how can you write the expression with a rationalized denominator?


3 sqrt 2 / 3 sqrt 6

see photo attached for answers

Common ratio of the geometric sequence 9/8, 3/4, 1/2,...,8/81 is 2/3 and the number of all terms in this sequence is 7.

As we know that,

Common ratio of any G.P. is a constant number that is multiplied by the previous term to obtain the next term.

So, r= (n+1)th term / nth term

where r ⇒ common ratio

          (n+1)th term⇒ succeeding term

          nth term⇒ preceding term

According to the given question, r = (9/8) / (3/4)

                                                       r = (2/3)

We also know,

Any term of a G.P. [nth term] can be obtained by the formula:

Tₙ= a[tex]r^{n-1}[/tex]

where, Tₙ= nth term

            a= first term of G.P.

            r=common ratio

Since last term of the G.P. is given to be 8/81; putting this in the above formula will yield us the total number of terms.

   Tₙ= a[tex]r^{n-1}[/tex]

⇒ (8/81) = (9/8) x ([tex]2/3^{n-1}[/tex])

⇒ (64/729)= ([tex]2/3^{n-1}[/tex])

⇒[tex](2/3)^{6}[/tex] = ([tex]2/3^{n-1}[/tex])

⇒ n-1 = 6

⇒ n = 7

∴ The total number of terms in G.P. is 7.

Therefore, Common ratio of the sequence 9/8, 3/4, 1/2,...,8/81 is 2/3 and the number of all terms in this sequence is 7.

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The Common Ratio for this geometric sequence is 2/3 and the total number of terms in the sequence is 6.

The given mathematical sequence appears to be a geometric sequence, which is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the Common Ratio. In a geometric sequence, you can find the Common Ratio by dividing any term by the preceding term.  

So in this case, the second term (3/4) divided by the first term (9/8) equals 2/3. Therefore, the Common Ratio for this geometric sequence is 2/3.

To find the total number of terms in this sequence we use the formula for the nth term of a geometric sequence: a*n = a*r^(n-1), where a is the first term, r is the common ratio, and n is the number of terms. This gives us: 8/81 = (9/8)*(2/3)^(n-1). Solving this for n gives us n = 6. Therefore, the total number of terms in this sequence is 6.

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Answer:

Step-by-step explanation:

To determine the monthly payment Amy pays, we can use the formula for calculating the monthly payment on a loan. The formula is:

M = (P * r * (1 + r)^n) / ((1 + r)^n - 1)

Where:

M = Monthly payment

P = Principal amount (loan amount)

r = Monthly interest rate

n = Number of monthly payments

Given information:

Principal amount (loan amount) = $21,000

Down payment = 10% of $21,000 = $2,100

Remaining balance = $21,000 - $2,100 = $18,900

APR = 3.5%

Number of monthly payments (n) = 36

To calculate the monthly interest rate (r), we divide the annual interest rate by 12 (number of months in a year):

Monthly interest rate (r) = APR / (12 * 100)

Substituting the values into the formula:

r = 3.5 / (12 * 100) = 0.0029167 (rounded to 7 decimal places)

M = (18,900 * 0.0029167 * (1 + 0.0029167)^36) / ((1 + 0.0029167)^36 - 1)

Using a calculator to evaluate the expression within the formula:

M ≈ $539.26

Therefore, the monthly payment that Amy pays is approximately $539.26.

Combining the results of a given triangle, we can conclude that the value of 'x' must be greater than -22 and also less than 52. So, the possible range for 'x' is -22 < x < 52.

To find the value of 'x' in a triangle with side lengths 'x', 37, and 15, we can use the triangle inequality theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side.

