clara is building a triangular garden. she wants the length of the longest side to be three more than twice as long as the length of the shortest side, and the third side will be twelve feet long . what expression could she write to determine the perimeter of the triangle if s represents the length of the shortest side ?

Answer:

b = 4

Step-by-step explanation:

4/6 is equivalent to 2/3, so with the slopes and y-intercepts being the same, there will be infinitely many solutions

Answer:

9 square root 2 is correct answer

The square and the triangle have the same area, which is of 4 square units.

For a square of side length L, the area is:

A = L²

For a triangle of base B, and height H, the area is:

A = B*H/2

For the square, we can see that:

L = 2, then:

A = 2² = 4

For the triangle we can see that:

B = 2

H = 4

Then the area is:

A = 2*4/2 = 4

So yea, both figures have the same area.

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To find the height that the ladder reaches up the wall, we can use the Pythagorean theorem.

which relates the lengths of the sides of a right triangle. In this case, the ladder forms the hypotenuse, the distance from the base of the wall to the foot of the ladder is one side (4.5 m), and the height we want to find is the other side.

In this case, the ladder forms the hypotenuse, and the distance from the foot of the ladder to the base of the wall is one of the sides.

Let's denote the height of the ladder on the wall as 'h'. According to the problem, the length of the ladder is 12 m, and the distance from the foot of the ladder to the base of the wall is 4.5 m.

Using the Pythagorean theorem, we have:

a^2 + b^2 = c^2

where a represents the height, b represents the distance from the base of the wall, and c represents the length of the ladder.

Substituting the given values, we have:

a^2 + (4.5)^2 = (12)^2

Simplifying:

a^2 + 20.25 = 144

a^2 = 144 - 20.25

a^2 = 123.75

Taking the square root of both sides:

a ≈ 11.11

Therefore, the ladder reaches a height of approximately 11.11 meters up the wall.

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To determine the number of minutes you would have to exercise each day to have a resting heart rate of 60 beats per minute, we need to consider the relationship between exercise and heart rate.

Regular exercise can help lower resting heart rate as it strengthens the cardiovascular system. The American Heart Association recommends engaging in moderate-intensity aerobic exercise for at least 150 minutes per week to maintain cardiovascular health.

If we assume that you exercise evenly throughout the week, we can calculate the daily exercise time as follows:

150 minutes per week ÷ 7 days = approximately 21.43 minutes per day.

Therefore, if you exercise for approximately 21.43 minutes per day, it can contribute to maintaining a healthy resting heart rate. It's important to note that individual results may vary, and consulting with a healthcare professional is always recommended before starting or modifying an exercise routine.

However, it's crucial to understand that exercise alone may not be the sole factor affecting resting heart rate. Other factors, such as genetics, overall health, stress levels, and lifestyle choices, can also influence heart rate. Additionally, achieving a resting heart rate of 60 beats per minute may not be feasible or suitable for everyone, as the ideal range can vary based on individual circumstances.

Therefore, while regular exercise can be beneficial for cardiovascular health, it's essential to consider personalized factors and consult with a healthcare professional for tailored advice on achieving and maintaining a healthy resting heart rate.

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Using the formula for the standard deviation and solving for x, we will get:

x = 7.6

To find the positive value of 'x' in the given set of numbers (3, 6, x, 7, 5) when the standard deviation is √2, we can use the formula for standard deviation.

