Let R be the region in the first quadrant bounded by the graph of y=x3 the line x=2 and the x-axis. R is the base of a solid whose cross sections perpendicular to the x-axis are equilateral triangles. What is the volume of the solid?

The maximum length of each prism is equal to 7.99 centimeter.

Maximum surface area = 275 square centimeter (cm²).

Given, Light fixtures of regular hexagonal prism .

Determine the maximum surface area of this regular hexagonal prism by using this mathematical expression:

Maximum surface area (quantity) = Cost/unit price

Maximum surface area (quantity) = $11/$0.04

Maximum surface area (quantity) = 275 square centimeter (cm²).

Mathematically, the surface area of a regular hexagonal prism can be calculated by using this formula:

[tex]A = 6al + 3\sqrt{3} a^2[/tex]

Where:

A represents the surface area of a regular hexagonal prism.

a represents the edge length (apothem) of a regular hexagonal prism.

l represents the length of a regular hexagonal prism.

Substituting the given parameters into the formula, we have;

[tex]275 = 6 \times 4l + (3\sqrt{3} \times 4^2)[/tex]

[tex]275 = 24l + 48\sqrt{3} \\24l = 275 - 48\sqrt{3}\\ 24l = 191.8616\\l = 191.8616/24[/tex]

Length, l = 7.99 centimeter.

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The vector field for  7 (x,y) = xî + xyî (1,3), (-1,2), (3,-2) can be plotted with arrows with magnitude 4 at point (1,3), magnitude 3 at point (-1,2), and magnitude 3 at point (3,-2).

The  given vector field 7 (x,y) = xî + xyî (1,3), (-1,2), (3,-2) can be drawn by plotting arrows at each point (x,y) in the plane with the direction and magnitude of each arrow presented  by the vector

7 (x,y) = xî + xyî.

over point (1,3),

the vector is 7(1,3) = 1î + 3î = 4î.

over point (-1,2),

the vector is 7(-1,2) = -1î - 2î = -3î.

Over point (3,-2),

the vector is 7(3,-2) = 3î - 6î = -3î.

The vector field for  7 (x,y) = xî + xyî (1,3), (-1,2), (3,-2) can be plotted with arrows with magnitude 4 at point (1,3), magnitude 3 at point (-1,2), and magnitude 3 at point (3,-2).

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Optimize crate cost by expressing material cost in terms of x and y, calculating cost of all four sides, and finding minimum cost. Optimal dimensions: x = 16 ft, y = 48 ft. Minimum cost: $8704.

To find the dimensions of the crate that will minimize the total cost of material, we need to use optimization techniques, we need to express the cost of materials in terms of x and y then calculate the cost of all four sides, then find the minimum cost.
Let's start by defining the variables we need to work with:

Let x be the length of one side of the square base (in feet),  Let y be the height of the crate (in feet).
From the given volume, we know that:
V = x^2 * y = 12288 ft^3
We can use this equation to solve for one of the variables in terms of the other:
y = 12288 / (x^2)
Now we need to express the cost of materials in terms of x and y.
The area of the bottom is x^2, so the cost of the bottom is:
[tex]C_b = 10 * x^2[/tex]
The area of each side is x * y, and there are four sides, so the cost of the sides is:
[tex]C_s = 4 * 2 * x * y = 8xy[/tex]
The area of the top is also x^2, so the cost of the top is:
[tex]C_t = 2 * x^2[/tex]
The total cost of materials is the sum of these three costs:
[tex]C = C_b + C_s + C_t = 10x^2 + 8xy + 2x^2[/tex]
Now we can substitute y = 12288 / (x^2) into this equation:
[tex]C = 10x^2 + 8x * (12288 / x^2) + 2x^2[/tex]
Simplifying this expression, we get:
[tex]C = 12x^2 + 98304 / x[/tex]
To find the minimum cost, we need to find the value of x that minimizes this expression. We can do this by taking the derivative of C with respect to x and setting it equal to zero:
C' = 24x - 98304 / x^2 = 0
Solving for x, we get:
x = 16 ft
Now we can use this value of x to find y:
y = 12288 / (16^2) = 48 ft
Therefore, the dimensions of the crate that will minimize the total cost of material are:
- Length of one side of the square base = x = 16 ft
- Height of the crate = y = 48 ft
To check that this is indeed the minimum cost, we can plug these values back into the expression for C and calculate the cost:
C = 10 * 16^2 + 8 * 16 * 48 + 2 * 16^2 = 8704
Therefore, the minimum cost of material for the crate is $8704.

