Exercise 4. Let n ≥ 2 be an even integer. Determine in how many ways we can color an nxn floor (split into a grid of 1 x 1 tiles) with k colors; we consider two colorings to be the same if we obtain one from the other by rotating the grid.

a. E[Y] = (0)(0.2) + (2)(0.5) + (4)(0.3) = 1.8 is the expected value for Y.

b. The cumulative distribution function of X+Y is P(X+Y = k) = Σ P(X=i)P(Y=k-i).

c. P(X=Y) is 0.3654.

d. Calculation is not correct. 3.6 is the correct value of E[XY].

In mathematics, a variable is defined as an alphabetic character that expresses a numerical value or number. A variable is used to represent an unknown quantity in algebraic equations.

(a) The expected value of Y can be calculated as E[Y] = (0)(0.2) + (2)(0.5) + (4)(0.3) = 1.8.

(b) To find the cumulative distribution function of X+Y, we first note that the sum of two independent random variables has a probability mass function given by the convolution of their respective probability mass functions. That is,

P(X+Y = k) = Σ P(X=i)P(Y=k-i)

where the sum is taken over all possible values of i such that both P(X=i) and P(Y=k-i) are nonzero. Using this formula, we can compute the cumulative distribution function of X+Y as:

F(x) = P(X+Y ≤ x) = Σ P(X+Y = k) for k ≤ x

     = Σ Σ P(X=i)P(Y=k-i) for k ≤ x

     = Σ P(X=i) Σ P(Y=k-i) for k ≤ x

     = Σ P(X=i) [tex]F_Y[/tex](x-i)

where [tex]F_Y[/tex](x) is the cumulative distribution function of Y. Since X has a Poisson(2) distribution, we can compute the cumulative distribution function of X+Y as:

F(x) = Σ P(X=i) F_Y(x-i)

     = Σ [tex]e^{(-2)} (2^i / i!) (0.2P(Y=x-i=0) + 0.5P(Y=x-i=2) + 0.3P(Y=x-i=4))[/tex]

where P(Y=x-i=k) is the probability mass function of Y.

(c) P(X=Y) can be calculated as:

P(X=Y) = Σ P(X=i, Y=i)

         = Σ P(X=i)P(Y=i) (since X and Y are independent)

         = Σ [tex]e^{(-2)} (2^i / i!) (0.2)(0) + (0.5)(e^(-2))(2^i / i!) + (0.3)(e^{(-2)})(2^i / i!)^2[/tex]

         = [tex]e^{(-4)} (0 + 0.5(2e^2/2) + 0.3(4e^2/4))[/tex]

         = 0.3654

(d) The student's calculation is not correct. To see why, let's first note that E[XY] can be computed as:

E[XY] = E[E[XY|X]] = E[XE[Y|X]]

where E[Y|X] is the conditional expected value of Y given X. Since X and Y are independent, we have E[Y|X] = E[Y] = 1.8. Therefore,

E[XY] = E[XE[Y|X]] = E[X(1.8)] = 2(1.8) = 3.6

So the correct value of E[XY] is 3.6, which is twice the value calculated by the student. The mistake the student made was in assuming that E[XY] is equal to the product of E[X] and E[Y]. This is only true if X and Y are uncorrelated, which is not the case here since X and Y are independent but not identically distributed.

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Answer:

m = +/- 5

Step-by-step explanation:

We can solve the equation by taking the fourth root of both sides.  

[tex]+/-\sqrt[4]{m^4}=+/-\sqrt[4]{625} \\m=5\\m=-5[/tex]

(5)(5)(5)(5) = 625

(-5)(-5)(-5)(-5) = 625

The +/- in the answers come from the fact that whenever you have a even exponent (e.g., x^2 or m^4), you always get a positive answer, even if the base you're raising to the particular exponent is negative

The  area of the lateral face of cylinder = 150.79 ft²

The  area of the two bases of the cylinder = 56.55 ft²

The total surface area of the cylinder =  207.34  ft²

We know that the formula for the surface area of cylinder is:

A = 2πrh + 2πr²

where r is the radius of the cylinder

and h is the height of the cylinder

Here, r = 3 ft and h = 8 ft

The area of the lateral face of cylinder would be,

A₁ = 2 × π × r × h

A₁ = 2 × π × 3 × 8

A₁ = 48 × π

A₁ = 150.79 sq. ft.