The formula for calculating the sample standard deviation is as follows:

σ = √[(Σ(xi - m)²) / (n - 1)]

Where:

First, let's calculate the mean (m) of the given set of numbers:

Mean (m) = (3 + 6 + x + 7 + 5) / 5 = (21 + x) / 5

Next, let's substitute the values into the standard deviation formula:

√2 = √[( (3 - (21 + x) / 5)² + (6 - (21 + x) / 5)² + (x - (21 + x) / 5)² + (7 - (21 + x) / 5)² + (5 - (21 + x) / 5)² ) / 4]

Simplifying the equation:

2 = [( (3 - (21 + x) / 5)² + (6 - (21 + x) / 5)² + (x - (21 + x) / 5)² + (7 - (21 + x) / 5)² + (5 - (21 + x) / 5)² ) / 4]

Multiplying both sides of the equation by 4:

8 = (3 - (21 + x) / 5)² + (6 - (21 + x) / 5)² + (x - (21 + x) / 5)² + (7 - (21 + x) / 5)² + (5 - (21 + x) / 5)²

Expanding and simplifying:

8 = (15 - (21 + x) / 5)² + (30 - (21 + x) / 5)² + (5x/5)² + (35 - (21 + x) / 5)² + (25 - (21 + x) / 5)²

8 = (15 - (21 + x) / 5)² + (30 - (21 + x) / 5)² + x² + (35 - (21 + x) / 5)² + (25 - (21 + x) / 5)²

Expanding and simplifying further:

8 = (225 - 2(21 + x) + (21 + x)² / 25) + (900 - 2(21 + x) + (21 + x)² / 25) + x² + (1225 - 2(21 + x) + (21 + x)² / 25) + (625 - 2(21 + x) + (21 + x)² / 25)

Combining like terms:

8 = (225 + 900 + 1225 + 625) / 25 - 10(21 + x) + 5(21 + x)² / 25 + x²

8 = 2975 / 25 - 10(21 + x) + 5(21 + x)² / 25 + x²

Simplifying:

8 = 119 - 10(21 + x) + (21 + x)² / 5 + x²

Rearranging the terms:

8 - 119 = -10(21 + x) + (21 + x)² / 5 + x²

-111 = -10(21 + x) + (21 + x)² / 5 + x²

Multiplying through by 5 to eliminate the fraction:

-555 = -50(21 + x) + (21 + x)² + 5x²

Expanding and simplifying:

-555 = -1050 - 50x + x² + 441 + 42x + x² + 5x²

Combining like terms:

0 = 7x² - 8x - 114

Now, we can solve this quadratic equation to find the value of 'x'.

Using the quadratic formula:

x = (-b ± √(b² - 4ac)) / (2a)

In this case, a = 7, b = -8, and c = -114.

Plugging the values into the formula:

x = (-(-8) ± √((-8)² - 4(7)(-114))) / (2(7))

Simplifying:

x = (8 ± √(64 + 3192)) / 14

x = (8 ± 2√(814)) / 14

x = (4 ± √(814)) / 7

Therefore, the values of 'x' that satisfy the given equation are approximately:

x ≈ (4 + √(814)) / 7 ≈ 7.62

x ≈ (4 - √(814)) / 7 ≈ -0.29

Since we are looking for the positive value of 'x', the solution is:

x ≈ 7.6

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The five-number summary for the height of 18 sunflowers include:

In order to determine the five-number summary for the height of 18 sunflowers, we would arrange the data set in an ascending order:

40, 45, 47, 50, 50, 51, 51, 51, 55, 55, 55, 55, 55, 55, 58, 58, 62, 63

From the data set above, we can logically deduce that the minimum (Min) is equal to 40.

For the first quartile (Q₁), we have:

Q₁ = [(n + 1)/4]th term

Q₁ = (18 + 1)/4

Q₁ = 4.75th term

Q₁ = 4th term + 0.75(5th term - 4th term)

Q₁ = 50 + 0.75(50 - 50)

Q₁ = 50 + 0.75(0)

Q₁ = 50.

From the data set above, we can logically deduce that the median (Med) is given by:

Median = (9th term + 10th term)/2

Median = (55 + 55)/2

Median = 55

For the third quartile (Q₃), we have:

Q₃ = [3(n + 1)/4]th term

Q₃ = 3 × 4.75

Q₃ = 14.25th term

Q₃ = 14th term + 0.25(15th term - 14th term)

Q₃ = 55 + 0.25(58 - 55)

Q₃ = 55 + 0.25(3)

Q₃ = 55.75

In conclusion, the maximum height is equal to 63.

