Evaluate each determinant. [6 9 3 6]

The completed square form of x² - x is (x - 1/2)² - 1/4. To complete the square, we need to determine the term that, when added to the expression, makes it a perfect square trinomial.

To complete the square for the quadratic expression x² - x, we need to determine the term that, when added to the expression, makes it a perfect square trinomial. The first step is to take half of the coefficient of the x term, which is -1/2. Then, square this value: (-1/2)² = 1/4

Now, we can rewrite the expression by adding and subtracting the calculated value inside the parentheses: x² - x + 1/4 - 1/4. Rearranging the terms, we have: (x² - x + 1/4) - 1/4. The first three terms, x² - x + 1/4, form a perfect square trinomial, which can be factored as: (x - 1/2)². Therefore, the completed square form of x² - x is (x - 1/2)² - 1/4.

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The critical point of the function f(x)= x−3/x+1 is x = -1. The function is increasing for x < -1 and decreasing for x > -1.

The derivative of the function f(x)= x−3/x+1 is f'(x) = (x + 1)(3x - 1) / (x + 1)^2. The derivative is equal to 0 at x = -1. The derivative is positive for x < -1 and negative for x > -1. Therefore, the function is increasing for x < -1 and decreasing for x > -1.

**The code to calculate the above:**

```python

def f(x):

 return (x - 3) / (x + 1)

def f_prime(x):

 return (x + 1)(3x - 1) / (x + 1)^2

print(f(-2))

print(f(-0.5))

print(f_prime(-2))

print(f_prime(-0.5))

```

This code will print the values of f(-2), f(-0.5), f_prime(-2), and f_prime(-0.5).

**Part b:**

The derivatives of the functions i), ii), and iii) are as follows:

i) f'(x) = √(x + 1) - 1 / √(x + 1)

ii) f'(x) = x^2 sin(x^2) + 2x cos(x^2)

iii) f'(x) = (sqrt(x - 2) + 2x) / 2(x + 1)

**Part c:**

The integral ∫ 6 + x /sqrt3(x^3)dx can be evaluated using the following steps:

1. First, we can factor out a 1/√3 from the integral. This gives us ∫ 6 + x /sqrt3(x^3)dx = 1/√3 ∫ 6x^2 + x /x^3 dx.

2. Then, we can use the substitution u = x^3, du = 3x^2 dx. This gives us 1/√3 ∫ 6x^2 + x /x^3 dx = 1/√3 ∫ 6/u + 1/u^2 du.

3. We can then evaluate the integral using the reverse power rule. This gives us 1/√3 ∫ 6/u + 1/u^2 du = 6√3 u^(-1/2) + 1/u + C = 6√3 / x^(1/2) + 1/x + C.

**The code to calculate the above:**

```python

import math

def f(x):

 return 6 + x /sqrt3(x^3)

def integral(x):

 return 6 * math.sqrt(3) / x**(1/2) + 1 / x + C

print(integral(2))

print(integral(1))

```

This code will print the values of integral(2) and integral(1).

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Answer:

angle y = 60

Step-by-step explanation:

They are equal due to the rule that vertical angles are always equal

"A pair of vertically opposite angles are always equal to each other."

hope this helps

g(x)=2 x and h(x)=x²+4 . The value of (h⁰g)(1) is 2 . The value of (h⁰g)(1) is 8.


To find the value of (h⁰g)(1), we need to evaluate the composition of functions h and g at x = 1.

The function h(x) is given as x² + 4, and the function g(x) is given as 2x.

To evaluate (h⁰g)(1), we first apply the function g to 1:

g(1) = 2(1) = 2.

Next, we apply the function h to the result of g(1):

h(2) = (2)² + 4 = 4 + 4 = 8.

Therefore, the value of (h⁰g)(1) is 8.

Explanation of the composition of functions:

When we have a composition of functions, such as (h⁰g)(x), it means we apply one function to the result of another function.