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The equation of the axis of symmetry for the given function is x = 1.

To find the equation of the axis of symmetry for the function f(x) = -4x² + 8x - 28, we can use the formula:

x = -b / (2a)

where "a" and "b" are coefficients in the quadratic equation ax² + bx + c.

In this case, a = -4 and b = 8. Plugging these values into the formula, we get:

x = -8 / (2*(-4))

x = -8 / (-8)

x = 1.

The axis of symmetry for a quadratic function in the form of [tex]f(x) = ax^2 + bx + c[/tex]  can be found using the formula x = -b / (2a).

In the case of the given quadratic function f(x) = -4x² + 8x - 28, the coefficient of [tex]x^2[/tex] is a = -4 and the coefficient of x is b = 8.

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Answer:

Yes

Step-by-step explanation:

9/25

√9/√25 = 3/5 = 0.6

so it is a rational number because it has an integer as a denominator also because the decimal is not reoccurring

Answer: Yes

Step-by-step explanation:

Yes.

9/25 = .36  

Because the decimal stop it is rational

Only decimals that have no pattern and go on infinitely then it is irrational like [tex]\pi[/tex]  or √7   if you plug those into a calculator they go on forever and have no pattern

Roy should cut the pizza at an angle of 4.7 radians if the total diameter of the pizza is 12 inches.

Diameter of the pizza = 12 inches

The area he can eat = 84. 8 inches

The area of the pizza for 12 inches diameter can be calculated by using the formula:

Area = π*[tex]r^2[/tex]

Area = π* ([tex]6^2[/tex])

Area = 36π

The angle of the slice of pizza he can eat about 84.8 square inches can be calculated as:

Area of sector = (θ/2) ×[tex]r^2[/tex]

84.8 = (θ/2) × [tex]6^2[/tex]

84.8 = 18*θ

θ = 84.8 / 18

θ = 4.71 radians

Therefore, we can conclude that Roy should cut the pizza at an angle of  4.7 radians.

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In Roman Numerals, smaller numbers can be subtracted from larger numbers, but only to a certain extent.

The number VCX is incorrect in Roman Numerals because V (5) cannot be subtracted from X (10) to make 5, and C (100) cannot be subtracted from X (10) to make 90.

In Roman Numerals, smaller numbers can be subtracted from larger numbers, but only to a certain extent.

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Step-by-step explanation:

volume of a cylinder is pi × r² × height

r = 2.5

height= 4

volume = pi × 2.5² × 4

volume = 25 pi

The general solution to the differential equation is, |3x^4/(y(x+y))|x + |x^3| ln|y| + |x^3| ln|x+y| + h(x) = C.

The partial derivative of (3xy + y^2) with respect to y is 6xy + 2y, and the partial derivative of (x^2 + xy) with respect to x is 2x + y. Since these are not equal, the differential equation is not exact.

To make it exact, we need to find an integrating factor μ(x, y) such that μ(x, y)(3xy + y^2) dx + μ(x, y)(x^2 + xy) dy = 0 is exact. We can find μ(x, y) by using the formula:

μ(x, y) = e^(∫(∂M/∂y - ∂N/∂x)/N dx)

where M = 3xy + y^2 and N = x^2 + xy. We have:

(∂M/∂y - ∂N/∂x)/N = (6xy + 2y - 2x - y)/(x^2 + xy) = (6xy - x - y)/(x^2 + xy)

We can now find the integrating factor μ(x, y) by integrating this expression with respect to x:

μ(x, y) = e^(∫(6xy - x - y)/(x^2 + xy) dx) = e^(3ln|x| - ln|y| - ln|x+y| + C) = e^(ln|x^3/(y(x+y))| + C) = |x^3/(y(x+y))|e^C

where C is the constant of integration.

Now we multiply the original differential equation by the integrating factor μ(x, y) to obtain:

|3x^4/(y(x+y))| dx + |x^3/(y(x+y))| dy = 0

This is now an exact differential equation, and we can find its solution by integrating with respect to x or y. Integrating with respect to x, we get:

|3x^4/(y(x+y))|x + g(y) = C

where g(y) is the constant of integration. To find g(y), we integrate the coefficient of dy:

g(y) = ∫|x^3/(y(x+y))| dy = |x^3| ln|y| + |x^3| ln|x+y| + h(x)

where h(x) is another constant of integration. Substituting g(y) back into the solution, we have:

|3x^4/(y(x+y))|x + |x^3| ln|y| + |x^3| ln|x+y| + h(x) = C

This is the general solution to the differential equation.