And the area of two bases is,

A₂ = 2πr²

A₂ = 2 × π × 3²

A₂ = 18 × π

A₂ = 56.55 sq. ft.

The total surface area of cylinder would be,

A = A₁ + A₂

A = 150.79 + 56.55

A = 207.34 sq. ft.

Therefore, the required surface area of cylinder = 207.34 ft²

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Answer:

The type of study conducted is an observational study. The response variable of the study is the concentration of the chemical.

PART A: The component form of vector is: u = <4, 8> and v = <4, 7>

PART B: u + v = <8, 15>

PART C: 5u - 2v = <12, 26>

The component form of a vector is <x, y>.

PART A:

Looking at the graph, vector u is the displacement from (2, -6) to (6, 2). Thus,

u = (6, 2) - (2, -6)

u = (6-2, 2-(-6))

u = (4, 8)

In component form, u = <4, 8>

Vector v is the displacement from (7, 8) to (11, 1). Thus,

v = (11, 1) - (7, 8)

v = (11-7, 8-1)

v = (4, 7)

In component form, v = <4, 7>

PART B:

u + v = <4, 8> + <4, 7>

u + v = <4+4, 8+7>

u + v = <8, 15>

PART C:

5u - 2v = 5*<4, 8> - 2*<4, 7>

5u - 2v = <20, 40> - <8, 14>

5u - 2v = <20-8, 40-14>

5u - 2v = <12, 26>

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The 90% confidence interval for the average fee for checking a single bag is $21.71 to $28.29, which corresponds to option B) $25 + $3.29

To calculate a 90% confidence interval for the average fee for checking a single bag.

To calculate a 90% confidence interval, we need the sample mean (X), the standard deviation (σ), and the sample size (n). From your question, we have:

X = $25
σ = $6
n = 9

Since we know the standard deviation, we can use the z-score for a 90% confidence interval, which is 1.645 (you can find this in a standard z-table).

Next, we need to calculate the standard error (SE), which is the standard deviation divided by the square root of the sample size:

[tex]SE=\frac{σ}{\sqrt{n} }[/tex]
[tex]SE=  \frac{ 6}{\sqrt{9} }[/tex]
[tex]SE=  \frac{ 6}{3 }[/tex]
SE = $2

Now, multiply the z-score by the standard error:

Margin of Error (ME) = 1.645 × SE
ME = 1.645 × $2
ME = $3.29

Finally, construct the 90% confidence interval by adding and subtracting the margin of error from the sample mean:

Lower Limit: X - ME = $25 - $3.29 = $21.71
Upper Limit: X + ME = $25 + $3.29 = $28.29

Thus, the 90% confidence interval for the average fee for checking a single bag is $21.71 to $28.29, which corresponds to option B) $25 + $3.29.

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Answer:

(1,-1)

Step-by-step explanation:

Substituting the given point into the equation

[tex]-5/6x=y+1/6\\-5/6(1) = -1 + 1/6\\-5/6 = -5/6[/tex]

Therefore, the area of the surface S is approximately 1.14 square units.

Here We can parametrize the  hyperbolic-paraboloid surface S as follows:

r(u,v) = (u, v, [tex]u^2[/tex] - v)

Here u is restricted to the interval [−1, 1] and v is restricted to the interval [−5, 5].

The area of the surface, we need to compute the magnitude of the cross product of the partial derivatives of r with respect to u and v:

|ru x rv| = |(1, 0, 2u) x (0, 1, -1)| = |(2u, 1, 0)| = [tex]\sqrt{(4u^2 + 1)}[/tex]

Therefore, the area of the surface is given by the double integral:

A = ∬S dS = ∫[tex]-5^5 * -1^1 \sqrt{ (4u^2 + 1)}[/tex] du/dv

We can evaluate this integral by making the substitution w = [tex]2u^2 + 1,[/tex] ,which gives:

A = ∫[tex]1^2 *1/4 \sqrt{w} dw[/tex]

= [tex](2/3) * w^{({3/2)}} |1^2 *1/4[/tex]

= [tex](2/3)(2 * \sqrt{5} - 1)[/tex]

So the area of the surface S is approximately 1.14 square units.

