Three factories produce the same tool and supply it to the market. Factory A produces 30% of the tools for the market and the remaining 70% of the tools are produced in factories B and C. 98% of the tools produced in factory A, 95% of the tools produced in factory B and 97% of the tools produced in factory C are not defective. What percent of tools should be produced by factories B and C so that a tool picked at random in the market will have a probability of being non defective equal to 96%?

If x = [tex]t^3[/tex] - t and y = [tex]\sqrt{3t + 1}[/tex], then dy/dx is 3/8 at t = 1.

To find dy/dx at t=1, we need to first find dx/dt and dy/dt, and then use the chain rule.
1. Find dx/dt:
x = [tex]t^3[/tex] - t
Differentiate with respect to t:
dx/dt = [tex]3t^2[/tex] - 1
2. Find dy/dt:
y = sqrt(3t + 1)
Differentiate with respect to t:
dy/dt = 1/2 * [tex](3t + 1)^{(-1/2)}[/tex] * 3
dy/dt = 3/(2 * [tex]\sqrt{(3t + 1)}[/tex])
3. Use the chain rule to find dy/dx:
dy/dx = (dy/dt) / (dx/dt)
4. Plug in the values at t = 1:
dx/dt (t=1) = [tex]3(1)^2[/tex] - 1 = 2
dy/dt (t=1) = 3/(2 * [tex]\sqrt{(3(1) + 1)}[/tex]) = 3/4
5. Calculate dy/dx at t = 1:
dy/dx = (3/4) / 2 = 3/8
So, the answer is 3/8.

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The first four nonzero terms in the power series expansion option (D) - [tex]x^{3/6} + x^{5/120} - x^{7/5040[/tex]

In mathematics, a function is a unique arrangement of the inputs (also referred to as the domain) and their outputs (sometimes referred to as the co-domain), where each input has exactly one output and the output can be linked to its input.

The power series expansion of the function f(x) = sin(x) about x = 0 is given by:

f(x) = [tex]x - x^{3/3}! + x^{5/5}! - x^{7/7}! + ...[/tex]

To find the first four nonzero terms, we substitute x = 0 into the series and discard all the terms that are zero:

f(0) = 0 - 0 + 0 - 0 + ... = 0

f'(0) = 1 - 0 + 0 - 0 + ... = 1

f''(0) = 0 - 3!/2! + 0 + 0 + ... = -3/2

f'''(0) = 0 + 0 + 5!/3! - 0 + ... = 5/6

Therefore, the first four nonzero terms in the power series expansion of sin(x) about x = 0 are:

sin(x) ≈ [tex]x - x^{3/3}! + x^{5/5}! - x^{7/7}![/tex]

Simplifying this expression, we get:

sin(x) ≈ [tex]x - x^{3/6} + x^{5/120} - x^{7/5040[/tex]

So, the correct answer is option (D) - [tex]x^{3/6} + x^{5/120} - x^{7/5040[/tex].

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A. There are 37 terms in the sum.

B. The sum of the given series is 7,696.

A. Using arithmetic sequences, We can observe that each term in the sum is obtained by adding 11 to the previous term. Therefore, the nth term can be expressed as:

[tex]a_n = 10 + 11(n-1)[/tex]

We want to find the number of terms in the sum up to [tex]a_n[/tex] = 406. Setting [tex]a_n[/tex]= 406 and solving for n, we get:

406 = 10 + 11(n-1)

396 = 11(n-1)

n = 37

Therefore, there are 37 terms in the sum.

B. We can use the formula for the sum of an arithmetic series:

[tex]S_n = n/2 * (a_1 + a_n)[/tex]

where [tex]S_n[/tex] is the sum of the first n terms, [tex]a_1[/tex] is the first term, and[tex]a_n[/tex] is the nth term.

In this case, we have:

[tex]a_1[/tex]= 10

[tex]a_n[/tex]= 406

n = 37

Substituting these values, we get:

[tex]S_{37}[/tex] = 37/2 * (10 + 406)

[tex]S_{37}[/tex] = 18.5 * 416

[tex]S_{37}[/tex] = 7,696

Therefore, the sum of the given series is 7,696.

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The final answer will be an approximation.