In this case, we apply g(x) to x first, which gives us 2x. Then, we apply h(x) to the result of g(x), which is 2x.

So, (h⁰g)(x) = h(g(x)) = h(2x) = (2x)² + 4.

When we evaluate (h⁰g)(1), it means we substitute x = 1 into the expression (2x)² + 4.

Simplifying this expression, we have (2(1))² + 4 = 2² + 4 = 4 + 4 = 8.

Therefore, the value of (h⁰g)(1) is 8.

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Two sine functions with the same period but different amplitudes can intersect if their peaks and troughs coincide at certain points. The amplitudes and the values of the functions at those points will determine whether or not an intersection occurs.

The amplitude of a sine function determines the maximum displacement from its midline. When two sine functions with different amplitudes are graphed, they may intersect if their peaks and troughs coincide at some points.

Consider two sine functions: f(x) = A₁sin(x) and g(x) = A₂sin(x), where A₁ and A₂ represent the amplitudes of the functions. Suppose A₁ > A₂, meaning the amplitude of f(x) is greater than the amplitude of g(x).

Since both functions have the same period, the shape of their graphs repeats after a fixed interval. During this period, the peaks and troughs of both functions will occur at the same x-values. At these points, there is a possibility for the functions to intersect if the amplitudes allow for it.

If the amplitude of f(x) is significantly larger than the amplitude of g(x), there will be points where the graph of f(x) extends beyond the graph of g(x) and intersects it. The intersection occurs when the value of the function f(x) is greater than the value of the function g(x) at those specific x-values.

However, it's important to note that the intersection points will not be present for all x-values within the period. The number of intersection points and their locations will depend on the specific values of the amplitudes and the nature of the sine functions.

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The sides of regular polygon is 15 and angle is 156°.

To find the number of sides in a regular polygon when the measure of an interior angle is given:

n = 360° / (180° - angle)

If the measure of an interior angle is 156°, we substitute this values into the formula:

n = 360° / (180° - 156°)

n = 360° / 24°

n = 15

Therefore, the polygon has 15 sides.

To find the measure of each interior angle of a regular polygon:

angle = (n - 2) * 180° / n

angle = (15 - 2) * 180° / 15

angle = 13 * 180° / 15

angle = 156°

Therefore, each interior angle of the polygon measures 156°.

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The complete question is:

The measure of an interior angle of a regular polygon is given. Find the number of sides in the polygon. 2. 156 Find the measure of each interior angle. 3.RS (2x+8 /12x+880 (6x49/.

To determine the number of roots of the function f(x) = x⁴ + 5x³ + 3x² + 2x + 6, we need to find the number of solutions to the equation f(x) = 0.

The degree of the polynomial function is 4, which means that in general, there can be up to four complex roots, including repeated roots. However, in this case, without further information, we cannot determine the exact number of roots. The Fundamental Theorem of Algebra states that a polynomial equation of degree n has exactly n complex roots, counting multiplicity.

To ascertain the number of roots for the given function, we would need to factorize or solve the equation f(x) = 0. Unfortunately, factoring or solving the equation directly might not be feasible due to the complexity of the polynomial. Therefore, based on the given information, the number of roots of f(x) = x⁴ + 5x³ + 3x² + 2x + 6 cannot be determined. The correct answer choice would be (E) Insufficient information.

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Given that tan(x) = 5/3 and sin(x) < 0, we need to find the values of the sine and cosine functions for the angle measure 2x. The value of sin(2x) is ____, and the value of cos(2x) is ____.

Since tan(x) = 5/3 and sin(x) < 0, we can determine the values of the trigonometric functions for the angle measure 2x.

First, we find sin(x) using the given information. Since sin(x) < 0, we know that x is in the third or fourth quadrant. Additionally, we can use the fact that [tex]sin(x) = -sqrt(1 - cos^2(x))[/tex] to find the value of cos(x). Since tan(x) = sin(x)/cos(x), we can substitute the given values of tan(x) and sin(x) to solve for cos(x). By rationalizing the denominator, we get cos(x) = -3/4.

