a farmer plans to enclose a rectangular region, using part of his barn for one side and fencing for the other three sides. if the side parallel to the barn is to be twice the length of an adjacent side, and the area of the region is to be 242 ft2, how many feet of fencing should be purchased?

The volume of the sphere with a radius of 8.9 m is given as follows:

V = 2,951.5 m³.

The volume of a sphere of radius r is given by the equation presented as follows:

V = 4πr³/3.

The radius for this problem has the measure given as follows:

r = 8.9m.

Hence, considering that π is rounded to 3.14, the volume of the sphere is obtained as follows:

V = 4 x 3.14 x (8.9)³ x 1/3

V = 2,951.5 m³.

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the boat will coast for a distance of 400 ln(2) ft, which is approximately 277.3 ft.

To find how far the boat will coast, we need to use the concept of differential equations. The equation that describes the velocity of the boat at any time "t" after its motor quits is given by:

dv/dt = -k*v

where "v" is the velocity of the boat at time "t", "k" is the proportionality constant, and the negative sign indicates that the velocity is decreasing with time.

We can solve this differential equation to obtain the velocity of the boat as a function of time:

v(t) = v(0) * e^(-k*t)

where "v(0)" is the initial velocity of the boat (40 ft/sec).

Using the given information, we can set up two equations:

v(10) = 20 (velocity after 10 sec)

v(0) = 40 (initial velocity)

Plugging these values into the equation, we get:

20 = 40 * e^(-10k)

Solving for "k", we get:

k = ln(2)/10

Now, we can integrate the velocity equation to find the distance the boat travels during the coasting period:

d = [tex]\int\limits^{\infty}_{10} {v(t)} \, dt[/tex]

Plugging in the values, we get:

d = (40/k) * [e^(-k*10) - 0] = 400 ln(2) ft

To explain the solution, we used the concept of differential equations to model the motion of the boat as it coasts after its motor quits. We used the given information to solve for the proportionality constant "k", which allowed us to find the velocity of the boat at any time.

Finally, we integrated the velocity equation to find the distance the boat travels during the coasting period. The key to this problem was using the concept of differential equations to model the motion of the boat and setting up the appropriate equations to solve for the unknowns.

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Complete question is:

Suppose that a motorboat is moving at 40 ft/sec when its motor suddenly quits, and then 10 sec later the boat has slowed to 20 ft/sec. Assume, that the resistance it encounters while coasting is proportional to its velocity. How far will the boat coast in all?

Answer: The equation X + y = -5 represents a straight line in a two-dimensional plane. To determine whether a specific point (X, y) is a solution of the equation, you can substitute the values of X and y into the equation and see if it holds true. If the equation is satisfied, then the point (X, y) is a solution. If the equation is not satisfied, then the point (X, y) is not a solution.

For example, if X = 2 and y = -7, then the equation becomes:

2 + (-7) = -5

-5 = -5

Since the equation holds true, the point (2, -7) is a solution of the equation X + y = -5.

Step-by-step explanation:

Answer:

Step-by-step explanation:

The volume of the three tennis balls combined is approximately 4.88 cubic inches. The volume of the packaging cylinder is approximately 50.33 cubic inches. Thus, the volume of the three tennis balls is about 9.7% of the volume of their packaging cylinder.

To calculate this, we can use the formula for the volume of a sphere (V = 4/3 x π x r3) and the formula for the volume of a cylinder (V = π x r2 x h).

The radius of each tennis ball is half the diameter, or 1.35 inches. So, the volume of each tennis ball is 4/3 x π x (1.35)3 = 4.88 cubic inches.

The radius of the cylinder is the same as the diameter of each tennis ball, or 2.7 inches. The height of the cylinder is 8.1 inches. So, the volume of the cylinder is π x (2.7)2 x 8.1 = 50.33 cubic inches.

The volume of the three tennis balls is 4.88 x 3 = 14.64 cubic inches. The volume of the packaging cylinder is 50.33 cubic inches. Thus, the volume of the three tennis balls is 14.64 ÷ 50.33 = 0.097 = 9.7% of the volume of their packaging cylinder.

The perimeter of one square which combined with other to form a rectangle of 288 cm² is 12 cm.

The area of rectangle is given by the formula -

Area = length × width

Now we know that all the sides of square are same. Also, the identical squares have been kept side by side. So, let us represent one dimension of square with b. Thus, length of rectangle will be (b + b)

Length of rectangle = 2b

Width of rectangle will be b. Now, finding the value of b first.

288 = 2b×b

Performing multiplication

2b² = 288

b² = 288/2

Performing division

b² = 144

b = ✓144

Taking square root

b = 12 cm

The perimeter of one square = 2(length + width)

Perimeter of one square = 2 ( 12 + 12)

Perimeter = 2 × 24

Performing multiplication

Perimeter = 48 cm

Thus, perimeter of one square is 48 cm.

