/26)^2)

Note that we cannot simplify these expressions any further without more information about the value of F.

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Using a t-distribution, you can then calculate the lower and upper bounds of the 95% CI. This will give you an interval where you can be 95% confident that the true average Breakdown voltage falls within.

Based on the provided information, it appears that the mean breakdown voltage, u, has been precisely estimated. This is because the interval is quite narrow relative to the scale of the data values themselves. A narrow interval indicates a higher level of precision in the estimate.
To determine an appropriate sample size for a 95% confidence interval (CI) with a width of 1 kV (so that u is estimated within 0.5 kV with 95% confidence), the investigator needs to consider the range of breakdown voltage values (40-70) and the desired level of precision. Calculating sample size depends on the standard deviation and the desired margin of error. However, without the standard deviation, it's not possible to provide an exact sample size.
For constructing a boxplot, you will need to find the quartiles, median, and outliers of the given data set. Once these values are determined, you can plot them on a graph ranging from 40 to 70 to visualize the breakdown voltage distribution.
Lastly, to calculate a 95% CI for the true average breakdown voltage u, you will need to find the mean and standard deviation of the given data set. Using a t-distribution, you can then calculate the lower and upper bounds of the 95% CI. This will give you an interval where you can be 95% confident that the true average breakdown voltage falls within.

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The two variables x and y from the given equation shows that they are inverse variations.

Two variables are said to be inverse variations of themselves if the increase in one variable, say for example variable (x) leads to a decrease in another variable (y).

They are usually represented in reciprocal also knowns as inverse of one another. From the given information, we have xy = 12, where x and y are the two variables and 12 is the constant.

To make x the subject of the formula, we have:

x = 12/y

To make y the subject of the formula, we have:

y = 12/x

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the equation of the curve in the form y = f(x) is:

y = 4x^(-4/3)

We can eliminate the parameter t by expressing it in terms of x and substituting into the equation for y.

From the equation x = e^(-3t), we have:

t = -(1/3)ln(x)

Substituting this expression for t into the equation y = 4e^(4t), we get:

y = 4e^(4(-(1/3)ln(x))) = 4(x^(-4/3))

what is parameter?

In mathematics, a parameter is a quantity that defines the characteristics of a mathematical object or system, and whose value can be changed. It is typically denoted by a letter, such as a, b, c, etc., and is often used in mathematical equations or models to express the relationships between different variables.

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The distance from the point (1,2) to the line 4x - 3y = 0 is 2/5 units.

To find the distance between a point and a line, we need to use the formula:

distance = |ax + by + c| / √(a^2 + b^2)

where a, b, and c are the coefficients of the equation of the line in the form ax + by + c = 0. In this case, the equation of the line is 4x - 3y = 0, so a = 4, b = -3, and c = 0.

To apply the formula, we need to find the values of x and y that correspond to the point (1,2) when they are plugged into the equation of the line. Solving for y in terms of x, we get:

4x - 3y = 0

-3y = -4x

y = (4/3)x

Now we can plug in the coordinates of the point (1,2) and find the distance:

distance = |4(1) - 3(2) + 0| / √(4^2 + (-3)^2)

= |-2| / √(16 + 9)

= 2 / √25

= 2/5

Therefore, the distance from the point (1,2) to the line 4x - 3y = 0 is 2/5 units.

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The procedure that should be conducted to directly address this research question is the one sample proportion z test. This is because the research question is about the proportion of ring-tailed lemurs that are left hand dominant, which is a categorical variable. The sample size is greater than 30, so the central limit theorem can be applied and the distribution of the sample proportion can be assumed to be approximately normal. Therefore, a one sample proportion z test can be used to test whether the proportion of left hand dominant ring-tailed lemurs is greater than 0.5.

The one sample proportion z test is a statistical test used to determine whether a sample proportion is significantly different from a hypothesized population proportion. This test requires a categorical variable and a sample size greater than 30 in order to apply the central limit theorem and assume normality of the distribution of the sample proportion. The test statistic is calculated by subtracting the hypothesized population proportion from the sample proportion and dividing by the standard error of the sample proportion.

To directly address the research question of whether more than half of all ring-tailed lemurs are left hand dominant, a one sample proportion z test should be conducted. This test is appropriate for a categorical variable with a sample size greater than 30 and assumes normality of the distribution of the sample proportion. The test will determine whether the proportion of left hand dominant ring-tailed lemurs is significantly different from 0.5, which is the null hypothesis.

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The car accelerates uniformly at 5.0 m/s² from rest. To determine the time it takes for the car to reach a speed of 32 m/s, we can use the equation of motion for uniformly accelerated motion. The time elapsed is approximately 6.4 seconds.

We can use the equation of motion for uniformly accelerated motion to find the time it takes for the car to reach a speed of 32 m/s. The equation is:

v = u + at

Where:

v is the final velocity (32 m/s in this case),

u is the initial velocity (0 m/s since the car starts from rest),

a is the acceleration (5.0 m/s²),

t is the time elapsed.

