To save for retirement, Ed is investing 14.286 thousand dollars this year. Write the function R(t) that shows the stream of money flowing into Ed's investment account under each of the following circumstances. Assume a continuous income stream. Do NOT round any values (a) Ed continues to invest the same amount each year for the next 20 years. R(t)-14.286thousand dollars per year (b) Ed increases his investment by 3.75 thousand dollars per year over the next 20 years. A(t)-18.036 | X thousand dollars per year (c) Ed increases his investment by 3.75% each year over the next 20 years. R(t)- thousand dollars per year

Answer:

  10.309 in²/s

Step-by-step explanation:

Given A³ = 36πV² and V' = 18 in³/s, you want to know A' when A=153.24 in² and V=178.37 in³.

Using implicit differentiation, we have ...

  3A²·A' = 36π·2V·V'

  A' = (36π·2)/3·V/A²·V' = 24πV/A²·V'

  A' = 24π·(178.37 in²/(153.24 in²)²·18 in³/s

  A' ≈ 10.309 in²/s

The surface area is increasing at about 10.309 square inches per second.

__

Additional comment

There are at least a couple of ways a calculator can be used to find the rate of change. The first attachment shows evaluation of the expression we derived above. The second attachment shows the rate of change when the area is expressed as a function of the volume.

The result rounded to 5 significant figures is the same for both approaches.

The y intercept of line A is -2 times the y intercept of line B.

The line's intersection with the y-axis on a graph is known as the y-intercept. In every case, the associated x-coordinate is 0. This is accomplished by calculating the difference between the y- and x-coordinates and then dividing this difference.

standard form of equation of line:

y = mx + c

m is the slope and c is the y intercept.

Equation of line A:

3x + 2y = -6

Convert equation in standard form

2y = -3x -6

y = -3/2 x - 3

On comparing; y intercept = -3

Equation of line B. point taken (-3, 0) and (0,6)

y - y1 = m(x - x1)

slope m = (6 - 0)/(0 + 3) = 2

y - 0 = 2(x +3)

y = 2x  + 6

y intercept of line B is 6.

Thus,  y intercept  of line A is -2 times the y intercept of line B.

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Answer:

Parallelogram

Step-by-step explanation:

a quadrilateral (4 sides figure) whose opposite sides are parallel.  The opposite sides have the same length and opposite angles are equal.

Helping in the name of Jesus.

The solution is, (a/2) ^4  or a^4/16 is in Simplest Form of (8a^3)^-4/3.

An exponent refers to the number of times a number is multiplied by itself. For example, 2 to the 3rd (written like this: 23) means: 2 x 2 x 2 = 8. 23 is not the same as 2 x 3 = 6. Remember that a number raised to the power of 1 is itself.

here , we have,

(8a^-3)^-4/3

split into two parts

8^ -4/3   *  (a^-3)^-4/3

using the power to the power rule we can multiply the exponents

8^(-4/3)  *a^(-3*-4/3)

8^ (-4/3) * a^(4)

replace 8 with 2^3

(2^3)^(-4/3) * a^(4)

using the power to the power rule we can multiply the exponents

2^(3*-4/3) * a^(4)

2 ^ (-4) * a^4

the negative exponent means it goes in the denominator if it is in the numerator

a^4/2^4

make a fraction

(a/2) ^4

or a^2/16

Hence, The solution is, (a/2) ^4  or a^4/16 is in Simplest Form of (8a^3)^-4/3.

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Answer: The father is 44 and the son is 22 I think.

Step-by-step explanation:

Son is x then father's age is 2x so 20 years ago,12(x-20)=2x-20 so 10x=240–20=220 or x=22, and father's age is 44 years.

Answer:

Step-by-step explanation:

[tex](x-20)^2=x^2-20[/tex]

Expanding LHS gives:

  [tex]x^2-40x+400=x^2-20[/tex]

         [tex]-40x+400=-20[/tex]          (subtracted [tex]x^2[/tex] from both sides)

                   [tex]-40x=-420[/tex]        (subtracted 400 from both sides)

                         [tex]x=\frac{-420}{-40}[/tex]          (divided both sides by-40)

                         [tex]x=\frac{21}{2} =10\frac{1}{2}[/tex]

Answer:

Step-by-step explanation:

24=100°

28=x

24x=2800(cross multiply)

x=200

Point KLMN in the quadrilateral has coordinates that are [tex]N' (-3, 2)[/tex].

