the volume of a cube decreases at a rate of 0.4 ft^3/min. what is the rate of change on the side length when the side lengths are 12 feet?

The dimension of the row space of matrix A is 2, the dimension of the column space of A is 4, and the dimension of the null space of A is 3.

To find the dimension of the row space of A, we can count the number of nonzero rows in the echelon form. Since there are two zero rows, the echelon form has 4 - 2 = 2 nonzero rows. Therefore, the dimension of the row space of A is 2.

To find the dimension of the column space of A, we can count the number of pivot columns in the echelon form. Since there are two zero rows, there are at most 5 pivot columns. However, since A is a 4 times 7 matrix, there must be exactly 4 pivot columns. Therefore, the dimension of the column space of A is 4.

To find the dimension of the null space of A, we can use the rank-nullity theorem. The rank of A is the dimension of the column space, which we found to be 4. The nullity of A is the dimension of the null space, which is given by nullity(A) = n - rank(A), where n is the number of columns of A. In this case, n = 7.

Therefore, nullity(A) = 7 - 4 = 3. Therefore, the dimension of the null space of A is 3.

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The values of the missing parts of the triangles are shown below.

Trigonometry is used to solve problems involving angles and distances, and it has many practical applications in fields such as engineering, physics, and astronomy.

DE = 5/Sin 30

= 10

DF = 10 Cos 30

= 8.6

JK = 2√6/Sin 60

= 2√6/√3/2

JK = 2√6 * 2/√3

JK = 4√6 /√3

LK = Cos 60 *  4√6 /√3

= 1/2 * 4√6 /√3

LK = 2√6 /√3

Thus the missing parts have been filled in by the use of the trigonometric ratios.

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Step-by-step explanation:

The diagram is not included:

18 ft  x  10 ft  =  180 ft^2

180 ft^2  *  $ 3.19 / ft^2 = $  574.20     for a rectangular pool cover

If it is triangular    1/2 * 10 * 18   * $3.19 = $287.10

Answer:

SA = [2π(1.4)² + 2.8π(4.2)] ft²   (Answer B)

Step-by-step explanation:

d = 2.8 ft; r = 1.4 ft

h = 4.2 ft

SA = area of 2 circular bases + lateral area

SA = 2πr² + 2πrh

SA = 2π(1.4)² + 2π(1.4)(4.2)

SA = 2π(1.4)² + 2.8π(4.2)

Jim can arrange the 11 songs on the playlist in 39,916,800 different ways. This can be answered by the concept of Permutation and Combination.

To determine the number of ways Jim can arrange the 11 songs on the playlist, we need to use the formula for permutations. The formula for permutations is n! / (n-r)!, where n is the total number of items and r is the number of items being chosen at a time. In this case, Jim has 11 songs and he wants to arrange all 11 of them on the playlist, so n = 11 and r = 11. Plugging these values into the formula, we get:

11! / (11-11)! = 11!

Simplifying 11!, we get:

11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 39,916,800

Therefore, Jim can arrange the 11 songs on the playlist in 39,916,800 different ways.

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The degree of freedom for the t statistic, in this case, is 18.

Hi! To answer your question about testing the difference between the means of two normally distributed independent populations with sample sizes of n1 = 10 and n2 = 10, we will use the formula for degrees of freedom (df) in a two-sample t-test:

df = (n1 - 1) + (n2 - 1)

Plug in the sample sizes, n1 = 10 and n2 = 10:

df = (10 - 1) + (10 - 1)

df = 9 + 9

df = 18
The degrees of freedom for the t statistic in this case is 18.

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Answer:

90 degrees counterclockwise

Step-by-step explanation:

Step-by-step explanation:

See image below:

The equation of the plane in general form is: 2x - 16y - 16z + 16t + 416 = 0

How to find the equation of a plane?

To find the equation of a plane passing through a point and perpendicular to a vector, we can use the point-normal form of the equation of a plane:

Ax + By + Cz = D

where (A, B, C) is the normal vector to the plane, and (x, y, z) is any point on the plane.