We have the iterated integral:

∫(0 to √16) ∫(0 to √[tex]16-x^2) e^(-x^2-y^2) dy dx[/tex]

To convert to polar coordinates, we use the substitution x = r cos(θ) and y = r sin(θ), where r is the radial distance and θ is the angle with the positive x-axis.

The limits of integration also need to be changed accordingly. For the inner integral, we have:

0 ≤ y ≤ √[tex](16-x^2)[/tex]

Substituting y = r sin(θ), we get:

0 ≤ r sin(θ) ≤ √(16 - [tex]r^2 cos^2[/tex](θ))

Squaring both sides, we get:

0 ≤ [tex]r^2 sin^2[/tex](θ) ≤ 16 - [tex]r^2 cos^2[/tex](θ)

Rearranging, we get:

[tex]r^2 (cos^2[/tex](θ) + [tex]sin^2[/tex](θ)) ≤ 16

[tex]r^2[/tex] ≤ 16

0 ≤ r ≤ 4

For the outer integral, we have:

0 ≤ x ≤ √16

Substituting x = r cos(θ), we get:

0 ≤ r cos(θ) ≤ √16

0 ≤ r ≤ 4 cos(θ)

Thus, the integral in polar coordinates becomes:

∫(0 to π/2) ∫(0 to 4 cos(θ)) [tex]e^(-r^2[/tex]) r dr dθ

Evaluating the inner integral, we get:

∫(0 to 4 cos(θ)) e^(-r^2) r dr = -1/2 e^(-r^2) |0 to 4 cos(θ)) = 1/2 (1 - e^(-16 cos^2(θ)))

Substituting back into the outer integral, we get:

∫(0 to π/2) 1/2 (1 - e^(-16 cos^2(θ))) dθ

Simplifying, we get:

1/2 ∫(0 to π/2) (1 - e^(-16 cos^2(θ))) dθ

To evaluate this integral, we need to use numerical methods or approximations. Therefore, the final answer will be an approximation.

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Answer:

87.96

Step-by-step explanation:

Where pi is approximately equal to 3.14 and r is the radius of the circle.

In this case, the radius is 14 yards.

So the circumference is:

2pi14 = 87.96 yards.

Answer:

x = -2 + sqrt(4 + 77c*m^2) cm.

Step-by-step explanation:

Let's denote the side of the square by "x".

When the strip of width 4 cm is attached to one side of the square, the resulting rectangle has dimensions of (x+4) cm by x cm.

The area of the rectangle is given by:

(x+4) * x = 77c * m^2

Expanding the left-hand side and simplifying, we get:

x^2 + 4x = 77c * m^2

Moving all the terms to one side, we get:

x^2 + 4x - 77c * m^2 = 0

Now we can use the quadratic formula to solve for x:

x = [-4 ± sqrt(4^2 - 41(-77cm^2))] / (21)

x = [-4 ± sqrt(16 + 308cm^2)] / 2

x = [-4 ± 2sqrt(4 + 77cm^2)] / 2

x = -2 ± sqrt(4 + 77c*m^2)

Since the length of a side of a square must be positive, we can discard the negative solution and get:

x = -2 + sqrt(4 + 77c*m^2)

Therefore, the length of the side of the square is x = -2 + sqrt(4 + 77c*m^2) cm.

The length of the side of the square is x = -2 + 2√(1 + 19c) cm.

The area of Rectangle is length times of width.

Let the side of the square be x cm.

When a strip of width 4 cm is attached to one side of the square, the resulting rectangle will have dimensions (x + 4) cm and x cm.

The area of the rectangle is given as 77c m² so we have:

(x + 4)x = 77c

Expanding the left side, we get:

x² + 4x = 77c

Bringing all the terms to one side, we have:

x² + 4x - 77c = 0

Now, we can use the quadratic formula to solve for x:

x = [-4 ± √(4² - 4(1)(-77c))] / 2(1)

x = [-4 ± √(16 + 308c)] / 2

x = [-4 ± √(16(1 + 19c))] / 2

x = [-4 ± 4√(1 + 19c)] / 2

x = -2 ± 2√(1 + 19c)

Since x must be positive, we take the positive root:

x = -2 + 2√(1 + 19c)

Therefore, the length of the side of the square is x = -2 + 2√(1 + 19c) cm.

