Sam has a deck that is shaped like a triangle with a base of 18 feet and a height of 7 feet. He plans to build a 2:5 scaled version of the deck next to his horse's water trough.

Part A: What are the dimensions of the new deck, in feet? Show every step of your work. (4 points)

Part B: What is the area of the original deck and the new deck, in square feet? Show every step of your work. (4 points)

Part C: Compare the ratio of the areas to the scale factor. Show every step of your work. (4 points)

Answer:

1244 DCD

Step-by-step explanation:

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 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

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

Step-by-step explanation:

Answer:

70

Step-by-step explanation:

360

120+100

360-220

140/2

=70

when finding the angle you want to find the arc first. You use the add the arcs given and subtract by 360 because a circle is 360. Then you divide by 2 because the angle will always be half of the arc.

Based on the line of best fit in the provided image, it appears that for y = 14, the estimated x-value is approximately 6.

By examining the line of best fit, we can estimate the x-value corresponding to y = 14. In the image provided, the line of best fit appears to be a straight line passing through several data points. Let's assume this line can be approximated by the equation y = mx + b, where m represents the slope and b represents the y-intercept.

To find the x-value when y = 14, we can substitute y = 14 into the equation and solve for x. However, since we don't have the equation explicitly, we will have to estimate the x-value based on the visual representation of the line of best fit.

Looking at the image, we can observe that the line of best fit intersects the y = 14 mark at approximately x ≈ 6. This is an estimation based on the position of the line relative to the given point.

Please note that this estimation is subject to the accuracy of the plotted points and the line of best fit in the image. For a more precise answer, the actual equation of the line of best fit or additional data would be required.

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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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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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Using simultaneous equation, the solution to the system of linear equations are 1223 $10 tickets, 1332 $20 tickets, and 763 $30 tickets were sold.

Let's solve the problem step by step.

Let:

x = number of $10 tickets sold

y = number of $20 tickets sold

z = number of $30 tickets sold

From the given information, we can form the following equations:

Equation 1: x + y + z = 3318 (Total number of tickets sold)

Equation 2: y = x + 109 (109 more $20 tickets than $10 tickets were sold)

Equation 3: 10x + 20y + 30z = 61760 (Total sales from ticket sales)

We can use these three equations to solve for the values of x, y, and z.

First, let's substitute Equation 2 into Equation 1:

x + (x + 109) + z = 3318

2x + 109 + z = 3318

2x + z = 3209 (Equation 4)

Now, let's substitute the value of y from Equation 2 into Equation 3:

10x + 20(x + 109) + 30z = 61760

10x + 20x + 2180 + 30z = 61760

30x + 30z = 59580

x + z = 1986 (Equation 5)

We now have a system of equations (Equations 4 and 5) with two variables (x and z). We can solve this system to find the values of x and z.

Multiplying Equation 4 by 30, and Equation 5 by 2, we get:

60x + 30z = 96270 (Equation 6)

2x + 2z = 3972 (Equation 7)

Now, subtract Equation 7 from Equation 6:

(60x + 30z) - (2x + 2z) = 96270 - 3972

58x + 28z = 92298

Simplifying, we have:

29x + 14z = 46149 (Equation 8)

Now, we can solve Equations 5 and 8 simultaneously:

x + z = 1986 (Equation 5)

29x + 14z = 46149 (Equation 8)

Multiplying Equation 5 by 14, and Equation 8 by 1, we get:

14x + 14z = 27804 (Equation 9)

29x + 14z = 46149 (Equation 8)

Now, subtract Equation 9 from Equation 8:

(29x + 14z) - (14x + 14z) = 46149 - 27804

15x = 18345

Divide both sides of the equation by 15:

x = 18345 / 15

x = 1223

Substituting the value of x into Equation 5, we can find z:

1223 + z = 1986

z = 1986 - 1223

z = 763

Now that we have the values of x and z, we can substitute them back into Equation 1 to find y:

1223 + y + 763 = 3318

y + 1986 = 3318

y = 3318 - 1986

y = 1332

Therefore, the solution to the problem is:

x = 1223 (number of $10 tickets sold)

y = 1332 (number of $20 tickets sold)

z = 763 (number of $30 tickets sold)

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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]

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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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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Using chain rule, the derivative of f(x) = 4√(8x³ + 2x⁴) is

f'(x) = (2/√(8x³ + 2x⁴)) * (24x² + 8x³)

To find the derivative of the function f(x) = 4√(8x³ + 2x⁴), we can use the chain rule.