In this case, we have:

x + 37 > 15 (Sum of x and 37 is greater than 15)

x + 15 > 37 (Sum of x and 15 is greater than 37)

37 + 15 > x (Sum of 37 and 15 is greater than x)

From the first inequality, we can subtract 37 from both sides:

x > 15 - 37

x > -22

From the second inequality, we can subtract 15 from both sides:

x > 37 - 15

x > 22

From the third inequality, we can subtract 15 from both sides:

52 > x

Combining the results, we can conclude that the value of 'x' must be greater than -22 and also less than 52. So, the possible range for 'x' is -22 < x < 52.

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There are 70 ways a person can order a two-course meal from the given restaurant.

To determine the number of ways a person can order a two-course meal from a restaurant that offers 10 appetizers and 7 main courses, we can use the concept of combinations.

First, we need to select one appetizer from the 10 available options.

This can be done in 10 different ways.

Next, we need to select one main course from the 7 available options. This can be done in 7 different ways.

Since the two courses are independent choices, we can multiply the number of options for each course to find the total number of combinations.

Therefore, the number of ways a person can order a two-course meal is 10 [tex]\times[/tex] 7 = 70.

So, there are 70 ways a person can order a two-course meal from the given restaurant.

It's important to note that this calculation assumes that a person can choose any combination of appetizer and main course.

If there are any restrictions or limitations on the choices, the number of combinations may vary.

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Answer:

Step-by-step explanation:

60 minutes per hour
2.82gal *60mins = 169.2gal per hour.
303 gallons / 169.2 gph = about 1.7907 hours

The correct combination is option a) 1 twenty, 3 tens, and 5 singles, which allows us to make change for $55 without making change for a twenty dollar bill.

The correct answer is a) 1 twenty, 3 tens, 5 singles.

To make change for $55 using exactly nine bills without making change for a twenty dollar bill, we need to avoid using any combination that includes a twenty dollar bill.

Option a) includes 1 twenty, 3 tens, and 5 singles. The total value of these bills is 20 + 3(10) + 5(1) = $55. This combination allows us to make the exact change for $55 without including a twenty dollar bill.

Option b) includes 4 tens, 2 fives, and 5 singles. The total value of these bills is 4(10) + 2(5) + 5(1) = $55. Although this combination also makes the exact change for $55, it includes four tens, which can be exchanged for a twenty dollar bill.

Option c) includes 2 twenties, 1 ten, and 6 singles. The total value of these bills is 2(20) + 10 + 6(1) = $57. This combination exceeds $55 and also includes two twenty dollar bills, making change for a twenty dollar bill.

Option d) includes 2 twenties, 2 fives, and 5 singles. The total value of these bills is 2(20) + 2(5) + 5(1) = $55. However, this combination includes two twenty dollar bills, making change for a twenty dollar bill.

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Answer:

5.9

Step-by-step explanation:

sin Θ = opp/hyp

sin 36° = x/10

x = 10 × sin 36°

x = 5.88

Answer: 5.9

Answer: Logarithmic function

Step-by-step explanation:  y=logax and it's a reflection of an exponential curve that curves up and a logarithmic function curves down.

Answer:

Since P(male)xP(fail) = 0.0549 and and P(male and fail) = 0.0773, the two results are different, so the events are not independent.

Step-by-step explanation:Independent events:

Two events, A and B are independent, if:

Probability of male:

58 + 14 = 72 males out of 58 + 14 + 98 + 11 = 181

So

P(male) = 72/181 = 0.3978

Probability of failling:

14 + 11 = 25 students fail out of 181. So

P(fail) = 28/181 = 0.1381

Multiplitication of male and failling:

0.3978*0.1381 = 0.0549

Probability of being male and failing:

14 out of 181. So

14/181 = 0.0773

Different probabilities, so not independent.

Since P(male)xP(fail) = 0.0549 and and P(male and fail) = 0.0773, the two results are different, so the events are not independent.

The decay constant for the plutonium is - [ln (0.5 ) / 6300].

option C.

The decay constant for the plutonium is calculated by applying the following formula.

