What is Limit of StartFraction 4 minus 3 x + x squared Over 2 x squared + 1 EndFraction as x approaches infinity? 0 One-half 2 nonexistent

Answer:

- 40

Hope this helps!

Step-by-step explanation:

( 4d - 6 )( 4d - 4 )  :  Cross Multiply

( 4d × 4d ) + ( 4d × (-4)) + ((-6) × 4d ) + ((-6) × (-4))

[tex]16d^{2}[/tex] - 16d - 24d + 24  :  ( Combine like terms )

[tex]16d^{2}[/tex] - 40d + 24

The length of other leg of the triangle is 64√2 cm.

What is Pythagoras theorem?

For right angle triangle,

Hypotenuse² = Base²+Height ²

Here we have a right angle triangle.

It is given that height and base of the triangle is 8 cm.

Now we want to find other leg of the triangle.

It is clear that other leg means hypotenuse here because this is a right angle triangle.

We know,

Hypotenuse ²= Base²+Height ²

[tex] = {64}^{2} + {64}^{2} \\ = 4096+ 4096 \\ = 2 \times 4096[/tex]

So,

[tex]hypotenuse = \sqrt{2 \times 4096} \\ = \sqrt{64 \times 64 \times2 } \\ = 64\sqrt{2} [/tex]

Therefore, length of the third leg is 64√2 cm.

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Correct question is " There is a right angle triangle. Height and base of the triangle is 8 cm.What is the length of other leg of the triangle?

64 cm

64√2 cm

128 cm

128√2 cm"

Answer:

How many times can 3 go into 64?

21 times

3 x 21

Step-by-step explanation:

You're welcome.

Answer: 21.3333333333

Step-by-step explanation:

64 / (Divided by) 3 equals 21.3333333...(Recurring, Meaning to go on for infinity)  

The equilateral triangle ABC have the value for x as 4√3, the measure of angle B is equal to 60° and the area is derived to be equal to 16√3.

An equilateral triangle have all its sides and angle to be equal, and each interior angles is equal to 60°

using the trigonometric ratio of sin for the angle C;

recall sin 60° = √3/2

sin 60° = x/8 {opposite/hypotenuse}

√3/2 = x/8

x = 4√3 {cross multiplication}

angle B = 60° {one interior angle of an equilateral triangle}

area of triangle = 1/2 × base × height

area of triangle ABC = 1/2 × 8 × 4√3

area of triangle ABC = 16√3.

Therefore, the equilateral triangle ABC have the value for x as 4√3, the measure of angle B is equal to 60° and the area is derived to be equal to 16√3.

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226 integers are present between 100 and 999, inclusive, and have the property that some permutation of its digits is a multiple of 11 between 100 and 999.

Here, we have,

The problem statement is to find the number of multiples of 11 between 100 and 999 inclusive, where some multiples may have digits repeated twice and some may not.

To solve this problem, we can first count the number of multiples of 11 between 100 and 999 inclusive, which is 81.

Some of these multiples may have digits repeated twice, and each of these can be arranged in 3 permutations.

Other multiples of 11 have no repeated digits, and each of these can be arranged in 6 permutations.

However, we must account for the fact that switching the hundreds and units digits of these multiples also yields a multiple of 11, so we must divide by 2, giving us 3 permutations for each of these multiples.

Thus, we have a total of 81 × 3 = 243 permutations.

However, we have overcounted because some multiples of 11 have 0 as a digit.

Since 0 cannot be the digit of the hundreds place, we must subtract a permutation for each of these multiples.

There are 9 such multiples (110, 220, 330, ..., 990), yielding 9 extra permutations.

Additionally, there are 8 multiples (209, 308, 407, ..., 902) that also have 0 as a digit, yielding 8 more permutations.

Therefore, we must subtract these 17 extra permutations from the total of 243, giving us 226 permutations in total.

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There are a total of 36 boxes of popcorn with each box containing 2 popped kernels except for the last box which contains 1 popped kernel.