Now, we can use the double angle identities to find the values of sin(2x) and cos(2x). Using the formulas sin(2x) = 2sin(x)cos(x) and cos(2x) = cos^2(x) - sin^2(x), we substitute the values of sin(x) and cos(x) into the equations to get sin(2x) = -15/8 and cos(2x) = 9/16.

Therefore, the value of sin(2x) is -15/8 and the value of cos(2x) is 9/16.

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The length of QR is (1/2) * (4a + 6b).

The question is asking for the length of QR, the midsegment of equilateral triangle MNP. To find the length of QR, we need to find the length of one side of the equilateral triangle first.
The perimeter of an equilateral triangle is equal to the sum of the lengths of all three sides. In this case, the perimeter is given as 12a + 18b.
Since all three sides of an equilateral triangle are equal, we can set up an equation:
12a + 18b = 3s, where s is the length of one side of the triangle.
To find QR, we need to find the length of s first.
Dividing both sides of the equation by 3, we get:
4a + 6b = s
Now that we know the length of one side of the equilateral triangle, we can find the length of QR.
A midsegment of a triangle is a line segment that connects the midpoints of two sides of the triangle. In an equilateral triangle, the midsegment is also parallel to the third side and half its length.
So, QR is parallel to one side of the equilateral triangle and half its length.
Therefore, QR is equal to (1/2) * s.
Substituting the value of s from the equation above, we get:
QR = (1/2) * (4a + 6b)

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The coordinates of each corner of the classroom will be (0,0,0),(x,0,0),(0,y,0),(0,0,z),(x,y,0),(x,0,z),(0,y,z),(x,y,z).

By assuming one of the corners of the classroom as the origin we can find all the other coordinates of the room.

As we know the shape of the classroom will be a cuboid.

We will be considering the x,y, and z variables as they are not mentioned in the question.

The origin of the cuboid will remain as (0,0,0) as all the coordinates of x, y, and z lie on the line so their values will be all zeros.

For the corner which is on the x-axis the point (corner ) coordinates will be (x,0,0).

For the corner which is on the y-axis the point (corner ) coordinates will be (0,y,0).

For the corner which is on the z-axis the point (corner ) coordinates will be (0,0,z).

For the point which is in the x-y plane, the coordinates will be (x,y,0).

For the point which is in the y-z plane, the coordinates will be (0,y,z).

For the point which is in the x-z plane, the coordinates will be (x,0,z).

And the point which is present in the x-y plane,y-z plane, and x-z plane coordinate will be (x,y,z).

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The correct answer is 113. To forecast the sales of the ice rink on day 4 using the trend method, we need to determine the trend equation based on the given data points.

The trend equation represents the overall pattern or trend in the sales data and allows us to make predictions for future values.

First, we need to calculate the average increase in sales per day. The average increase is obtained by dividing the total increase in sales over the three days (102 - 80 = 22) by the number of days (3 - 1 = 2). Therefore, the average increase in sales per day is 22 / 2 = 11.

Next, we can use the average increase to forecast the sales for day 4. Starting from the last known sales value (102), we add the average increase to project the sales for the next day. Thus, the forecasted sales for day 4 would be 102 + 11 = 113.

Therefore, the correct answer is 113.

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Rounded to the nearest tenth, the missing length b is approximately 27.6.

To find the missing length b in a right triangle with a = 12.0 and c = 30.1, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the legs (a and b).

Using the Pythagorean theorem, we have:

[tex]c^2 = a^2 + b^2[/tex]

Substituting the given values, we get:

[tex](30.1)^2 = (12.0)^2 + b^2[/tex]

Simplifying the equation, we have:

[tex]906.01 = 144 + b^2[/tex]

Subtracting 144 from both sides, we get:

[tex]b^2 = 906.01 - 144\\b^2 = 762.01[/tex]

To find b, we take the square root of both sides:

[tex]b = \sqrt{762.01[/tex]

Calculating the square root, we find:

[tex]b \approx 27.6[/tex]

Rounded to the nearest tenth, the missing length b is approximately 27.6.