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Answer:

Step-by-step explanation:

its is 14

[tex] \sf{\blue{«} \: \pink{ \large{ \underline{A\orange{N} \red{S} \green{W} \purple{E} \pink{{R}}}}}}[/tex]

To find the value of 'x', we can use the property of parallel lines that states when a transversal intersects two parallel lines, the corresponding angles are equal.

In triangle ABC, we have DE parallel to BC. Therefore, we can conclude that triangle ADE is similar to triangle ABC.

Using the property of similar triangles, we can set up the following proportion:

[tex]\displaystyle\sf \dfrac{AD}{DB} = \dfrac{AE}{EC}[/tex]

Substituting the given values:

[tex]\displaystyle\sf \dfrac{6x - 7}{4x - 3} = \dfrac{3x - 3}{2x - 1}[/tex]

To solve this proportion for 'x', we can cross-multiply:

[tex]\displaystyle\sf (6x - 7)(2x - 1) = (4x - 3)(3x - 3)[/tex]

Expanding both sides:

[tex]\displaystyle\sf 12x^{2} - 6x - 14x + 7 = 12x^{2} - 9x - 12x + 9[/tex]

Combining like terms:

[tex]\displaystyle\sf 12x^{2} - 20x + 7 = 12x^{2} - 21x + 9[/tex]

Moving all terms to one side:

[tex]\displaystyle\sf 12x^{2} - 12x^{2} - 20x + 21x = 9 - 7[/tex]

Simplifying:

[tex]\displaystyle\sf x = 2[/tex]

Therefore, the value of 'x' is 2.

[tex]\huge{\mathfrak{\colorbox{black}{\textcolor{lime}{I\:hope\:this\:helps\:!\:\:}}}}[/tex]

♥️ [tex]\large{\underline{\textcolor{red}{\mathcal{SUMIT\:\:ROY\:\:(:\:\:}}}}[/tex]

Answer:

Step-by-step explanation:

The basic proportionality theorem states that if a line is drawn parallel to one side of a triangle and it intersects the other two sides at two distinct points then it divides the two sides in the same ratio.

It is given that AD=4x−38, BD=3x−1, AE=8x−7 and CE=5x−3. Let AC=x

Using the basic proportionality theorem, we have

BD

AD

​

=

AC

AE

​

⇒

3x−1

4x−3

​

=

x

8x−5

​

⇒x(4x−3)=(3x−1)(8x−5)

⇒4x

2

−3x=3x(8x−5)−1(8x−5)

⇒4x

2

−3x=24x

2

−15x−8x+5

⇒4x

2

−3x=24x

2

−23x+5

⇒24x

2

−23x+5−4x

2

+3x=0

⇒20x

2

−20x+5=0

⇒5(4x

2

−4x+1)=0

⇒4x

2

−4x+1=0

⇒(2x)

2

−(2×2x×1)x+1

2

=0(∵(a−b)

2

=a

2

+b

2

−2ab)

⇒(2x−1)

2

=0

⇒(2x−1)=0

⇒2x=1

⇒x=

2

1

​

Hence, x=

2

1

​

.

Answer:Yes, because 11 − 4 < 8

Step-by-step explanation:

Answer:

C)  $1.93

Step-by-step explanation:

To calculate the additional amount paid for using the credit card instead of cash, we need to consider the interest charges incurred over one month.

The annual interest rate is 16.99%.

Calculate the monthly interest rate by dividing the annual interest rate by 12 (number of months in a year):

[tex]\textsf{Monthly interest rate} = \dfrac{16.99\%}{12} =1.4158333...\%=0.014158333...[/tex]

Now we can calculate the interest charged on the annual subscription fee for one month:

[tex]\begin{aligned}\textsf{Interest charged}& = \textsf{Annual subscription fee} \times \textsf{Monthly interest rate}\\\\&=\$135.99 \times 0.014158333...\\\\&=1.92539175\\\\&= \$1.93\; \textsf{(nearest cent)}\end{aligned}[/tex]

Therefore, the additional amount paid instead of using cash is approximately $1.93.