Let's break down the function into its components:

u(x) = 8x³ + 2x⁴ (inside function)

v(u) = 4√u (outer function)

To find the derivative, we apply the chain rule, which states:

(f(g(x)))' = f'(g(x)) * g'(x)

In this case, f(g(x)) = v(u(x)), and g(x) = u(x).

First, let's find the derivative of the inner function u(x):

u'(x) = 24x² + 8x³ (using the power rule for differentiation)

Next, let's find the derivative of the outer function v(u):

v'(u) = 4 * (1/2) * 1/√u = 1/√2u = 2/√u

Now, we can apply the chain rule:

f'(x) = v'(u(x)) * u'(x)

f'(x) = (2/√u) * (24x² + 8x³)

f'(x) = (2/√(8x³ + 2x⁴)) * (24x² + 8x³)

The derivative of the function is  (2/√(8x³ + 2x⁴)) * (24x² + 8x³)

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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 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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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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Using simultaneous method to solve the system of linear equations, 56 $10 tickets, 1310 $20 tickets, and 1902 $30 tickets were sold.

Let's solve the problem step by step.

Let:

x = number of $10 tickets sold

y = number of $20 tickets sold

z = number of $30 tickets sold

From the given information, we can form the following equations:

Equation 1: x + y + z = 3268 (Total number of tickets sold)

Equation 2: y = x + 259 (259 more $20 tickets than $10 tickets were sold)

Equation 3: 10x + 20y + 30z = 63920 (Total sales from ticket sales)

We can use these three equations to solve for the values of x, y, and z.

First, let's substitute Equation 2 into Equation 1:

x + (x + 259) + z = 3268

2x + 259 + z = 3268

2x + z = 3009 (Equation 4)

Now, let's substitute the value of y from Equation 2 into Equation 3:

10x + 20(x + 259) + 30z = 63920

10x + 20x + 5180 + 30z = 63920

30x + 30z = 58740

x + z = 1958 (Equation 5)

We now have a  (Equations 4 and 5) with two variables (x and z). We can solve this system to find the values of x and z.

Multiplying Equation 4 by 30, and Equation 5 by 2, we get:

60x + 30z = 60270 (Equation 6)

2x + 2z = 3916 (Equation 7)

Now, subtract Equation 7 from Equation 6:

(60x + 30z) - (2x + 2z) = 60270 - 3916

58x + 28z = 56354

Simplifying, we have:

29x + 14z = 28177 (Equation 8)

Now, we can solve Equations 5 and 8 simultaneously:

x + z = 1958 (Equation 5)

29x + 14z = 28177 (Equation 8)

Multiplying Equation 5 by 14, and Equation 8 by 1, we get:

14x + 14z = 27332 (Equation 9)

29x + 14z = 28177 (Equation 8)

Now, subtract Equation 9 from Equation 8:

(29x + 14z) - (14x + 14z) = 28177 - 27332

15x = 845

Divide both sides of the equation by 15:

x = 56

Substituting the value of x into Equation 5, we can find z:

56 + z = 1958

z = 1958 - 56

z = 1902

Now that we have the values of x and z, we can substitute them back into Equation 1 to find y:

56 + y + 1902 = 3268

y + 1958 = 3268

y = 3268 - 1958

y = 1310

Therefore, the solution to the problem is:

x = 56 (number of $10 tickets sold)

y = 1310 (number of $20 tickets sold)

z = 1902 (number of $30 tickets sold)

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To identify the appropriate graph displaying the line of best fit, one would need to examine the slope, position, and dispersion of the data points in relation to the line.

To determine which graph displays the line of best fit for the relationship between the number of football games played and the total number of rushing yards, we would need to analyze the given graphs and identify the one that best represents the trend in the data.

The line of best fit is a straight line that represents the average trend of the data points in a scatterplot. It is calculated using regression analysis to find the line that minimizes the overall distance between the line and the data points.

Without the specific graphs to examine, it is not possible to identify the exact graph that displays the line of best fit. However, in a scatterplot of the number of football games played versus the total number of rushing yards, the line of best fit would be a straight line that best represents the general trend or relationship between the two variables.

The line of best fit may have a positive slope if there is a positive correlation between the number of games played and rushing yards, indicating that more games played result in higher rushing yards. Conversely, it may have a negative slope if there is a negative correlation, suggesting that more games played lead to lower rushing yards.

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Note:- This questions is not available in search engine

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: 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.

The 98% confidence interval for the true mean statistics anxiety score μ is approximately (39.023, 39.777).

To calculate the 98% confidence interval for the true mean statistics anxiety score μ, we can use the formula:

[tex]\[\text{CI} = \bar{X} \pm Z \cdot \left(\frac{\sigma}{\sqrt{n}}\right)\][/tex]

where [tex]\bar{X}[/tex] is the sample mean (39.4), Z is the Z-score corresponding to the desired confidence level (in this case, 98% corresponds to a Z-score of 2.33), σ is the population standard deviation (0.9), and n is the sample size (33).