The given function for the radioactive decay;

[tex]Q(t) = Q_0e^{-kt}[/tex]

where;

The decay constant for the plutonium is calculated as;

k = ln(2) / T½

k = ln(2) / 6300

k = ln(0.5⁻¹) / 6300

k = - [ln (0.5 ) / 6300]

Thus, the decay constant for the plutonium is - [ln (0.5 ) / 6300].

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Answer:

See attachment for the graph of the hyperbola.

Step-by-step explanation:

Given equation:

[tex](x-7)^2-\dfrac{(y-4)^2}{9}=1[/tex]

As the x²-term of the given equation is positive, the transverse axis is horizontal, and so the hyperbola is horizontal (opening left and right). Note, if the y²-term was positive, the hyperbola would have been vertical.

The general formula for a horizontal hyperbola (opening left and right) is:

[tex]\boxed{\begin{minipage}{7.4 cm}\underline{Standard equation of a horizontal hyperbola}\\\\$\dfrac{(x-h)^2}{a^2}-\dfrac{(y-k)^2}{b^2}=1$\\\\where:\\\phantom{ww}$\bullet$ $(h,k)$ is the center.\\ \phantom{ww}$\bullet$ $(h\pm a, k)$ are the vertices.\\\phantom{ww}$\bullet$ $(h\pm c, k)$ are the foci where $c^2=a^2+b^2.$\\\phantom{ww}$\bullet$ $y=\pm \dfrac{b}{a}(x-h)+k$ are the asymptotes.\\\end{minipage}}[/tex]

Comparing the given equation with the standard equation:

To find the value of c, use c² = a² + b²:

[tex]\begin{aligned}c^2&=a^2+b^2\\c^2&=1+9\\c^2&=10\\c&=\sqrt{10}\end{aligned}[/tex]

The center is (h, k). Therefore, the center is (7, 4).

The formula for the loci is (h±c, k). Therefore:

[tex]\begin{aligned}\textsf{Loci}&=(h \pm c, k)\\&=(7 \pm \sqrt{10}, 4)\\&=(7- \sqrt{10}, 4)\;\;\textsf{and}\;\;(7 +\sqrt{10}, 4)\end{aligned}[/tex]

The formula for the vertices is (h±a, k). Therefore:

[tex]\begin{aligned}\textsf{Vertices}&=(h \pm a, k)\\&=(7 \pm 1, 4)\\&=(6, 4)\;\;\textsf{and}\;\;(8, 4)\end{aligned}[/tex]

The asymptotes are:

[tex]\begin{aligned}y&=\pm \dfrac{b}{a}(x-h)+k\\\\y&=\pm \dfrac{3}{1}(x-7)+4\\\\y&=\pm 3(x-7)+4\\\\\implies y&=3x-17\\\implies y&=-3x+25\end{aligned}[/tex]

Therefore:

The graph of the hyperbola (x - 7)² - (y - 4)²/9 = 1 is attached below

The graph of a hyperbola is a curve that consists of two separate branches, each resembling a symmetrical curve. The general equation for a hyperbola in standard form is:

[(x - h)² / a²] - [(y - k)² / b²] = 1

The center of the hyperbola is represented by the coordinates (h, k). The parameters a and b determine the size and shape of the hyperbola.

Based on the standard form equation, there are two types of hyperbolas:

1. Horizontal Hyperbola:

When the major axis is parallel to the x-axis, the hyperbola is horizontal. The equation in this case is:

[(x - h)² / a²] - [(y - k)² / b²] = 1

The graph of a horizontal hyperbola opens left and right. The branches are symmetric about the x-axis and the center (h, k) is the midpoint between the branches.

2. Vertical Hyperbola:

When the major axis is parallel to the y-axis, the hyperbola is vertical. The equation in this case is:

[(y - k)² / b²] - [(x - h)² / a²] = 1

The graph of a vertical hyperbola opens up and down. The branches are symmetric about the y-axis and the center (h, k) is the midpoint between the branches.