Answer:

$1.93

Step-by-step explanation:

The first step is to find out how much interest will accrue in one month on the annual fee of $135.99.

[tex]\rm\implies{Interest = \dfrac{APR \times Balance}{12}}[/tex]

Substitute the given values into the formula:

[tex]\begin{aligned}\rm\implies Interest& =\rm \dfrac{16.99 \times 135.99}{12}\\&=\rm\dfrac{23.104701}{12}\\& \approx \boxed{\rm{\$1.93}}\end{aligned}[/tex]

[tex]\therefore[/tex] The subscriber will have paid an additional $1.93 in interest charges by using their credit card instead of paying in cash.

Answer:

A) 12 inches by 20 inches

Step-by-step explanation:

Dilation of a scale factor means to increase by a factor of 4.
That basically mean multiply the object by 4.

Therefore 3 inches x 4 = 12 inches

And 5 inches x 4 = 20

The product of two numbers is minus 28/27 if one number is (-4/9), then the other number is -7/3.

let x=(-4/9) be the first no. and y be the second no.

then, according to the question,

x × y=-28/27

-4/9 × y=-28/27

y=-28/27 × -9/4

y= -7/3

then second no. i.e y=-7/3

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Answer: (1,2)(2,2)

Step-by-step explanation:

Answer:

Step-by-step explanation:

Perimeter:  

     [tex]P=(x+y+z) \ cm[/tex]

Semi-perimeter:

     [tex]SP=\frac{1}{2} (x+y+z) \ cm[/tex]

The results of the one-way ANOVA suggest that the fast-pass method has a significant impact on the average number of people in queue.

The F-statistic for this example is 3.53. The p-value for the one-way ANOVA is calculated using the F-distribution. The p-value for this example is 0.029.

The p-value is less than the significance level of 0.05, so we can reject the null hypothesis. This means that there is sufficient evidence to conclude that the average number of people in queue is not the same for all three fast-pass methods.

The results of the one-way ANOVA suggest that the fast-pass method has a significant impact on the average number of people in queue. Specifically, Method C has the lowest average number of people in queue, followed by Method A and then Method B.

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The quadratic function for the value of David's investment indicates;

(i) $45,000

(ii) 9.375 months

A quadratic function is a function that can be expressed in the form; f(x) = a·x² + b·x + c, where a ≠ 0, and a, b, and c are numbers.

The model of the value of the investment in the bank obtained from the amount of his retirement funds David invested in the bank can be presented as follows;

a = 45 + 75·t - 4·t²

Where;

a = The value of the investment in thousand of dollars after t months

t = The number of months of the investment

(i) The initial amount David invested can be found by plugging in t = 0, in the function for the amount David invested in the bank, as follows;

a = 45 + 75 × 0 - 4 × 0² = 45

The initial amount David invested is; a = $45,000

(ii) The number of months it takes for David investment to reach a maximum value can be found from the quadratic function as follows;

The number of months t(max) at the maximum amount is; t(max) = -75/(2 × (-4)) = 9.375

Therefore, it will take 9.375 months for David's investment to reach a maximum value

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Answer:

g(- 5) = 3

Step-by-step explanation:

to find g(- 5) substitute x = - 5 into g(x)

g(- 5) = - (- 5) - 2 = 5 - 2 = 3

The function [tex] \sf f(x) [/tex] is defined as:

[tex] \sf f(x) = x^3 + \frac{3}{2}x^2 - 6x + 10 [/tex]

To find the stationary points of [tex] \sf f [/tex], we need to find the values of [tex] \sf x [/tex] where the derivative of [tex] \sf f(x) [/tex] is equal to zero.