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To calculate the constrained minimum, we need to use the method of Lagrange multipliers. We define the Lagrangian function L as L(x,y,λ) = xy^2 - λ(d^2 - x^2 - y^2 - z^2), where d represents the distance between the origin and the nearest point on the curve.

Taking the partial derivatives of L with respect to x, y, and λ, we get:

dL/dx = y^2 + 2λx = 0
dL/dy = 2xy - 2λy = 0
dL/dλ = d^2 - x^2 - y^2 - z^2 = 0

Solving these equations simultaneously, we get:

x = ± 3√6, y = ± √6, and λ = 3/2

Therefore, the points on the curve xy^2 = 54 nearest to the origin are:

(3√6, √6) and (-3√6, -√6)

These points are the constrained minimum because they are the closest points on the curve to the origin.
To find the constrained minimum and the points on the curve xy^2 = 54 nearest to the origin, we can use the method of Lagrange multipliers. Let f(x, y) = x^2 + y^2 be the distance squared from the origin, and g(x, y) = xy^2 - 54 as the constraint.

First, calculate the gradients:
∇f(x, y) = (2x, 2y)
∇g(x, y) = (y^2, 2xy)

Now, set ∇f(x, y) = λ ∇g(x, y):
(2x, 2y) = λ(y^2, 2xy)

This gives us two equations:
1) 2x = λy^2
2) 2y = λ2xy

From equation (2), we can get:
λ = 1/y

Now, substitute λ into equation (1):
2x = (1/y)y^2
2x = y

Using the constraint equation g(x, y) = xy^2 - 54 = 0, we can substitute y = 2x:
2x(2x)^2 = 54
8x^3 = 54
x^3 = 27/4
x = ∛(27/4) = 3/√4 = 3/2

Now we can find y using y = 2x:
y = 2(3/2) = 3

Thus, the point nearest to the origin on the curve xy^2 = 54 is (3/2, 3).

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Yes the opposite sides of the figure are congruent because:

Both WX and YZ have a length of 3.6

Both XY and WZ have a length of 16.8

Pythagoras Theorem is the way in which you can find the missing length of a right angled triangle.

The triangle has three sides, the hypotenuse (which is always the longest), Opposite (which doesn't touch the hypotenuse) and the adjacent (which is between the opposite and the hypotenuse).

Pythagoras Theorem is in the form of;

a² + b² = c²

Using Pythagoras theorem, we have:

WX = √(3² + 2²)

WX = √13 = 3.6

YZ = √(3² + 2²)

YZ = √13 = 3.6

Similarly:

XY = √(5² + 16²)

XY = √281

XY = 16.8

WZ = √(5² + 16²)

WZ = √281

WZ = 16.8

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Yes the opposite sides of the figure are congruent because:

Both WX and YZ have a length of 3.6

Both XY and WZ have a length of 16.8

Pythagoras Theorem is the way in which you can find the missing length of a right angled triangle.

The triangle has three sides, the hypotenuse (which is always the longest), Opposite (which doesn't touch the hypotenuse) and the adjacent (which is between the opposite and the hypotenuse).

Pythagoras Theorem is in the form of;

a² + b² = c²

Using Pythagoras theorem, we have:

WX = √(3² + 2²)

WX = √13 = 3.6

YZ = √(3² + 2²)

YZ = √13 = 3.6

Similarly:

XY = √(5² + 16²)

XY = √281

XY = 16.8

WZ = √(5² + 16²)

WZ = √281

WZ = 16.8

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Answer:

(the length of leg)^2=87^2-60^2=3969

so when take the root :

the length of leg=63ft

The mean of sample mean for sample size of 10 is e. 3.05.

To find the mean of sample means for a sample size of 10 from a normally distributed population with mean 30.5 and standard deviation 3.5, we use the formula:

mean of sample means = population mean = 30.5

So the answer is not affected by the sample size or standard deviation. The mean of the sample means will always be equal to the population mean.

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For x = 3, the expression 5x - 19 is equivalent to -4.

Here, we have an expression, 5x - 19, which is not an equation yet because it doesn't have an equal sign (=). An equation requires that both sides of the equal sign be equivalent.