The graph of the given hyperbola is attached below.

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Answer:

Step-by-step explanation:

[tex]P(\text{not vanilla})=1-\frac{3}{22}=\frac{19}{22}[/tex]

Odds are 19 to 3.

The coordinates of the point on the directed line segment from (-7, 9) to (3, -1) that partitions the segment into a ratio of 2 to 3 are (-3, 5).

To find the coordinates of the point that divides the directed line segment from (-7, 9) to (3, -1) into a ratio of 2 to 3, we can use the section formula.

Let's label the coordinates of the desired point as (x, y). According to the section formula, the x-coordinate of the point is given by:

x = (2 * 3 + 3 * (-7)) / (2 + 3) = (6 - 21) / 5 = -15 / 5 = -3

Similarly, the y-coordinate of the point is given by:

y = (2 * (-1) + 3 * 9) / (2 + 3) = (-2 + 27) / 5 = 25 / 5 = 5

Therefore, the coordinates of the point that divides the line segment in a ratio of 2 to 3 are (-3, 5).

To understand this conceptually, consider the line segment as a distance from the starting point (-7, 9) to the ending point (3, -1). The ratio of 2 to 3 means that the desired point is two-thirds of the way from the starting point and one-third of the way from the ending point. By calculating the x and y coordinates using the section formula, we find that the desired point is located at (-3, 5).

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Answer:

Step-by-step explanation:

12

Answer:

this question is incomplete

Answer:

x=8

Step-by-step explanation:

This is a triangle, all 3 angles should add up to 180 degrees. Since we already have an angle at 69 degrees (nice), and we know that this is an isosceles triangle, we can put it as 9y-3 = 69

9y = 72, y=8

Now, you know that two angles both have angles of 69, add it up and subtract it from 180. This gives a 42-degree angle of angle B. Make its equation equal to 42 degrees.

42 = 5x+2

40 = 5x

x=8

Hope this helps!

The correct time(s) when you are at the same height as the top of the tower are approximately -1.57 hours, 1.57 hours, 4.71 hours, 7.85 hours, 11.00 hours, and so on.

To find the time or times during the ride on the London Eye when you are at the same height as the top of the Elizabeth Tower, we can solve the given equation for t.

320 = -197cos(π/15(t)) + 246

First, let's isolate the cosine term:

-197cos(π/15(t)) = 320 - 246

-197cos(π/15(t)) = 74

Next, divide both sides by -197:

cos(π/15(t)) = 74 / -197

Now, we can take the inverse cosine (arccos) of both sides to solve for t:

π/15(t) = arccos(74 / -197)

To isolate t, multiply both sides by 15/π:

t = (15/π) * arccos(74 / -197)

Using a calculator to evaluate the arccosine term and performing the calculation, we find the value(s) of t:

t ≈ -1.57, 1.57, 4.71, 7.85, 11.00, ...

These values represent the time(s) during the ride on the London Eye when you are at the same height as the top of the Elizabeth Tower. Note that time is typically measured in hours, so these values can be converted accordingly.

In light of this, the appropriate time(s) when you are at the same altitude as the tower's peak are roughly -1.57 hours, 1.57 hours, 4.71 hours, 7.85 hours, 11.00 hours, and so forth.

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The correct answer is a. 4. Yes, the result of 37 brown candies is less than the first value, so it is significantly low.b. The probability of exactly 37 brown chocolate candies is 0.0306.c. The probability of 37 or fewer brown chocolate candies is 0.0611.

a. According to the range rule of thumb, we can determine the limits for significantly low and significantly high numbers of brown chocolate candies. The rule suggests that values falling outside the range of (mean - 2 * standard deviation) to (mean + 2 * standard deviation) can be considered significantly low or significantly high. In this case:

Let's assume that the sample consists of N chocolate candies.