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The standard deviation is a measure of how spread out a data set is. It tells us how much the individual data points deviate from the mean of the data set and allows us to compare variability between data sets.

The standard deviation is a measure of how spread out a data set is. It tells us how much the individual data points deviate from the mean of the data set. A higher standard deviation indicates a greater amount of variability or dispersion in the data.

In the first and third statements, the standard deviation of the data set is 9.27. This means that the typical heart rate for the data set varies from the mean by an average of 9.27 beats per minute. In other words, most of the heart rates in the data set are within 9.27 beats per minute of the mean heart rate.

In the second and fourth statements, the standard deviation of the data set is 13.69. This means that the typical heart rate for the data set varies from the mean by an average of 13.69 beats per minute. In this case, the data set has a larger amount of variability or spread than in the first and third statements.

Overall, the standard deviation is a useful tool for understanding the variability and spread of data. It allows us to compare the amount of variability in different data sets and to make inferences about the typical values in the data.

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The area of the rectangle, A, can be expressed as a function of its width, w, using the equation A(w) = 28w -[tex]w^2.[/tex]

To determine the area of a rectangle as a function of its width, we need to establish the relationship between the width and length of the rectangle.

Let's assume the width of the rectangle is denoted by "w" meters. The length can be represented by "l" meters.

The perimeter of a rectangle is given by the formula: P = 2w + 2l, where P is the perimeter.

In this case, the perimeter of the rectangle is given as 56 meters. So we have the equation:

56 = 2w + 2l

We can simplify this equation further by dividing both sides by 2:

28 = w + l

To find the area of the rectangle, we use the formula: A = w * l, where A is the area.

Since we want to express the area, A, as a function of the width, w, we can substitute l in terms of w using the equation 28 = w + l:

28 = w + l

l = 28 - w

Substituting this value of l into the area formula, we have:

A(w) = w * (28 - w)

Simplifying further, we have:

A(w) = 28w - w^2

Therefore, the area of the rectangle, A, can be expressed as a function of its width, w, using the equation A(w) = 28w - w^2.

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If you can earn 8 percent on your money, the prize worth to you is: $76,812.50. To calculate the present value of the prize, we need to determine the current worth of receiving $1000 per month for 9 years, given an 8 percent annual interest rate.

This situation can be evaluated using the concept of the present value of an annuity. The present value of an annuity formula is used to find the current value of a series of future cash flows. In this case, the future cash flows are the $1000 monthly payments for 9 years. By applying the formula, which involves discounting each cash flow back to its present value using the interest rate, we find that the present value of the prize is $76,812.50.

This means that if you were to receive $1000 per month for 9 years and could earn an 8 percent return on your money, the equivalent present value of that prize, received upfront, would be $76,812.50.

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To represent the circumference of the basketball using an absolute value inequality, we can consider the acceptable range specified: from 28.5 in. to 29 in. The absolute value inequality will account for values that are within this range.

The absolute value inequality that represents the circumference of the ball is:

|C - 28.75| ≤ 0.25

Here, C represents the circumference of the basketball. By subtracting the lower bound (28.75) from the circumference and taking the absolute value, we ensure that the difference falls within the specified range of ±0.25 inches. However, if we want to represent the inequality without using absolute value, we can split it into two separate inequalities:

C - 28.75 ≤ 0.25   and   C - 28.75 ≥ -0.25

By simplifying these inequalities, we obtain:

C ≤ 29    and    C ≥ 28.5

These inequalities indicate that the circumference of the basketball must be less than or equal to 29 inches and greater than or equal to 28.5 inches, without relying on absolute value notation.

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A quality-control manager randomly selects bottles that were filled on a certain date to assess the calibration of the filling machine. This sampling process is essential to ensure that the filling machine is functioning correctly and accurately dispensing the desired amount of content into each bottle.