The enclosed figure of a quadrilateral has four sides. Raphael created the quadrilaterals in these geometric forms. a shape in quadrilaterals. The shape contains no right angles and just single set of parallel sides.

A closed form noted for having sides with various widths and lengths is a quadrilateral. It is a closed, two-dimensional polygon with four sides, four angles, and four vertices. The trapezium, parallelogram, rectangular, square, rhombus, and kite are just a few examples of quadrilaterals.

When a point is reflected across the [tex]y-axis[/tex], its [tex]x[/tex]-coordinate becomes its opposite while its [tex]y[/tex]-coordinate remains the same.

So to find the coordinates of [tex]N[/tex]', we need to reflect the point [tex]N(3, 2)[/tex]across the y-axis, which means we change the sign of its x-coordinate:

[tex]N' = (-3, 2)[/tex]

Therefore, the coordinates of point [tex]N'[/tex] are [tex](-3, 2)[/tex].

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The classmate's method is not correct because dividing the 1-week growth factor by 7 assumes that the growth rate is constant over each day of the week, which may not be the case so the correct 1-day correct growth factor is 1.036 and it is verified.

To calculate the correct 1-day growth factor, we can take the 7th root of the 1-week growth factor.

This is because if the height of the bamboo plant grows by a factor of x in one week, then the daily growth factor is the same as the weekly growth factor raised to the power of 1/7.

So the correct 1-day growth factor would be:

1-day growth factor = (1-week growth factor) raise to (1/7) = 1.27 raise to (1/7) ≈ 1.036

To verify that this is correct, we can raise the 1-day growth factor to the 7th power to obtain the 1-week growth factor:

1-week growth factor = (1-day growth factor)⁷ ≈ (1.036)⁷≈ 1.27

Therefore, the 1-day growth factor is approximately 1.036, and using this to find the 1-week growth factor yields approximately 1.27, which matches the given growth factor.

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Instead of putting aed and dec equal to one another angle, he should have set the total of aed and dec to 180°.

Vernon made the error of presuming that the two angles aed and dec are equal to one another, but in reality, they are on different sides of a straight line, adding up to 180°. Vernon calculated the wrong number for x by assuming the angles to be equal to one another. This error is frequent because it is simple to forget that these two angles complement one another. Before attempting to solve for any unknown values, it is crucial to thoroughly examine the provided information and the connections between the angles. One can find the right answer for x by understanding the connection between the two angles in this issue.

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Using the definitions of trigonometric functions for x = 0, we have tan(0) = 0 and cos(0) = 1, which simplifies to 0 * 1 = 0, and sin(0) = 0,

therefore tan(0) cos(0) = sin(0).

Define trigonometric.

Trigonometric refers to the branch of mathematics that deals with the relationships between the sides and angles of triangles, particularly right triangles. It involves the study of trigonometric functions such as sine, cosine, tangent, cotangent, secant, and cosecant, and their properties and applications in various fields including mathematics, science, engineering, and navigation.

For any angle x, we have the identity:

tan(x) = sin(x) / cos(x)

Setting x = 0, we get:

tan(0) = sin(0) / cos(0)

Since tan(0) = 0 and cos(0) = 1, we can simplify this equation to:

0 = sin(0)

which is true, since the sine of 0 degrees is 0. Therefore, we have shown that:

tan(0) cos(0) = sin(0)

Therefore, the identity tan(0) cos(0) = sin(0) is proven.