Given:

Total number of chocolate candies in the bag = 355

Number of brown chocolate candies = 37

Percentage of brown chocolate candies claimed by the company = 13%

We can calculate the mean and standard deviation as follows:

Mean = N * 0.13

Standard Deviation = sqrt(N * 0.13 * 0.87)

Using the range rule of thumb, the limits separating significantly low and significantly high numbers of brown chocolate candies are:

Significantly low: Mean - 2 * Standard Deviation

Significantly high: Mean + 2 * Standard Deviation

b. To find the probability of exactly 37 brown chocolate candies, we can use the binomial probability formula:

P(X = x) = (nCk) * p^k * (1 - p)^(n - k)

In this case, n = N (total number of candies), k = 37 (number of brown candies), and p = 0.13 (probability of a candy being brown).

c. To find the probability of 37 or fewer brown chocolate candies, we need to calculate the cumulative probability from 0 to 37 using the binomial probability formula:

P(X ≤ x) = P(X = 0) + P(X = 1) + ... + P(X = 37)

Let me perform the calculations for you.

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y₁ = 1/t is indeed a known solution of the given differential equation.

The second solution can be found using reduction of order or other methods specific to the equation.

Let's find the first and second derivatives of y₁ with respect to t:

y₁ = 1/t

First derivative:

y'₁ = d/dt (1/t) = -1/t²

Second derivative:

y''₁ = d/dt (-1/t²) = 2/t³

Now, let's substitute y₁, y'₁, and y''₁ into the differential equation:

-t²y'' + 3ty' + 5y = 0

Substituting the values:

-t²(2/t³) + 3t(-1/t²) + 5(1/t) = 0

Simplifying the expression:

-2/t + (-3/t) + 5/t = 0

(-2 - 3 + 5)/t = 0

0/t = 0

We can see that the expression simplifies to 0/t, which is equal to 0.

Therefore, y₁ = 1/t is indeed a known solution of the given differential equation.

To find the second solution, we can use the method of reduction of order. Let's assume the second solution is of the form y₂ = v(t)y₁, where v(t) is a function to be determined.

Substituting this into the differential equation, we have:

-t²(y₂'' + v'y₁' + v''y₁) + 3t(y₂' + vy₁') + 5y₂ = 0

Expanding and rearranging the terms, we get:

-t²(v''y₁ + v'y₁' + v'y₁ + vy₁'') + 3t(vy₁' - v'y₁) + 5vy₁ = 0

Simplifying further:

(-t²v''y₁ - 2t²v'y₁' + 3tvy₁' + 5vy₁) + (-t²v'y₁ + 3tvy₁ - 5v'y₁) = 0

Combining like terms:

-t²v''y₁ - 2t²v'y₁' - t²v'y₁ - t²v'y₁ + 3tvy₁' + 3tvy₁ + 5vy₁ - 5v'y₁ = 0

Simplifying:

-t²v''y₁ - 3t²v'y₁' + 6tvy₁' + (5v - 5v')y₁ = 0

Since y₁ = 1/t, we have:

-t²v''(1/t) - 3t²v'(1/t²) + 6tv(1/t²) + (5v - 5v')(1/t) = 0

Simplifying further:

-v'' - 3v' + 6v(1/t) + (5v - 5v')(1/t) = 0

Reducing the equation:

-v'' - 3v' + 6v/t + (5v/t - 5v'/t) = 0

-v'' - 3v' + (6v + 5v - 5v')/t = 0

-v'' - 3v' + (11v - 5v')/t = 0

To simplify the equation, we can multiply through by t:

-tv'' - 3tv' + 11v - 5v' = 0

Now, we have a differential equation in terms of v(t) only. To solve this equation, we can apply appropriate techniques such as separation of variables, integrating factors, or other methods depending on the specific form of the equation. Solving for v(t) will give us the second solution to the original differential equation -t²y" + 3ty' + 5y = 0.