First, let's find the derivative of [tex] \sf f(x) [/tex]:

[tex] \sf f'(x) = 3x^2 + 3x - 6 [/tex]

To find the stationary points, we set [tex] \sf f'(x) = 0 [/tex] and solve for [tex] \sf x [/tex]:

[tex] \sf 3x^2 + 3x - 6 = 0 [/tex]

We can factor the quadratic equation as follows:

[tex] \sf 3(x^2 + x - 2) = 0 [/tex]

Now, we solve for [tex] \sf x [/tex] by factoring further:

[tex] \sf 3(x + 2)(x - 1) = 0 [/tex]

This gives us two solutions: [tex] \sf x = -2 [/tex] and [tex] \sf x = 1 [/tex].

So, the stationary points of [tex] \sf f(x) [/tex] are [tex] \sf x = -2 [/tex] and [tex] \sf x = 1 [/tex].

To determine the intervals where [tex] \sf f(x) [/tex] is increasing, we need to analyze the sign of the derivative [tex] \sf f'(x) [/tex] in different intervals. We can use the values of [tex] \sf x = -2 [/tex], [tex] \sf 1 [/tex], and any other value between them.

For [tex] \sf x < -2 [/tex], we choose [tex] \sf x = -3 [/tex] as a test point:

[tex] \sf f'(-3) = 3(-3)^2 + 3(-3) - 6 = 12 > 0 [/tex]

For [tex] \sf -2 < x < 1 [/tex], we choose [tex] \sf x = 0 [/tex] as a test point:

[tex] \sf f'(0) = 3(0)^2 + 3(0) - 6 = -6 < 0 [/tex]

For [tex] \sf x > 1 [/tex], we choose [tex] \sf x = 2 [/tex] as a test point:

[tex] \sf f'(2) = 3(2)^2 + 3(2) - 6 = 18 > 0 [/tex]

From the above analysis, we can conclude that [tex] \sf f(x) [/tex] is increasing in the intervals [tex] \sf (-\infty, -2) [/tex] and [tex] \sf (1, \infty) [/tex].

To find the inflection point of [tex] \sf f [/tex], we need to determine where the concavity changes. This occurs when the second derivative of [tex] \sf f(x) [/tex] changes sign.

The second derivative of [tex] \sf f(x) [/tex] is:

[tex] \sf f''(x) = 6x + 3 [/tex]

To find the inflection point, we set [tex] \sf f''(x) = 0 [/tex] and solve for [tex] \sf x [/tex]:

[tex] \sf 6x + 3 = 0 [/tex]

[tex] \sf 6x = -3 [/tex]

[tex] \sf x = -\frac{1}{2} [/tex]

Therefore, the inflection point of [tex] \sf f(x) [/tex] is [tex] \sf x = -\frac{1}{2} [/tex].

Entry Point 1 focuses on pattern recognition and rule identification, while Entry Point 2 emphasizes pattern continuation and application.

One mathematical task suitable for intermediate phase learners that can have at least two entry points is a problem involving geometric patterns and sequences. Here's an example:

Problem: Consider the following geometric pattern:

□

□ □

□ □ □

□ □ □ □

Entry Point 1: Identifying the Rule

Ask students to examine the pattern and determine the rule for the number of squares in each row. Encourage them to look for patterns, count the number of squares in each row, and think about how it changes as the rows progress. The entry point here is to observe the pattern and identify the rule that governs the number of squares in each row.

Entry Point 2: Extending the Pattern

Provide students with the first few rows of the pattern and ask them to continue the pattern for a certain number of rows. For example, give them the first four rows and ask them to extend the pattern for three more rows. The entry point here is to extend the pattern by applying the identified rule.

By providing these two entry points, students can engage in different levels of thinking. Entry Point 1 focuses on pattern recognition and rule identification, while Entry Point 2 emphasizes pattern continuation and application.

