100 Points! Geometry question. Photo attached. Please show as much work as possible. Thank you!

The solution to the equation log2(3x - 7) = 3 is x = 5.

To find the solution of the equation log2(3x - 7) = 3, we can use logarithmic properties to rewrite the equation in exponential form. The logarithmic equation states that log(base 2) of (3x - 7) equals 3. In exponential form, this can be expressed as:

2^3 = 3x - 7

Simplifying the left side of the equation, we have:

8 = 3x - 7

To isolate the variable term, we add 7 to both sides of the equation:

8 + 7 = 3x

15 = 3x

Next, we divide both sides of the equation by 3 to solve for x:

15/3 = x

5 = x

Therefore, the solution to the equation log2(3x - 7) = 3 is x = 5. By substituting x = 5 back into the original logarithmic equation, we can verify the solution:

log2(3(5) - 7) = 3

log2(15 - 7) = 3

log2(8) = 3

Simplifying further:

[tex]2^3 = 8[/tex]

8 = 8

Both sides are equal, confirming that x = 5 is indeed the solution to the given equation.

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The number of vans, small truck and large truck they can buy is 130, 65 and 65 respectively.

Total cost of vehicles = $10 million

Total number of vehicles = 260

Cost of each commercial van = $25,000

Cost of each small truck = $50,000

Cost of each large truck = $50,000

Let

c = commercial van

s = small truck

l = large truck

v = 2s

s = l

So,

v + s + l = 260

2s + s + s = 260

4s = 260

divide both sides by 4

s = 260/4

s = 65

Therefore,

v = 2s

= 2(65)

= 130

s = l

l = 65

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The calculated values of x and y in the figure are x = 2 and y = 4

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

The figure

Where, we have

equal side lengths

This means that

2y - 1 = 3y - 5

Evaluate

y = 4

Next, we have

x + 3 = 3/2x + 2

So, we have

1/2x = 1

This gives

x = 2

Hence, the values of x and y in the figure are x = 2 and y = 4

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The calculated length x of the right triangle is 16

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

The right triangle

The length x of the right triangle can be calculated using the following Pythagoras theorem

x² = difference of squares of the other sides

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

x² = (20)² - (12)²

Evaluate

x² = 256

Take the square roots

x = 16

Hence, the length x of the right triangle is 16

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Answer:

C.  5x(-x + 2)

Step-by-step explanation:

To factor the expression -5x² + 10x, we need to look for a common factor that can be factored out.

Finding a common factor involves identifying a term or expression that can be factored out from each term of a given expression.

Both terms have the common factor of 5x, so we can factor out 5x:

5x(-x + 2)

Therefore, the factored form of -5x² + 10x is -5x(x - 2).

[tex]\hrulefill[/tex]

Additional notes:

If we expand the expressions in the given answer options, we get:

  A.  −5x(x + 2) = -5x² - 10x

  B.  5(−x² + 10x) = -5x² + 50x

  C.  5x(−x + 2) = -5x² + 10x

  D.  x(5x + 10) = 5x² + 10x

Hence confirming that the correct answer is option C.

To factor [tex]-5x^2+10x[/tex], we can begin by factoring out the greatest common factor, which is [tex]-5x[/tex]:

We can check our answer by distributing [tex]-5x[/tex] to the expression inside the parentheses:

[tex]\therefore[/tex] The answer is [tex]-5x(x-2)[/tex].

[tex]\blue{\overline{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}}[/tex]

Average speed of the humpback whale: 27 kilometers per hour.

To calculate the average speed of the humpback whale, we can use the formula:

Average speed = Total distance / Total time

Given that the humpback whale traveled a distance of 13.5 kilometers in 30 minutes, we need to convert the time to hours. There are 60 minutes in an hour, so 30 minutes is equal to 0.5 hours.

Now, we can substitute the values into the formula:

Average speed = 13.5 kilometers / 0.5 hours

Average speed = 27 kilometers per hour

Therefore, the average speed of the humpback whale is 27 kilometers per hour.

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The sum diverges to negative infinity.