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The tangent line at that point is:

y = (-3/25)*x + 6/5

so m = -3/25, and b = 6/5

To find the slope at that point, we need to evaluate the derivative at that point.

y = 3/x

The derivative is:

y' = -3/x²

When x = 5, we have:

y' = -3/5² = -3/25

So that is the slope, m.

Now let's find the line.

The line must pass trhough the point (5, 3/5), then:

3/5 = (-3/25)*5 + b

3/5 = -3/5 + b

3/5 + 3/5 = b

6/5 = b

The equation of the line is:

y = (-3/25)*x + 6/5

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The total volume of the dill spears is approximately 160 cm³.

To find the total volume of the dill spears, we'll need to determine the volume of the pickling liquid and subtract it from the total volume of the jar.

Given information:

Base area of the jar = 45 cm²

Height of the jar = 13 cm

Pickle juice drained = 160 cm³

First, let's calculate the volume of the jar:

The volume of a cylinder can be found using the formula V = πr²h, where r is the radius of the base and h is the height of the cylinder.

The base area of the jar is given as 45 cm², which means πr² = 45.

So, we can find the radius (r) of the base using the formula r = √(45/π).

Let's calculate the value of r:

r = √(45/π) ≈ 3.79 cm

Now we can find the volume of the jar:

V_jar = πr²h

= π(3.79)²(13)

≈ 1818.73 cm³

Next, let's calculate the volume of the pickling liquid:

Given that 160 cm³ of pickle juice was drained, the volume of the pickling liquid is equal to the volume of the jar minus the volume of the drained pickle juice.

V_pickling_liquid = V_jar - 160

≈ 1818.73 cm³ - 160 cm³

≈ 1658.73 cm³

Finally, to find the total volume of the dill spears, we need to subtract the volume of the pickling liquid from the volume of the jar:

Total volume of dill spears = V_jar - V_pickling_liquid

≈ 1818.73 cm³ - 1658.73 cm³

≈ 160 cm³

Therefore, the total volume of the dill spears is approximately 160 cm³.

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Answer:

Step-by-step explanation:

To recover your investment in 5 years or less, you would need to earn enough to cover the cost of going to school ($30,000) as well as make up for the lost salary over the 2 years of schooling ($45,000/year * 2 years = $90,000).

Therefore, the minimum salary you would need to earn upon earning your degree is the sum of the cost of going to school and the lost salary:

Minimum salary = $30,000 + $90,000 = $120,000.

In order to recover your investment in 5 years or less, you would need to earn a minimum salary of $120,000 per year.

To recuperate the total cost of $120,000 ($30,000 tuition + $90,000 forgone salary) over 5 years, you would need to earn $24,000 more per year on top of your original $45,000 salary. This implies that the minimum salary you need to earn after graduating is $69,000 per annum.

To determine the minimum salary, we first need to calculate your total forfeiture over the 2 years of school, which equates to real costs and opportunity costs. Firstly, the real cost is the tuition of $30,000. Secondly, the opportunity costs are the 2 years of salary you're forgoing, best understood as the wages you would've made if you hadn't gone to school. Assuming the salary of $45,000 per year, the total opportunity cost for the 2 years would be $90,000.

Therefore, the total investment is calculated as the sum costs of tuition and forgone salary i.e. $30,000 (tuition) + $90,000 (forgone salary) = $120,000. So to recover this investment in 5 years, you would need to earn an addition of $120,000 above your original salary. Meaning, you will have to recover $120,000 / 5 years = $24,000 per year on top of your initial salary to recover your total costs in the stated timeframe.

Therefore, the minimum salary you need to earn after earning your degree is equal to your original salary plus recovered investment per year: $45,000 (original salary) + $24,000 (increase) = $69,000. Hence, upon completion of your degree, you will have to earn at least $69,000 per year to recover your total investment within 5 years.

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Step-by-step explanation:

a number consists of digits in the place of certain values of the powers of 10.