If there are 85 kernels in a pound of popcorn, then we can assume that there are 84 boxes of popcorn with each box containing one kernel less than the previous box. The last box will contain only one kernel.

Following formula should be used to find the sum of the first n natural numbers:

n(n+1)/2.

Therefore, the number of boxes required to hold all the popped kernels is given by solving the equation:

n(n+1)/2 = 84

Solutions for the quadratic equation can be found as:

n = -9 or n = 8

Since we cannot have a negative number of boxes, therefore, we take:

n = 8.

Therefore, there are a total of 36 boxes of popcorn with each box containing 2 popped kernels except for the last box which contains 1 popped kernel.

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The number of boxes required for the popped corn can be calculated by dividing the total number of popped kernels by the capacity of each box. To do this, we need to know the yield of the popcorn, which means the ratio of popped kernels to unpopped kernels.

Let's assume that the yield of the popcorn is 10:1, which means that for every 10 unpopped kernels, 1 kernel pops. Therefore, there would be 10 x 16 = 160 unpopped kernels in a pound of popcorn.

Assuming that each box can hold 100 popped kernels, we can calculate the total number of boxes required as follows: Total number of popped kernels = 1 lb of popcorn x 10 (yield) = 1600 popped kernels Number of boxes required = 1600 popped kernels ÷ 100 kernels per box = 16 boxes

Therefore, 16 boxes are needed to hold all the popped kernels. The last box will be partially filled, and the number of popped kernels in it can be calculated by subtracting the number of kernels in the other 15 boxes from the total number of popped kernels:

Number of kernels in the last box = 1600 popped kernels - (15 boxes x 100 kernels per box) = 100 popped kernels Hence, there are 16 boxes required to hold all the popped kernels, with the last box having 100 popped kernels in it.

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Check the picture below.

so hmmm the square inscribed in the circle, we can see it as two congruent triangles, whose base is 28 and height is 14.  Now, if we get the whole area of the circle with radius of 14, and subtract the area of those triangles, what's leftover is the shaded area.

[tex]\stackrel{ \textit{\LARGE Areas} }{\stackrel{ circle }{\pi (14)^2}~~ - ~~\stackrel{ \textit{two triangles} }{2\left[\cfrac{1}{2}(\underset{b}{28})(\underset{h}{14}) \right]}}\implies 196\pi -392 ~~ \approx ~~ \text{\LARGE 223.75}[/tex]

Answer:

B 18√3

Step-by-step explanation: 1. Calculate the base and height of the triangle.

2. Multiply the base and height of the triangle.

3. Divide the result by 2.

4. Multiply the result by the square root of 3.

5. The answer is 18√3.

The missing value in the given figure is 120 degree.

An isosceles trapezoid is a trapezoid where the non-parallel sides are congruent.

The base angles of an isosceles trapezoid are congruent.

The converse is also true: If a trapezoid has congruent base angles, then it is an isosceles trapezoid.

In CFED,

As it is and isosceles trapezoid,

m∠C=m∠F

m∠D=m∠E

∴m∠F = 60 degrees

Now

m∠D+m∠E+m∠F+m∠C = 360 degree

2m∠D+120=360

m∠D=(360-120)/2

∴m∠D=120 degree

Thus the value of missing angle is 120 degree.

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With given expression error=2%,  the filling machine is not working within specifications for this container of oats.

What exactly are expressions?

In mathematics, an expression is a combination of symbols, numbers, and operators (such as + and -) that represents a value or a quantity. It can contain variables, which are symbols that represent unspecified values, and constants, which are fixed values. Expressions can be simple, like "5 + 3", or more complex, like "3x² - 2xy + 7".

Now,

The oat-filling machine must fill each container with 16 ounces of oats with a 2% margin of error, according to the requirements. This means that the acceptable range of oat quantity in a container is from 16 - 0.0216 = 15.68 ounces to 16 + 0.0216 = 16.32 ounces.