Rearranging the equation to solve for t:

t = (v - u) / a

Substituting the given values:

t = (32 m/s - 0 m/s) / 5.0 m/s²

t = 32 m/s / 5.0 m/s²

t = 6.4 seconds

Therefore, it takes approximately 6.4 seconds for the car to reach a speed of 32 m/s under uniform acceleration at a rate of 5.0 m/s².

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f does not satisfy the additivity and homogeneity conditions, it is not a linear transformation.

To check if f is a linear transformation, we need to verify if the following two conditions hold for all x and y in R and all scalars c:

f(x + y) = f(x) + f(y) (additivity)

f(cz) = c f(x) (homogeneity)

Let's test these conditions:

Additivity:

f(x + y) = -7(x + y), -2(x + y), 5(x + y) + 5

= (-7x - 7y, -2x - 2y, 5x + 5y + 5)

On the other hand,

f(x) + f(y) = (-7x, -2x, 5x + 5) + (-7y, -2y, 5y + 5)

= (-7x - 7y, -2x - 2y, 5x + 5y + 10)

These two expressions are not equal, so f is not additive.

Homogeneity:

f(cz) = -7cz, -2cz, 5cz + 5

= c(-7x, -2x, 5x + 5)

However, this does not hold for all c, since the scalar c only affects the x-component of the vector f(z), and not the other two components. Hence, f is not homogeneous.

Since f does not satisfy the additivity and homogeneity conditions, it is not a linear transformation.

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The value of tobt for the test of the significance of r is 3.10 option B.

To find the value of tobt for the test of the significance of r, we can use the formula:

tobt = (r * √(N - 2)) / √(1 - r²)

Given r = 0.84 and N = 6, we can plug the values into the formula:

tobt = (0.84 * √(6 - 2)) / √(1 - 0.84²)

tobt = (0.84 * √4) / √(1 - 0.7056)

tobt = (0.84 * 2) / √0.2944

tobt = 1.68 / 0.542

tobt ≈ 3.10

The answer is (B) 3.10.

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The probability that it is in the shaded region of the rectangle is 0.5.

Option G is the correct answer.

We have,

The figure is a rectangle where a rhombus is inside the circle.

Now,

Rectangle:

Length = 48 in

Width = 12 in

Area = 48 x 12 = 576 in²

And,

Rhombus.

We can consider it to be two triangles.

Base = 12 in

Height = 24 in

So,

Area = 2 x (1/2 x base x height)

= 12 x 24

= 288 in²

Now,

The probability that it is in the shaded region of the rectangle.

= Area of the rhombus / Area of the rectangle

= 288/576

= 0.5

Thus,

The probability that it is in the shaded region of the rectangle is 0.5.

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Answer:

Step-by-step explanation:

Answer:

A. 1,800≤250+2i .

Step-by-step explanation:

To find f(x+h), we simply replace every occurrence of x in the expression for f(x) with x+h:

f(x+h) = 5(x+h) - 13

Simplifying this expression, we get:

f(x+h) = 5x + 5h - 13

Therefore, the simplified expression for f(x+h) is f(x+h) = 5x + 5h - 13.

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The matrix A:  [[0, 2], [2, 1]] has two eigen value i.e. λ1 = (1 + sqrt(17))/2,

λ2 = (1 - sqrt(17))/2 and their eigen values are [2/(1 + sqrt(17)), 1] , [2/(1 - sqrt(17)), -1] respectively and similarly the eigen value of the matrix

A^-1 is λ1 = (1 + sqrt(3))/2 ,  λ2 = (1 - sqrt(3))/2 and their eigen vector is

[2/(1 + sqrt(17)), 1] and [2/(1 - sqrt(17)), -1] respectively and the trace of the matrix  A and A-1 is 1 and 1/2 respectively.

To compute the eigenvalues and eigenvectors of matrix A, we need to solve the characteristic equation det(A - λI) = 0, where I is the 2x2 identity matrix.

STEP 1:-This gives us:

det(A - λI) = (0 - λ)(1 - λ) - 4 = λ^2 - λ - 4 = 0

Using the quadratic formula, we can solve for the eigenvalues:

λ1 = (1 + sqrt(17))/2

λ2 = (1 - sqrt(17))/2

STEP 2 :-To find the eigenvectors, we can solve the system of equations (A - λI)x = 0 for each eigenvalue. This gives us:

For λ1:

-λ1x1 + 2x2 = 0

2x1 - (λ1 - 1)x2 = 0

Solving this system, we get the eigenvector [2/(1 + sqrt(17)), 1].

For λ2:

-λ2x1 + 2x2 = 0

2x1 - (λ2 - 1)x2 = 0

Solving this system, we get the eigenvector [2/(1 - sqrt(17)), -1].