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Answer:

3 / (7-5)

Step-by-step explanation:

"Subtract 5 from 7"

When you're subtracting from something, the reverse the order of the numbers.  So, the expression here would be "7 - 5"

"Then, divide 3 by the result."

Here, you're dividing 3 by the result, so the 3 must be in the numerator (on top of the fraction), and the "result from the previous step must be in the denominator (on the bottom of the fraction).  So, the expression here would be "3 / result"

Since order of operations forces division to happen before subtraction, we'll need parentheses around the first result to force the subtraction to happen first, as instructed.

So, the final expression would be "3 / (7-5)"

The area of a sector with a central angle of 45° and a diameter of 5.6 in. is 1.23 square inches.

To see why, you can use the formula for the area of a sector, which is:

A = (θ/360) x π x r^2

where θ is the central angle in degrees, r is the radius, and π is approximately 3.14.

First, you need to find the radius of the sector, which is half of the diameter:

r = d/2 = 5.6/2 = 2.8 in.

Next, you can plug in the values for θ and r into the formula:

A = (45/360) x 3.14 x 2.8^2 = 1.23 square inches

Therefore, the area of the sector is 1.23 square inches.

The area of a sector with a central angle of 120° and a radius of 18.4 m is 1908.57 square meters.

To see why, you can use the same formula for the area of a sector:

A = (θ/360) x π x r^2

First, you need to convert the radius from meters to centimeters, since π is in terms of centimeters:

r = 18.4 m x 100 cm/m = 1840 cm

Next, you can plug in the values for θ and r into the formula:

A = (120/360) x 3.14 x 1840^2 = 1908.57 square meters

Therefore, the area of the sector is 1908.57 square meters.

To find the volume of a rectangle APR  prism, you need to multiply its length, width, and height. In this case, the length is 3 1/3 (or 10/3) units, the width is 4 2/3 (or 14/3) units, and the height is 25 units.

So, the volume of the rectangle can be calculated as:

Volume = length x width x height
Volume = (10/3) x (14/3) x 25
Volume = 1166.67 cubic units (rounded to two decimal places)

Therefore, the volume of the rectangle with a length of 3 1/3, a width of 4 2/3, and a height of 25 is approximately 1166.67 cubic units.

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We can see here that an equation to show the total length of the bandages if they are placed end-to-end is:

([tex]1\frac{1}{4}[/tex] × [tex]2) + (1\frac{2}{4}[/tex] × 1) + ([tex]1\frac{3}{4}[/tex] × 3) + (2 × 4) + (3 × 6) = [tex]35\frac{1}{4}[/tex]

An equation in mathematics is a claim made regarding the equality of two expressions. Normally, it has two sides that are separated by an equal sign (=).

Variables, constants, and mathematical operations including addition, subtraction, multiplication, division, exponentiation, and more can be used on each side of the equation.

We can see here that the above answer is correct because on the number line:

[tex]1\frac{1}{4}[/tex] has 2 Xs on it.

[tex]1\frac{2}{4}[/tex] has 1 X on it.

[tex]1\frac{3}{4}[/tex] has 3 Xs on it.

2 has 4 Xs on it.

3 has 6 Xs on it.

And multiplying and adding the variables, we arrived at:  [tex]35\frac{1}{4}[/tex]

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Tricia has a total of $6.35. The answer is D.

Tricia has 10 quarters, which is equivalent to $2.50 (since 1 quarter is $0.25).

She also has 17 dimes, which is equivalent to $1.70 (since 1 dime is $0.10).

Tricia has 40 nickels, which is equivalent to $2.00 (since 1 nickel is $0.05).

Finally, she has 15 pennies, which is equivalent to $0.15 (since 1 penny is $0.01).

To find the total amount of money Tricia has, we can add up the values of each coin:

$2.50 + $1.70 + $2.00 + $0.15 = $6.35

Therefore, Tricia has a total of $6.35. The answer is option D.

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The volume of the 4th doll is approximately [tex]130.1 cm^3.[/tex]

The diameter of the smallest doll is 1.9 cm, so its radius is 0.95 cm (half of the diameter).

Similarly, the radius of the second smallest doll is (2.85/2) = 1.425 cm.

Since the scale factor is the same from one doll to the next, the ratio of the radius of the second smallest doll to the radius of the smallest doll is:

1.425 cm / 0.95 cm = 1.5

Similarly, the ratio of the radius of the third smallest doll to the radius of the second smallest doll is also 1.5.