Since the container in question has been filled with 16.5 ounces of oats, we can see that it is outside the acceptable range specified by the machine's specifications. Therefore, the filling machine is not working within specifications for this container of oats.

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Therefore, the value of F is approximately 18.92 degrees.

An angle is a geometric figure formed by two rays that share a common endpoint, called the vertex of the angle. The rays are also known as the sides or legs of the angle. Angles are typically measured in degrees or radians and can be classified based on their measure as acute (less than 90 degrees), right (exactly 90 degrees), obtuse (between 90 and 180 degrees), or straight (exactly 180 degrees). Angles are an important concept in geometry and have many real-world applications, including in physics, engineering, and architecture.

Here,

If we look at the diagram, we can see that angles d° and e° are vertical angles, which means they are equal in measure. Similarly, angles (9-86)° and 45° are supplementary angles, which means they add up to 180°. Using this information, we can set up an equation:

d = e (since they are vertical angles)

(9-86) + 45 + e + (6f) + 15 = 180 (since the sum of angles in a triangle is 180°)

Simplifying the equation, we get:

-32 + e + 6f + 60 = 180

e + 6f + 28 = 180

e + 6f = 152

Substituting d = e, we get:

d + 6f = 152

But we know that d + e + (9-86) = 180 (since the sum of angles in a straight line is 180°)

Substituting d = e, we get:

2d + (9-86) = 180

2d = 77

d = 38.5

Substituting d = e = 38.5 in the equation e + 6f = 152, we get:

38.5 + 6f = 152

6f = 113.5

f = 18.92

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2x + 35 = 180 is the correct equation that could be used to determine the degree measure of one of the congruent angles.

What are congruent angles?

Congruent angles are angles that have the same measure. Two angles are said to be congruent if they have the same degree measurement.

2x + 35 = 180 can be used to determine the degree measure of one of the congruent angles.

Let x be the degree measure of each of the two congruent angles.

Then, the sum of the interior angles of a triangle is 180 degrees, so we can set up the equation:

35 + x + x = 180

Simplifying and solving for x, we get:

2x + 35 = 180

Therefore, 2x + 35 = 180 is the correct equation that could be used to determine the degree measure of one of the congruent angles.

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Answer:

2x + 35 = 180 is the correct equation that could be used to determine the degree measure of one of the congruent angles.

Step-by-step explanation:

working on notes right now also the other guy is error free.

The conclusion of hypothesis testing through the t-test is that the technical engineer have not statistically significant evidence to present to the university budget committee to purchase Krapple because it has, on average, a longer battery life. The p-value for t-test is equals to the 0.0074.

We have study related to battery life of two different laptops for student usage at a community college in california. There are two models, Krapple and Madroid. Engineer randomly assigned students to one of the laptop models and recorded the number of minutes the students were able to use the computer until the battery ran out. Here we have paired data so paired t test should be used.

Let d = Krapple - Madroid ( Sample sum of difference). Above table shows the calculations, Sample size, n = 15

Sample sum of difference, d = 1243

Sample mean of differences: Σα

= 1243/15 = 82.87

Sample standard deviation = [tex]\sqrt{\frac{488317.73}{15-1}}[/tex]

= 186.76

The Null and alternative Hypotheses are defined as [tex]H_0 : \mu_1- \mu_2 = 0[/tex]

[tex]H_a : \mu_1 - \mu_2 < 0[/tex]

Level of significance, α = 0.05

Test is one tailed (right tailed) df = n- 1

=> Degree of freedom = 14

Using the t - test, [tex]t = \frac{\sum d }{ \sqrt{\frac{n\sum d² - (\sum d)²}{n - 1}}}[/tex]

so, [tex] t = \frac{ 1243}{\sqrt{\frac{15× 488317.73 - (1243)²}{ 15 -1}}}[/tex]

=> [tex]t =\frac{1243}{\sqrt{412836.925}}[/tex]

=> t = 1.93

Now, using distribution table, P- value for t = 1.93, is 0.074. See, p-value > 0.05 , so result is not statistically significant. So, there is no evidence to reject the null hypothesis.