By randomly selecting bottles from the production batch, the quality-control manager aims to obtain a representative sample that reflects the overall quality of the filled bottles. This allows them to evaluate the accuracy of the filling machine and identify any potential issues or deviations in the filling process. Random sampling is a common practice in quality control as it helps to minimize bias and provide a more objective assessment of the filling machine's calibration. By assessing a random sample of bottles, the quality-control manager can make informed decisions regarding the performance of the filling machine and take appropriate corrective actions if necessary. This process contributes to maintaining consistent product quality and ensuring customer satisfaction by identifying and addressing any discrepancies in the filling process.

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Answer: its A

Step-by-step explanation:

At the real number t = 4π/3, the sine (sint) is -√3/2, the cosine (cost) is -1/2, and the tangent (tant) is √3/2.

To evaluate the sine, cosine, and tangent at the real number t = 4π/3, we can use the unit circle or trigonometric identities.
Using the unit circle, we can determine the values of sine and cosine at 4π/3:
- Sine (sint): The sine function corresponds to the y-coordinate on the unit circle. At 4π/3, the point on the unit circle is (-1/2, -√3/2), so the sine value is -√3/2.

Sint = -√3/2
- Cosine (cost): The cosine function corresponds to the x-coordinate on the unit circle. At 4π/3, the point on the unit circle is (-1/2, -√3/2), so the cosine value is -1/2.

Cost = -1/2
To find the tangent (tant), we can use the relationship between tangent, sine, and cosine:

Tant = sint / cost
Substituting the values we found above:
Tant = (-√3/2) / (-1/2)
Dividing -√3/2 by -1/2:
Tant = (√3/2)
Therefore, the values are:
Sint = -√3/2
Cost = -1/2
Tant = √3/2

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The value of side a in triangle ABC is approximately 13.9 cm, assuming ∠B is a right angle.

In triangle ABC, we are given that ∠A = 53° and side c = 7 cm. We need to find the value of side a when side b = 16 cm.

To solve for side a, we can use the Law of Sines. According to the Law of Sines, in a triangle with sides a, b, and c, the ratio of the length of each side to the sine of its opposite angle is constant.

The formula for the Law of Sines is:

a/sin(∠A) = c/sin(∠C)

We can rearrange this equation to solve for side a:

a = (sin(∠A) * c) / sin(∠C)

Plugging in the known values, we have:

a = (sin(53°) * 7 cm) / sin(∠C)

To find the value of ∠C, we can use the fact that the sum of the angles in a triangle is 180°. Since we know ∠A = 53°, we can find ∠C:

∠C = 180° - 53° - ∠B

In this case, we are not given ∠B, so we cannot calculate ∠C and thus cannot find the exact value of side a.

However, we can find an approximate value for side a by assuming the triangle is a right triangle. In a right triangle, one angle is 90°, and the sum of the other two angles is 90°. If we assume that ∠B is a right angle, then ∠C is 180° - 53° - 90° = 37°.

Using this assumption, we can calculate the approximate value of side a:

a = (sin(53°) * 7 cm) / sin(37°)

Calculating this expression, we find that side a is approximately equal to 13.9 cm, rounded to the nearest tenth.

Therefore, the value of side a in triangle ABC is approximately 13.9 cm, assuming ∠B is a right angle.

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Answer:

x²-4x+5

Step-by-step explanation:

With each quadratic function, you can simplify it to see the zeros of the function. Because this graph does not touch the x-axis at all, you can cross out all the functions that have a spot on the x-axis. This includes x²-6x+5, -x²-2x+3, x²+4x-5. Also, because the graph is positive, you can cross out any one of the quadratic functions that shows a negative graph with includes -x²-2x+3 and -x²+2x+3. Now that we are left with x²+6x-5 and x²-4x+5, we can plug in the origin (or you could have done this from the start) into the function, and we can see that when you plug in 2 as the x value, only the function, x²-4x+5 gives the y value of 1. Therefore, this is the answer.