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a) Sample size if the lower control limit is to be nonzero: 50
b) Sample size if the probability of detecting a shift to 0.04 is to be 0.50: 100

a) How large should the sample size be if the lower control limit is to be nonzero?

n = (2σ / d)²We know that:

Center line (CL) = 0.01

Sigma (σ) = LCL = 0.005

d = Centerline - LCL = 0.01 - 0.005 = 0.005

Substituting the values in the formula, we get

n = (2 * 0.005 / 0.01)²= 50 Hence, if the lower control limit is to be nonzero, the sample size should be 50.

b) How large should the sample size be if we wish the probability of detecting a shift to 0.04 to be 0.50?

The probability of detecting a shift to 0.04 is denoted by β and is calculated using the following formula:

β = Φ [(-Zα/2 + Zβ) / √ (p₀q₀/n)], Where, Φ is the standard normal distribution function, Zα/2 is the critical value for the normal distribution at the (α/2)th percentile, Zβ is the critical value for the normal distribution at the βth percentile, p₀ is the assumed proportion of nonconforming items, q₀ is 1 – p₀, and n is the sample size.

In order to determine the sample size, we must first select a value for β. If we select a value for β of 0.50, then β = 0.50. This implies that we have a 50% chance of detecting a shift if one occurs. Since the exact value for p₀ is unknown, we assume that p₀ = 0.01, which is equal to the center line.

n = (Zα/2 + Zβ)² p₀q₀ / β², Substituting the values in the formula, we get,

n = (Zα/2 + Zβ)² p₀q₀ / β²= (1.96 + 0.674)² (0.01) (0.99) / 0.50²= 99.7 ≈ 100

Hence, if we wish the probability of detecting a shift to 0.04 to be 0.50, the sample size should be 100.

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Answer:

1

Step-by-step explanation:

Given values are:

x1 y1=(2,4)

x2 y2=( 4,6)

slop=(6-4)divide (4-2)=1

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thanks

Answer:

To find the population in the year 2001, we need to work backwards from the given population of 54,000 in 2003.

Let P be the population in the year 2001. From 2001 to 2003, there are two years, during which the population grows at a rate of 5% per annum. We can calculate the population in 2003 using the formula:

P * (1 + r)^n = 54,000

where r is the annual growth rate (5% or 0.05) and n is the number of years (2). Plugging in the values, we get:

P * (1 + 0.05)^2 = 54,000

Simplifying the equation, we get:

P = 54,000 / (1.05)^2

P = 48,543 (rounded to the nearest whole number)

Therefore, the population in the year 2001 was approximately 48,543.

To find the population in the year 2005, we can use the same formula with n = 2 + 2 = 4 (since we want to find the population four years after 2001):

P * (1 + 0.05)^4 = ?

Plugging in the value of P we just found, we get:

48,543 * (1 + 0.05)^4 = 60,723 (rounded to the nearest whole number)

Therefore, the population in the year 2005 would be approximately 60,723

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To find the total cost of purchasing the video game at the local Big Box store with a [tex]10[/tex] % discount and  [tex]6.5[/tex] % sales tax, we can use the following formula:

Total cost = (Sale amount—Discount) x (1 + Tax rate)

Plugging in the values we have:

Total cost [tex]= (49.95–4.995) \times 1.065[/tex]

Total cost  [tex]= 44.955 \times 1.065[/tex]

Total cost [tex]= 47.83[/tex]

Therefore, the total cost of purchasing the video game at the local Big Box store would be $ [tex]47.83[/tex].

At the online store, the game costs $  [tex]44.95[/tex] plus a shipping charge of $ [tex]4.00[/tex] , which comes to a total of $ [tex]48.95.[/tex]

So, purchasing the game at the online store would be slightly cheaper.

To calculate the total cost and interest earned for each principal amount and time period, we can use the following formula:

Total cost = Principal + Interest earned

Interest earned = Principal x Annual rate x Time period (in years)

Plugging in the values we have:

For the first row:

Interest earned [tex]= 2400 \times 0.049 \times0.5[/tex]

Interest earned = $58.80

Total cost [tex]= 2400 + 58.80[/tex]

Total cost = $ [tex]2,458.80[/tex]

For the second row:

Interest earned [tex]= 9460.12 \times 0.022 \times 2[/tex]

Interest earned = $ [tex]415.24[/tex]

Total cost  [tex]= 9460.12 + 415.24[/tex]