To make this expression into an equation, we need to set it equal to something. Let's say we want to find an expression that is equivalent to 5x - 19. We can represent this unknown expression as "y". So, our equation will be:

5x - 19 = y

This equation is now saying that the expression 5x - 19 is equivalent to y. We can also say that "y" is a function of "x", meaning that the value of "y" depends on the value of "x".

We can use this equation to find the value of "y" for any given value of "x". If we substitute x = 3, we get:

5(3) - 19 = y

y = 15 - 19

y = -4

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Complete Question:

what is equivalent to 5x - 19 when x = 3?

A graph of the function [tex]f(x) = 2^x[/tex] is shown in the image below.

If f(x) is translated 4 units down, the equation of the new function g(x) is [tex]g(x) = 2^x-4[/tex]

The transformed function g(x) is shown on the same grid below.

In Mathematics and Geometry, the vertical translation a geometric figure or graph downward simply means subtracting a digit from the value on the y-coordinate of the pre-image or function.

In Mathematics and Geometry, a horizontal translation to the right is modeled by this mathematical equation g(x) = f(x - N) while a vertical translation to the positive y-direction (downward) is modeled by this mathematical equation g(x) = f(x) - N.

Where:

In this scenario, we can logically deduce that the graph of the parent function f(x) was translated or shifted downward (vertically) by 4 units as shown below.

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The sets of natural numbers {1, 2, 3, 4, ...} and even numbers {2, 4, 6, 8, ...} have the same cardinality because there exists a bijective function between the two sets. A bijective function is a one-to-one correspondence that pairs each element in one set with exactly one element in the other set. In this case, the function f(n) = 2n pairs each natural number n with an even number 2n, ensuring that the two sets have the same cardinality.

The two sets, the set of natural numbers {1,2,3,4,...} and the set of even numbers {2, 4, 6, 8, . . .}, have the same cardinality because we can create a one-to-one correspondence between the two sets. To do this, we can simply map each natural number to its corresponding even number (i.e., 1 maps to 2, 2 maps to 4, 3 maps to 6, and so on). This mapping covers all elements of both sets, without skipping any, and without duplicating any. Thus, the two sets have the same number of elements, which means they have the same cardinality.

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The middle value between 26.27 and 26.89 is 26.58.

To find the middle value between 26.27 and 26.89, you can follow these steps:

1. Add the two numbers together: 26.27 + 26.89 = 53.16
2. Divide the sum by 2 to find the average: 53.16 ÷ 2 = 26.58

The middle value between 26.27 and 26.89 is 26.58.

The middle value of a set of numbers is commonly referred to as the "median". The median is a statistical measure of central tendency that represents the value that separates the lower and upper halves of a dataset.

To find the median of a set of numbers, the numbers must first be arranged in order from lowest to highest (or highest to lowest). If the dataset contains an odd number of values, the median is the middle value.

For example, if we have the set of numbers {1, 3, 5, 7, 9}, the median is 5, which is the value that separates the lower half {1, 3} from the upper half {7, 9}.

If the dataset contains an even number of values, the median is the average of the two middle values. For example, if we have the set of numbers {2, 4, 6, 8}, the median is (4 + 6) / 2 = 5, which is the average of the two middle values that separate the lower half {2, 4} from the upper half {6, 8}.

The median is a useful measure of central tendency because it is not affected by extreme values or outliers in the dataset, unlike the mean, which can be skewed by such values.

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There is an error in Step 1. The correct answer should be: a) Step 1, replace "angle bisector" with "perpendicular bisector"

An inscribed figure is, in essence, one shape drawn inside of another. A form drawn outside another shape is referred to as a circumscribed figure. All of a polygon's corners, or vertices, need to touch the circle in order for it to be enclosed inside of it.

The appropriate procedures for drawing a circle around a triangle are as follows:

Step 1: Create the perpendicular bisector of one triangle side .

Step 2: Create the perpendicular bisector of a different triangle side .

Step 3: Mark the place where the bisectors cross a point that represents the centre of the circle.

Step 4: Center the compass and adjust its length to reach any triangle vertex before drawing your circle's circumference.

Hence, option a is correct answer.

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B. The vector w is in Col(A) because Ax = w is a consistent system. One solution is x is the correct option.