Conclusion: The technical engineer do not have statistically significant evidence to present to the university budget committee to purchase Krapple because it has, on average, a longer battery life.

Hence, required p-value is 0.074.

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Complete question:

the above first figure complete the question.

a technical engineer is interested in understanding the battery life of two different laptops for student usage at a community college in california. the two models he has are madroid and krapple. he randomly assigned students to one of the laptop models and recorded the number of minutes the students were able to use the computer until the battery ran out. use data file: laptops.csv download laptops.csv does the technical engineer have statistically significant evidence to present to the university budget committee to purchase krapple because it has, on average, a longer battery life? provide the p-value from your analysis.

The indefinite integral of [tex]xsin(x^2)[/tex] with respect to x is [tex]-(1/2)cos(x^2) + C[/tex], where C is the constant of integration.

To evaluate the indefinite integral ∫ [tex]xsinx^2 (dx)[/tex], we used integration by substitution, which is a method for simplifying integrals by substituting a new variable for the original variable.

We can use integration by substitution to evaluate this integral.

Let u = [tex]x^2[/tex], then du/dx = 2x, or equivalently, dx = du/(2x).

Substituting, we have:

∫ [tex]xsinx^2 dx[/tex] = ∫ sin(u) * (du/2)

Now we can integrate with respect to u:

∫ sin(u) * (du/2) = -(1/2)cos(u) + C

We then integrated the resulting expression with respect to u, obtaining -(1/2)cos(u) + C, where C is the constant of integration.

∫ [tex]xsinx^2 dx = -(1/2)cos(x^2) + C[/tex]

Finally, we substituted back u = [tex]x^2[/tex], obtaining [tex]-(1/2)cos(x^2) + C[/tex] as the indefinite integral of [tex]xsinx^2 (dx)[/tex] with respect to x.

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Answer:

20 cars

Step-by-step explanation:

.4x = 8

x = 20 cars

To verify the answer, 40% of 20 is 8.

On solving the provided question we can say that

1. circle centered at the origin with a radius of 10 is:

[tex]x^{2} +y^{2} =100[/tex]

2. circle with center R(-1, 8) and a radius of 5 is:

[tex](x + 1)^2 + (y - 8)^2 = 25[/tex]

3. center P(8, -5) and a radius of 24.5 is:

[tex](x -8)^2 + (y +5)^2 = 24.5[/tex]

4. the origin and passing through the point (9, -2) is:

[tex]x^{2} +y^{2} -18x+4y+85=0[/tex]

5. center B(0, -2) and passing through the point (-6, 0) is:

[tex](x )^2 + (y +2)^2 = 40[/tex]

6. center F(11, 4) and passing through the point (-2, 5) is:

[tex](x -11)^2 + (y - 4)^2 = 1708[/tex]

A circle seems to be a two-dimensional component that is defined as the collection of all places in a jet that are equidistant from the hub. A circle is typically shown with a capital "O" for the centre and a lower portion "r" for the radius, which represents the distance from the origin to any point on the circle. The formula 2r gives the girth (the distance from the centre of the circle), where (pi) is a proportionality constant about equal to 3.14159. The formula r2 computes the circumference of a circle, which relates to the amount of space inside the circle.

circle centered at the origin with a radius of 10 is:

[tex]x^{2} +y^{2} =100[/tex]

circle with center R(-1, 8) and a radius of 5 is:

[tex](x + 1)^2 + (y - 8)^2 = 25[/tex]

center P(8, -5) and a radius of 24.5 is:

[tex](x -8)^2 + (y +5)^2 = 24.5[/tex]

the origin and passing through the point (9, -2) is:

[tex]x^{2} +y^{2} -18x+4y+85=0[/tex]

center B(0, -2) and passing through the point (-6, 0) is:

[tex](x )^2 + (y +2)^2 = 40[/tex]

center F(11, 4) and passing through the point (-2, 5) is:

[tex](x -11)^2 + (y - 4)^2 = 1708[/tex]

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We can conclude that C. the neighborhood is significantly less likely to be Democrat.