Using this pattern, we can find the radius of the 4th doll as:

Radius of 4th doll = 1.5 × (Radius of 3rd doll) = 1.5 × 2.1375 cm = 3.2063 cm (rounded to 4 decimal places)

The volume of the 4th doll can then be calculated as:

Volume of 4th doll = (4/3) × π ×[tex](Radius of 4th doll)^3[/tex]

                           = (4/3) × π × [tex](3.2063 cm)^3[/tex]

                           ≈ [tex]130.1 cm^3[/tex]

Therefore, the volume of the 4th doll is approximately [tex]130.1 cm^3.[/tex]

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To find the length of the opposite side and the adjacent side, we can use the ratios of the sides in a 30-60-90 degree triangle.

The ratio of the opposite side to the hypotenuse is 1:2, and the ratio of the adjacent side to the hypotenuse is √3:2.

Using these ratios, we can find the length of the opposite side and the adjacent side as follows:

Opposite side = 1/2 x hypotenuse = 1/2 x 10 = 5 units

Adjacent side = √3/2 x hypotenuse = √3/2 x 10 = 5√3 units

Given a right triangle with an acute angle of 60 degrees and an adjacent side of 5 units, find the length of the hypotenuse and the opposite side.

To find the length of the hypotenuse and the opposite side, we can use the ratios of the sides in a 30-60-90 degree triangle.

The ratio of the hypotenuse to the adjacent side is 2:1, and the ratio of the opposite side to the adjacent side is √3:1.

Using these ratios, we can find the length of the hypotenuse and the opposite side as follows:

Hypotenuse = 2 x adjacent side = 2 x 5 = 10 units

Opposite side = √3 x adjacent side = √3 x 5 = 5√3 units

Given a right triangle with an acute angle of 45 degrees and an opposite side of 7 units, find the length of the hypotenuse and the adjacent side.

To find the length of the hypotenuse and the adjacent side, we can use the ratios of the sides in a 45-45-90 degree triangle.

In this type of triangle, the opposite side and the adjacent side are equal, and the hypotenuse is √2 times the length of the legs.

Using these ratios, we can find the length of the hypotenuse and the adjacent side as follows:

Opposite side = Adjacent side = 7 units

Hypotenuse = √2 x opposite side = √2 x 7 = 7√2 units

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Answer:

Step-by-step explanation

Domain :

x

>

4

, in interval notation :

(

4

,

∞

)

Range:

g

(

x

)

∈

R

, in interval notation :

(

−

∞

,

∞

)

Explanation:

g

(

x

)

=

ln

(

x

−

4

)

;

(

x

−

4

)

>

0

or

x

>

4

Domain :

x

>

4

, in interval notation :

(

4

,

∞

)

Range: Output may be any real number.

Range:

g

(

x

)

∈

R

, in interval notation :

(

−

∞

,

∞

)

graph{ln(x-4) [-20, 20, -10, 10]} [Ans]  x>4

Answer:

Step-by-step explanation:

The Domain of g(x) = -|x| is all real numbers (no restrictions on what values x can take).

The Range of g(x) = -|x| is all real numbers less than or equal to zero. Absolute value of any real number is always greater than or equal to zero, and multiplying by a negative sign, that flips the sign of the result. So, g(x) will always be less than or equal to zero.

Domain:  (-∞, ∞), {x|x ∈ R}

Range: (-∞, 0), {y ≤ 0}

Using the ratio, we solve and get, Ben needs 0.8 litres of red paint, 0.64 litres of yellow paint, and 0.06 litres of white paint to make 1.5 litres of orange paint.

To make 1.5 litres of orange paint, Ben needs to mix the red, yellow, and white paint in the ratio of 20 : 16 : 1.5.

To calculate the amount of each color needed, you first need to find the total number of parts in the ratio.

The total number of parts is 20 + 16 + 1.5 = 37.5.

Next, you can calculate the amount of each color needed by dividing 1.5 litres by the total number of parts in the ratio, and then multiplying by the respective parts for each color.

So, the amount of red paint needed is:

(1.5/37.5) x 20 = 0.8 litres

The amount of yellow paint needed is:

(1.5/37.5) x 16 = 0.64 litres

And the amount of white paint needed is:

(1.5/37.5) x 1.5 = 0.06 litres

Therefore, Ben needs 0.8 litres of red paint, 0.64 litres of yellow paint, and 0.06 litres of white paint to make 1.5 litres of orange paint.