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The numbers that are arranged from greatest to least are

Let's first rewrite the given options in a clearer way and compare the numbers:

A) 1.6 x 10^0; 1.62 x 10^1; 1.7 x 10^(-1)

B) 1.62 x 10^4; 1.6 x 10^1; 1.7 x 10^1

C) 1.6 x 10^0; 1.7 x 10^0; 1.62 x 10^4

D) 1.62 x 10^0; 1.7 x 10^0; 1.6 x 10^1

Simplifying the numbers

Option A:

1.6 x 10^0 = 1.6; 1.62 x 10^1 = 16.2; 1.7 x 10^(-1) = 0.17

not in descending order

Option B:

1.62 x 10^4 = 16200; 1.6 x 10^1 = 16; 1.7 x 10^1 = 17

in descending order

Option C:

1.6 x 10^0 = 1.6; 1.7 x 10^0 = 1.7; 1.62 x 10^4 = 16200

in descending order

Option D:

1.62 x 10^0 = 1.62; 1.7 x 10^0 = 1.7; 1.6 x 10^1 = 16

not in descending order

Hence we can say that options B and C correctly sorted numbers from highest to lowest as follows:

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If Sheila’s test had a z-score of 2.2 then her test score was 89.6%.

We can use the formula for calculating the z-score of a value,

z = (x - μ) / σ, value we want to convert to a z-score is x, mean of the distribution is μ,  standard deviation of the distribution is σ and z score is z. In this case, we know that the average mark on the test was 72%, which means μ = 72. We also know that the standard deviation was 8%, which means σ = 8. We know that Sheila's z-score was 2.2,

We can rearrange the formula to solve for x,

x = μ + zσ

Substituting in the values we know,

x = 72 + 2.2 * 8

x = 72 + 17.6

x = 89.6

Therefore, Sheila's test score was 89.6%.

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a) Let A denote the event that 4 candies taken have different tastes, B denote the event that 2 candies have the same taste, and C denote the event that 6 candies include 2 candies with marzipan, 2 candies with nuts and 2 candies with caramel.

b) The probability of event A is 1/210

The probability of event B is 5/126

The probability of event C is 3/70

To find the probability of event A, we need to count the number of ways to choose 4 candies out of 10, where each candy has a different taste. Thus, the probability of event A is given by:

To find the probability of event B, we need to count the number of ways to choose 2 candies of the same taste and 2 candies of different tastes out of 10. There are 4 choices for the taste of the 2 candies that are the same, and 6 choices for the taste of the other 2 candies. Thus, the probability of event B is given by:

To find the probability of event C, we need to count the number of ways to choose 2 candies with marzipan, 2 candies with nuts, and 2 candies with caramel out of 10. There are (3 choose 2) = 3 ways to choose 2 candies with marzipan, (2 choose 2) = 1 way to choose 2 candies with nuts, and (4 choose 2) = 6 ways to choose 2 candies with caramel. Thus, the probability of event C is given by:

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Shannon found a stack of 100 collectible fantasy horse cards in her desk that she had forgotten about. She randomly looked at 20 cards and got 8 centaur, 7 pegasus, and 5 unicorn cards.

Based on the data, estimate how many unicorn cards are in the stack.

Experiment: "Favorite Cake Flavor"

Hypothesis: Different people have different favorite cakes flavors.

Now, Data collection enables us to identify the cakes flavors both popular and unpopular amongst users.

Furthermore, data analysis based on factors such as age, gender, and location allows for determining if disparities exist between consumer flavor preferences.

The findings of this study would be beneficial to cakes manufacturers and retailers, as they could enhance comprehension of preferred buyer choices and eventuate their products accordingly.  

Conclusively, improved knowledge of preferred flavor profiles would facilitate refining marketing methodologies to attract a broader target audience.

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The value of correct option for maps ΔABC to a triangle that is similar, but not congruent, to ΔABC are,

⇒  dilation with scale factor 2 about the origin

We have to given that;

To find correct option for maps ΔABC to a triangle that is similar, but not congruent, to ΔABC

Since, We know that;

For any translation the condition of congruency is not change.