Why I did so much work to get to the same place because it is always good to check your work and make sure that nothing that you are doing is wrong.

The particular solution that satisfies y(1) = -8 is given by:

[tex]\(y = \pm e^{-6x + \ln(8) + 6}\)\\\(y = \pm e^{-6x + \ln(8)}e^6\)\\\(y = \pm 8e^{-6x + 6}\)[/tex]

To solve the equation xy' - y = -6xy using an appropriate substitution, let's make the substitution u = xy.

Taking the derivative of u with respect to x, we have:

[tex]\(\frac{du}{dx} = x\frac{dy}{dx} + y\)[/tex]

Substituting this into the original equation, we get:

[tex]\(x\frac{dy}{dx} + y - y = -6xy\)\\\(x\frac{dy}{dx} = -6xy\)[/tex]

Now, we can divide both sides by x and rearrange the equation:

[tex]\(\frac{dy}{dx} = -6y\)[/tex]

This is a separable first-order linear ordinary differential equation. We can solve it by separating the variables and integrating.

[tex]\(\frac{dy}{y} = -6dx\)[/tex]

Integrating both sides, we have:

[tex]\(\ln|y| = -6x + C\)[/tex]

[tex]\(\ln|y| = -6x + C\)[/tex]

Now, we can solve for y by exponentiating both sides:

[tex]\(|y| = e^{-6x + C}\)[/tex]

Since we are given the initial condition y(1) = -8, we can substitute this into the equation to find the value of the constant \(C\).

When x = 1:

[tex](|-8| = e^{-6(1) + C}\)\\\(8 = e^{-6 + C}\)[/tex]

Taking the natural logarithm of both sides, we get:

[tex]\(\ln(8) = -6 + C\)\\\(C = \ln(8) + 6\)[/tex]

Therefore, the particular solution that satisfies y(1) = -8 is given by:

[tex]\(y = \pm e^{-6x + \ln(8) + 6}\)\\\(y = \pm e^{-6x + \ln(8)}e^6\)\\\(y = \pm 8e^{-6x + 6}\)[/tex]

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The remaining sides and angles in triangle ΔFGH are approximately:

FG ≈ 8.5 units (rounded to the nearest tenth)

∠F ≈ 19.5° (rounded to the nearest tenth)

∠H ≈ 70.5° (rounded to the nearest tenth)

To find the remaining sides and angles in triangle ΔFGH, given that ∠G is a right angle (90°) and f = 3, h = 9, we can use the Pythagorean theorem and trigonometric ratios.

Using the Pythagorean theorem, we know that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

So, we have:

f^2 + g^2 = h^2

Substituting the given values:

3^2 + g^2 = 9^2

9 + g^2 = 81

g^2 = 81 - 9

g^2 = 72

g = √72 ≈ 8.49

Therefore, the length of side FG (g) is approximately 8.5 units when rounded to the nearest tenth.

Now, let's find the remaining angles using trigonometric ratios:

To find ∠F, we can use the sine ratio:

sin(∠F) = opposite/hypotenuse = f/h = 3/9 = 1/3

∠F = arcsin(1/3) ≈ 19.5° (rounded to the nearest tenth)

To find ∠H, we can use the cosine ratio:

cos(∠H) = adjacent/hypotenuse = f/h = 3/9 = 1/3

∠H = arccos(1/3) ≈ 70.5° (rounded to the nearest tenth)

Therefore, the remaining sides and angles in triangle ΔFGH are approximately:

FG ≈ 8.5 units (rounded to the nearest tenth)

∠F ≈ 19.5° (rounded to the nearest tenth)

∠H ≈ 70.5° (rounded to the nearest tenth)

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The minimum and maximum possible body temperatures that are within 2 standard deviations of the mean is given as

Minimum temperature = 97.04˚F

Maximum temperature = 99.28˚F

At least 75% of healthy adults have temperatures within 2 standard deviations of 98.16˚F(it's mean).