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The volume of the box is 160 cubic inches. Using the volume of a rectangular prism formula the volume of the box is 160 cubic inches. Using another formula base area times height, the area is the same.

The volume of James' rectangular prism box can be calculated using both unit cubes and a formula. Part (a) involves counting the unit cubes in the model and multiplying the number of cubes by their volume, which is 1 cubic inch. In this case, there are 160 unit cubes, so the volume of the box is 160 cubic inches.

Part (b) involves using the formula for volume of a rectangular prism, which is length times width times height. Plugging in the given dimensions, we get 8 x 4 x 5 = 160 cubic inches, which is the same as the answer in part (a) using unit cubes.

Part (c) involves using a different formula for volume, which is base area times height. In this case, the base of the rectangular prism is a rectangle with length 8 inches and width 4 inches, so the base area is 8 x 4 = 32 square inches. Multiplying by the height of 5 inches, we get 160 cubic inches, which is again the same as the answers in parts (a) and (b).

Using the volume formulas is much quicker and more efficient than counting unit cubes, especially for larger boxes. However, counting unit cubes can provide a more concrete visual representation of the volume and can be helpful for students who are just learning about volume. In this case, the answers obtained using the formulas were the same as the answer obtained by counting unit cubes, which reinforces the accuracy of the formulas.

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70 degrees to radian is 1.22 radian.

In mathematics,, both degree and radian represent the measure of an angle. One complete anticlockwise revolution can be represented by 2π (in radians) or 360° (in degrees).

Therefore,

360 degrees = 2π radian

where

Therefore, let's find 70 degrees in radian.

Hence,

360 degrees = 2π radian

70 degrees = ?

cross multiply

angle in radian = 70 × 2π / 360

angle in radian = 140π / 360

angle in radian = 0.38888888888 × 3.14

angle in radian = 1.22 radian

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By applying the Consecutive Angles Theorem, Jim can confirm that the angled section for the steps is indeed a parallelogram, even without accurately proving that any two sides are parallel.

Jim building a deck with an angled section in the shape of a parallelogram. To be assured that he has an actual parallelogram, Jim can use the Consecutive Angles Theorem.

This theorem states that if the consecutive angles of a quadrilateral are supplementary (add up to 180 degrees), then the quadrilateral is a parallelogram.

To use the Consecutive Angles Theorem, Jim should follow these steps:

1. Measure the four angles of the quadrilateral he has created for the angled section of the deck.
2. Check if the consecutive angles are supplementary (i.e., the sum of each pair of consecutive angles is equal to 180 degrees).
3. If all consecutive angles are supplementary, he can be assured that the quadrilateral is a parallelogram.

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Using the federal list, the total amount of exemptions that Jerry would be allowed is $25,150.

Jerry's assets include a house with equity of $15,000, a car with equity of $2,500, household goods worth $6,000 (no single item over $400), and tools worth $5,800 that he needs for his business.

Using the state list, the total amount of exemptions that Jerry would be allowed varies depending on the state in which he resides. Without knowing the state, it is impossible to provide an accurate answer. However, it is worth noting that the state list is often more favorable for individuals than the federal list.

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Latoya may experience some fluctuations in her winnings and losses in the short run, in the long run, her average winnings will approach $0.50 per selection.

What is the expected value of playing the game and what Latoya can expect after playing the game many times?

(a) To find the expected value of playing the game, we need to multiply the amount that Latoya can win or lose by the probability of each outcome, and then add up the results. Let p(i) be the probability of selecting the ball with the number i. Since there are 8 balls in total and each ball is equally likely to be selected, we have:

p(i) = 1/8 for i = 1, 2, ..., 8

Now we can calculate the expected value:

E(X) = ∑[i=1 to 5] (i * p(i)) + ∑[i=6 to 8] (-1 * p(i))

= (1/8)(1) + (1/8)(2) + (1/8)(3) + (1/8)(4) + (1/8)(5)

- (1/8)(1) - (1/8)(1) - (1/8)(1)

= 0.5

Therefore, the expected value of playing the game is $0.50.

(b) In the long run, after playing the game many times, Latoya can expect to break even (neither gain nor lose money) on average per selection. This means that over a large number of selections, she can expect to win some money on some selections and lose some money on others, but on average, her total winnings and losses will balance out to zero.

To see why this is the case, consider that the expected value of a single selection is $0.50. If Latoya plays the game many times, the law of large numbers tells us that the average winnings per selection will converge to the expected value of $0.50. So even though Latoya may experience some fluctuations in her winnings and losses in the short run, in the long run, her average winnings will approach $0.50 per selection.

