Consider the construction of a pen to enclose an area. you have 400 ft of fencing to make a pen for hogs. if you have a river on one side of your property, what are the dimensions (in ft) of the rectangular pen that maximize the area? shorter side ft longer side ft

67 is the sum of handshakes given by the first and second participants.

Let n be the total number of participants in the workshop venue.

i.e, here n= 35

For the first participant, the number of handshakes is = (n-1)

= (35-1)

= 34

The number of handshakes by the second participant is also same as that of the first participant = 34

The number of handshakes given by the first and second participants together = (first participant handshake + second participant handshake - 1)

= (34 + 34 -1)

= 68-1

= 67

Hence 67 is the sum of handshakes given by the first and second participants.

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The complete question is =

A workshop venue has 35 participants . Each participant shakes hands with each each and every other participant . How is the sum of. Handshakes that will bemade bythe first and second participant

Therefore , the solution of the given problem of equation comes out to be  x must be less than -5 for any value of x that causes 5 + x = n to be a negative number.

In order to demonstrate consistency between two opposing statements, variable words are frequently used in sophisticated algorithms. Equations are academic phrases that are used to demonstrate the equality of different academic figures. Consider expression the details as y + 7 offers. In this case, elevating produces b + 7 when partnered with building y + 7.

Here,

If n is a negative number and 5 + x = n, then x must be less than -5.

This is due to the fact that n would be greater than or equal to 5, which is not a negative number, if x were greater than or equal to -5, which would lead 5 + x to be greater than or equal to 0.

However,

if x is less than -5, then 5 + x will be less than 0, and n will be a negative number because n will be less than 5.

Therefore, x must be less than -5 for any value of x that causes 5 + x = n to be a negative number.

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Answer is: 3,984.375 cubic inches

To help you calculate the volume of the aquarium. Using the formula

V = L x W x H, where V is volume, L is length, W is width, and H is height:

Length (L) = 25 inches
Width (W) = 12.5 inches (12 + 0.5)
Height (H) = 12.75 inches (12 + 3/4)

Now, plug these values into the formula:

Volume (V) = 25 x 12.5 x 12.75
V = 3,984.375 cubic inches

The volume of the aquarium is 3,984.375 cubic inches.

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Answer: yes

Step-by-step explanation:

1 ft = 12 in so 3ft = 36 in

36in + 10in = 46in

John would have the option of taking 10 different cones

The questions says that there is the option of having the flavors that are available ice cream flavors are: chocolate (C), mint chocolate chip (M), strawberry (S), rainbow sherbet (R), and vanilla (V).

The available flavors are then 5 in number

Then the number of scoops that he can have from each of the cone is said to be 2

Hence we would have 5 x 2

= 10

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After plugging the derivatives of f(x) we get, dx f-'(4) = f'(√(2/3)) = 4 + cos(Ф)

To find dx f-'(4), we need to take the derivative of f(x) and then solve for x when f'(x) equals 4.

First, let's find the derivative of f(x):

f'(x) = 6x² + cos(Ф)

Next, we need to solve for x when f'(x) equals 4:

6x² + cos(Ф) = 4

cos(Ф) = 4 - 6x²

Now, we can use the given value of πα d to solve for x:

πα d = -1/2

α = -1/2πd

α = -1/2π(-1)

α = 1/2π

d = -1/2πα

d = -1/2π(1/2π)

d = -1/4

So, we have:

cos(Ф) = 4 - 6x²

cos(πα d) = 4 - 6x²   (substituting in the given value of πα d)

cos(-π/2) = 4 - 6x²    (evaluating cos(πα d))

0 = 4 - 6x²

6x² = 4

x² = 2/3

x = ±√(2/3)

Since we're looking for the derivative at x = 4, we can only use the positive root:

x = √(2/3)

Now, we can plug this value of x back into the derivative of f(x) to find dx f-'(4):

f'(√(2/3)) = 6(√(2/3))² + cos(Ф)

f'(√(2/3)) = 4 + cos(Ф)

dx f-'(4) = f'(√(2/3)) = 4 + cos(Ф)

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(a) The triangle is a right triangle when x is equal to 24.

(b) The side lengths do not form a Pythagorean triple.

(a) To determine when the triangle is a right triangle, we can apply the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides.