Total cost = $ [tex]9,875.36[/tex]

For the third row:

Interest earned = [tex]3923.87 \times 0.02 \times5[/tex]

Interest earned = $ [tex]392.39[/tex]

Total cost  [tex]= 3923.87 + 392.39[/tex]

Total cost = $ [tex]4,316.26[/tex]

Let's assume Jorge's commission for the month was C. We can set up the following equation:

[tex]C = 0.09 \times 89400[/tex]

We know that Harris sold the same amount, but made $447 more in commission. So we can set up another equation:

[tex]C + 447 = \times89400[/tex]

Where x is the commission rate for Harris.

We can substitute the value of C from the first equation into the second equation:

[tex]0.09 \times 89400 + 447 = \times 89400[/tex]

Simplifying:

[tex]8046 + 447 = 89400x[/tex]

[tex]8493 = 89400x[/tex]

[tex]x = 0.095[/tex] or  [tex]9.5[/tex] %

Therefore, Harris' commission rate is [tex]9.5%[/tex]%  .

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The surface area of the object in the image is 80 cm². To find the surface area of the object in the image, we need to add up the areas of all the faces.

Most of the lateral faces are rectangular. Sometimes, it might even be a parallelogram. Here is the Prism Formula: A prism has a surface area of (2BaseArea) + Lateral Surface Area. Prism volume equals Base Area + Height.

The length of the rectangle is 4 cm and the width is 3 cm. Therefore, the area of the rectangular base is:

Area of rectangular base = length × width = 4 cm × 3 cm = 12 cm²

Area of each side = length × width = 5 cm × 3 cm = 15 cm²

Since there are four identical sides, the total area of all four sides is:

Total area of four sides = 4 × Area of each side = 4 × 15 cm² = 60 cm²

Area of each triangular face = 1/2 × base × height = 1/2 × 4 cm × 2 cm = 4 cm²

Since there are two triangular faces, the total area of both triangular faces is:

Total area of both triangular faces = 2 × Area of each triangular face = 2 × 4 cm² = 8 cm²

Now we can add up all the areas to get the total surface area:

Total surface area = Area of rectangular base + Total area of four sides + Total area of both triangular faces

= 12 cm² + 60 cm² + 8 cm²

= 80 cm²

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T(U) is a subspace of W Since V is a subspace of U, the zero vector of V, 0, is in U. As T is linear, [tex]T(0) = 0_w[/tex] , where [tex]0_w[/tex]  is the zero vector of W. Therefore, [tex]0_{w}[/tex] is in T(U).

It can be proved  Since U is a subspace of V, the zero vector of V, 0y, is in U. Since T is linear, [tex]T(0_{y}) = 0_{w}[/tex] , where [tex]0_{w}[/tex] is the zero vector of W. So[tex]0_{w}[/tex]  is in T(U).

To show that T(U) is a subspace of W, we also need to show that T(U) is closed under vector addition and scalar multiplication. Let u1, u2 be vectors in U, and let c be a scalar. Then we have: [tex]T(u1 + u2) = T(u1) + T(u2)[/tex]  (since T is linear)

[tex]T(cu1) = cT(u1)[/tex]  (since T is linear)

Since U is a subspace of V, we have [tex]u1 + u2[/tex] and [tex]cu1[/tex]  are also in U. Therefore, [tex]T(u1 + u2)[/tex]  and [tex]T(cu1)[/tex]  are both in T(U), which shows that T(U) is closed under vector addition and scalar multiplication.

Thus, T(U) is a subspace of W.

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Answer:

I ug hi my name is princess

Answer:

1. Vertex on the y-axis (correct, since the x-coordinate of the vertex is 0)

2. Concave down (correct, since the coefficient of the x^2 term is negative)

3. One positive x-intercept (correct, since the discriminant is positive, indicating one real x-intercept)

4. Negative y-intercept (incorrect, since the y-intercept is positive)

5. Line of symmetry at y=3 (incorrect, since the line of symmetry is x=3)

Therefore, the correct answers are 1, 2, and 3.