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Answer:

Step-by-step explanation:

Since the two cylinders are similar, their corresponding dimensions (radius and height) are proportional. Let the radius of the smaller cylinder be r.

Then, we can write:

r / 6 = R / 30

where R is the radius of the larger cylinder.

Simplifying this equation, we get:

R = 5r

Now, we can use the formula for the volume of a cylinder to find the volume of the larger cylinder:

Volume of smaller cylinder = πr^2h = 90 cm^3

Volume of larger cylinder = πR^2H = π(5r)^2(30) = 750πr^2 cm^3

Substituting R = 5r, we get:

Volume of larger cylinder = 750πr^2 cm^3

Therefore, the volume of the larger cylinder is 750π times the volume of the smaller cylinder:

Volume of larger cylinder = 750π(90 cm^3) = 67,500π/ cm^3 (approx. 211,239.74 cm^3 rounded to five decimal places).

Answer: 3/8

Step-by-step explanation:

7/8-1/2

Find common denominator which is 8 and you get that from 1/2 by multiplying 4 on the top and bottom and so you wind up with 4/8 and 7/8-4/8 is 3/8

The maximum error of estimate is 4.75%.

To calculate the maximum error of estimate for the given problem, we will use the formula for margin of error:

Margin of Error = Z-score * (Standard Deviation / √n)

Where:
- Z-score corresponds to the 90% confidence level, which is 1.645
- Standard Deviation is 0.5 hours
- n is the sample size, which is 300 students

Margin of Error = 1.645 * (0.5 / √300) ≈ 0.0475

To express this as a percentage, multiply by 100:

0.0475 * 100 ≈ 4.75%

Thus, the maximum error of estimate with a 90% confidence level is 4.75%.

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The perimeter is indeed 26 feet, which means they have used all the extra fencing.

Let's start by assuming that the length and width of the garden patch are whole numbers of feet, since we are asked to use only whole numbers.

Let's call the length of the garden patch "L" and the width "W".

We know that Liam has 26 feet of fencing left over. This fencing will be used to make the perimeter of the garden patch, which is given by:

Perimeter = 2L + 2W

We can substitute the value of the perimeter with the amount of fencing that Liam has:

26 = 2L + 2W

Simplifying this equation, we get:

13 = L + W

Since we want to use all the extra fencing, we know that the perimeter of the garden patch must be 26 feet. We can use this information to write another equation:

Perimeter = 2L + 2W = 26

We can substitute the value of 13 for L + W in this equation:

2L + 2W = 26

2L + 2(13-L) = 26

2L + 26 - 2L = 26

26 = 26

This equation is true, which means that our assumption that L and W are whole numbers is correct.

Therefore, the dimensions of the garden patch that Liam and his brother can make for their sister are 6 feet by 7 feet.

To check, we can calculate the perimeter:

Perimeter = 2L + 2W = 2(6) + 2(7) = 12 + 14 = 26

So the perimeter is indeed 26 feet, which means they have used all the extra fencing.

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The solution to the system of linear equations is (x, y) = (13, -4).

To solve the system of linear equations using Gaussian elimination, we need to eliminate one variable from one of the equations. Here, we can eliminate x from the second equation by subtracting twice the first equation from the second equation:

x + 2y = 5    (equation 1)

2x + 3y = 6    (equation 2)

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  -2x - 4y = -10   (2 * equation 1)

        y = -4

Now, we can substitute the value of y into the first equation to solve for x:

x + 2(-4) = 5

x - 8 = 5

x = 13

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The results of the survey suggest that the student's claim of three-quarters of 17-year-olds having their driver's licenses may be accurate, but further research would be necessary to confirm this with a larger and more representative sample.

Based on the computer output, there is no convincing evidence that the true proportion of 17-year-olds at this high school with driver's licenses is not 0.75. This is because the P-value of 0.9315 is very large, indicating that the results of the survey are not statistically significant. Additionally, the 95% confidence interval for the sample proportion of 0.755556 includes 0.75, further supporting the claim that the true proportion may be close to 0.75.

It is important to note that the correct significance test was performed, testing the null hypothesis that p = 0.75 against the alternative hypothesis that p ≠ 0.75. This is the appropriate test when the claim being tested is about a specific value of the proportion, as in this case. The alternative hypothesis being p > 0.75 or p < 0.75 would be incorrect, as it assumes a one-sided test rather than a two-sided test.

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