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Caleb can choose which 20 items to buy in 1140 ways if he wants all 3 cheeses.

To choose 20 items at the grocery store, Caleb has narrowed down his selections to 5 vegetables, 8 fruits, 3 cheeses, and 7 whole grain breads. Since he wants all 3 cheeses, he only needs to choose 17 more items.

The number of ways he can choose these 17 items from the remaining vegetables, fruits, and breads is given by the combination: C(5+8+7, 17) = C(20, 17) = 1140.

This formula means that there are 20 items to choose from, and Caleb wants to choose 17. The order doesn't matter, so we use a combination.

Therefore, Caleb can choose which 20 items to buy in 1140 ways if he wants all 3 cheeses.

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After 13 hours the population of bacteria in the culture is approximately 5,905,820, rounded to the nearest whole number.

we can use the formula for exponential growth

P = P0 * 2^(t/k)

Where:

P = population at time t

P0 = initial population = 74000

t = time = 13 hours

k = time for population to double = 3 hours

Plugging in the values:

P = 74000 * 2^(13/3)

P = 74000 * 2^4.33

P = 74000 * 79.83

P = 5905820

Answer:

Step-by-step explanation:

Top problem:

A = πr² = (22/7)(21)² = 1386 ft²

Bottom problem:

r = D/2 = 20/2 = 10 in

A = (22/7)(10)² = 314 in²

The probability expressed as P(x ≥ 92) in the normal distribution is; C: 0.84

We are given that the normal distribution has;

Population mean: μ = 98

Population standard deviation; σ = 6

Sample mean; x' = 92

Thus, z-score is;

z = (x' - μ)/σ

z = (92 - 98)/6

z = -1

Thus, from z-score tables we have;

P(z ≥ -1) = 0.84

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Answer:

I don't have a way to explain, but it is an Equilateral and Isosceles :)

Hope it helps!

Given parallelogram points are (4,3),(-4 3), and (2, -3).

Let's say A= (4,3) , B=(-4 3) ,C=(2, -3) and D=(x, y)

Parallelogram defined as opposite sides are equal and parallel to each other.

From the properties of parallelogram :

We can say diagonals bisect each other.

From this we can easily find 4th point of parallelogram by using mid point concept.

Mid point formula

 (X,Y)=[tex](\frac{x_{1} +x_{2}}{2},\frac{y_{1} +y_{2}}{2})[/tex]

Find mid point between AC line

midpoint (X,Y)= [tex](\frac{4 +2}{2},\frac{3-3}{2})[/tex]

                (X,Y)=[tex](\frac{6}{2},\frac{0}{2})[/tex]

                 (X,Y)=(3,0)

Find mid point between BD Line

Mid point (X,Y)= [tex](\frac{-4 +x}{2},\frac{3+y}{2})[/tex]

Mid point of AC and BD are equal to each other

(3,0)=[tex](\frac{-4 +x}{2},\frac{3+y}{2})[/tex]

Equate x-coordinate and y-coordinate.

[tex]3=(\frac{-4 +x}{2})[/tex]        and      [tex]0=(\frac{3+y}{2})[/tex]

6=-4+x             and       0=3+y

x=10                 and        y=-3

Therefore ,the coordinates of the 4th point is D(x, y)=(10,-3)

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The coordinates of the fourth point that would complete the parallelogram are (10, -3).

A parallelogram is defined as a special type of quadrilateral that has both pairs of opposite sides parallel and equal.

We know A, B, and C, where A = (4, 3), B = (-4, 3), and C = (2, -3).

To determine the fourth point, which represents D.

First, we need to find the vector representing side AB.

Subtracting the coordinates of point B from the coordinates of point A. So:

AB = A - B = (4, 3) - (-4, 3) = (8, 0)

Next, we need to find a point that is on the line that is parallel to AB and passes through point C.

The vector CD (where D is the point) is equal to AB. So:

CD = AB = (8, 0)

We know that C = (2, -3),

We can start by adding CD to C to get D:

D = C + CD = (2, -3) + (8, 0) = (10, -3)

Therefore, the coordinates of the fourth point are (10, -3).

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Limits function lim p(x) can be calculated if p is a polynomial function by:

To calculate the limits of a polynomial function, p(x), first identify the degree of the polynomial. Then, use the formula Lim p(x) = aₙ xⁿ + aₙ₋₁ xⁿ⁻¹ + ... + a₀, where an is the leading coefficient. For example, if the polynomial is of degree 3, then the formula would be:

Lim p(x) = a₃ x³ + a₂ x² + a₁ x + a₀.