But for any type of dilation condition of congruency for triangles are change.

Thus, The value of correct option for maps ΔABC to a triangle that is similar, but not congruent, to ΔABC are,

⇒  dilation with scale factor 2 about the origin

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The surface area of a surface parametrized by a function f(s, t), we use the formula:

Surface Area = ∫∫ √[f_s(s,t)^2 + f_t(s,t)^2 + 1] ds dt

The formula above calculates the surface area by integrating the square root of the sum of the squares of the partial derivatives of f with respect to s and t, plus one, over the surface.

Essentially, the formula is finding the magnitude of the gradient of the surface, which gives the rate of change of the surface in all directions.

Surface Area = ∫∫ √[f_s(s,t)^2 + f_t(s,t)^2 + 1] ds dt

The surface area formula can be used to find the surface area of various types of surfaces, such as parametric surfaces, implicit surfaces, and surfaces of revolution.

However, the integration required to evaluate the formula can be quite challenging, especially for complex surfaces. In such cases, numerical methods may be used to approximate the surface area.

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all four properties are satisfied, we can conclude that (M,d) is a metric space.

What is metric space?

In mathematics, a metric space is a set of objects called points, together with a function called the distance function or metric, that defines a notion of distance between any two points in the space. The metric satisfies certain conditions to ensure that it is a useful measure of the "distance" between points, such as being non-negative, symmetric, and satisfying the triangle inequality. Metric spaces are used to study properties of objects that can be thought of as having a notion of distance, such as Euclidean space, graphs, and networks.

Let's check each of these properties:

Non-negativity: This property holds since d(x, y) is defined to be 0 or 1, both of which are non-negative.

Identity of indiscernibles: This property also holds since d(x, y) is defined to be 0 if and only if x = y.

Symmetry: This property holds since d(x, y) = d(y, x) for any x, y in M.

Triangle inequality: For any x, y, z in M, there are three cases to consider:

If x = y or y = z, then d(x, y) + d(y, z) = d(x, z) = 1 by definition, and the inequality holds.

If x = z, then both sides of the inequality are 0.

If x, y, and z are all distinct, then d(x, y) + d(y, z) = 2 and d(x, z) = 1, so the inequality holds.

Since all four properties are satisfied, we can conclude that (M,d) is a metric space.

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Answer:

x = -6

Step-by-step explanation:

you have given

10x + 8y = 12 and 4x + y = -15

so you must put the one var. x or y by same cofficent and in opposite sighn

so 10x + 8y = 12

- 8 (4x + y = -15 ) ......... i multiplied by -8

10x + 8y = 12

-32x - 8y = 120

then you will add the equation

(10x + 8y) + (-32x - 8y) = 12 + 120

afeter you simlify it u will get

-22x = 132

-22x = 132

x = -6 .... by dividing both sides by -22

Answer:

(-6,9)

Step-by-step explanation:

Multiply 4x + y = -15 all the say through by -8 and then add to 10x + 8y = 12

-32x -8y =  120

10x + 8y = 12

-22x =  132  Divide both sides by -22

x = -6

Substitute -6 for x

4x + y = -15

4(-6) + y = -15

-24 + y = -15  Add 24 to both sides

y = 9

Check

10x + 8y = 12

10(-6) + 8(9) = 12

-60 + 72 = 12

12 = 12  checks

4x + y = -15

4(-6) + 9 = -15

-24 + 9 = -15

-15 = -15 Checks.

Helping in the name of Jesus.

The critical region would be in the tail of the null distribution corresponding to the alpha level (0.05), which is the upper 5%.

We have,

The critical region would be in the upper 5% of the null distribution.

This is because alpha is the probability of making a type I error (rejecting the null hypothesis when it is actually true),

In this case,

We are looking for evidence that the population mean is greater than 1.2.

Therefore,

The critical region would be in the tail of the null distribution corresponding to the alpha level (0.05), which is the upper 5%.

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The answer is: True. When matrix A is upper triangular, its eigenvalues are located on its main diagonal. If you find the eigenvectors corresponding to each eigenvalue, you can construct an eigenvector matrix S.