Step-by-step explanation:

The derivative of the given function [tex]y = 5x^2 sec^{(-1)(2x-3)^2}[/tex] is [tex]dy/dx=-20x\sqrt{((2x-3)^2-1)}[/tex]

It can be derived as:

We can use the chain rule and the derivative of [tex]sec^{(-1)x}[/tex] which is [tex]-1/(x*\sqrt{(x^2-1)})[/tex]

First, we apply the chain rule to the function.

Let  [tex]u = (2x-3)^2[/tex], then:

[tex]y = 5x^2 sec^{(-1)u}[/tex]

[tex]dy/dx = d/dx [5x^2 sec^{(-1)u}][/tex]

[tex]dy/dx = d/dx [5x^2 sec^{(-1)[(2x-3)^2]}][/tex]

[tex]dy/dx= 5x^2 d/dx[sec^{(-1)u}][/tex]     (Using the chain rule)

Now, let [tex]v = u^{(1/2)} = (2x-3)[/tex].

Then:

[tex]dy/dx = 5x^2 d/dv [sec^{(-1)v}] dv/dx[/tex]          (Using the chain rule again)

We have:

[tex]d/dv [sec^{(-1)v}] = -1/(v*\sqrt{(v^2-1)}) = -1/[(2x-3)*\sqrt{((2x-3)^2-1)}][/tex]

Also, [tex]dv/dx = 2[/tex]

Substituting these back into the equation:

[tex]dy/dx = 5x^2 d/dv [sec^{(-1)v}] dv/dx[/tex]

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

Simplifying this expression gives:

[tex]dy/dx = -20x (2x-3)/[(2x-3)*\sqrt{((2x-3)^2-1)}][/tex]

[tex]dy/dx = -20x\sqrt{((2x-3)^2-1)}[/tex]

Therefore, the derivative of y with respect to x is:

[tex]dy/dx = -20x\sqrt{((2x-3)^2-1)}[/tex]

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The researcher can use the one-sample t-test to test the null hypothesis that the mean body temperature of residents in a nursing home is 98.6 F.

The one-sample t-test is a statistical test used to compare the mean of a sample to a known value or a hypothesized value. It is also known as the single-sample t-test.

The null hypothesis is that the mean body temperature of residents in a nursing home is 98.6 F, while the alternative hypothesis is that the mean body temperature of residents in a nursing home is not 98.6 F.

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The car sticker information indicates a highway average of 27 mpg, while the sample mean's standard deviation is around 1.455 mpg. The mean estimate's margin of error is roughly 3.06 mpg.

The simplest approach to figure out your gas mileage is to simply divide the distance driven by the quantity of petrol your car required to fill up. That amounts to kilometers traveled divided by petrol consumed.

a) The formula for the standard deviation of the sample mean, σx¯, is:

σx¯ = σ / sqrt(n)

where σ is the standard deviation of the population, and n is the sample size. In this case, σ = 6.5 mpg and n = 20. So,

σx¯ = 6.5 / sqrt(20) ≈ 1.455 mpg

b) To find the margin of error for the mean estimate at a 95% confidence level, we can use the following formula:

Margin of error = z* (σx¯)

where z* is the critical value for a 95% confidence interval, which is 1.96.

Therefore, the margin of error = 1.96 * 1.454 = 2.85 mpg.

c) To find the 95% confidence interval for the mean highway mpg for this vehicle, we can use the following formula:

Confidence interval = x¯ ± Margin of error

where x¯ is the sample mean, which can be calculated by adding up all the mpg readings and dividing by the sample size:

x¯ = (37.2+21.0+17.4+29.2+7.4+27.0+36.9+38.8+35.3+32.3+19.0+26.1+25.8+41.4+34.4+32.5+26.5+28.2+22.1) / 20

= 29.231 mpg

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A negative exponent indicates that the base should be inverted or flipped, and the exponent should become positive. Therefore, -45 in positive exponent form would be: 1/ (45²1) or 1/45

What is a negative component?

A negative exponent is a mathematical notation indicating that the number or variable should be inverted or flipped, and the exponent should be made positive. In other words, a negative exponent represents the reciprocal of the number or variable raised to the corresponding positive exponent.

