hem t
Q4.
Mr Jones has two sizes of square paving stones.
He uses them to make a path.
3.72m
large
1.55m
small
ASKED
The path measures 1.55 metres by 3.72 metres.
Calculate the width of a small paving stone.

a) To find the volume of the region A bounded by the sphere x^2 + y^2 + z^2 = 12 and the paraboloid z = x^2 + y^2, we can use cylindrical coordinates.

In cylindrical coordinates, the equations of the surfaces become:Sphere: ρ^2 + z^2 = 12Paraboloid: z = ρ^2The region A is bounded by the sphere and the paraboloid, so we need to integrate over the range of ρ, φ, and z that satisfies both equations. The limits for ρ are 0 to √(12 - z^2), the limits for φ are 0 to 2π, and the limits for z are 0 to 4. So the triple integral for the volume of region A in cylindrical coordinates is:∫∫∫ ρ dρ dφ dz, where the limits of integration are ρ: 0 to √(12 - z^2), φ: 0 to 2π, and z: 0 to 4.b) To find the volume of the region B in the first octant bounded by the surfaces z = x^2 and x^2 + y^2 + z = 1, we can again use cylindrical coordinates. In cylindrical coordinates, the equations of the surfaces become:z = ρ^2 (since we are in the first octant where x and y are non-negative)z = 1 - ρ^2The limits for ρ are 0 to 1, and the limits for φ are 0 to π/2. So the triple integral for the volume of region B in cylindrical coordinates is:∫∫∫ ρ dρ dφ dz, where the limits of integration are ρ: 0 to 1, φ: 0 to π/2, and z: ρ^2 to 1 - ρ^2.c) To find the volume of the region C inside both spheres x^2 + y^2 + (z - 2)^2 = 16 and x^2 + y^2 + 2 = 16, we can once again use cylindrical coordinates. In cylindrical coordinates, the equations of the surfaces become:Sphere 1: ρ^2 + (z - 2)^2 = 16Sphere 2: ρ^2 = 12The region C is bounded by both spheres, so we need to integrate over the range of ρ, φ, and z that satisfies both equations. The limits for ρ are 0 to 2√3, the limits for φ are 0 to 2π, and the limits for z are 2 - √(16 - ρ^2) to 2 + √(16 - ρ^2). So the triple integral for the volume of region C in cylindrical coordinates is:∫∫∫ ρ dρ dφ dz, where the limits of integration are ρ: 0 to 2√3, φ: 0 to 2π, and z: 2 - √(16 - ρ^2) to 2 + √(16 - ρ^2).

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2X (2² + sin 3) can be simplified as 8X + 2X sin 3

To solve the expression, we will first have to compute the values inside the parentheses and then apply the given operations. so we Calculate the values inside the parentheses by multiplying across the bracket:

2² is equal to 4, and sin 3 is a trigonometric function that returns the sine of the angle 3

Therefore, the expression 2X (2² + sin 3) simplifies to:

2X (4 + sin 3)

or

8X + 2X sin 3

where X is an unknown variable and sin 3 is a trigonometric function

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Answer:

Solution

verified

Verified by Toppr

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2x

sin3xdx

I=sin3x∫e

2x

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dx

d

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cos3xe

2x

dx

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e

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−

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​

∫cos3x∫e

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dx−∫(

dx

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​

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​

−

2

3

​

[

2

cos3xe

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−∫(−sin3x

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2

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2x

​

−

4

3

​

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4

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I=

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Solve any question of Integrals with:-

Patterns of problems

Patterns of problems

>

Solve :∫e xe e xe e e x

dx

Medium

View solution>Solve:-∫ a 2b 2(a 2 −b 2) 2

Step-by-step explanation:

pls brain

Answer:

40 meters

Step-by-step explanation:

Let's assume the width of the garden is x meters, then the length of the garden would be (x+10) meters since we know that the length is 10 meters more than the width.