An upper triangular matrix is a square matrix in which all entries below the main diagonal are zero. An eigenvector of a matrix A is a nonzero vector x such that Ax is a scalar multiple of x. That is, there exists a scalar λ such that Ax = λx.
For a complete upper triangular matrix, all of its eigenvalues are on the diagonal. To see this, consider the characteristic polynomial of a complete upper triangular matrix:
p(λ) = det(A - λI)
where I is the identity matrix. Since A is upper triangular, its determinant is the product of its diagonal entries, and det(A - λI) is a polynomial of degree n (the size of the matrix) in λ. Therefore, there are n roots of p(λ), which correspond to the eigenvalues of A. Since A is completely upper triangular, all of its eigenvalues are on the diagonal.

Now, let's consider the eigenvector matrix S of A. This is a matrix whose columns are the eigenvectors of A. Since A is upper triangular, any eigenvector of A must also be upper triangular (or zero). Therefore, the eigenvector matrix S must also be upper triangular. In summary, if a is a complete upper triangular matrix, then all of its eigenvalues are on the diagonal, and its eigenvector matrix S is upper triangular. Therefore, the statement is true.
"If A is a complete upper triangular matrix, then it has an upper triangular eigenvector matrix S." When a matrix A is upper triangular, its eigenvalues are located on its main diagonal. If you find the eigenvectors corresponding to each eigenvalue, you can construct an eigenvector matrix S. Since A is upper triangular, the eigenvector matrix S will also be upper triangular.

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We need to solve the equation 1 - Φ((L - Cm)/0.025) = 0.9265 for L. This can be done using a standard normal table or calculator.

(a) Let X be the length of the cylindrical bar. Then X ~ N(Cm, 0.25^2). We need to find P(X < 11.75).

Z = (X - Cm)/0.25 follows standard normal distribution.

P(X < 11.75) = P((X-Cm)/0.25 < (11.75-Cm)/0.25) = P(Z < (11.75-Cm)/0.25)

Using a standard normal table or calculator, we get P(Z < (11.75-Cm)/0.25) = Φ((11.75-Cm)/0.25)

where Φ is the cumulative distribution function of the standard normal distribution.

(b) The sample mean length, X, follows normal distribution with mean Cm and standard deviation σ/√n, where n = 100 is the sample size. So, X ~ N(Cm, 0.25/√100) = N(Cm, 0.025). Therefore, the mean of the sample mean length is Cm and the standard deviation of the sample mean length is 0.025.

(c) We need to find P(10.99 < X < 11.01), where X is the sample mean length.

Z = (X - Cm)/(0.025) follows standard normal distribution.

P(10.99 < X < 11.01) = P((10.99 - Cm)/(0.025) < Z < (11.01 - Cm)/(0.025))

Using a standard normal table or calculator, we get P((10.99 - Cm)/(0.025) < Z < (11.01 - Cm)/(0.025)) = Φ((11.01 - Cm)/(0.025)) - Φ((10.99 - Cm)/(0.025))

(d) Let L be the length such that 92.65% of the sample means are more than L. This means we need to find the value of L such that P(X > L) = 0.9265.

Z = (X - Cm)/(0.025) follows standard normal distribution.

P(X > L) = P(Z > (L - Cm)/0.025) = 1 - Φ((L - Cm)/0.025)

Therefore, we need to solve the equation 1 - Φ((L - Cm)/0.025) = 0.9265 for L. This can be done using a standard normal table or calculator.

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Answer:

-8.5

Step-by-step explanation:

First we need to get the -4x by it self which means moving the 8 so what we need to do is subtract 8 from its self and what we do on one side we do to the other so 42-8=34 then the last thing to do is just divided 34 divided by -4 and we get

-8.5 as the answer.

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HAVE A GREAT DAY!!

We do not have sufficient evidence to conclude that 20-year-olds have more cavities than 30-year-olds.

First, we need to calculate the sample mean and standard deviation of the given data:

x = (6+7+7+8+7+8+9+6+5+6+7+8+7+6+9+8)/16 = 7

s = sqrt((Σ(x - x)²)/(n-1)) = sqrt((Σ(x²) - n(x)²)/(n-1)) = 1.247

Now, we can set up the hypothesis test:

H0: μ = 7 (20-year-olds have the same average number of cavities as 30-year-olds)

Ha: μ > 7 (20-year-olds have more cavities than 30-year-olds)

We will use a t-test since the population standard deviation is unknown and we have a small sample size (n = 16). The test statistic is:

t = (x - μ) / (s/sqrt(n)) = (7 - 7) / (1.247/sqrt(16)) = 0

The degrees of freedom is n-1 = 15. Using a t-table with α = 0.05 and df = 15, we find the critical value to be 1.753.