Here we want to find the value of the sum:

[tex]\sum_{m=1}^{ \infty}} (-11/2)*(3/2)^{m + 1}[/tex]

So, that sum goes for infinite values of m, that is bad because you can see that the term with an exponent is larger than 1.

So when m is a really large value, then the term will also be a really large value, which means that the fuction eventually diverges to negative inifnity.

The usual rule that we need to check is that, for large values of m, as m increases, the absolute value of each term decreases.

Here this cleraly does not happen, so the sum diverges.

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The domain of the function is all real numbers less than 5, and the range is all real numbers greater than or equal to 9.

The given function is f(x) = -log(5-x) + 9.

To determine its domain and range, we need to consider the restrictions on x and the possible values of f(x).

Domain:

The domain refers to the set of all valid inputs or values of x for which the function is defined.

In this case, the function contains a logarithmic term, -log(5-x), which is only defined for positive values inside the logarithm.

Therefore, we must ensure that the expression 5-x is greater than zero:

5 - x > 0

Solving this inequality, we find:

x < 5

Hence, the domain of the function is (-∞, 5).

Range:

The range refers to the set of all possible output values or values of f(x). Since the logarithm term can take any positive value, and we add 9 to it, the range of the function is shifted upward by 9 units.

The range can be defined as all real numbers greater than or equal to 9:

Range: [9, +∞)

To summarize, the domain of the function f(x) = -log(5-x) + 9 is (-∞, 5), and the range is [9, +∞), indicating that f(x) can take any value greater than or equal to 9.

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keeping in mind that parallel lines have exactly the same slope, let's check for the slope of the second line

[tex](\stackrel{x_1}{4}~,~\stackrel{y_1}{5})\qquad (\stackrel{x_2}{6}~,~\stackrel{y_2}{4}) ~\hfill~ \stackrel{slope}{m}\implies \cfrac{\stackrel{\textit{\large rise}} {\stackrel{y_2}{4}-\stackrel{y1}{5}}}{\underset{\textit{\large run}} {\underset{x_2}{6}-\underset{x_1}{4}}} \implies \cfrac{ -1 }{ 2 } \implies - \cfrac{1}{2}[/tex]

so we're really looking for the equation of a line whose slope is -1/2 and it passes through (2 , -2)

[tex](\stackrel{x_1}{2}~,~\stackrel{y_1}{-2})\hspace{10em} \stackrel{slope}{m} ~=~ - \cfrac{1}{2} \\\\\\ \begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{(-2)}=\stackrel{m}{- \cfrac{1}{2}}(x-\stackrel{x_1}{2}) \implies y +2 = - \cfrac{1}{2} ( x -2) \\\\\\ y+2=- \cfrac{1}{2}x+1\implies {\Large \begin{array}{llll} y=- \cfrac{1}{2}x-1 \end{array}}[/tex]

Answer:

Line passing through (4, 5) and (6, 4):

[tex]m = \frac{4 - 5}{6 - 4} = - \frac{1}{2} [/tex]

Line passing through (2, -2) and with slope -1/2:

-2 = (-1/2)(2) + b

-2 = -1 + b, so b = -1

y = (-1/2)x - 1

-2y = x + 2

-x - 2y = 2

x + 2y = -2

The slope of the line represents the speed in miles per hour

The y intercept represents the initial distance

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

The graph

Where, we have

x = time in hours

y = distance in miles

The slope is

slope = change in y/x

So, we can conclude that the slope is the speed

By definition, the y intercept is the initial value of the graph

In this case;

y intercept = initial distance

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Answer: 1/7

Step-by-step explanation:

There are seven balls in Shawn's bag, and he has to choose one ball out of them.

Since all the balls are equally likely to be chosen, the total number of outcomes can be found using the following formula:

Total number of outcomes = number of balls in the bag = 7

Now, Shawn has only one purple ball in his bag.

Hence, the number of favorable outcomes will be one.

Therefore, the probability of Shawn choosing the purple ball out of the bag can be calculated using the following formula:

Probability = Number of favorable outcomes/ Total number of outcomes

Probability of Shawn choosing the purple ball = 1/7 or 0.143 (rounded off to three decimal places).