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The answer is:

x =8

Work/explanation:

For now, I will focus on the left side, and use the distibutive property:

[tex]\sf{4(x+3)+2x=60}[/tex]

[tex]\sf{4x+12+2x=60}[/tex]

[tex]\sf{6x+12=60}[/tex]

Subtract 12 on each side

[tex]\sf{6x=48}[/tex]

Divide:

[tex]\sf{x=8}[/tex]

The value of the result of the expression from the computation is [tex]7.54 * 10^-1[/tex]

Standard form refers to a specific format or notation used to represent mathematical equations or numbers.

When referring to the standard form of a number, it usually means expressing a number in scientific notation or standard index form. In scientific notation, a number is written as a product of a decimal number between 1 and 10 and a power of 10.

We can write the given problem as;

[tex]3.77 * 10^-7 * 1.4 * 10^3/7 * 10^4\\= 7.54 * 10^-1[/tex]

This is the required format.

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Answer:

See below

Step-by-step explanation:

Part A

[tex]f(g(x))=f(\frac{x}{3})=3(\frac{x}{3})=x\\g(f(x))=g(3x)=\frac{3x}{3}=x[/tex]

Since BOTH [tex]f(g(x))=x[/tex] and [tex]g(f(x))=x[/tex], then [tex]f[/tex] and [tex]g[/tex] are inverses of each other

Part B

[tex]f(g(x))=f(\frac{x+1}{2})=2(\frac{x+1}{2})+1=x+1+1=x+2\\g(f(x))=g(2x+1)=\frac{(2x+1)+1}{2}=\frac{2x+2}{2}=x+1[/tex]

Since BOTH [tex]f(g(x))\neq x[/tex] and [tex]g(f(x))\neq x[/tex], then [tex]f[/tex] and [tex]g[/tex] are NOT inverses of each other

Answer:

Step-by-step explanation:

The bell boy added $270 and $20 incorrectly. The $20 bill was something that was returned due to overcharching. On the other hand, $270 was the amount that they paid for their room. This only means that $20 should be deducted from $270 and that's the amount that they paid for their room while $30 and $20 are the amount that the bell boy received as a tip

Total money of the three men: 3($100) = $300

They paid $270 for the room: $300 - $270 = $30

Tip for the bell boy: $30 - $30 = $0

Amount overcharged to them: $0 + $20 = $20

Tip to the bell boy: $20 - $20 = $0

They were left with no more money from the original $300.

Answer:don’t know how to answer

Step-by-step explanation: by the looks of it ur just connecting dots there are three dots you have to connect

The sprinter will complete the race in approximately 17.07 seconds.

To calculate the time for the race, we need to consider two parts: the acceleration phase and the constant velocity phase.

Acceleration Phase:

The acceleration of the sprinter is 1.64 m/s², and the distance covered during this phase is 25 m. We can use the equation of motion to calculate the time taken during acceleration:

v = u + at

Here:

v = final velocity (which is the velocity at the end of the acceleration phase)

u = initial velocity (which is 0 since the sprinter starts from rest)

a = acceleration

t = time

Rearranging the equation, we have:

t = (v - u) / a

Since the sprinter starts from rest, the initial velocity (u) is 0. Therefore:

t = v / a

Plugging in the values, we get:

t = 25 m / 1.64 m/s²

Constant Velocity Phase:

Once the sprinter reaches the end of the acceleration phase, the velocity remains constant. The remaining distance to be covered is 100 m - 25 m = 75 m. We can calculate the time taken during this phase using the formula:

t = d / v

Here:

d = distance

v = velocity

Plugging in the values, we get:

t = 75 m / (v)

Since the velocity remains constant, we can use the final velocity from the acceleration phase.