Using the given side lengths, we have:

5^2 + 7^2 = x^2

Simplifying the equation:

25 + 49 = x^2

74 = x^2

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

x = √74

Approximating the square root of 74, we get:

x ≈ 8.60

Therefore, when x is approximately 8.60, the triangle is a right triangle.

(b) For the side lengths to form a Pythagorean triple, they must satisfy the condition of the Pythagorean theorem. In this case, we have:

5^2 + 7^2 = x^2

Simplifying the equation:

25 + 49 = x^2

74 = x^2

Since the sum of the squares of the two smaller sides (25 + 49 = 74) is not equal to the square of the longest side (x^2 = 74), the side lengths of 5, 7, and x do not form a Pythagorean triple.

In conclusion, the triangle is a right triangle when x is equal to approximately 8.60, and the side lengths do not form a Pythagorean triple.

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There were initially 8 people on Team One.

Let's assume the number of people on Team Two to be "x". According to the given condition, the number of people on Team One is one-quarter of Team Two. Therefore, the number of people on Team One is x/4.

Now, 12 people are transferred from Team Two to Team One. So, the new number of people on Team Two is x-12, and the new number of people on Team One is x/4+12.

As per the given condition, the new number of people on each team is equal. So, we can write the following equation:

x/4+12 = x-12

Solving this equation, we get:

3x/4 = 24

x = 32

So, the initial number of people on Team One is x/4, which is:

32/4 = 8

Therefore, there were initially 8 people on Team One.

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the price per foot of the copper wire is $1.62.

What is the arithmetic operation?

The basic mathematical operations are addition, subtraction, multiplication, and division, which involve manipulating two or more quantities. They are essential to the study of numbers, including the order of operations, and are fundamental to other mathematical areas such as algebra, data management, and geometry. Understanding the rules of arithmetic operations is crucial for solving problems that involve these operations.

There are 3 feet in one yard, so 2 yards of copper wire is equal to 6 feet.

To find the price per foot, we need to divide the total price by the number of feet:

Price per foot = Total price ÷ Number of feet

Price per foot = $9.72 ÷ 6

Price per foot = $1.62

Therefore, the price per foot of the copper wire is $1.62.

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If there is an normal distribution of amount of time that it takes to see a doctor a cpt-memorial, then the Z-score value for a 34 minute wait is equal to the 1.1.

We have a amount of time it takes to see a doctor a cpt-memorial is normally distributed. Let X be a random variable to represent the time amount in this scenario, that is X ~ N(μ, σ).

Mean, [tex], \mu[/tex] = 23 minutes

Standard deviations, [tex] \sigma[/tex]

= 10 minutes

We have to calculate Z-score for 34 a minute wait. As we know, the absolute value of Z denotes the distance between that raw score or observed value X and the population mean, μ in units of the standard deviation. Mathematically, formula for Z-score is [tex]Z = \frac{ X - \mu } {\sigma} [/tex]

where, Z --> Z-score

X--> observed value

σ --> standard deviations

μ --> population mean

Here, X = 34 minutes so plug all known values, [tex]Z = \frac{34 - 23}{10}[/tex]

=> Z = 11/10 = 1.1

Hence, the required Z-score value is 1.1.

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The sector angle formed by the cone when it is opened is 54°.

V = 95.4 cm³

r = 2.4 cm

Calculating the height of the cone using the volume formula,

V= 1/3 x π x r² x h

Substituting the values -

95.4 = 1/3 x 3.14 x 2.4² x h

95.4 = 6.03 x h

h = 15.8 cm

Calculating the slant height using  the Pythagoras theorem -

l = √(h² + r²)

Substituting the values -

l = √(15.8² + 2.4²)

l = 16

Calculating the curved surface area of the cone -

= πrl

= π(2.4)(16)

= 120.6 cm².

Calculating the sector angle of the sector formed -

The curved surface area of the cone = area of the sector formed by the cone

= 120.6 cm².

Area of a sector in a circle = ∅/360 × πr²,

120.6 = ∅/360 × (3.14)(16²)

120.6 = ∅/360 × 803.84

(120.6)(360) = (∅)(803.84)

43,416/803.84 = ∅

∅ = 54°

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The lengths of the sides are: AB = CD = 4.5667 inches; BC = 9.1333 inches and AD = 9.1333 inches

An isosceles trapezoid is a four-sided figure with two parallel sides and two non-parallel sides that are equal in length. In this problem, we are given that the perimeter of the isosceles trapezoid ABCD is 27.4 inches, and that BC is twice as long as AB.