For example, 2²-3 is read as "2 to the power of negative 3" and represents 1/(2²3), which is equal to 1/8. Similarly, x²-2 is read as "x to the power of negative 2" and represents 1/(x²2), which is the reciprocal of x squared.

Negative exponents are used in various mathematical and scientific applications, such as in calculations involving very small or large numbers, in scientific notation, and in the rules of exponents.

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Complete question:

Write -45 in positive exponent form.

Answer:

Step-by-step explanation:

a) To determine if the ball passes over the net, we need to find the height of the ball when it reaches x=12m (where the net is located).

y = -0.05x - 0.005x^2

y = -0.05(12) - 0.005(12)^2

y = -0.6 - 0.72

y = -1.32m

The ball reaches a height of -1.32m when it passes over the net, which is lower than the height of the net (0.9m). Therefore, the ball does pass over the net.

b) To determine if the serve is allowed, we need to find the horizontal distance where the ball lands and check if it is between the net and the service line (18m away from the origin).

We can find the distance the ball travels horizontally by finding the value of x when y = 0 (i.e., the height of the ball is 0 when it lands).

0 = -0.05x - 0.005x^2

0.005x^2 + 0.05x = 0

x(0.005x + 0.05) = 0

x = 0 (when the ball is hit) or x = -10 (when the ball lands)

The negative value of x indicates that the ball lands behind the server. Therefore, the serve is not allowed.

The value of μV is 40 cm³.

Given,The area of bottom of cylindrical container = 10 cm²The height of container = h = Mean height = 4 cm Standard deviation of height = σ = 0.2 cm We are supposed to find the mean volume of fluid in the container.In order to calculate the mean volume, first we need to calculate the volume of fluid in the container.Volume of a cylindrical container = πr²h Where, r is the radius of the base of the container.So, we need to calculate the value of r.The area of the bottom of the container is given as 10 cm².

We know that the area of the base of a cylinder is given as:Area of base of cylinder = πr² We are given that area of the base is 10 cm². So,10 = πr²r² = 10/πr = √(10/π) We can find the volume of fluid using the values we have.Volume of fluid = πr²h = π(√(10/π))² x 4 = 40 cm³We know that mean volume, μV is given as:μV = πr²μh So, we need to calculate the value of μh. We know that standard deviation σh is given as:σh = 0.2 cm So,μh = h = 4 cm So,μV = πr²μh = π(√(10/π))² x 4 = 40 cm³

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In the vector space model, the terms in the documents represent the dimensions in the multi-dimensional space.

A vector space model is a mathematical model that is utilized in information retrieval to represent a document as a multidimensional array of numbers (terms or concepts), in which each dimension corresponds to a specific term, and each value in the array represents the weight or importance of the corresponding term in the document. This representation is used to evaluate the relevance of documents to a given search query.

The dimensions in the multi-dimensional space of a vector space model correspond to the terms that the documents contain. In other words, each dimension represents a term, and each document is represented as a vector of weights or scores associated with each term.

The more frequently a term appears in a document, the higher the weight assigned to it. The goal of the vector space model is to represent documents in a way that facilitates comparison and ranking based on the similarity between the document and the query.

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The greatest number of 24 inch pieces the carpenter can cut from 6 boards of wood is 51.

A carpenter needs to cut 24-inch pieces of wood from a board that is 17 feet in length.

Therefore, the greatest number of 24 inches pieces the carpenter can cut from 6 of these boards of wood can be calculated as follows:

let's convert from feet to inches.

17 feet = 204 inches

1 board = 204 inches

6 board = ?

cross  multiply

length of 6 board = 204 × 6

length of 6 board = 1224 inches

Hence,

greatest number of 24 inch that can be cut from 6 boards = 1224 / 24

greatest number of 24 inch that can be cut from 6 boards = 51

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The types of inferences that will be made about population parameters are causation, estimation, and regression on the basis of relationship.

Causation is the process of showing the cause-and-effect relationship between two variables. In this case, one variable influences the other variable. This type of inference is significant when making decisions because it helps us understand how a change in one variable leads to a change in another variable.