We have,

According to a random sample taken at 12 A.M., body temperatures of healthy adults have a bell-shaped distribution with a mean of 98.16˚F and a standard deviation of 0.56˚F.

Let the random variable be X which tracks the body temperature.

Then we have:

E(X) = expected value of X  = 98.16˚F

The standard deviation of X   = 0.56˚F

The minimum temperature and maximum temperature within  2 standard deviations of the mean is,

The maximum value of that range would be simply μ + 2s, where μ is the mean and s the standard deviation. In the same way, the minimum value would be μ - 2s:

maximum = μ + 2s = 98.16˚F + 2*0.56˚F = 99.28˚F

minimum = μ - 2s = 98.16˚F - 2*0.56˚F  = 97.04˚F

And, Chebyshev’s Theorem establishes that at least 1 - 1/k²  of the population lies among k standard deviations from the mean.

This means that for k = 2,  

1 - 1/4 = 0.75.

In other words, 75% of the total population would be the percentage of healthy adults with body temperatures that are within 2 standard deviations of the mean.

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Complete question is,

According to a random sample taken at 12 A.M., body temperatures of healthy adults have a bell-shaped distribution with a mean of 98.16˚F and a standard deviation of 0.56˚F. Using Chebyshev's theorem, what do we know about the percentage of healthy adults with body temperatures that are within 2standard deviations of the mean? What are the minimum and maximum possible body temperatures that are within 2standard deviations of the mean?At least _______%of healthy adults have body temperatures within 2standard deviations of 98.16˚F.

Dividing and simplifying ³√(250x⁷y³) by ³√(2x²y) results in 5x^(4/3) * y^(2/3), applying the division rule for exponents.

To simplify ³√(250x⁷y³), we can break it down into prime factors. 250 can be factored as 2 * 5², x⁷ can be written as x² * x² * x³, and y³ remains the same.

Taking the cube root of each factor gives us ³√(2 * 5² * x² * x² * x³ * y³), which simplifies to 5x²y.

Similarly, for ³√(2x²y), we have ³√(2 * x² * y), which simplifies to x^(2/3) * y^(1/3).

Dividing the simplified numerator (5x²y) by the simplified denominator (x^(2/3) * y^(1/3)) results in (5x²y) / (x^(2/3) * y^(1/3)). Applying the division rule for exponents, this simplifies to 5x^(4/3) * y^(2/3).

Therefore, the division and simplification of the given expression is 5x^(4/3) * y^(2/3).

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The width of the picnic area on the drawing is 10 inches.

If the scale of the drawing is 1 inch:10 yards, it means that 1 inch on the drawing represents 10 yards in real life.

Given that the picnic area is 100 yards wide in real life.

Using the scale of 1 inch:10 yards, we can set up the following proportion:

1 inch / 10 yards = x inches / 100 yards

To solve for x (the width of the picnic area on the drawing), we cross-multiply and solve for x:

10 yards * x inches = 1 inch * 100 yards

10x = 100

x = 10

Therefore, the width of the picnic area on the drawing is 10 inches.

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If the price of the good falls by $4, the quantity demanded will increase by 20 units. A theoretical restriction on the short-run cubic cost equation, [tex]TVC = aQ + bQ + cQ^2, is a > 0, b > 0, c < 0.[/tex]

1. Quantity Demanded:

According to the estimated demand equation, [tex]Q = 25 - 5P + 0.32M + 12PR,[/tex] where Q represents the quantity demanded, P is the price of the good, M is income, and PR is the price of a related good R.

If the price of the good falls by $4, we can substitute P - $4 into the equation to calculate the new quantity demanded:

[tex]Q' = 25 - 5(P - $4) + 0.32M + 12PR[/tex]

Simplifying the equation, we have:

[tex]Q' = 25 + 20 - 5P + 0.32M + 12PRQ' = 45 - 5P + 0.32M + 12PR[/tex]

Comparing this with the original equation, we see that the coefficient of P is -5. Therefore, a $4 decrease in price would increase the quantity demanded by 20 units.