To determine if w is in Col(A), we need to check if there exists a solution x such that Ax = w. If such a solution exists, then w is in Col(A).

However, we cannot determine if w is in Nul(A) without knowing more information about matrix A. The nullspace of a matrix is the set of all solutions x to the equation Ax = 0. It is possible for a vector to be in the column space but not in the nullspace, and vice versa.

So, the correct choice is:

B. The vector w is in Col(A) because Ax = w is a consistent system. One solution is x = (-2,4)^T.

We cannot determine if w is in Nul(A) without more information about A.

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A. The number of solutions to the equation x1 + x2 + x3 + x4 + x5 + x6 = 29, where xi is a nonnegative integer, is: a) 22C6, b) 29C5, c) 23C5, and d) 6C122C5 - 1 - 6C114C5.

B.

a) Since xi>1, we can subtract 2 from each xi, which gives us the equation x1' + x2' + x3' + x4' + x5' + x6' = 17, where xi' = xi - 2. Then, we can use stars and bars to get the solution as (17 + 6 - 1)C(6 - 1) = 22C5.

b) Since x2>1, x5>5, and x6>0, we can subtract 2 from x2, subtract 6 from x5, and subtract 1 from x6, which gives us the equation x1 + x2' + x3 + x4 + x5' + x6' = 20, where xi' = xi - 2 for i = 2, 5, and xi' = xi - 1 for i = 6. Then, we can use stars and bars to get the solution as (20 + 6 - 1)C(6 - 1) = 29C5.

c) Since x3≥5, we can subtract 5 from x3, which gives us the equation x1 + x2 + x3' + x4 + x5 + x6 = 24, where x3' = x3 - 5. Then, we can use stars and bars to get the solution as (24 + 6 - 1)C(6 - 1) = 23C5.

d) To find the number of solutions where xi<8 for all i and xi>8 for at least one i, we can subtract 1 from x5, which gives us the equation x1 + x2 + x3 + x4 + x5' + x6 = 21, where x5' = x5 - 1. Then, we can use stars and bars to get the total number of solutions where xi<8 for all i as (21 + 6 - 1)C(6 - 1) = 26C5. To find the number of solutions where xi<8 for all i and xi>8 for at least one i, we can subtract 9 from xi for the i that is greater than 8 and subtract 1 from x5, which gives us the equation x1 + x2 + x3 + x4 + x5' + x6' = 12, where x5' = x5 - 1 and x_i' = x_i - 9 for i>8. Then, we can use stars and bars to get the total number of solutions where xi<8 for all i as (12 + 6 - 1)C(6 - 1) = 16C5. Therefore, the number of solutions where xi<8 for all i except xi>8 for at least one i is 26C5 - 16C5 = 6C122C5 - 1 - 6C114C5.

Note: In all cases, we use the stars and bars technique to count the number of nonnegative integer solutions to an equation of the form x1 + x2 + ... + xn = k. The answer

The set is not a vector space because it is not closed under addition.

To determine whether the given set is a vector space, we need to check if it satisfies the vector space axioms. Let's consider the closure under addition axiom. Let A = [a1 c1 b1 0] and B = [a2 c2 b2 0] be two 4x4 matrices in the set. Adding A and B, we get:

A + B = [a1+a2 c1+c2 b1+b2 0+0] = [a1+a2 c1+c2 b1+b2 0]

The resulting matrix is not guaranteed to be in the set because the sum of b1 and b2 (b1+b2) might not be equal to the sum of a1 and a2 (a1+a2). Therefore, the set is not closed under addition, and it is not a vector space.

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Answer:

b=32

Step-by-step explanation:

a^2+b^2=c^2

24^2+b^2=40^2

576+b^2=1600

b^2=1600-576

b^2=1024

b=32

The Euclidean distance between the two points (5,8,-2) and (-4,5,3) is approximately 10.72 (rounded to two decimal places).