Hence, the probability of Shawn choosing the purple ball out of the bag is 1/7.

Answer:

The probability that on a particular week, the hospital will receive on high BP pregnant woman is 0.0068

Step-by-step explanation:

We use the Exponential distribution,

Since we are given that on average, 5 pregnant women with high blood pressure come per week,

So, average = m = 5

Now, on average, 5 people come every week, so,

5 women per week,

so, we get 1 woman per (1/5)th week,

Hence, the mean is m = 1/5 for a woman arriving

and λ = 1/m = 5 = λ

we have to find the probability that it takes higher than a week for a high BP pregnant woman to arrive, i.e,

P(X>1) i.e. the probability that it takes more than a week for a high BP pregnant woman to show up,

Now,

P(X>1) = 1 - P(X<1),

Now, the probability density function is,

[tex]f(x) = \lambda e^{-\lambda x}[/tex]

And the cumulative distribution function (CDF) is,

[tex]CDF = 1 - e^{-\lambda x}[/tex]

Now, CDF gives the probability of an event occuring within a given time,

so, for 1 week, we have x = 1, and λ = 5, which gives,

P(X<1) = CDF,

so,

[tex]P(X < 1)=CDF = 1 - e^{-\lambda x}\\P(X < 1)=1-e^{-5(1)}\\P(X < 1)=1-e^{-5}\\P(X < 1) = 1 - 6.738*10^{-3}\\P(X < 1) = 0.9932\\And,\\P(X > 1) = 1 - 0.9932\\P(X > 1) = 6.8*10^{-3}\\P(X > 1) = 0.0068[/tex]

So, the probability that on a particular week, the hospital will receive on high BP pregnant woman is 0.0068

Rounded to the nearest tenth of a percent, the probability is approximately 27.8%.

To determine if the event of rolling a pair of dice and getting doubles or a sum of 6 is mutually exclusive, we need to assess whether the two outcomes can occur simultaneously or if they are mutually exclusive.

Let's analyze the two outcomes separately:

1. Getting doubles: When rolling a pair of dice, getting doubles means that the numbers on both dice are the same. The possible outcomes for getting doubles are (1,1), (2,2), (3,3), (4,4), (5,5), and (6,6). There are a total of 6 outcomes that satisfy this condition.

2. Getting a sum of 6: To get a sum of 6, the numbers on the two dice should add up to 6. The possible outcomes for this condition are (1,5), (2,4), (3,3), (4,2), and (5,1). There are a total of 5 outcomes that satisfy this condition.

Now, to determine if the two outcomes are mutually exclusive, we need to check if there are any outcomes that satisfy both conditions. In this case, the outcome (3,3) satisfies both conditions (doubles and sum of 6).

Since there is one outcome that satisfies both conditions, we can conclude that the event of rolling a pair of dice and getting doubles or a sum of 6 is not mutually exclusive.

To find the probability of this event, we need to determine the total number of possible outcomes when rolling a pair of dice. Each die has 6 sides, so there are 6 possible outcomes for each die, resulting in a total of 6 * 6 = 36 possible outcomes when rolling a pair of dice.

The probability is calculated by dividing the number of favorable outcomes (satisfying the condition) by the total number of possible outcomes.

Favorable outcomes = 6 (for doubles) + 5 (for sum of 6) - 1 (for the overlapping outcome) = 10

Probability = Favorable outcomes / Total number of outcomes = 10 / 36 ≈ 0.2778

Rounded to the nearest tenth of a percent, the probability is approximately 27.8%.

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The answer is:

C. circumference

Work/explanation:

Here's the correct answer:

The distance around a circle is called the circumference.

How to find the circumference :

When summarizing the plot, it is important to pay attention to the key ideas and how the conflict develops over time. The given pattern suggests a narrative structure where the plot unfolds gradually, revealing information and building tension.

The plot starts with an introduction (line 1) and then introduces the conflict or problem (lines 2 and 3). As the plot progresses, the conflict continues to develop, possibly reaching a climax or a turning point (line 4). Afterward, the plot moves towards a resolution or conclusion (lines 5, 6, and 7).