In this case, the point given is (8t, 8 – t, 9 + 3t), and the vector perpendicular to the plane is (2, 0, 1).

First, we need to find the normal vector to the plane. We can do this by taking the cross product of the given vector and the vector formed by the line:

(2, 0, 1) x ((8, -1, 0) - (0, 8, 9)) = (2, -16, -16)

Now we can use the point-normal form with the given point and the normal vector we just found:

2x - 16y - 16z = D

To find the value of D, we can substitute in the coordinates of the given point:

2(8t) - 16(8 - t) - 16(9 + 3t) = D

16t - 128 + 16t - 288 - 48t = D

-16t - 416 = D

So the equation of the plane in general form is: 2x - 16y - 16z + 16t + 416 = 0

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A subspace of V containing all vectors that are parallel to the vector (1, 2) is { (k, 2k) | k ∈ R }. A subspace of V containing all vectors that are perpendicular to the vector (1, 1) is { (x, -x) | x ∈ R }.

(a) To construct a subspace of V containing all vectors parallel to the vector (1, 2), we need to find a scalar multiple of the given vector.

A vector is parallel to another vector if it is a scalar multiple of that vector.

Step 1: Let k be a scalar in R (real numbers).
Step 2: Multiply the given vector (1, 2) by k:

k(1, 2) = (k, 2k).
Step 3: The subspace of V containing all vectors parallel to (1, 2) is given by the set { (k, 2k) | k ∈ R }.

(b) To construct a subspace of V containing all vectors perpendicular to the vector (1, 1), we need to find vectors that have a dot product of 0 with the given vector.

Step 1: Let the vector we are looking for be (x, y).
Step 2: Calculate the dot product:

(1, 1) · (x, y) = 1*x + 1*y = x + y.
Step 3: To find the vectors perpendicular to (1, 1), set the dot product to 0:

x + y = 0.
Step 4: Rearrange the equation to isolate y:

y = -x.
Step 5: The subspace of V containing all vectors perpendicular to (1, 1) is given by the set { (x, -x) | x ∈ R }.

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A subspace of V containing all vectors that are parallel to the vector (1, 2) is { (k, 2k) | k ∈ R }. A subspace of V containing all vectors that are perpendicular to the vector (1, 1) is { (x, -x) | x ∈ R }.

(a) To construct a subspace of V containing all vectors parallel to the vector (1, 2), we need to find a scalar multiple of the given vector.

A vector is parallel to another vector if it is a scalar multiple of that vector.

Step 1: Let k be a scalar in R (real numbers).
Step 2: Multiply the given vector (1, 2) by k:

k(1, 2) = (k, 2k).
Step 3: The subspace of V containing all vectors parallel to (1, 2) is given by the set { (k, 2k) | k ∈ R }.

(b) To construct a subspace of V containing all vectors perpendicular to the vector (1, 1), we need to find vectors that have a dot product of 0 with the given vector.

Step 1: Let the vector we are looking for be (x, y).
Step 2: Calculate the dot product:

(1, 1) · (x, y) = 1*x + 1*y = x + y.
Step 3: To find the vectors perpendicular to (1, 1), set the dot product to 0:

x + y = 0.
Step 4: Rearrange the equation to isolate y:

y = -x.
Step 5: The subspace of V containing all vectors perpendicular to (1, 1) is given by the set { (x, -x) | x ∈ R }.

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To find the absolute minimum and maximum values of the function f(x) = x^4 - 72x^2 within the interval [-5, 13], we'll first identify critical points and then evaluate the function at the endpoints.

The absolute minimum value is equal to -93911 at x = 13, and the absolute maximum value is equal to 31104 at x = 6.

Step 1: Find the derivative of f(x) with respect to x:
f'(x) = 4x^3 - 144x

Step 2: Find the critical points by setting f'(x) equal to 0:
4x^3 - 144x = 0
x(4x^2 - 144) = 0
x(x^2 - 36) = 0

The critical points are x = -6, 0, and 6.

However, x = -6 is not in the given interval, so we'll only consider x = 0 and x = 6.