Explanation: -

To calculate the Euclidean distance between two points (5,8,-2) and (-4,5,3), you can use the following formula:

Euclidean Distance = √((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2)

Here, (x1, y1, z1) = (5, 8, -2) and (x2, y2, z2) = (-4, 5, 3). Now, plug in the values:

Step 1. Calculate the differences between the co-ordinates;
  (x2 - x1) = (-4 - 5) = -9
  (y2 - y1) = (5 - 8) = -3
  (z2 - z1) = (3 - (-2)) = 5

Step 2. Square the differences:
  (-9)^2 = 81
  (-3)^2 = 9
  (5)^2 = 25

Step 3. Add the squared differences:
  81 + 9 + 25 = 115

Step 4. Take the square root:
  √115 ≈ 10.72

So, the Euclidean distance between the two points is approximately 10.72 (rounded to two decimal places).

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p is a prime number

p^2 divides ab with gcd(a, b) = 1,

then p^2 divides a or p^2 divides b.

1. Since gcd(a, b) = 1, we know that a and b are coprime, meaning they have no common factors other than 1.

2. Given that p is a prime number and p^2 divides ab, this implies that p divides either a or b (or both) due to the Fundamental Theorem of Arithmetic.

3. Let's assume p divides a. Then, we can write a = pk for some integer k.

4. Now, we know that p^2 divides ab, which means ab = p^2m for some integer m.

Substitute a with pk from step 3: ab = (pk)b.

5. Thus, p^2m = pkb. Since p is a prime number, by Euclid's Lemma, we know that p must divide either kb or b itself. We already assumed p divides a, so p cannot divide b (as gcd(a, b) = 1). Therefore, p must divide kb.

6. As p divides a (a = pk) and p divides kb, we can conclude that p^2 divides a. So, p^2 divides a.

7. If we instead assumed p divides b, we would arrive at a similar conclusion: p^2 divides b.

In summary, if p is a prime number and p^2 divides ab with gcd(a, b) = 1, then either p^2 divides a or p^2 divides b.

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The graph is given in the image below:

To plot a full cycle of y = 7sin(6x), begin by dividing the period (2π) by six to obtain π/3, which is then used to mark every increment of π/6 along the x-axis.

Additionally, since y ranges from -7 to 7, label the y-axis in increments of either 1 or 2.

Plot the key points at (0,0), (π/12,7), (π/6,0), (π/4,-7), and (π/3,0), and finally connect them smoothly with a curve to complete the plot of one full cycle.

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The normal vector to the tangent plane is the opposite of this gradient, which is:

[tex]n = (-14e^{64}, 4, -1)[/tex]

The gradient of the surface is given by:

∇z = ( ∂z/∂x, ∂z/∂y, ∂z/∂z )

where ∂z/∂x and ∂z/∂y are the partial derivatives of z with respect to x and y, respectively.

The gradient of the surface at that point must first be determined before we can determine the normal vector to the tangent plane of the surface [tex]z=7e^{x^{2} } -4y[/tex] at the point (8, 16, 7).

Taking the partial derivatives, we get:

[tex]∂z/∂x = 14e^{x^{2} }[/tex]

∂z/∂y = -4

Plugging in the values x=8 and y=16, we get:

∂z/∂x = [tex]14e^{(8)^2} = 14e^{64}[/tex]

∂z/∂y = -4

Therefore, the gradient of the surface at the point (8, 16, 7) is:

∇z = ( [tex]14e^{64}, -4,[/tex]∂z/∂z )

The last component of the gradient (∂z/∂z) is always equal to 1, so we have:

∇z = ( [tex]14e^{64},[/tex] -4, 1 )

This gradient is perpendicular to the tangent plane of the surface at the point (8, 16, 7). Therefore, the normal vector to the tangent plane is the opposite of this gradient, which is:

n =[tex](-14e^{64}, 4, -1)[/tex]

The component of the normal vector in the z-direction is -1, as given in the problem statement.

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Answer:

x= - 20

y= 6

Step-by-step explanation:

2x + 5y = -10

-2x + y = 46

if u solve it

it is 6y= 36 so y =6

and substitute y value into the equation:

-2x+ 6 = 46

-2x= 40

so x= -20

The period of the function represent the given context is (8.5, 2900).

The period of this function is 8.5 seconds, and it represents the time it takes for the person's lungs to fill up with air, then empty out again.

The graph shows that the volume of air in the person's lungs is at its maximum (2900 ml) at the start of each period, and then decreases over time until it reaches its minimum (1100 ml) at the end of each period.