We also know that the area of the garden is 100 square meters, therefore:

Area = Length x Width

100 = (x+10) x x

Expanding the equation we get:

100 = x^2 + 10x

Rearranging the terms we have:

x^2 + 10x - 100 = 0

Solving for x using the quadratic formula, we get:

x = 5 or x = -20

Since the width cannot be negative, we discard the negative solution and conclude that the width of the garden is 5 meters. Therefore, the length of the garden is (5+10) = 15 meters.

The perimeter of the garden is the sum of the four sides, which is:

Perimeter = 2 x (Length + Width)

Perimeter = 2 x (15 + 5)

Perimeter = 2 x 20

Perimeter = 40 meters

Therefore, the perimeter of the garden is 40 meters.

Answer:

Let's start by using algebra to solve for the width of the garden:

- Let w be the width of the garden.

- Then the length of the garden is w + 10.

- The area of the garden is length x width, so we can write the equation: (w + 10)w = 100.

- Expanding the left side of the equation, we get: w^2 + 10w = 100.

- Rearranging the equation, we get: w^2 + 10w - 100 = 0.

- Factoring the left side of the equation, we get: (w + 20)(w - 10) = 0.

- Solving for w, we get: w = -20 or w = 10. Since the width cannot be negative, we have w = 10.

Now that we know the width of the garden is 10 meters, we can find the length by adding 10 meters:

- Length = width + 10 = 10 + 10 = 20 meters.

Finally, we can find the perimeter of the garden by adding up the lengths of all four sides:

- Perimeter = 2(length + width) = 2(20 + 10) = 2(30) = 60 meters.

Therefore, the perimeter of the garden is 60 meters.

The effective utilization for the XYZ medical facility provides medical services to patients in Russellville is 81.63%, option B.

The percentage of your available resources that are now being used is known as resource utilization. To make sure your business is as productive as possible, it might assist to plan how to use your resources more wisely.

Employers and workers may both benefit from efficient resource management. On one side of the spectrum, it may also avoid overworking and burnout, resulting in a more balanced work life overall. On the other, it makes sure that workers have enough work to keep their position sustainable and lucrative.

Design capacity = 40 patients per hour,

Effective capacity is 35 patients per hour

Actual capacity is given by,

200/7 patients per hour

Effective utilization is given by = 200 x 100 /7 x 35

= 20000/245

= 81.63%.

Therefore, the effective utilization is 81.63%.

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The particle is at rest at t = 3. Therefore, the particle is at rest at t = 2 and t = 3.

To find when the particle is at rest, we need to find the values of t where the velocity of the particle is zero.

The velocity function is obtained by taking the derivative of the position function: v(t) = x'(t) = x²(t) - 5x(t) + 6

Setting v(t) = 0, we get a quadratic equation in x(t): x²(t) - 5x(t) + 6 = 0. Factoring the quadratic, we get: (x(t) - 2)(x(t) - 3) = 0

Therefore, x(t) = 2 or x(t) = 3. We now need to check which values of t correspond to these values of x(t).

At x(t) = 2, we get: v(t) = x²(t) - 5x(t) + 6 = 4 - 10 + 6 = 0. Thus, the particle is at rest at t = 2. At x(t) = 3, we get: v(t) = x²(t) - 5x(t) + 6 = 9 - 15 + 6 = 0

Thus, the particle is at rest at t = 3. Therefore, the particle is at rest at t = 2 and t = 3.

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The expression (2x-3)(2x-3) is equal to (2x-3)^2.

To expand the expression (2x-3)(2x-3), we can use the FOIL method (which stands for First, Outer, Inner, Last).

Multiplying the first terms of each binomial, we get 2x times 2x, which is 4x^2.

Multiplying the outer terms, we get -3 times 2x, which is -6x.

Multiplying the inner terms, we get -3 times 2x again, which is also -6x.

Multiplying the last terms of each binomial, we get -3 times -3, which is 9.

Combining like terms, we get 4x^2 - 12x + 9.

Therefore, (2x-3)(2x-3) is equal to (2x-3)^2, which is equivalent to 4x^2 - 12x + 9.