The p-value is the probability of getting a t-value as extreme or more extreme than the calculated t-value under the null hypothesis. Since our null hypothesis is that μ = 7 and our alternative hypothesis is that μ > 7, we have a one-tailed test. Using a t-table with df = 15, we find the p-value to be 0.5.

Since our p-value (0.5) is greater than α (0.05), we fail to reject the null hypothesis. We do not have sufficient evidence to conclude that 20-year-olds have more cavities than 30-year-olds.

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Given: The solution to [tex]x^3[/tex] = [tex]-2-i[/tex] In polar form Is:

[tex]2 < 75^o, 2 < 195^o, 2 < 315^o[/tex]

Answer:

[tex]\large \boxed{\mathrm{ion \ even \ no \ fr}}[/tex]

Step-by-step explanation:

DO IT YOUR SELF [tex]\large \boxed{\mathrm{BOZO}}[/tex]

The number of blocks needed to complete the big or full cube depends on size of cube. So, number of blocks required to complete it equals to 45.

A cube is a three-dimensional geometry, which may be solid or hollow and containing six equal squares. According to the shape of the cube, we can make a cube from any cubes. Now look at the cube image above. There is one more layer of blocks to fill in because the blocks are still 5 blocks by 4 blocks (not make a cube). We need to add another 25 blocks at the top to meet the cube definition. Some times we would often just count the missing blocks which is 20 here but after adding all 20 blocks in figure still it isn't a cube. So, 20 is wrong because the sides won’t be equal. The width and the length is made of 5 blocks but the height is just four blocks. So, it's need to add another 25 blocks the top to make it a cube. Hence, the correct answer is 45.

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Complete question:

The above figure complete the question.

How many blocks are needed to complete the full cube ?

The CPMP mathematics methodology seems to be more effective on average at helping students solve word problems compared to the traditional method.

There is a greater variation in performance among students in the CPMP group. We can compare the CPMP and traditional math program groups' performance on solving word problems by looking at the mean and standard deviation.

1) Observe the mean scores:

- CPMP: 57.4

- Traditional: 53.9

The CPMP group has a higher mean score than the traditional group, indicating that students in the CPMP group performed better on average.

2) Observe the standard deviations:

- CPMP: 32.1

- Traditional: 28.5

The CPMP group has a higher standard deviation than the traditional group, meaning that the scores in the CPMP group are more spread out. This suggests there's a greater range of performance levels among students in the CPMP group.

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Answer:

Step-by-step explanation:

Why did many Americans consider César Chávez a hero in the '60s and '70s?

Answer:

The surface area is 70 yards

Step-by-step explanation:

The formula for surface area is (SA)=2lw+2lh+2hw. Meaning it would be 2 times (8 times 3 + 8 times 1 + 3 times 1) which equals 70 yards.

The number of frames that the welders would be able to complete in 4 hours would be 10 frames.

The time taken by one welder is 8 hours for one frame which means the work rate would be:

= 1 / 8

= 1 / 8 frames per hour

If we have 20 welders therefore, the work rate per hour would be :

= 1 / 8 x 20

= 2. 5 frames per hour

Given 4 hours, the number of frames they would make is:

= 2. 5 x 4 hours

= 10 frames

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First part of question is:

A welding company produces burglar frames for windows and doors. In order to complete one frame, one welder needs 8 hours.

Answer:

z = 10

Step-by-step explanation:

Lets explain this to you,

Step one: Solve with the parenthesis first (2 times z is 2z and 2 times -8.5 is

-17)

Step two: Subtract 2z from both sides (0.3z - 2z = - 1.7) (2z - 2z = 0)

Step three: Divide negative 1.7 on both sides (-1.7 = -17)

Step four: Simplify - 1.7 / 17 which will finally be 10

There is an easier way but im also in class so I'll do it later!!!!