Now, let's calculate the time for each phase and sum them up to get the total race time:

Acceleration Phase:

t1 = 25 m / 1.64 m/s²

Constant Velocity Phase:

t2 = 75 m / v

Total race time:

Total time = t1 + t2

Let's calculate the values:

t1 = 25 m / 1.64 m/s² = 15.24 s (rounded to two decimal places)

Now, we need to calculate the final velocity (v) at the end of the acceleration phase. We can use the formula:

v = u + at

Here:

u = initial velocity (0 m/s)

a = acceleration (1.64 m/s²)

t = time (25 m)

Plugging in the values, we get:

v = 0 m/s + (1.64 m/s²)(25 m) = 41 m/s

Now, let's calculate the time for the constant velocity phase:

t2 = 75 m / 41 m/s ≈ 1.83 s (rounded to two decimal places)

Finally, let's calculate the total race time:

Total time = t1 + t2 = 15.24 s + 1.83 s ≈ 17.07 s (rounded to two decimal places)

Therefore, the sprinter will complete the race in approximately 17.07 seconds.

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The equation of the parabola is:

y = (x + 4)² - 3

We can see that we have the graph of a parabola.

Remember that the vertex form of a parabola whose leading coefficient is a and whose vertex is (h, k) is:

y = a*(x - h)² + k

Here we can see that the vertex is at (-4, -3)

So we can write:

y = a*(x + 4)² - 3

We also can see that the function passes through (-3, -2), replacing that we will get:

-2 = a*(-3 + 4)² - 3

-2 = a - 3

-2 + 3 = a

1 = a

The equation is:

y = (x + 4)² - 3

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To find the percentage of racers who did not complete the marathon, we need to calculate the proportion of racers who gave up or were disqualified out of the total number of racers who started the marathon.

The number of racers who did not complete the marathon is the sum of those who gave up and those who were disqualified:

Number of racers who did not complete = Number who gave up + Number who were disqualified

                                     = 23 + 3

                                     = 26

Now, we can calculate the percentage using the formula:

Percentage = (Number who did not complete / Total number who started) * 100

Percentage = (26 / 240) * 100

Percentage ≈ 10.83%

Therefore, approximately 10.83% of the racers did not complete the marathon.

This percentage represents the portion of racers who were unable to finish the race due to various reasons such as fatigue, injury, or disqualification. It highlights the challenges and demands of participating in a marathon and the determination required to complete the race.

The percentage provides a quantitative measure of the proportion of racers who were not able to reach the finish line, giving an understanding of the attrition rate in the marathon.

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The number of triangles in the diagram that can be mapped to one another by similarity transformations include the following: D. 3.

In Mathematics and Geometry, two triangles are said to be similar when the ratio of their corresponding side lengths are equal and their corresponding angles are congruent.

Additionally, the lengths of three (3) pairs of corresponding sides or corresponding side lengths are proportional to the lengths of corresponding altitudes when two (2) triangles are similar.

Based on the side, side, side (SSS) similarity theorem, we can logically deduce the following congruent and similar triangles:

ΔABC

ΔDEF

ΔGHI

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The arborist will make approximately $28.65.

First, let's draw an image of the problem:

Using the Pythagorean Theorem, we can find the diagonal measurement (d) from the top of the tree to the end of the shadow:

d² = h² + s²

Substituting the given values:

d² = 15² + 8²

d² = 225 + 64

d² = 289

Taking the square root of both sides:

d = √289

d = 17 feet

Now, let's convert the measurements from feet to meters using the conversion ratio:

1 meter / 3.3 feet

Height in meters: 15 feet [tex]\times[/tex] (1 meter / 3.3 feet) ≈ 4.55 meters

Shadow in meters: 8 feet [tex]\times[/tex] (1 meter / 3.3 feet) ≈ 2.42 meters

Diagonal in meters: 17 feet [tex]\times[/tex] (1 meter / 3.3 feet) ≈ 5.15 meters

To calculate the arborist's earnings, we need to find the square area (in square meters) of the tree and its shadow:

Area = Height [tex]\times[/tex] Shadow

Area = 4.55 meters [tex]\times[/tex] 2.42 meters

Area ≈ 11.02 square meters

Finally, the arborist will earn:

Earnings = Area [tex]\times[/tex] $2.60 per square meter

Earnings = 11.02 square meters [tex]\times[/tex] $2.60 per square meter

Earnings ≈ $28.65  

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