Let's start by assigning variables to the lengths of the sides. Let AB = x, BC = 2x, CD = x, and AD = y. Since the perimeter of the trapezoid is the sum of all four sides, we can write the equation:

x + 2x + x + y = 27.4

Simplifying the equation, we get:

4x + y = 27.4

We also know that the non-parallel sides of an isosceles trapezoid are equal in length, so we can write:

AB = CD = x

Now we can use the fact that BC is twice as long as AB to write:

BC = 2AB

Substituting x for AB, we get:

2x = BC

Now we can use the Pythagorean theorem to find the length of AD. The Pythagorean theorem states that in a right triangle, the sum of the squares of the legs (the shorter sides) is equal to the square of the hypotenuse (the longest side). Since AD is the hypotenuse of a right triangle, we can write:

AD^2 = BC^2 - (AB - CD)^2

Substituting the values we know, we get:

y^2 = (2x)^2 - (x - x)^2

Simplifying, we get:

y^2 = 4x^2

Taking the square root of both sides, we get:

y = 2x

Now we can use the equation we found earlier to solve for x:

4x + y = 27.4

4x + 2x = 27.4

6x = 27.4

x = 4.5667

Now we can find the lengths of the other sides:

AB = CD = x = 4.5667

BC = 2AB = 2x = 9.1333

AD = y = 2x = 9.1333

So the lengths of the sides are:

AB = CD = 4.5667 inches

BC = 9.1333 inches

AD = 9.1333 inches

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Answer:

147 mm^2

Step-by-step explanation: The surface area has to be more, but not double the smaller figures surface area therefore the answer is 147 mm^2

The requried surface area of the larger solid is approximately 147 mm². Option A is correct.

The surface area of any shape is the area of the shape that is faced or the Surface area is the amount of area covering the exterior of a 3D shape.

Let's call the larger solid's surface area S.

Since the two solids are similar, their volumes have a ratio of (side length)³. Let's call the ratio of the side lengths of the larger to the smaller solid as k. Then:

[tex](k^3)(540 mm^3) = 857.5 mm^3[/tex]

Simplifying the above equation, we get:

[tex]k = (857.5/540)^{(1/3)}[/tex] =7/6

So, the larger solid is about 7/6=1.183 times bigger than the smaller solid in terms of side length. Since the surface area has a ratio of [tex](side length)^2[/tex], we can find the surface area of the larger solid by:

[tex]S = (1.183^2)(108 mm^2) \approx 147 mm^2[/tex]

Therefore, the surface area of the larger solid is approximately 147 mm². Answer: A.

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The LHS is equal to the RHS, and the given equation is verified as an identity. We have to verify that the following equation is an identity:

2cos(x) 2x / sin2(x) = cot(x) - tan(x)

Starting from the left-hand side (LHS):

2cos(x) 2x / sin2(x) = 2cos(x) 2x / (1 - cos2(x)) (using the identity sin2(x) = 1 - cos2(x))

= 2cos(x) 2x / (1 - cos(x)) (1 + cos(x))

= 2cos(x) 2x / (1 - cos(x)) (1 + cos(x)) (multiplying the denominator by (1 + cos(x)))

= 2cos(x) 2x / (1 - cos2(x))

= 2cos(x) 2x / sin2(x) (using the identity 1 - cos2(x) = sin2(x))

= 2cos(x) / sin(x) (simplifying by canceling out the common factor of 2 and cos(x))

= 2cos(x) / sin(x) * (cos(x) / cos(x)) (multiplying by 1 in the form of cos(x)/cos(x))

= 2cos2(x) / (sin(x)cos(x))

= 2cos(x)/sin(x) * cos(x)

= cot(x) * cos(x)

Now, moving to the right-hand side (RHS):

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

= cos2(x)/sin(x)cos(x) - sin2(x)/sin(x)cos(x)

= (cos2(x) - sin2(x))/sin(x)cos(x)

= cos(x)/sin(x) * cos(x)/cos(x) - sin(x)/cos(x) * sin(x)/sin(x) (using the identity cos2(x) - sin2(x) = cos(x)cos(x) - sin(x)sin(x))

= cot(x) * cos(x)

Therefore, the LHS is equal to the RHS, and the given equation is verified as an identity.