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Answer:

Step-by-step explanation:

The original image is the the black one.  The dilated image is blue.  It got bigger.  So it is an enlargment.

Each side got bigger by times 2

So the dilation is 2

Solving real-life problems involving mean or expected value can be quite useful in various situations, such as finance, statistics, and decision-making.

To begin, identify the problem that requires the calculation of mean or expected value.

The mean is the average of a set of numbers, while expected value is the anticipated result based on probability distribution.

Next, collect the necessary data for the problem.

In calculating the mean, gather all values in the data set.

For expected value, you'll need the probability of each outcome and its corresponding value.

To calculate the mean, add all the values together and divide by the total number of values. For expected value, multiply each outcome's value by its probability and then sum up the results.

Once you have the mean or expected value, apply it to the real-life problem to make informed decisions or predictions. This can help in areas such as budgeting, risk assessment, and determining the likelihood of future events.

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Based on the cost and revenue function, if they want to make a profit, the range of pulp in tons per day that they should produce is between 152.6 and 152.8 tons per day.

To find the range of pulp in tons per day that will generate profit, we need to set the profit function equal to zero and solve for x. The profit function P(x) is given by:

P(x) = R(x) - C(x)

Substituting the given revenue and cost functions, we get:

P(x) = -3x^2 + 1000x - 400

Setting P(x) = 0, we can solve for x using the quadratic formula:

x = [-b ± sqrt(b^2 - 4ac)] / 2a

Plugging in the values from our profit function, we get:

x = [-(1000) ± sqrt((1000)^2 - 4(-3)(-400))] / 2(-3)

Simplifying, we get:

x = [1000 ± sqrt(1000000 + 4800)] / 6

x = [1000 ± sqrt(1004800)] / 6

x ≈ 152.7 or x ≈ 55.6

Since we're looking for the range of pulp in tons per day that will generate profit, we only want the positive solution, which is approximately 152.7 tons per day. Therefore, the company should produce between 152.6 and 152.8 tons per day to generate profit, rounded to the nearest tenth of a ton.

Note: The question is incomplete. The complete question probably is: A paper pulp company has discovered their cost and revenue functions for each day: C(x) = 2x^2 - 250x + 525 and R(x) = -3x^2 + 750x + 125, where x is the amount of pulp in tons. If they want to make a profit, what is the range of pulp in tons per day that they should produce? Round to the nearest tenth of a ton which would generate profit.

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The equation of the ellipse is x^2/9 + y^2/6.75 = 1

To find the equation of an ellipse, we need to know the center, the major and minor axis, and the foci.

Since we are given the eccentricity and foci, we can use the following formula:

c = (1/2)a

Since the foci are (0, +/-3), the center is at (0, 0). We know that c = 3/2, so we can find a:

c = (1/2)a

3/2 = (1/2)a

a = 3

The distance from the center to the end of the minor axis is b, which can be found using the formula:

b = √(a^2 - c^2)

b = √(3^2 - (3/2)^2)

b = √6.75

So the equation of the ellipse is:

x^2/a^2 + y^2/b^2 = 1

Plugging in the values we found, we get:

x^2/3^2 + y^2/6.75 = 1

Simplifying:

x^2/9 + y^2/6.75 = 1

Therefore, the equation of the ellipse is x^2/9 + y^2/6.75 = 1

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In the given binary operation, a and b are two numbers, and the operation is defined as a * b = a - 2b.

Binary operations are mathematical operations that take two operands and produce a single result. In this problem, we are given a binary operation "*". The operation is defined such that for any two numbers a and b, a * b = a - 2b.

We are then asked to evaluate 5 * 2 using this operation. To do so, we substitute a = 5 and b = 2 into the expression a * b = a - 2b:

5 * 2 = 5 - 2(2)

Simplifying the right-hand side of the equation, we get:

5 * 2 = 5 - 4

5 * 2 = 1

Therefore, 5 * 2 equals 1 when the binary operation is defined as a * b = a - 2b.