Plugging in the values, we get:

CI = 39.4 ± 2.33 * (0.9/√33).

Calculating this expression, we find:

CI = 39.4 ± 2.33 * (0.162),

CI ≈ 39.4 ± 0.377.

Therefore, the 98% confidence interval for the true mean statistics anxiety score μ is approximately (39.023, 39.777).

This means that we are 98% confident that the true average statistics anxiety score for all undergraduate students in the U.S. falls within this interval.

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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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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 correct statement that describes the transformation of the graph of f(x) = x when considering g(x) = f(9x) is: "It is the graph of f(x) shrunk horizontally by a factor of 9."

To understand this transformation, let's analyze the function g(x) = f(9x). When we substitute x with 9x inside f(x), we effectively compress the graph horizontally.

Since the input x is multiplied by 9, the graph is squeezed horizontally by a factor of 9. This means that the x-values on the graph of g(x) are condensed compared to those on the graph of f(x).

It is important to note that the vertical scaling remains unchanged since the coefficient multiplying x remains 1 in both functions. Therefore, there is no vertical stretching or shrinking.

In summary, the graph of g(x) = f(9x) is the graph of f(x) shrunk horizontally by a factor of 9, while the vertical scaling remains unaffected.

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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 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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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

The limit of the given expression is -3/4.

To find the limit of the given expression, we can apply the properties of limits and the limit theorem.

Let's break down the expression step by step:

We have the expression [tex](-6x^3 + 7x + 7) / (8x^3 - 8x + 5).[/tex]

First, we notice that both the numerator and denominator are polynomials, and the degree of the denominator is greater than the degree of the numerator.

In such cases, we can use the fact that as x approaches either positive or negative infinity, the highest power term dominates the expression. Therefore, we can simplify the expression by dividing every term by[tex]x^3:(-6x^3/x^3 + 7x/x^3 + 7/x^3) / (8x^3/x^3 - 8x/x^3 + 5/x^3).[/tex]

This simplifies to:

[tex](-6 + 7/x^2 + 7/x^3) / (8 - 8/x^2 + 5/x^3).[/tex]

Now, we can take the limit as x approaches infinity.

As x becomes infinitely large, the terms with x in the denominator tend to zero:

((-6 + 0 + 0) / (8 - 0 + 0)).

Thus, the limit of the given expression as x approaches infinity is:

-6/8 = -3/4.

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The half-life and exponential decay formulas indicates that the the half-life and amounts are;

31. 60 days

32. A(t) = 0.5·[tex]e^{(-0.0115\cdot t)}[/tex]

0.3 grams

33. A(t) = 0.5·[tex]e^{(-0.00822\cdot t)}[/tex]

84 minutes

The half-life of a radioactive substance is the duration it takes for half the amount of the substance to decay.

31. The exponential growth formula indicates that we get;

0.5 = 1 × [tex]e^{(-0.0115\cdot t)}[/tex]

Where t = The time in days, which is the half life of the Iodine-125

-0.0115·t = ln(0.5)

The half life of the Iodine-125, [tex]t_{1/2}[/tex] = ln(0.5)/(-0.0115) ≈ 60.3

Therefore, the number of days for half of the Iodine-125 to decay is about 60 days

32. The amount of Iodine-125 remaining after t days, A(t) is can be represented as follows;

The amount of Iodine-125 remaining after 60 days which is the half life for the Iodine-125 is therefore;

33. The initial amount of the radioactive substance, A(0) = 250 g

The amount after 250 minutes, A(250) = 32 g

Therefore, we get;

A(0) = 250·[tex]e^{(k\times 0)}[/tex] = 250

A(250) = 250·[tex]e^{(k\times 250)}[/tex] = 32

32/250 = [tex]e^{(k\times 250)}[/tex]

ln(32/250) = ln([tex]e^{(k\times 250)}[/tex])

k × 250 = ln(32/250)

k = ln(32/250)/250 ≈ -0.00822

The exponential equation is; A(t) = 250·[tex]e^{(-0.00822\cdot t)}[/tex]

The half life can therefore, be found as follows;

125 = 250·[tex]e^{(-0.00822\times t)}[/tex]

[tex]e^{(-0.00822\times t)}[/tex] = 125/250 = 1/2

ln([tex]e^{(-0.00822\times t)}[/tex]) = ln(1/2)

-0.00822·t = ln(1/2)

t = ln(1/2)/(-0.00822) ≈ 84

The half life of the substance is about 84 minutes

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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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