Step 3: Evaluate f(x) at the critical points and endpoints:
f(-5) = (-5)^4 - 72(-5)^2 = 3125 - 18000 = -14875
f(0) = 0^4 - 72(0)^2 = 0
f(6) = 6^4 - 72(6)^2 = 46656 - 15552 = 31104
f(13) = 13^4 - 72(13)^2 = 28561 - 122472 = -93911

Step 4: Determine the minimum and maximum values:
The absolute minimum value is equal to -93911 at x = 13, and the absolute maximum value is equal to 31104 at x = 6.

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The probability of B from k is 0.655

The probability tree of the distribution is given as

A = 0.5k

B = 7k - 0.15

C  = 2.5k

By definition, we have

Sum of probabilities = 1

This means that

A + B  C = 1

substitute the known values in the above equation, so, we have the following representation

0.5k + 7k - 0.15 + 2.5k = 1

When evaluated, we have

10k - 0.15 = 1

So, we have

10k = 1.15

Divide

k = 0.115

Recall that

B = 7k - 0.15

So, we have

B = 7(0.115) - 0.15

Evaluate

B = 0.655

Hence, the probability of B is 0.655

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The three factors that affect the size of the margin of error when constructing a confidence interval for the population proportion are Sample size, Confidence level, and Population proportion.

1. Sample size (n): Larger sample sizes generally result in smaller margins of error, as the estimates become more precise.

2. Confidence level: Higher confidence levels (e.g., 95% vs 90%) lead to wider confidence intervals and larger margins of error, as they cover a greater range of potential values for the population proportion.

3. Population proportion (p): The margin of error is affected by the population proportion itself. When the proportion is close to 0.5, the margin of error is largest, while it is smaller when the proportion is near 0 or 1.

These factors are important to consider when constructing confidence intervals to ensure accurate and reliable results.

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The required computations are: G(-1) = -3 and G(G(-1)) = -9

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Answer:

m=147°

Step-by-step explanation:

Sum of interior angles of an n-sided polygon = (n-2)×180°, where n is the number of sides.

Sum of interior angles = (5-2) × 180°

= 3 × 180°

= 540°

m = 540-132-84-101-76

= 147°

We need to multiply the numerator and denominator by 3-√5 to rationalize the denominator of the fraction. Therefore, the correct answer is option B

To rationalize the denominator of the fraction 6−2√3+√5, we need to eliminate any radicals present in the denominator. We can do this by multiplying both the numerator and denominator by an expression that will cancel out the radicals in the denominator.

In this case, we can observe that the denominator contains two terms with radicals: -2√3 and √5. To eliminate these radicals, we need to multiply both the numerator and denominator by an expression that contains the conjugate of the denominator.

The conjugate of the denominator is 6+2√3-√5, so we can multiply both the numerator and denominator by this expression, giving us:

(6−2√3+√5)(6+2√3-√5) / (6+2√3-√5)(6+2√3-√5)

Simplifying the numerator and denominator, we get:

(6 * 6) + (6 * 2√3) - (6 * √5) - (2√3 * 6) - (2√3 * 2√3) + (2√3 * √5) + (√5 * 6) - (√5 * 2√3) + (√5 * -√5) / ((6^2) - (2√3)^2 - (√5)^2)

This simplifies to:

24 + 3√3 - 7√5 / 20

Therefore, the correct answer is option B.

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The numerical value of A2 = 10, A3 = 1, A4 = 10, A5 = 1, A6 = 10, A7 = 1, A8 = 10, and so on and the general form of An is 10.

The pattern is that A2, A4, A6, A8, etc. are all 10, while A3, A5, A7, A9, etc. are all 1. Therefore, the general formula for An is An = 10 if n is even, and An = 1 if n is odd. This pattern is a result of the alternating values of 1 and 10 in the original sequence.

By squaring any odd number (i.e., A2, A4, A6, etc.), we always get 100, and by squaring any even number (i.e., A3, A5, A7, etc.), we always get 1. This pattern continues indefinitely, and the general formula for An allows us to easily determine any term in the sequence without having to calculate all of the previous terms.