Therefore, the period of the function represent the given context is (8.5, 2900).

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The probability that William is the first to throw a 6 is approximately 125/1296 or roughly 0.096.

You asked for the probability that William is the first to throw a 6 when Nicola, Stephanie, Kathryn, and William take turns rolling a fair die.

To calculate this probability, we need to consider the following events:
1. William rolls a 6.
2. Nicola, Stephanie, and Kathryn all roll a number other than 6.

The probability of each person rolling a 6 is 1/6, while the probability of not rolling a 6 is 5/6.

So, the probability that William is the first to roll a 6 is:

P(William rolls 6) = P(Nicola rolls not 6) × P(Stephanie rolls not 6) × P(Kathryn rolls not 6) × P(William rolls 6)
= (5/6) × (5/6) × (5/6) × (1/6)

Now, multiply the fractions:

= (5 × 5 × 5 × 1) / (6 × 6 × 6 × 6)
= 125 / 1296

So, the probability that William is the first to throw a 6 is approximately 125/1296, or roughly 0.096 (rounded to 3 decimal places).

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a) The ratio of the photography students to the chess students is 2 : 1

b) The ratio of photography students to all students is 1 : 4

c) The ratio of chess students to all students is 1 : 8

Given data ,

Let the number of cooking students = 9

Let the number of chess students = 4

Let the number of photography students be = 8

Let the number of robotics students = 11

So , the total number of students = 32

a)

The ratio of the photography students to the chess students = 8 / 4

On simplifying the proportion , we get

The ratio of the photography students to the chess students = 2 : 1

b)

The ratio of photography students to all students = 8 / 32

The ratio of photography students to all students = 1 : 4

c)

The ratio of chess students to all students = 4 / 32

The ratio of chess students to all students = 1 : 8

Hence , the proportion is solved

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The complete question is attached below :

The table shows the number of students who signed up for different after school activities. Each student signed up for exactly one activity.

An expression for the number of bacteria after t hours is [tex]N(t) = e^{(10 + 2t)}[/tex].

To find an expression for the number of bacteria after t hours, given that a bacteria culture starts with e¹⁰ bacteria and grows at a rate proportional to its size, we can use the exponential growth formula:
[tex]N(t) = N(0) * e^{(kt)}[/tex]
where N(t) is the number of bacteria after t hours, N(0) is the initial number of bacteria, k is the growth constant, and t is the time in hours.

First, we are given that after 3 hours there are e¹⁶ bacteria.

Let's use this information to find the growth constant k:
[tex]e^{16} = e^{10} * e^{(3k)} \\e^{6} = e^{(3k)}[/tex]

Now, we can find k by dividing both sides by 3:
6 = 3k
k = 2

Now that we have the growth constant k, we can plug it into the exponential growth formula:
[tex]N(t) = e^{10} * e^{(2t)}[/tex]
which is equivalent to:
[tex]N(t) = e^{(10 + 2t)}[/tex]

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The total number of ways to partition the set of n+1 distinct objects into k nonempty subsets is {n+1,k} = k{n,k} + {n,k-1}, as required.

To show that {n+1,k}=k{n,k}+{n,k-1}, we can use a combinatorial argument.

Consider a set of n+1 distinct objects. We want to partition this set into k nonempty subsets. We can do this in two ways

Choose one of the n+1 objects to be the "special" object. Then partition the remaining n objects into k-1 nonempty subsets. This can be done in {n,k-1} ways.

Partition the n+1 objects into k nonempty subsets, and then choose one of the subsets to be the subset that contains the special object. There are k ways to choose the subset that contains the special object, and once we have chosen it, we need to partition the remaining n objects into k-1 nonempty subsets. This can be done in {n,k-1} ways.

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The null hypothesis (H0) for this probability would be that there is no effect of taking the vitamin on the chances of getting high blood pressure.

The alternative hypothesis (Ha) would be that taking the vitamin reduces the chances of getting high blood pressure.

Therefore, the hypotheses for this claim can be written as:

H0: The proportion of Americans with hypertension who take the vitamin is the same as the proportion of Americans with hypertension who do not take the vitamin.

Ha: The proportion of Americans with hypertension who take the vitamin is lower than the proportion of Americans with hypertension who do not take the vitamin.

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