Next, we need to subtitute the variable x with the defined limits. For example, if the defined limit of the function is 2, then the limit function would be:

Lim p(2) =  a₃ (2)³ + a₂ (2)² + a₁ (2) + a₀

Then, we just need to simplify our finding:

Lim p(2) =  8a₃ + 4a₂ + 2a₁ + a₀

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Answer:

Reflection across the x-axis followed by dilation by a scale factor of with the center of dilation at the origin

Step-by-step explanation:

Rectangle EFGD has vertices at points E(-6,8), F(0,8), G(0,1) and D(-6,1).

1 transformation - reflection across the x-axis with the rule

So, the image rectangle E''F''G''D'' has vertices with coordinates

2 transformation - dilation by a scale factor of with the center of dilation at the origin. This transformation has the rule

Thus,

These are exactly vertices of rectangle E'F'G'D'.

In mathematical terms, each point (x, y) on Shape A is mapped to a point (2 + (2 - x), y) on Shape B as reflected line equation and explained as

When a shape is reflected in a line, it means that each point on the original shape is mapped to a corresponding point on the reflected shape that is symmetrical with respect to the reflecting line.

The line of reflection can be any straight line, and in this case, the line of reflection is given by the equation x = 2. To understand how the reflection works, imagine holding a mirror along the line x = 2 and placing Shape A on the mirror. The reflected shape, Shape B, would be the image that you see in the mirror. In mathematical terms, each point (x, y) on Shape A is mapped to a point (2 + (2 - x), y) on Shape B. This means that for each point on Shape A, we take its x-coordinate, subtract it from 2 to find the distance from the line of reflection, and then add that distance to 2 to find the x-coordinate of the corresponding point on Shape B. The y-coordinate remains the same.

The line of reflection acts as a dividing line between Shape A and Shape B, and the points on Shape B are exactly the same distance from the line of reflection as the points on Shape A, but on the opposite side. This means that Shape B is a perfect mirror image of Shape A, reflected across the line x = 2.

In summary, the reflection of Shape A in the line x = 2 maps each point on Shape A to a corresponding point on Shape B that is symmetrical with respect to the line of reflection. The result is a perfect mirror image of the original shape, reflected across the line.

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The area of the shaded region found using trigonometric ratios is about 86.6 square centimeters

Trigonometric ratios are expressions of the relationship between the interior angle of a right triangle and the ratio of two of the sides of the triangle.

Whereby, the inscribing figure of the quadrilateral is a regular hexagon, we get;

Interior angle of a regular hexagon = 120°

The 10 cm segment bisects the top vertex angle of the regular hexagon, which together with the 5·√3 cm side and trigonometric ratios, indicates that we get;

sin(60) = s/(5·√3) = (√3)/2

Where;

s = The side length of the regular hexagon

(5·√3)/s = (√3)/2

s = (5·√3)/((√3)/2) = 10

The side length of the regular hexagon, s = 10

The area of the regular hexagon, A = ((3·√3)/2) × s² = (1/2)×6 ×s ×a

Where;

a = The apothem = 5·√3

Therefore; A = ((3·√3)/2) × 10² = 150·√3

A  = (1/2)×6 ×10 × 5·√3 = 150·√3

Area of the quadrilateral = 4 × (1/2) × 5·√3 × 10 = 100·√3

Area of the shaded region = Area of the hexagon - Area of the quadrilateral

Therefore;

Area of the shaded region = 150·√3 - 100·√3 = 50·√3 ≈ 86.6

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Answer:

Step-by-step explanation:

4x + 9 = x + 30

3x + 9 = 30

3x = 21

x = 7

The percentage increase in her taxes from 2010 to 2011 is 44.30%.

A percentage is a value per hundredth. Percentages can be converted into decimals and fractions by dividing the percentage value by a hundred.

Given, Sophia paid $3,360,00 in taxes in 2010 and her taxes rose to $4,848,60 in 2011.

So, The amount of tax increase is $(484860 - 336000) = $148860.

Therefore, The percentage increase is,

= (148860/336000)×100%.

= 44.30%.

The net amount change will now be divided by the starting total, and the result will be multiplied by 100% to determine the percentage change.

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1. The function for this problem is defined as follows: y = 2x + 3.

2. The numeric value of the function at x = 2 is given as follows: y = 7.

To find the numeric value of a function or of an expression, we replace each instance of the variable in the function or in the expression by the value at which we want to find the numeric value.

We define a function as follows:

y = 2x + 3.

At x = 2, the numeric value is found replacing the lone instance of x by 2, hence:

y = 2(2) + 3

y = 7.