Estimation: In statistical analysis, estimation refers to determining the possible value of an unknown population parameter. It is impossible to calculate the population parameters directly, and hence we use sample statistics to estimate them.

Regression analysis is the statistical technique used to identify the relationship between two variables. It involves estimating the coefficients of the model that best fit the data.

This type of inference helps us predict the value of a dependent variable based on an independent variable.

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When a point is rotated, it must be rotated around a point.

See attachment for the image of the rotation about point K

We should note the following:

In order to construct triangles, you will need a protractor, a pair of compasses and a ruler. To draw the triangle, three properties must be taken into account: length, angle and shape

The given parameters are:

ΔEFG

The angle of rotation is

∅ = 180⁰

The above angle of rotation means that:

The translated triangle will be 180 degrees from ΔEFG about point K.

It also means that:

ΔEFG and ΔE'F'G' will be equidistant from point K

See attached image for ΔE'F'G'

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The temperature of the cup of water is approximately 180°F after 6 minutes.

Using the given formula, we can write:

T = Ta + (To - Ta) * e^(-kt)

where Ta = 73°F (the temperature of the room), To = 201°F (the initial temperature of the water), and T = 189°F (the temperature of the water after 3 minutes).

We can solve for the decay constant k as follows:

(T - Ta) / (To - Ta) = e^(-kt)

ln[(T - Ta) / (To - Ta)] = -kt

k = -ln[(T - Ta) / (To - Ta)] / t

Substituting the given values, we get:

k = -ln[(189°F - 73°F) / (201°F - 73°F)] / 3 minutes

k = -ln[116 / 128] / 3 minutes

k ≈ 0.0434 minutes^-1 (rounded to the nearest thousandth)

Now we can use this value of k to find the temperature of the water after 6 minutes:

T = Ta + (To - Ta) * e^(-kt)

T = 73°F + (201°F - 73°F) * e^(-0.0434 minutes^-1 * 6 minutes)

T ≈ 180°F (rounded to the nearest degree)

Therefore, the temperature of the cup of water is approximately 180°F after 6 minutes.

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The relation O is not reflexive, symmetric, and not transitive is one of the following statements that is true about the relation O.  which is option (C).


Given, [tex]\forall m, n \in Z, m O n \longleftrightarrow \exists k \in Z \mid(m-n)=2 k+1[/tex]
Let's verify for the following relations :
Reflexive relation:
[tex]\forall a\in Z, a O a \longrightarrow \exists k\in Z \mid (a-a)= 2k+1[/tex]
[tex]0\neq 2k+1[/tex] for all k [tex]\in[/tex] Z
Since 2k+1 can never be zero for any k [tex]\in[/tex] Z, hence we conclude that the relation O is not reflexive.
Symmetric relation:
Suppose a, b [tex]\in[/tex] Zsuch that a O b i.e. (a-b)=2k+1, where k[tex]\in[/tex] Z.
Now, we need to check whether b O a is true or not i.e. (b-a)=2j+1 for some j[tex]\in[/tex] Z
We have,
[tex](a-b) = 2k+1 \longrightarrow (b-a) = -2k-1 = 2(-k) - 1[/tex]
Let j=-k-1, then we have j[tex]\in[/tex] Z and 2j+1 = -2k-1
Hence, (b-a) = 2j+1, and we conclude that the relation O is symmetric.
Transitive relation:
Suppose a, b, c[tex]\in[/tex] Z such that a O b and b O c.
Now, we need to check whether a O c is true or not.
We have,
(a-b)=2k_1+1 and (b-c)=2k_2+1 for some k_1,k_2[tex]\in[/tex] Z
(a-b)+(b-c) = 2k_1+1 + 2k_2+1
a-c = 2k_1+2k_2+2
Let j=k_1+k_2+1, then we have j[tex]\in[/tex] Z and a-c=2j
Hence, (a-c) is even and we conclude that the relation O is not transitive.
Therefore, the relation O is not reflexive, symmetric, and not transitive. Hence, option (C) is the correct answer.

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