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Step-by-step explanation:

a probability is always the ratio

desired cases / totally possible cases.

we have here a total of 6 + 8 + 6 = 20 marbles.

to not pull a blue or yellow marble is in this context the same event as pulling a red marble.

so, the desired cases are 6 (red).

which we can get directly from the 6 red marbles, or by counting off the undesired cases : 20 - 8 - 6 = 6.

and the probabilty for not pulling a blue or yellow marble (or simply pulling a red marble) is

6/20 = 3/10 = 0.3

Jane has 12 coins and Jim has 29 coins.

We know that this is a word problem and the first thing that we have to do is to form the equation from the problem that have been given to us here. This is what we shall now proceed to do below.

Let the number of coins that Jane has be x

Number of coins that Jim has = 5 + 2x

Total number of coins = 41

Thus we have that;

x + 5 + 2x = 41

3x + 5 = 41

3x = 41 - 5

3x = 36

x = 12

This implies that Jane has 12 coins and Jim has 5 + 2(12) = 29 coins

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Hi! I'd be happy to help you with your question. To find the volume of two laundry detergent boxes, each measuring 12 inches by 8 inches by 3 inches,

you can follow these steps:

Formula: Volume = Length × Width × Height

Step 1: Plug in the values for one box:
Volume = 12 inches × 8 inches × 3 inches

Step 2: Calculate the volume of one box:
Volume = 288 cubic inches

Step 3: Multiply the volume of one box by 2 to find the volume of two boxes:
Total volume = 288 cubic inches × 2

Solution: The total volume of two laundry detergent boxes is 576 cubic inches.

Your answer: The volume of two boxes of detergent is 576 cubic inches.

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The student did not solve the problem correctly. The student only discovered one of two solutions and made the mistake of presuming that the absolute value of -3 was the other solution without investigating the second situation.

When solving absolute value equations, we have to consider both cases:

|2-9x| = 29 can be rewritten as

2-9x = 29 or 2-9x = -29

Solving the first equation as the student did:

2-9x = 29

Subtracting 2 from both sides:

-9x = 27

Dividing both sides by -9:

x = -3

This is one solution, but we also need to solve the second equation:

2-9x = -29

Subtracting 2 from both sides:

-9x = -31

Dividing both sides by -9:

x = 31/9

So the two solutions are x = -3 and x = 31/9.

Taking the absolute value of -3 gives us 3, which is one of the solutions the student found. However, the other solution is x = 31/9, not |-3|.

Therefore, the student only found one of the two solutions and made an error in assuming that the absolute value of -3 was the other solution without considering the second case.

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The size  of square tablecloth needed to cover the entire square tabletop is 25 ft^2 (Option B).

To determine the size of the tablecloth needed to cover a square tabletop with a perimeter of 20 feet, we will first find the side length of the square and then calculate the area.

Step 1: Determine the side length of the square.
Since the perimeter of a square is equal to 4 times its side length (P = 4s), we can find the side length by dividing the perimeter by 4.
Side length (s) = Perimeter (P) / 4 = 20 ft / 4 = 5 ft

Step 2: Calculate the area of the square.
The area of a square is equal to the side length squared (A = s^2).
Area (A) = Side length (s) ^ 2 = 5 ft ^ 2 = 25 ft^2

So, the size of the tablecloth needed to cover the entire square tabletop is 25 ft^2 (Option B).

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The quilt's area is approximately 60.74 square feet.

To calculate the area of the quilt, we need to multiply the width by the length.

First, we need to convert the mixed numbers to improper fractions:

Width: 6 5/7 ft = (7 x 6 + 5)/7 = 47/7 ft

Length: 7 3/5 ft = (5 x 7 + 3)/5 = 38/5 ft

Now, we can multiply the width by the length:

Area = width x length

Area = (47/7) ft x (38/5) ft

Area = 2126/35 sq ft

Area ≈ 60.74 sq ft

Therefore, the area of the quilt is approximately 60.74 square feet.

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The form in which is easier to identify the x-intercepts is the one in option B. Intercept form.

If a quadratic equation has a leading coefficient a and x-intercepts x₁ and x₂, then the quadratic equation can be written as:

y = a*(x - x₁)*(x - x₂)

That is called the factored form or the intercept form.