STEP 3:-

To compute the eigenvalues and eigenvectors of matrix A^-1, we need to solve the characteristic equation det(A^-1 - λI) = 0. We can simplify this expression using the fact that det(A^-1) = 1/det(A), which gives us:

det(A^-1 - λI) = (1/2 - λ)(-λ) - (1/2)(1) = -λ^2 + (1/2)λ - (1/2) = 0

Using the quadratic formula, we can solve for the eigenvalues:

λ1 = (1 + sqrt(3))/2

λ2 = (1 - sqrt(3))/2

We can see that A^-1 has the same eigenvectors as A, since the equation (A - λI)x = 0 is equivalent to A^-1(Ax - λx) = 0. Therefore, the eigenvectors of A^-1 are [2/(1 + sqrt(17)), 1] and [2/(1 - sqrt(17)), -1].

We can also check that the trace of A is equal to the sum of its eigenvalues, and the trace of A^-1 is equal to the sum of its eigenvalues. We have:

trace(A) = 0 + 1 = 1

trace(A^-1) = 1/2 + 0 = 1/2

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The statement "The function is even because f(–x) = f(x)"  is true about the function [tex]f(x) = 5x^4[/tex] .

If the value of x is negative, then the resulting output is positive. Accordingly, option D would be deemed as the accurate answer.

Mathematics defines a function as a relationship that links inputs to assignable outputs within a given domain.  An essential characteristic of functions requires each input to have precisely one unique output designation.

These fundamental mathematical tools feature prominently in algebra, calculus, and statistics with implications extending across scientific and engineering fields.

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Complete question:

Which statement is true about the function [tex]f(x) = 5x^4[/tex]?

The function is odd because f(–x) = –f(x).

The function is odd because f(–x) = f(x).

The function is even because f(–x) = –f(x).

The function is even because f(–x) = f(x).

We can split this whole figure up into two separate shapes: a square and a triangle.

The square has a length and width of 20 meters, which means its area is 400m^2.

The triangle has a height of 20 meters, which we know from the side lengths of the square. But, we need to find the height. If we know that the entire left side of the figure is 32m and 20m of that is taken by the square, then what's left for the triangle must be 12m.

Therefore, the height of the triangle is 20m and the base is 12m.

1/2 x base x height = 1/2 x 20 x 12 = 120m^2

Area = square + triangle

Area = 400 + 120

Area = 520m^2

Answer: 520 m^2

Hope this helps!

Answer:520

Step-by-step explanation:

Hi! So to start/set up the problem, we start with the triangle. Since squares have all equal sides, 20 is the length of the sides is 20. 32-20 is 12, so 12 times 20= 240, but remember the formula you do base times height divided by 2 (240/2=120.). 20x20=400.

Last step:120+400=520.

The number of distinct ways to arrange 3 yellow marbles, 5 blue marbles, and 5 green marbles in a row will be 5625.

A permutation is an act of arranging items or elements in the correct order.

There are 3 yellow marbles, 5 blue marbles, and 5 green marbles.

The number of distinct ways to arrange 3 yellow marbles, 5 blue marbles, and 5 green marbles in a row will be

[tex]\Rightarrow (3 \times 5 \times 5)^2[/tex]

[tex]\Rightarrow 75^2[/tex]

[tex]\Rightarrow 5625[/tex]

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The parametric equations for the ellipse x^2/2^2 + y^2/4^2 = 1, traced clockwise as the parameter increases, are:
x = 2cos(t)
y = -2sin(t)

To find the corresponding y-value for a given x-value on the ellipse, we can rearrange the equation:

x^2/2^2 + y^2/4^2 = 1
y^2/4^2 = 1 - x^2/2^2
y^2 = 4^2(1 - x^2/2^2)
y = ±2sqrt(1 - x^2/2^2)

Since the curve is traced clockwise as the parameter t increases, we can set x = 2cos(t) and y = -2sqrt(1 - x^2/2^2) to trace the lower half of the ellipse:

x = 2cos(t)
y = -2sqrt(1 - (2cos(t))^2/2^2)
y = -2sqrt(1 - cos^2(t))

Using the identity sin^2(t) + cos^2(t) = 1, we can solve for sin(t):

sin^2(t) = 1 - cos^2(t)
sin(t) = ±sqrt(1 - cos^2(t))

Since we want the negative value to trace the lower half of the ellipse, we have:
y = -2sin(t)

Therefore, the parametric equations for the ellipse x^2/2^2 + y^2/4^2 = 1, traced clockwise as the parameter increases, are:
x = 2cos(t)
y = -2sin(t)

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A median in a right triangle joining the right angle to the hypothenuse has the same length as the segment connecting the midpoints of the legs.

The median equals half the hypotenuse

In triangle ABC where ∠B = 90° BD is median

AD = DC median divides into two equal part

DX ⊥ BC

BX = XC = BC/2

DX = AB/2

By Pythagorean theorem

BD² = DX² + BX²

BD² = BC²/4 + AB²/4

BD² = AC²/4

BD = AC/2

Now in triangles BXD and DXC

DX = DX ( common )

AB║ DX

∠BXD = ∠DXC (as corresponding angles )

BX = XC (corresponding side)

By SAS congruency

ΔBXD ≅ ΔDXC

BD = DC

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