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The score that he needs to acquire next time so that his average is 78 would be = 89.

The average of a set of values(scores) can be calculated by finding the total s of the values and dividing it by the number of the values.

That is ;

average = sum of the scores/number of scores

average = 78

sum of scores = 77+79+67+x

number of scores = 4

Therefore,X is solved as follows;

78 = 77+79+67+x/4

78×4 = 77+79+67+x

312 = 223+X

X = 312-223

= 89

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Answer: q=1

Step-by-step explanation: Because it connects

E(Z³) = 0.

Expected value of X E(X) is equal to its mean, μ.

We have two parts to answer:

(a) Calculate E(Z³), which is the third moment of Z
(b) Calculate E(X)

(a) Since Z ~ N(0, 1), it is a standard normal random variable. For standard normal random variables, all odd moments are equal to 0. This is because the standard normal distribution is symmetric around 0, and odd powers of Z preserve the sign, causing positive and negative values to cancel out when calculating the expectation. Therefore, E(Z³) = 0.

(b) To calculate E(X), recall that X = σZ + μ, where Z is a standard normal random variable, and X is a normal random variable with mean μ and variance σ². The expectation of a linear combination of random variables is equal to the linear combination of their expectations:

E(X) = E(σZ + μ) = σE(Z) + E(μ)

Since Z is a standard normal random variable, its mean is 0. Therefore, E(Z) = 0, and μ is a constant, so E(μ) = μ:

E(X) = σ(0) + μ = μ

So, the expected value of X is equal to its mean, μ.

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There exists a function f such that f(x) > 0, f 0 (x) < 0, and f 00(x) > 0 for all x. - True.

There exists a function f(x) that satisfies these conditions. To see why, consider the function f(x) = x^3 - 3x + 1.
First, note that f(0) = 1, so f(x) is greater than 0 for some values of x.
Next, f'(x) = 3x^2 - 3, which is negative for x < -1 and positive for x > 1. Therefore, f(x) has a local minimum at x = 1 and a local maximum at x = -1. In particular, f'(0) = -3, so f'(x) is negative for some values of x.
Finally, f''(x) = 6x, which is positive for all x except x = 0. Therefore, f(x) has a concave up shape for all x, including x = 0, and in particular f''(x) is positive for all x.
So we have found a function f(x) that satisfies all three conditions.
a function f with the properties f(x) > 0, f'(x) < 0, and f''(x) > 0 for all x. This statement is true.
An example of such a function is f(x) = e^(-x), where e is the base of the natural logarithm. This function satisfies the conditions as follows:
1. f(x) > 0: The exponential function e^(-x) is always positive for all x.
2. f'(x) < 0: The derivative of e^(-x) is -e^(-x), which is always negative for all x.
3. f''(x) > 0: The second derivative of e^(-x) is e^(-x), which is always positive for all x.

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Theorem 1 states that if the functions f and f' are continuous on an interval (a,b) containing the initial point, then there exists a unique solution to the initial value problem on that interval.

In this case, we can rewrite the given differential equation as y' = (eˣy - x⁴ + 3)/6, and notice that both eˣy and x⁴ are increasing functions. Therefore, for a unique solution to exist, we need to ensure that the denominator (6) is positive for all values of x in (a,b).

Solving for y'' using the differential equation and plugging in the given initial conditions, we get y''(6) = e⁶/2 - 6/6 = (e⁶ - 6)/2. Since y''(6) is positive, the function y is concave up at x = 6, which means the function is increasing and hence y'(6) > 0.

Therefore, we can choose a = 6 - ε and b = 6 + ε for any positive ε such that y'(x) > 0 for x in (a,6) and y'(x) < 0 for x in (6,b). Hence, the largest interval for which Theorem 1 guarantees the existence of a unique solution is (6-ε, 6+ε).

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The probability that a randomly selected bill will be at least $39.10 is 0.0344.

To calculate the probability, we need to standardize the value $39.10 using the mean and standard deviation provided.