The null hypothesis was that there is no difference between the proportion of Democrats in the sample and the population. Since the null hypothesis has been rejected, it means that there is a significant difference between the sample and the population proportions. In this case, 60% of the respondents in the random sample are Democrats, which is lower than the 75% Democrat proportion in the community as a whole. Therefore, we can conclude that the neighborhood from which the sample was drawn is significantly less likely to be Democrat compared to the overall community.

The other options are not supported by the given information:

A type I error might or might not have occurred. We cannot determine this based on the information provided. A type I error refers to the incorrect rejection of a true null hypothesis. Without knowing the true proportions of the neighborhood, we cannot determine if a type I error has been committed. Whether a one-tailed test or a two-tailed test was used is not specified in the question.

However, the conclusion that the neighborhood is significantly less likely to be Democrat can be derived from either type of test. Therefore, the correct option is C.

The question was incomplete, Find the full content below:

sixty percent of the respondents in a random sample drawn from a neighborhood are democrats. the community as a whole is 75% democrat. the difference between sample and population has been tested and the null hypothesis has been rejected. what might we conclude? select one:

a. a type i error has been committed

b. a one-tailed test has definitely not been used

c. the neighborhood is significantly less likely to be democrat

d. the difference is not significant

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According to the question we can say that it will take the artist 72 hours to paint 9 pictures for the art show.

If the artist can draw one-eighth of a picture in one hour, then he can draw one complete picture in 8 hours (since 8 x 1/8 = 1).

Therefore, to draw 9 pictures, the artist will need 9 x 8 = <<9*8=72>>72 hours.

So, it will take the artist 72 hours to paint 9 pictures for the art show.

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Answer:  [tex]a_n = 2n+11[/tex]

======================================

Explanation:

The gap between adjacent terms is 2.

We add 2 to each term to get the next. Therefore, d = 2 is the common difference.

The first term is [tex]a_1 = 13[/tex]

Use that and the value of d to determine the nth term formula of this arithmetic sequence.

[tex]a_n = a_1 + d(n-1)\\\\a_n = 13 + 2(n-1)\\\\a_n = 13 + 2n-2\\\\a_n = 2n+11 \textbf{ is the final answer}\\\\[/tex]

---------------

Check:

Plug in n = 1

[tex]a_n = 2n+11\\\\a_1 = 2*1+11\\\\a_1 = 2+11\\\\a_1 = 13\\\\[/tex]

That confirms 13 being the 1st term.

Now use n = 2

[tex]a_n = 2n+11\\\\a_2 = 2*2+11\\\\a_2 = 4+11\\\\a_2 = 15\\\\[/tex]

That matches with the fact 15 is the 2nd term.

I'll let you check the third term.

The area of the first triangle formed by points A, E, and B is 16 square units, the second triangle is formed by points B, C, and D, and the pentagon is formed by points B, C, and D the given points are 30 square units.

To plot and connect the given points in order, we first arrange them in a clockwise direction starting from point A(-5, 5).

The order is A, E, D, C, B, and A.

The rectangle is formed by the points C, D, and their respective y-coordinates. The length of the rectangle is 6 - 6 = 0 units, and the width is 5 - (-3) = 8 units.

The first triangle is formed by points A, E, and B. The base of the triangle is 4 units (the x-coordinate of E minus the x-coordinate of A), and the height is 8 units (the y-coordinate of A minus the y-coordinate of E).

The second triangle is formed by the points B, C, and D. The base of the triangle is 7 units (the x-coordinate of C minus the x-coordinate of B), and the height is 4 units (the y-coordinate of B minus the y-coordinate of D).