The pattern presented in the given example showcases the basic elements of a plot, including the exposition, rising action, climax, falling action, and resolution. The plot begins by setting the stage and introducing the characters and setting (line 1). Then, the conflict is introduced or developed, creating tension and driving the narrative forward (lines 2 and 3).

The plot further progresses with the development of the conflict, possibly reaching a high point or turning point (line 4). Finally, the plot moves towards its resolution or conclusion, providing closure to the story (lines 5, 6, and 7). By understanding these key ideas and the progression of the conflict, one can effectively summarize the plot and capture its essential elements.

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The slope, "m," indicates the rate of change of the line, while the y-intercept, "b," represents the value of y when x equals zero.

I apologize, but as a text-based AI model I'm unable to directly view or analyze any attached images or graphs. However, if you provide me with the equation of the line in the slope-intercept form or describe the relevant details from the graph, I can certainly help you with a response within the given word limit.

The slope-intercept form of a linear equation is given by y = mx + b, where "m" represents the slope of the line and "b" represents the y-intercept, the point where the line intersects the y-axis.

Using this equation, you can determine the slope and y-intercept by comparing the given equation with the slope-intercept form.

Once you provide me with the equation or necessary information, I'll be able to assist you further in understanding the line and its characteristics.

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1. Mr. John received more than 2 cords of wood based on the sizes of the stacked piles.

2. Based on the stacked piles, Mr. John received approximately 11,974 pieces of actual wood.

3. The cost per standard piece of wood is approximately $0.0417.

1. To determine if Mr. John received more or less than 2 cords of wood, we need to calculate the volume of each pile and compare it to the volume of a standard cord of wood.

Calculating the volume of each pile:

PILE #1:

Volume = 72 inches * 167 inches * 16 inches = 1,523,584 cubic inches

PILE #2:

Volume = 65 inches * 266 inches * 16 inches = 2,068,480 cubic inches

Calculating the volume of a standard cord of wood:

Volume of a standard cord = 4 feet * 4 feet * 8 feet = 128 cubic feet

Since 1 foot = 12 inches, the volume in cubic inches is 128 * 12 * 12 * 12 = 221,184 cubic inches.

Comparing the volumes:

Pile #1: 1,523,584 cubic inches

Pile #2: 2,068,480 cubic inches

Standard cord: 221,184 cubic inches

It is evident that both piles have significantly more volume than a standard cord of wood. Therefore, Mr. John received more than 2 cords of wood.

2. To determine the number of standard pieces of wood received, we need to calculate the number of pieces that can be obtained from the total volume of the piles.

Total volume of the piles = Volume of Pile #1 + Volume of Pile #2

Total volume = 1,523,584 cubic inches + 2,068,480 cubic inches = 3,592,064 cubic inches

Volume of a standard piece of wood = 5 inches * 5 inches * 12 inches = 300 cubic inches

Number of standard pieces of wood received = Total volume of the piles / Volume of a standard piece of wood

Number of pieces = 3,592,064 cubic inches / 300 cubic inches = 11,973.55 pieces (rounded to the nearest whole number)

Therefore, Mr. John received approximately 11,974 pieces of actual wood based on his stacked piles.

3. To calculate the cost per standard piece of wood, we need to divide the total cost of the two cords by the number of standard pieces received.

Total cost for both cords of wood = 2 cords * $250/cord = $500

Cost per standard piece of wood = Total cost / Number of standard pieces

Cost per piece = $500 / 11,974 = $0.0417 (rounded to four decimal places)

Therefore, the cost per standard piece of wood is approximately $0.0417.

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A. The volume of the hemisphere with a diameter of 48 yards is approximately 1001.1 cubic yards.

B. The volume of the sphere with a circumference of a great circle approximately equal to 26 meters is approximately 359.4 cubic meters.

A. To find the volume of a hemisphere, we first need to calculate the radius. The diameter is given as 48 yards, so the radius is half of that, which is 24 yards.