Notice that if the quadratic equation is written in that form, is really easy to identify the x-intercepts of the equation, then that would be the most beneficial form in order for Kristen to quickly identify the coordinates, the correct option is B.

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The median means that as many as friends have less than A. 1. 5 pets as those that have more than A. 1. 5 pets.

The median is a measure of central tendency in statistics that represents the middle value of a dataset when it is ordered from smallest to largest. The median is often used as a more robust measure of central tendency than the mean, because it is less affected by extreme values in the dataset.

From the box plot, the data set of friends with pets would be:

0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, ,2 ,2, 2, 2, 2, 3, 4, 4, 7.

The median here is:

= ( 10 th position + 11 th position ) / 2

= ( 1 + 2 ) / 3

= 1. 5

This therefore means that as many friends have more than 1. 5 pets as those with less than 1. 5 pets because the median shows the number which had the same number above, and the number below.

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The circumference of the given circle with TQ = 8 units is given by approximately 50.26 units.

We know that the formula for the circumference of a circle with radius of 'r' units is given by,

P = 2πr

Here in the given figure we can see that the length TQ is a radius for the given circle with center at point Q.

Given the value of TQ = 8 units.

So, radius = 8 units.

So the circumference of the circle is given by

= 2πr

= 2π*8

= 16π

= 50.26 units [taking π = 3.14 and approximating the value to the two decimal places]

Hence the circumference of circle is 50.26 units approximately.

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We can predict with 99% confidence that a new employee with 9 hours of training will produce between -4.16 and 3.49 defects per countertop.

To find the 99% prediction interval for the number of defects per countertop for an employee with 9 hours of training, we can use the estimated regression equation and the standard error provided.

The 99% prediction interval is given by:

Predicted value ± t(0.995, n-2) x SE

where t(0.995, n-2) is the t-score for the 99% confidence level with n-2 degrees of freedom (where n is the sample size), and SE is the standard error.

First, we need to calculate the predicted value:

Defects per Countertop = 6.717822 - 1.004950(Hours of Training)

Defects per Countertop = 6.717822 - 1.004950(9)

Defects per Countertop = -0.334578

Next, we need to find the t-score for the 99% confidence level with 8 degrees of freedom (n-2 = 10-2 = 8). Using a t-distribution table or calculator, we find that t(0.995, 8) = 3.355387.

Finally, we can calculate the 99% prediction interval:

-0.334578 ± 3.355387 x 1.2297787

This simplifies to:

-4.157722 < Defects per Countertop < 3.488566

Therefore, we can predict with 99% confidence that a new employee with 9 hours of training will produce between -4.16 and 3.49 defects per countertop. However, since the lower limit is negative, it does not have practical meaning in this context. Therefore, we can conclude that we can predict with 99% confidence that a new employee with 9 hours of training will produce between 0 and 3.49 defects per countertop.

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Answer:

y = 11x/13 + 36/13

Step-by-step explanation:

We can write the line using y = mx + b form.

To find the slope, m, we can use the formula (y1 - y2) / (x1 - x2):

(7-(-4)) / (5-(-8)) = (7+4) / (5+8) = 11 / 13.

To find b, we can plug in one of the points. Lets use (5, 7).

y = 11/13 * x + b

7 = 11/13 * 5 + b

7 - 55/13 = b

b = 91/13 - 55/13 = (91-55)/13 = 36/13.

Your equation is:

y = 11x/13 + 36/13.

Answer: y = [tex]\frac{11}{13}[/tex]x + [tex]\frac{36}{13}[/tex]

Step-by-step explanation:

First, we will find the slope.

       [tex]m=\displaystyle \frac{y_{2} -y_{1} }{x_{2} -x_{1} }=\frac{-4-7}{-8-5} =\frac{-11}{-13} =\frac{11}{13}[/tex]

Next, we will substitute this slope and a given point in and solve for our y-intercept (b).

       y = [tex]\frac{11}{13}[/tex]x + b

       (7) = [tex]\frac{11}{13}[/tex](5) + b

       (7) = [tex]\frac{11}{13}[/tex](5) + b

       7 = [tex]\frac{55}{13}[/tex] + b

       b = 7 - [tex]\frac{55}{13}[/tex]

       b = [tex]\frac{36}{13}[/tex]