Let X be the random variable representing the daily dinner bill. Then, X ~ N(30, 5^2). We want to find P(X ≥ 39.10).

We can standardize X as follows:

Z = (X - μ) / σ

where μ = 30 and σ = 5.

Substituting the given values, we get:

Z = (39.10 - 30) / 5 = 1.82

Now, we need to find the probability that Z is greater than or equal to 1.82. We can use a standard normal distribution table or calculator to find this probability.

Using a standard normal distribution table, we find:

P(Z ≥ 1.82) = 0.0344

Therefore, the answer is D. The probability that a randomly selected bill will be at least $39.10 is 0.0344, or approximately 3.44%.

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The probability that a randomly selected bill will be at least $39.10 is 0.0344.

To calculate the probability, we need to standardize the value $39.10 using the mean and standard deviation provided.

Let X be the random variable representing the daily dinner bill. Then, X ~ N(30, 5^2). We want to find P(X ≥ 39.10).

We can standardize X as follows:

Z = (X - μ) / σ

where μ = 30 and σ = 5.

Substituting the given values, we get:

Z = (39.10 - 30) / 5 = 1.82

Now, we need to find the probability that Z is greater than or equal to 1.82. We can use a standard normal distribution table or calculator to find this probability.

Using a standard normal distribution table, we find:

P(Z ≥ 1.82) = 0.0344

Therefore, the answer is D. The probability that a randomly selected bill will be at least $39.10 is 0.0344, or approximately 3.44%.

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According to the question the there will be 3 stacks, with each stack containing 120 books.

Height is the measure of vertical distance or length. It is most commonly measured in units of meters, centimeters, or feet and inches. Height is an important factor in many sports and everyday activities, such as determining the size of a person's clothing or the size of a person's house.

The number of stacks will be determined by the number of books in the set with the most books. In this case, that would be 336 books in the English set. Each stack must have the same number of books, so the total number of stacks will be 336 divided by the number of books in the other sets: 240 in mathematics and 96 in science. Therefore, there will be 3 stacks, with each stack containing 120 books.

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The percentage error is approximately 0.0042%.

To use linear approximation, we first need to find a "nice" point close to the values we want to multiply. Let's choose 3 and 2 as our nice points since they are easy to square.

We can then write:

(2.98)² ≈ (3 - 0.02)² = 3² - 2(3)(0.02) + (0.02)² = 9 - 0.12 + 0.0004 = 8.8804

(2.02)² ≈ (2 + 0.02)² = 2² + 2(2)(0.02) + (0.02)² = 4 + 0.08 + 0.0004 = 4.0804

Multiplying these two approximations, we get:

(2.98)²(2.02)² ≈ 8.8804 × 4.0804 ≈ 36.246

Using a calculator, we find the actual value to be 36.2444.

To compute the percentage error, we use the formula:

Error = |(approximation - actual value) / actual value| × 100%

Error = |(36.246 - 36.2444) / 36.2444| × 100% ≈ 0.0042%

Therefore, the percentage error is approximately 0.0042%.

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There is no value of implicit differentiation where the given situation can be satisfied.

The given equation is: xy * e^y = e

We want to find the value of y^n at the point where x=0.

1. Rewrite the equation: xy * e^y = e

We have an equation involving both x and y. To find the value of y^n at x=0, we need to implicitly differentiate the equation with respect to x.

2. Implicit differentiation:

To implicitly differentiate the equation, we treat y as a function of x and use the chain rule. Differentiating both sides of the equation with respect to x, we get:

d/dx (xy * e^y) = d/dx (e)

Using the product rule on the left side, we have:

y * d/dx (x * e^y) + x * d/dx (e^y) = 0

The derivative of e^y with respect to x can be found using the chain rule:

d/dx (e^y) = d/dy (e^y) * dy/dx = e^y * dy/dx

3. Plug in x=0:

Now, let's plug in x=0 into the equation. We have:

y * d/dx (0 * e^y) + 0 * d/dx (e^y) = 0

Simplifying, we get:

y * (0 * e^y) + 0 * (e^y * dy/dx) = 0

Since any number multiplied by 0 is 0, the equation becomes:

0 + 0 = 0

So, we have 0 = 0, which is a true statement.