The formula for the volume of a hemisphere is (2/3)πr^3, where r is the radius. Plugging in the values:

Volume = (2/3) * π * (24^3)

Volume ≈ 1001.1 cubic yards (rounded to the nearest tenth)

Therefore, the volume of the hemisphere is approximately 1001.1 cubic yards.

B. To find the volume of a sphere, we need to know the radius. Since the circumference of a great circle is given as approximately 26 meters, we can use the formula C = 2πr, where C is the circumference and r is the radius.

We can rearrange the formula to solve for the radius: r = C / (2π). Plugging in the circumference:

r = 26 / (2π)

r ≈ 4.134 meters (rounded to the nearest thousandth)

The formula for the volume of a sphere is (4/3)πr^3. Substituting the radius:

Volume = (4/3) * π * (4.134^3)

Volume ≈ 359.4 cubic meters (rounded to the nearest tenth)

Therefore, the volume of the sphere is approximately 359.4 cubic meters.

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Answer:

[tex]\textsf{8.} \quad \textsf{(A)}\;\;\overline{XB}[/tex]

[tex]\textsf{9.} \quad \textsf{(D)}\;\;\overleftrightarrow{BD}[/tex]

Step-by-step explanation:

The radius is the distance from the center of a circle to any point on its circumference.

The center of the given circle is point X.

Therefore, the radii in the given circle are line segments XB, XA and XC.

[tex]\hrulefill[/tex]

A tangent is a straight line that touches a circle at only one point.

The line BD touches the circle at point B.

Therefore, the tangent of the given circle is line BD.

Answer:

8. A

9. D

Step-by-step explanation:

The radius is a straight line from the midpoint to the circle's circumference.

A Tangent is a line going through the circumference of the circle.

Answer: The domain of the expression is all real numbers except where the expression is undefined.

Step-by-step explanation:

Therefore, Heather can make a maximum of 4 biscuits with the given ingredient quantities.

To determine how many biscuits Heather can make, we need to compare the amount of each ingredient required for a single biscuit to the total amount of ingredients she has.

Let's calculate the number of biscuits she can make based on the ingredient quantities provided:

First, we need to find the ratio of each ingredient required per biscuit:

Flour: 100g per biscuit

Butter: 120g per biscuit

Icing sugar: 80g per biscuit

Chocolate: 25g per biscuit

Next, we divide the total amount of each ingredient by the respective ratio to find the maximum number of biscuits she can make:

Flour: 2kg / 100g = 20 biscuits

Butter: 1kg / 120g = 8.33 biscuits (approximately)

Icing sugar: 340g / 80g = 4.25 biscuits (approximately)

Chocolate: 200g / 25g = 8 biscuits

Since we can only make a whole number of biscuits, the limiting factor is the icing sugar. Heather can only make 4 biscuits since she has 4.25 times the required amount of icing sugar.

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Answer:

(x + 9)(x - 8)

Step-by-step explanation:

Factorising the expression

x² + x - 72

consider the factors of the constant term (- 72) which sum to give the coefficient of the x- term (+ 1)

the factors are + 9 and - 8 , since

9 × - 8 = - 72 and 9 - 8 = + 1 , then

x² + x - 72 = (x + 9)(x - 8) ← in factored form

There can be multiple outputs (monthly charges) for a given input (number of credit cards), violating the definition of a function. hence, Situation 3 does not represent a function.

Situation 3: the total amount of money charged monthly on credit cards to the number of credit cards owned does not represent a function.

In a function, each input (or x-value) should have a unique output (or y-value). However, in this situation, the total amount of money charged monthly on credit cards depends on the number of credit cards owned. It is possible to have different credit card numbers but still have the same total amount of money charged monthly.

For example, two people could own different numbers of credit cards but have the same monthly charges.

As a result, the definition of a function can be violated by having many outputs (monthly charges) for a single input (number of credit cards).

Because of this, case 3 does not represent a function.

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Answer:

1244 DCD

Step-by-step explanation:

The probability that a point chosen at random lies on the shaded region is given as follows:

4/7.