Final equation:

       y = mx + b

     y = [tex]\frac{11}{13}[/tex]x + [tex]\frac{36}{13}[/tex]

Using the identity: sin(A + B) = sin A cos B + cos A sin B, we can rewrite the expression as follows:

Sin 7A * Cos 3A = (sin 4A + sin 10A)/2 * (cos 2A + cos A)/2

Expanding this expression using the same identity, we get:

= (sin 4A * cos 2A + sin 4A * cos A + sin 10A * cos 2A + sin 10A * cos A)/4

Now, using the identity sin 2A = 2 sin A cos A, we can simplify further:

= (1/2) sin 2A * cos 2A + (1/2) sin 2A * cos 8A + (1/2) sin 6A * cos 2A + (1/2) sin 6A * cos 8A

Therefore, Sin 7A * Cos 3A can be written as:

(1/2) sin 2A * cos 2A + (1/2) sin 2A * cos 8A + (1/2) sin 6A * cos 2A + (1/2) sin 6A * cos 8A

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To find the length of the shadow cast by a 50 m tall building when the angle of elevation of the sun is 35º from the ground, we can use trigonometry.

Step 1: Identify the known values and the unknown.
- Angle of elevation: 35º
- Building height: 50 m
- Unknown: Shadow length

Step 2: Recognize the trigonometric function to be used.
Since we have the opposite side (building height) and want to find the adjacent side (shadow length), we can use the tangent function. The formula is:

tan(angle) = opposite side/adjacent side

Step 3: Plug in the known values and solve for the unknown.
tan(35º) = 50 m / shadow length

Step 4: Rearrange the equation to isolate the shadow length.
shadow length = 50 m / tan(35º)

Step 5: Calculate the shadow length.
shadow length ≈ 50 m / 0.7002 ≈ 71.4 m

So, the length of the shadow cast by the building is approximately 71.4 m.

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We can follow the following strategies separated by comma's : Use implicit differentiation directly on the given equation, Multiply both sides of the given equation by the denominator of the left side, then use implicit differentiation
, Solve for y, then differentiate.

Strategy 1: Use implicit differentiation directly on the given equation Start by taking the derivative of both sides of the equation with respect to x:  dy/dx = (3x^2 + 2xy)/(2y - 3) . Now solve for 3r:
3r = (dy/dx)(2y - 3)/(2x)

3r = (3x^2 + 2xy)/(4x)

3r = (3/4)x + (1/2)y

Strategy 2: Multiply both sides of the given equation by the denominator of the left side, then use implicit differentiation
Start by multiplying both sides of the equation by (2y - 3):  (2y - 3)y = 3x^2 + 2xy . Simplify:
2y^2 - 3y = 3x^2 + 2xy

Now take the derivative of both sides with respect to x:

d/dx(2y^2 - 3y) = d/dx(3x^2 + 2xy)

4y(dy/dx) - 3(dy/dx) = 6x + 2y(dy/dx)

Solve for dy/dx:

dy/dx = (6x - 3y)/(2y - 4y) = (3x - y)/(y - 2)

Now solve for 3r:

3r = (dy/dx)(2y - 3)/(2x)

3r = ((3x - y)/(y - 2))(2y - 3)/(2x)

3r = (3/4)x + (1/2)y

Strategy 3: Solve for y, then differentiate Start by solving the given equation for y:  2y^2 - 3y = 3x^2 + 2xy
2y^2 - 2xy - 3y - 3x^2 = 0

Use the quadratic formula:

y = (2x ± sqrt(4x^2 + 24x^2))/4

Simplify:

y = (x ± sqrt(7)x)/2

Now take the derivative of y with respect to x:

dy/dx = (1 ± (1/2)sqrt(7))/(2)

Solve for 3r:

3r = (dy/dx)(2y - 3)/(2x)

3r = ((1 ± (1/2)sqrt(7))/(2))(2(x ± sqrt(7)x)/2 - 3)/(2x)

3r = (3/4)x + (1/2)y

All three strategies result in the same answer for 3r in terms of x and y, which is (3/4)x + (1/2)y. This can be shown by simplifying the expressions obtained in each strategy and verifying that they are equivalent. Unfortunately, we cannot proceed with the explanation as the given equation is missing from the student question. Please provide the equation involving x, y, and r to receive a detailed step-by-step explanation of the three strategies.

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