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We can start by finding the cost per gallon of gas for each person:

Javier: 5.6 gallons of gas for $26.96, so cost per gallon = 26.96/5.6 = $4.79/gallon

Ellema: 13.2 gallons of gas for $48.62, so cost per gallon = 48.62/13.2 = $3.68/gallon

Now we can find the cost of just the car wash for each person:

Javier: $26.96 - (5.6 gallons * $4.79/gallon) = $-1.344, which doesn't make sense. It's possible there was a mistake in the numbers given.

Ellema: $48.62 - (13.2 gallons * $3.68/gallon) = $1.958, which also doesn't make sense. There may be an error in the given information.

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The freezer chest has 20 cubic feet of space inside to hold frozen foods.

To find the amount of space inside the freezer chest, we need to calculate its volume. The formula for the volume of a rectangular prism is V = lwh, where l is the length, w is the width, and h is the height.

Using the measurements given, we can plug them into the formula and calculate:

V = 4 ft x 2.5 ft x 2 ft
V = 20 cubic feet

Therefore, there is 20 cubic feet of space inside the freezer chest to hold frozen foods.

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Answer:

Step-by-step explanation:

The role of the brackets, and how they effects the sum is given as the inside signs get changed after the opening of the bracket.

The associative property of addition is a mathematical statement that asserts that the arrangement of three or more integers does not affect their total. This indicates that no matter how the numbers are organised, the total of three or more integers remains the same.

The associative property of addition is a mathematical principle that asserts that when adding three or more integers, the amount obtained is constant regardless of how the numbers are grouped. Grouping here refers to where the brackets are positioned.

The sum for the 1st term is 10 + 7 – 5 + 3 = 15

The sum of the 2nd term is 10 + 7 – (5 + 3) = 17 - 8 = 9

Here, the bracket used made all the difference so after opening of the bracket the sign inside changed which impacted the summation of the terms above.

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The surface area of the square pyramid is approximately 41.83 cm².

To start with, let's define a square pyramid.

The slant height of the pyramid can be found using the formula:

l = √(h² + (s/2)²)

where h is the height of the pyramid and s is the length of one side of the base.

Plugging in the given values, we get:

l = √(7² + (4/2)²) = √(57)

Now that we know the slant height, we can find the area of each triangular face using the formula:

A = (1/2)bh

where b is the base of the triangle, and h is the height of the triangle.

Plugging in the given values, we get:

A = (1/2)(4)(√(57)) = 2√(57)

Since there are four triangular faces, the total area of all the triangular faces is:

4A = 8√(57)

Finally, we can find the total surface area of the pyramid by adding the area of the square base to the area of all the triangular faces:

surface area = 16 + 8√(57) = approximately 41.83 cm²

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Complete Question:

Find the surface area of the square pyramid where the base is 4cm and the  height is 7cm.

Answer:

Therefore, the product of 23.5 and 2.3 is 54.05.

Step-by-step explanation:

To determine the product of 23.5 and 2.3, we can use the following steps:

Align the numbers vertically with the ones digit of the second factor (2.3) under the tenths digit of the first factor (3 in 23.5).

   23.5

 x 2.3

 -----

Multiply the ones digit of the second factor by the first factor and write the result below, shifted one place to the right.

   23.5

 x 2.3

 -----

    71

Multiply the tenths digit of the second factor by the first factor and write the result below, shifted two places to the right.

   23.5

 x 2.3

 -----

    71

   470

Add the two partial products together.

   23.5

 x 2.3

 -----

    71

   470

 -----

   54.05

Therefore, the product of 23.5 and 2.3 is 54.05.

The dimensions of the rectangular pen that maximize the area are a shorter side of 100 ft and a longer side of 200 ft along the river.

To maximize the area of the rectangular pen using 400 ft of fencing, with a river on one side of the property, we need to determine the optimal dimensions. Let's denote the length of the pen along the river as 'x' and the width perpendicular to the river as 'y'.