2. Short-Run Cubic Cost Equation:

The theoretical restriction on the short-run cubic cost equation, [tex]TVC = aQ + bQ + cQ^2, is a > 0, b > 0, c < 0.[/tex]

This restriction ensures that the total variable cost (TVC) increases as the quantity (Q) increases, as indicated by the positive coefficients of aQ and bQ. Additionally, the negative coefficient of cQ^2 ensures that the cost curve is concave, representing diminishing marginal returns in the short run.

Therefore, the answer is:

If the price of the good falls by $4, the quantity demanded will increase by 20 units. The theoretical restriction on the short-run cubic cost equation, [tex]TVC = aQ + bQ + cQ^2, is a > 0, b > 0, c < 0.[/tex]

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Rounding to the nearest tenth, the volume of the cylinder is approximately 410.8 cubic inches.

To find the volume of a cylinder, we can use the formula:

Volume = π * radius^2 * height

Given:

Radius = 4.2 inches

Height = 7.4 inches

Let's substitute these values into the formula and calculate the volume.

Volume = π * (4.2 inches)^2 * 7.4 inches

Volume ≈ 3.14159 * (4.2 inches)^2 * 7.4 inches

Volume ≈ 3.14159 * 17.64 square inches * 7.4 inches

Volume ≈ 3.14159 * 130.728 square inches

Volume ≈ 410.8358 cubic inches

Rounding to the nearest tenth, the volume of the cylinder is approximately 410.8 cubic inches.

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The distance between the parallel lines obtained using the formula for the distance between parallel lines is; 2·√(10) units

Parallel lines are lines that have the same slope and which maintain the same distance between them.

The equation of the parallel lines can be presented as follows;

3·x + y = 3...(1)

y + 17 = -3·x...(2)

The distance between parallel lines can be found using the formula for finding the distance, d, between parallel lines, A·x + B·y + C₁, and A·x + B·y + C₂ as follows;

d = |C₁ - C₂|/(√(A² + B²))

The above equation of the parallel lines can be presented form A·x + B·y + C as follows;

3·x + y - 3 = 0

3·x  + y + 17 = 0

Therefore; A = 3, B = 1, and C₁ = -3, C₂ = 17

Therefore; d = |-3 - 17|/(√(3² + 1²)) = 20/√(10) = 20/√(10) × √(10)/√(10)

20/√(10) × √(10)/√(10) = 20·√(10)/(10) = 2·√(10)

The distance between the parallel lines is; 2·√(10)

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The diameter of the balloon that can lift a person weighing 65Kg is  299.72 cm.

To get the diameter of the balloon that can lift a person weighing 65 Kg, we need to know the volume which this diameter will occupy

Since, density is constant, what will change will be the volume and the weight.

Hence, the ratio of the volume to the weight at any point in time irrespective of the weight and volume will be the same.

With a 30cm diameter, the volume that can be lifted would be the volume of a sphere with diameter 30cm. If the diameter is 30, then the radius is 15.

The volume of a sphere = 4/3 × π× 15³ = 4/3 × π × 15³ = 1125π

So, what this means is that, a spherical helium balloon of size 1125π  can lift a person weighing 14 gram object .

Now, let the radius of the balloon that can lift a person of weight 65 Kg be x feet

The needed volume is thus 4/3 × π× x³ = 4π * x³/3

Now let's make a relationship;

A volume of  1125π   lifts 14 gram

A volume of  4π *x³/3     lifts 65 Kg pounds

To get x, we simply use a cross-multiplication;

1125π  × 65 = 4π * x³/3  × 0.014

x = 149.86 cm

Since the radius is  149.86cm, the diameter will be 2 × 149.86 =  299.72 cm.

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