Therefore, the area of the rectangle is:

0 x 8 = 0 square units.

The area of the first triangle is:

0.5 x 4 x 8 = 16 square units.

The area of the second triangle is:

0.5 x 7 x 4 = 14 square units.

The total area of the pentagon is:

area of the rectangle + area of the first triangle + area of the second triangle

16 + 14 + 0 = 30 square units.

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From the sign chart, we can see that the solution set is x < -5 or -5 < x < 7. Therefore, the solution to the original inequality is:

x < -5 or -5 < x < 7

In order to solve an inequality, you need to isolate the variable on one side of the inequality symbol (i.e., <, >, ≤, ≥) and determine the range of values for which the inequality holds true.

The process for solving an inequality depends on the specific type of inequality. Here are the general steps for solving different types of inequalities:

Solving linear inequalities:

a. Move all variable terms to one side of the inequality and all constant terms to the other side.

b. Divide or multiply both sides by a positive constant (i.e. not zero) to isolate the variable.

c. If you multiply or divide by a negative constant, flip the inequality symbol to maintain the inequality.

Solving quadratic inequalities:So, the inequality can be written as:

[tex](x - 7) (x + 5) < 0[/tex]

To solve this inequality, we need to consider the sign of the expression on the left-hand side for different values of x. We can do this by creating a sign table:

x      |   x - 7   |   x + 5   | (x - 7) (x + 5)

----------|-----------|-----------|----------------

-6 | -13 | -1 | +13

-4 | -11 | +1 | -11

-1 | -8 | +4 | -32

6 | -1 | +11 | -11

8 | +1 | +13 | +13

From the sign table, we can see that the expression (x - 7)(x + 5) is negative (i.e., less than zero) for values of x between -5 and 7. Therefore, the solution to the inequality is:

[tex]-5 < x < 7[/tex]

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Answer:1:3

Step-by-step explanation:

we can show this as 4:12=1:3

so ratio of bikes to children is 1:3

Mrs. Hernandez's section has sold 12 fruit pies and Mr. Kane's class had sold 17 bottles of fruit juice.

Let's use the variables 'x' and 'y' to represent the number of fruit pies sold by Mrs. Hernandez's class and the number of bottles of fruit juice sold by Mr. Kane's class, respectively.

From the problem, we know that:

Each fruit pie sold for $5, so the revenue from Mrs. Hernandez's class is 5x.

Each bottle of fruit juice sold for $2, so the revenue from Mr. Kane's class is 2y.

The total number of items sold is 29, so x + y = 29.

The total revenue earned is $94, so 5x + 2y = 94.

Our system of equations is:

x + y = 29

5x + 2y = 94

To solve the system, we can use the substitution method. Solving the first equation for y, we get:

y = 29 - x

Substituting this expression for y into the second equation, we get:

5x + 2(29 - x) = 94

Simplifying and solving for x, we get:

3x = 36

x = 12

Substituting x = 12 into the equation y = 29 - x, we get:

y = 29 - 12 = 17

Therefore, it can be concluded that Mrs. Hernandez's class sold 12 fruit pies and Mr. Kane's class sold 17 bottles of fruit juice.

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(a) The probability that a randomly selected customer spends less than 65 minutes at the gym is 0.4013. (b) The expected value and standard deviation (standard error) of the sample mean of the time spent at the gym is 2.857 minutes

a) The probability that a randomly selected customer spends less than 65 minutes at the gym is calculated using the standard normal distribution formula.

z = (x - μ) / σ

where,μ = 70 minutes, σ = 20 minutes, x = 65 minutes

Substituting the given values, we get

z = (65 - 70) / 20

z = -0.25

Using a standard normal table or calculator, the probability that a randomly selected customer spends less than 65 minutes at the gym is 0.4013.

b) The standard deviation (standard error) of the sample mean can be calculated using the formula:

SE = σ/√n

where,σ = 20 minutes, n = 49

Substituting the given values, we get

SE = 20/√49

SE = 2.857 minutes

Therefore, the expected value and standard deviation (standard error) of the sample mean of the time spent at the gym is 2.857 minutes, respectively.