The parameters that are needed to calculate a probability are listed as follows:

Then the probability is calculated as the division of the number of desired outcomes by the number of total outcomes.

The area of the shaded region in this problem is given as follows:

4² = 16.

The total area of the figure is given as follows:

16 + 2 x 1/2 x 3 x 4 = 28.

Hence the probability is given as follows:

16/28 = 4/7.

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Answer:

10: Zero-degree monomial.

8x^4: Quartic monomial.

5x² - 7x + 9: Quadratic trinomial.

2x³ + 3: Cubic binomial.

Step-by-step explanation:

10: This is a constant term, and it can be classified as a zero-degree monomial.

8x^4: This is a monomial with a degree of 4, and it contains one term. It can be classified as a quartic monomial.

5x² - 7x + 9: This is a polynomial with three terms. The highest power of the variable, x, is 2, so it is a quadratic polynomial. Therefore, it can be classified as a quadratic trinomial.

2x³ + 3: This is a polynomial with two terms. The highest power of the variable, x, is 3, so it is a cubic polynomial. Hence, it can be classified as a cubic binomial.

10: Zero-degree monomial.

8x^4: Quartic monomial.

5x² - 7x + 9: Quadratic trinomial.

2x³ + 3: Cubic binomial.

The line slants downward from left to right, indicating that as x increases, y decreases.

The given linear equation is -x = 2y + 1. To represent this equation graphically, we need to rearrange it into the slope-intercept form, which is y = mx + b, where m represents the slope and b represents the y-intercept.Rearranging the given equation, we get:2y = -x - 1

Dividing both sides by 2:y = (-1/2)x - 1/2

The slope of the line is -1/2, which means that for every unit increase in x, y decreases by 1/2 unit. The y-intercept is -1/2, indicating that the line intersects the y-axis at (0, -1/2).

Based on this information, the best graph that represents the linear equation y = (-1/2)x - 1/2 is a straight line with a negative slope of -1/2, starting at the point (0, -1/2) and extending infinitely in both directions.

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The value of the equation that defines the function is f(x) = 3(5/3)ˣ

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

(0, 3) and (1, 5)

An exponential function is represented s

y = abˣ

Where,

a = y when x = 0

So, we have

y = 3bˣ

Using the other point, we have

3b = 5

This gives

b = 5/3

So, we have

f(x) = 3(5/3)ˣ

Hence, the equation defining f is f(x) = 3(5/3)ˣ

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Statement 1: In triangle ADC and BCD, AD = BC (opposite sides of a rectangle are congruent).

Statement 2: Angle ADC = Angle BCD (angles of a rectangle are 90°).

Statement 3: DC = DC (reflexive property of equality).

Statement 4: Triangle ADC and BCD are congruent (by SAS postulate).

Statement 5: AC = BD (by CPCTC).

The statement that completes Jimmy's proof is:

DC = DC (reflexive property of equality)

The reflexive property of equality states that any quantity is equal to itself. In this case, statement 3 is stating that the diagonal DC is equal to itself, which is true by the reflexive property of equality.

Therefore, the completed proof is:

Statement 1: In triangle ADC and BCD, AD = BC (opposite sides of a rectangle are congruent).

Statement 2: Angle ADC = Angle BCD (angles of a rectangle are 90°).

Statement 3: DC = DC (reflexive property of equality).

Statement 4: Triangle ADC and BCD are congruent (by SAS postulate).

Statement 5: AC = BD (by CPCTC).

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The width of the cooling tower at the base is approximately 20 meters

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

[tex]\frac{x^2}{400} - \frac{(y - 110)^2}{2304} = 1[/tex]

The above equation is an equation of a hyperbols

The general equation for a hyperbola that has a center at (h, k) is

[tex]\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1[/tex]

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

a² = 400

So, we have

a = 20

Hence, the width is 20 meters

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Question

Cooling towers are used to remove or expel heat from a process. A cooling towers walls are modeled by. x^2/400 - (y - 110)^2/2304 = 1 where the measurements are in meters

What is the width of the cooling tower at the base of the structure? Round your answer to the nearest whole number