4. Conclusion:

From the result 0 = 0, we can conclude that the equation holds when x=0. However, this doesn't provide any specific value for y or y^n at x=0.

Therefore, the original question of finding the value of y^n at x=0 cannot be determined solely from the given equation. The equation does not provide enough information to solve for a specific value of y or y^n at x=0.

In summary, there is no value for y^n at the point where x=0 in the given situation because the equation cannot be satisfied at x=0.

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Next, you’re going to research the author. Write down notes that target specific facts about Cisneros in the box below. Your notes should be helpful in understanding her biases, experiences, and knowledge. List three well-developed ideas as opposed to three simple facts in the light blue area of the box (that will be four ideas including my sample for an “A” grade). Be sure you are not copying and pasting from a website and that your words are your own. Cite your sources by putting the author’s last name or the title of the website if there is not an author. I have an example as a model.

Sandra Cisneros was born 1954 in Chicago, USA making her 49 years old, thus she was writing about the 1960-90’s. She writes all different styles of pieces most of which are for pre-teens and teens, but she also writes for adults. She has won a lot of different writing awards throughout her life (Cisnero).

Step-by-step explanation:

answer. 804.25

the radius times two, times pie

Answer:

804.2496 square feet

Step-by-step explanation:

Just apply the formula for the area of a circle given the radius

A = π r²

where

A = area

r = radius

Given r = 16 ft

A = π x 16²

A= π x 256

Taking π as 3.1416 we get

A = 3.1416 x 256

A = 804.2496 square feet

The value of double integral is: (1/2) (1 - e) (1 - e).

To solve this integral, we will use the method of iterated integration. Let's first integrate with respect to x, treating y as a constant:

∫ e to 1 ( x ⋅ ln ( y ) / √ y y ⋅ ln ( x ) / √ x ) dx

Using substitution, let u = ln(x), du = 1/x dx, we get:

= ∫ e to 1 ( u / √ y y ) du

= [ ∫ e to 1 ( u / √ y y ) du ]

Now we integrate with respect to u:

= [ [ (1/2) u² ] from e to 1 ]

= (1/2) (1 - e)

Now, we integrate the remaining expression with respect to y:

= ∫ e to 1 (1/2) (1 - e) dy

= (1/2) (1 - e) [ y ] from e to 1

= (1/2) (1 - e) (1 - e)

So the value of given double integral is (1/2) (1 - e) (1 - e).

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The unit vector with positive x-component that is orthogonal to both p(0) and p′(0) is [tex]v_u_n_i_t[/tex] = v / ||v|| = ⟨-1,0,0⟩.

First, we need to find the vector that represents the position of the curve at t=0, which is p(0) = ⟨-1,0,0⟩.

Then we need to find the vector that represents the velocity of the curve at t=0, which is p'(t) = ⟨2t,2,4t⟩, so p'(0) = ⟨0,2,0⟩.

To find a unit vector that is orthogonal to both p(0) and p'(0), we can use the cross product:

v = p(0) x p'(0)

where "x" denotes the cross product. This will give us a vector that is perpendicular to both p(0) and p'(0), but it may not be a unit vector. To make it a unit vector, we need to divide by its magnitude:

[tex]v_u_n_i_t[/tex] = v / ||v||

where "||v||" denotes the magnitude of v.

So let's calculate v:

v = p(0) x p'(0) = ⟨0,0,2⟩ x ⟨0,2,0⟩ = ⟨-4,0,0⟩

And the magnitude of v is:

||v|| = sqrt((-4)^2 + 0^2 + 0^2) = 4

So the unit vector that is orthogonal to both p(0) and p'(0) and has a positive x-component is:

[tex]v_u_n_i_t[/tex] = v / ||v|| = ⟨-1,0,0⟩

Know more about vector here:

https://brainly.com/question/25705666

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