Since the river is on one side, we only need to use the fencing for the other three sides. The total fencing length is 400 ft, so the equation representing the fencing is:

x + 2y = 400

We need to find the maximum area of the pen, which is given by the product of its length and width, i.e., A = xy.

First, we need to express 'x' in terms of 'y' using the fencing equation. From the equation, we get:

x = 400 - 2y

Now, substitute this expression for 'x' in the area equation:

A(y) = (400 - 2y)y = 400y - 2y²

To find the maximum area, we need to find the critical points of this equation by taking the derivative with respect to 'y' and setting it to zero:

dA/dy = 400 - 4y = 0

Solve for 'y':

4y = 400
y = 100 ft

Now, find 'x' using the expression we derived earlier:

x = 400 - 2y
x = 400 - 2(100)
x = 400 - 200
x = 200 ft

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The difference in the sample proportions of likely voters who said they will vote for this candidate is 0.025.

The range of the 95% confidence interval for the actual difference between the proportions of probable voters who would support this candidate before and after the debate was (-0.014, 0.064). To find the difference (pre-debate minus post-debate) in the sample proportions of likely voters who said they will vote for this candidate, we need to find the midpoint of the confidence interval, which is the point estimate of the true difference.

In the given question, the interval is (-0.014, 0.064), then the expression for the likely difference is

(0.064 + (-0.014))/2 = 0.050/2

= 0.025

Hence, the difference in the sample proportions of likely voters who said they will vote for this candidate is 0.025.

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The value of x for the circle is,

⇒ x = 13.2

We have to given that;

In circle Y,

m arc WX = 142°

m ∠WZX = (8x - 35)°

Since, angle WZX is half the measure of arc WX,

Hence, We get;

⇒ (8x - 35)° = 142 / 2

⇒ 8x - 35 = 71

⇒ 8x = 35 + 71

⇒ 8x = 106

⇒ x = 13.2

Thus, The value of x for the circle is,

⇒ x = 13.2

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The correct statement is given as follows:

The function g(t) reveals the market value of the house increases by 3.6% each year.

An exponential function has the definition presented as follows:

[tex]y = ab^x[/tex]

In which the parameters are given as follows:

The parameter b for this problem is given as follows:

b = 1.036.

As the parameter b has an absolute value greater than 1, the function is increasing, with a rate given as follows:

1.036 - 1 = 0.036 = 3.6% a year.

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The solution of the graphs of the equations is; )(6, 3 2/3)

A system of equation consists of two or more equations that share the same variables.

The solution of a system of equations obtained graphically can be obtained from the point of intersection of the lines of the graph of the equations

Taking the axis as the lowermost and  leftmost white lines, we get;

The points on the function f are (0, 1), and (9, 5)

The slope is; (5 - 1)/(9 - 0) = 4/9

The y-intercept is; (0, 1)

The equation is; y = (4/9)·x + 1

The equation of the line g is; x = 6

Therefore, the point of intersection is; y = (4/9)×6 + 1 = 8/3 + 1 = 11/3 = 3 2/3

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The value of the scale factor for the similar figures is 1/2

From the question, we have the following parameters that can be used in our computation:

The similar figures

The corresponsing sides of the similar figures are

Original = 8

New = 4

Using the above as a guide, we have the following:

Scale factor = New /Original

substitute the known values in the above equation, so, we have the following representation

Scale factor = 4/8

Evaluate

Scale factor = 1/2

Hence, the scale factor for the similar figures is 1/2

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The angle at which the leaning tower of Pisa deviates from the vertical is approximately 5.31 degrees.

In order to calculate the angle at which the leaning tower of Pisa deviates from the vertical, we can use the concept of tangent function.