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When $2,000 is loaned for 5 years at 15% interest compounded monthly, the final amount (or future value) is $4,214.36.

The final amount is the future value of the present value investment.

The future value can be computed using the FV formula or an online finance calculator as follows:

N (# of periods) = 60 months (5 years x 12)

I/Y (Interest per year) = 15%

PV (Present Value) = $2,000

PMT (Periodic Payment) = $0

Results:

Future Value (Final amount)) $4,214.36

Total Interest = $2,214.36

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Step-by-step explanation:

The total cost of c pens is $cd. The total cost of e pencils is $0.01ef. Thus, the formula for the total cost is:

$T = cd + 0.01ef

Answer:

Step-by-step explanation:4(3x+y)+2(x-5y)+x²=12x+4y+2x-10y+x²

=14x-6y+x²

=x²+14x-6y

With the help of the volume and radius of the cylinder, we know that the height is 18.01 in.

A cylinder is one of the most basic curvilinear geometric shapes and has traditionally been solid in three dimensions.

In elementary geometry, it is regarded as a prism with a circle as its basis.

A cylinder can instead be described as an infinitely curved surface in a number of modern domains of geometry and topology.

So, in the given situation use the formula:

V=πr²h

Now, calculate as follows:

V=πr²h

h = V/πr²

h = 2772/π7²

h = 18.00724in

Rounding off: h = 18.01 in

Therefore, with the help of the volume and radius of the cylinder, we know that the height is 18.01 in.

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Therefore, the sides of the two triangles for x = 2 are approximately 106.099 and 22.317. After a few iterations, we get an approximate value of y for x = 2: y_1 = 9.618, y_2 = 9.614 and y_3 = 9.614

A triangle is a two-dimensional geometrical shape with three straight sides and three angles. It is one of the basic shapes in geometry and is commonly encountered in mathematics, science, and everyday life. Triangles can be classified based on the lengths of their sides and the measures of their angles. Some common types of triangles include equilateral triangles (all sides and angles are equal), isosceles triangles (two sides and two angles are equal), and scalene triangles (no sides or angles are equal). Triangles have many properties and relationships that are useful in solving problems involving angles, sides, and area.

Here,

Since the two triangles are similar and one triangle is inscribed in the other, the corresponding sides of the two triangles are proportional. That is,

(y² + 16) / (yx² + 9x) = (x² + 7x + y/2) / (3y/2 + 17)

Cross-multiplying and simplifying, we get:

(y² + 16)(3y/2 + 17) = (yx² + 9x)(x² + 7x + y/2)

Expanding and rearranging, we get a quadratic equation in y:

3y³ + 44y² + 102y - 816x³ - 5652x² - 7569x - 1836 = 0

This equation is difficult to solve explicitly for y, but we can use numerical methods or algebraic software to find an approximate value of y for a given value of x.

For example, let's assume x = 2. Using a numerical method such as Newton-Raphson, we can iteratively solve for y:

Let f(y) = 3y³ + 44y² + 102y - 816x³ - 5652x² - 7569x - 1836

Let f'(y) be the derivative of f(y): f'(y) = 9y² + 88y + 102

Let y_0 be an initial guess for y, such as y_0 = 10 (any positive value of y would work as an initial guess)

Iterate using the formula: y_n+1 = y_n - f(y_n) / f'(y_n), starting with n = 0

After a few iterations, we get an approximate value of y for x = 2:

y_1 = 9.618

y_2 = 9.614

y_3 = 9.614

Therefore, for x = 2, the positive value of y is approximately 9.614. We can then use this value to find the corresponding sides of the two triangles:

Side of big triangle: y² + 16

= 9.614² + 16

= 106.099

Side of small triangle: x² + 7x + y/2

= 2² + 7(2) + 9.614/2

= 22.317

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