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The set X ∪ Y is the union of sets X and Y, which is the set of all elements that are in either X or Y. In this case, X ∪ Y = X = {1, 2, 4, 8, 16, 32}.

At a price of $10 per square foot, it will cost $1920 to carpet the 12 by 16-foot space.

The rectangle's area in square meters The area's width and length should be measured in feet. To calculate area, multiply your length by your width. Use our length converter to convert your measures if they aren't already in feet.

Area is length times breadth, or 12 feet by 16 feet, or 192 square feet. The cost of the carpet is $10 per square foot. By dividing the square footage of the room by the price per square foot, we can get the overall cost to carpet it.

Total cost is equal to Area. Cost per square foot is $1920 or 192 square feet (around 18 m2) times $10 per square foot.

Therefore, at a price of $10 per square foot, it will cost $1920 to carpet the 12 by 16-foot space.

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The denominator 6 in 3/6 is should not be changed when adding fractions if other fractions have a denominator of 6, to make the addition easy because 6 will be the lowest common multiple.

One of the reasons why the denominator will always stay the same when adding fraction is because the size of the equal pieces does not change when you combine the two fractions together.

For example say you want to add 3/6 + 1/6 + 2/6,

you will notice that all the fractions have equal denominator, so adding the fractions together will have the following result.

3/6 + 1/6 + 2/6 = (3 + 1 + 2 ) / 6

= 6/6

= 1

So the 6 in 3/6 should not be changed if all other fractions you wish to add to 3/6 have 6 as their denominator, because the 6 is lowest common multiple.

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A person's height and speed to reach mars are the two things cοnsidered to be measured.

Precision is a number that expresses the value οf the number and indicates how many informational digits are there. Fοr instance, 3.14 is an approximation οf pi that is reasonable and accurate. Hοwever, the precision digit, which is 3.199, is less than the precise digit.

1. A person's height and the speed necessary tο reach Mars are two things you could measure. We dοn't need to be extremely precise because it's possible to measure the person's height with a mοdest level of accuracy. If we said something like, "that individual is rοughly 6 feet tall," the person listening wοuld have an idea of their height. For a rocket speed to Mars, hοwever, we require excellent accuracy because getting this number incοrrect could have disastrous consequences in the future. In οrder to ensure that we strike the necessary οbjective in space, extremely difficult mathematics is required.

2. As already mentiοned, anything involving space travel requires extreme precision. The spacecraft might sοmehow crash if the calculations are off. Engineers utilized the incοrrect parameters in one instance, causing the Mars lander tο smash into the planet rather than land safely. Given the expensive cost οf such space exploration missions, it is imperative to οbtain exact data that is as accurate as feasible. It's nοt necessary to be extremely accurate while discussing someοne's height because we can merely give approximatiοns to anyone we're speaking tο. Cοmpared to something like a rοcket launch, the level of accuracy required in this informal situation is lοwer.

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Answer:

Step-by-step explanation:

The ratio of the coordinate point on the line is  2 to 7.

The ratio of the coordinate point on the line is calculated as follows;

Let the coordinate points = (xp, yp)

Let the ratio of the line = a:b

xp = x₁  +  (a/(a+b) (x₂ - x₁)

yp = y₁  + (a/a+b) (y₂ - y₁)

-3 = -7 + ( a/a+b)(11 - - 7)

-3 = -7 + (a/a+b)(18)

-3(a + b)  = -7(a+b) + 18a

7(a + b ) - 3(a+b) = 18a

7a + 7b - 3a - 3b = 18a

4b = 14a

4/14 = a/b

2/7 = a/b

2 : 7 = a : b

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The blood platelet count is an illustration of normal distribution Approximately 95% of the data lies within 2 standard deviations of the mean. There are approximately 99.7% of women with platelet count between 65.2 and 431.8

The given parameters are:

μ = 248.5

[tex]\sigma = 61.1\\[/tex]

(a) The percentage within 2 standard deviation of mean or between 126.3 and 370.7

Start by calculating the z-score, when x = 126.3 and x = 370.7

[tex]Z = \frac{(x-u)}{\sigma}[/tex]

So, we have:

x = -2

Also,

x = 2

The empirical rule states that:

Approximately 95% of the data lies within 2 standard deviations of the mean.

Hence, there are approximately 95% of women with platelet count within 2 standard deviations of the mean.

(b) The percentage with platelet count between 65.2 and 431.8

Start by calculating the z-score, when x = 65.2 and x = 431.8

So, we have:

Z =-3

Also:

Z = 3

The empirical rule states that:

Approximately 99.7% of the data lies within 3 standard deviations of the mean.

Hence, there are approximately 99.7% of women with platelet count between 65.2 and 431.8

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