First, we need to calculate the distance that the top of the tower is displaced from the vertical axis. This can be done using the Pythagorean theorem, which states that the displacement (d) is equal to the square root of the height of the tower (h) squared plus the amount the tower is out of plumb (p) squared:

d = √(h² + p²)

d = √(179² + 16.5²)

d = 180.14 ft

Next, we can use the tangent function to find the angle of deviation (θ):

tan(θ) = p/h

tan(θ) = 16.5/179

θ = tan⁻¹(16.5/179)

θ ≈ 5.31 degrees

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The proportion that can be used to show that the slope of uw is equal to the slope of wy is B) 2-8 -1-2.

Let the angles opposite to sides UV, VW, and WU be denoted by theta1, theta2, and theta3, respectively, in triangle UVW. Similarly, let the angles opposite to sides WY, YX, and XW be denoted by theta4, theta5, and theta6, respectively, in triangle WXY. Since triangles UVW and WXY are similar right triangles, we have:

theta1 = theta4 = 90 degrees (both triangles are right triangles)

theta2 = theta5 (corresponding angles in similar triangles are equal)

theta3 = theta6 (corresponding angles in similar triangles are equal)

UW/VW = WY/YX (sides in similar triangles are proportional)

The slope of uw is given by the rise over run or (VU - UW)/(WV), and the slope of wy is given by the rise over run or (XY - WY)/(YW). Using the similar triangles proportion, we can substitute UW/VW = WY/YX and simplify to get (VU - UW)/(WV) = (XY - WY)/(YW). This equation shows that the slope of uw is equal to the slope of wy.

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Answer:

3/y

Step-by-step explanation:

15x÷5xy

15x/5xy....you will cancel x by x and simplify by 5(GCF)

3/y ...... it is the simplify form of the given equetion.

Answer:

60%

Step-by-step explanation:

Take 96 and divide it by 160.

(easier if done on a calculator.)

For example: Find A/B as a percentage: take "A" and divide it by "B"

Based on the information provided, James would have 20 different waffle options.

Since each waffle cone can hold two scoops of ice cream and James must choose a different flavor for each scoop, we can approach this problem by using the multiplication principle of counting.

There are 5 different ice cream flavors to choose from for the first scoop, and 4 different flavors remaining for the second scoop. This is because James must choose a different flavor for each scoop.

Therefore, the number of different waffle cone options that James has is:

5 x 4 = 20

So, James has 20 different waffle cone options if he chooses a different flavor of ice cream for each scoop.

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The matrix that represents a dilation by a factor of 3 followed by a reflection over the horizontal axis is given as follows:

[tex]\left[\begin{array}{cc}3&0\\0&-3\end{array}\right][/tex]

A dilation can be defined as a transformation that multiplies the distance between every point in an object and a fixed point, called the center of dilation, by a constant factor called the scale factor.

The scale factor for this problem is given as follows:

k = 3.

Hence the matrix is:

[tex]\left[\begin{array}{cc}3&0\\0&3\end{array}\right][/tex]

For the reflection over the horizontal axis, we have that the y-coordinate is multiplied by -1, hence the matrix rule is:

[tex]\left[\begin{array}{cc}3&0\\0&-3\end{array}\right][/tex]

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31.8 kilograms is equivalent to 31,800 grams.

The correct answer is (D) 31,800 grams.

To convert kilograms to grams, we multiply the number of kilograms by 1000. So, to convert 31.8 kilograms to grams, we can use the following formula:

31.8 kilograms x 1000 grams/kilogram = 31,800 grams

Therefore, 31.8 kilograms is equivalent to 31,800 grams.

To convert kilograms to grams, we need to multiply the number of kilograms by 1000 because there are 1000 grams in one kilogram. In this case, Eddie's dog weighs 31.8 kilograms. To find out how many grams this is, we simply multiply 31.8 by 1000, which gives us 31,800 grams. Therefore, 31.8 kilograms is equivalent to 31,800 grams. It's important to understand the basic metric system conversions, like kilograms to grams, as they are commonly used in everyday life, particularly when it comes to measuring weight. Knowing how to make these conversions can be helpful in many different situations, from cooking and baking to medical and scientific contexts.  

Learn more about Weight conversion.

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