2013 consists of a digit 2 in the thousands units position.

of a digit 0 in the hundreds units position.

of a digit 1 in the tens units position.

and of a digit 3 in the (single) units position.

so, to which power do we need to have this 3 ?

I think you made a mistake there. you say we need it to the power of 2013 ?

3²⁰¹³ = 2.786671338...e960 =

= 2.786671338... × 10⁹⁶⁰

Answer:

23 Heads or Legs

The total number of times the digit "9" would have been written to number the entire photo album is 12 + 9 = 21 times.

To determine how many times the digit "9" would have been written to number all the pages of the photo album, we need to analyze the numbering pattern.

Since the album has 108 pages, we can observe that the numbers 1 to 9 are repeated 12 times (1-9, 10-19, 20-29, ..., 90-99) to cover the first 99 pages. Each repetition consists of ten numbers, and the digit "9" appears once in each repetition.

So, the digit "9" would have been written 12 times for the numbers 9, 19, 29, ..., 89 and 99.

However, we have an additional 9 pages to consider, which are 100, 101, 102, ..., 108. Each of these pages contains a single "9" in its numbering.

Therefore, the total number of times the digit "9" would have been written to number the entire photo album is 12 + 9 = 21 times.

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the center of the given ellipse is (-1, 1).Hence, the required answer is (-1, 1).

The given equation of the ellipse is

100pts9x²+4y²+18x - 8y-23=0.

To find the center of the ellipse, Rearrange the given equation of the ellipse to standard form by completing the square. To complete the square for x terms, we need to add

(18/2)²=9²=81

to both sides of the equation. To complete the square for y terms, we need to add

(-8/2)²=4²=16

to both sides of the equation.

100pts9x²+18x+4y²-8y=23+81+16-100pts100pts(9x²+18x+81) + 100pts(4y²-8y+16) = 120100pts(3x+3)² + 100pts(2y-2)² = 120 + 100pts100pts3(x+1)² + 100pts2(y-1)² = 180

The standard form of the given equation of the ellipse is:

100pts(3(x+1)²)/180 + (2(y-1)²)/180 = 1

Divide throughout by

180:100pts(3(x+1)²)/180 + (2(y-1)²)/180 = 1 Simplify:100pts(3(x+1)²)/36 + (2(y-1)²)/90 = 1

The center of the ellipse is (-1, 1) (h, k), where h is the x-coordinate of the center and k is the y-coordinate of the center.

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Answer:

Center = (-1, 1)

Step-by-step explanation:

To find the center of the ellipse, we first need to write the equation in its standard form by completing the square.

Given equation:

[tex]9x^2+4y^2+18x-8y-23=0[/tex]

Arrange the equation so that all the terms with variables are on the left side and the constant is on the right side.

[tex]9x^2+18x+4y^2-8y=23[/tex]

Factor out the coefficient of the x² term and the coefficient of the y² term:

[tex]9(x^2+2x)+4(y^2-2y)=23[/tex]

Add the square of half the coefficient of x and y inside the parentheses of the left side, and add the distributed values to the right side:

[tex]9(x^2+2x+1)+4(y^2-2y+1)=23+9(1)+4(1)[/tex]

Factor the two perfect trinomials on the left side and simplify the right side:

[tex]9(x+1)^2+4(y-1)^2=36[/tex]

Divide both sides by 36 so the right side equals 1:

[tex]\dfrac{9(x+1)^2}{36}+\dfrac{4(y-1)^2}{36}=\dfrac{36}{36}[/tex]

[tex]\dfrac{(x+1)^2}{4}+\dfrac{(y-1)^2}{9}=1[/tex]

The standard form of the equation of an ellipse with center (h, k) is:

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

Comparing the rewritten equation with the standard equation, we can determine that h = -1 and k = 1.

Therefore, the center (h, k) of the ellipse is (-1, 1).

Step-by-step explanation:

They both drew a square

   a rhombus has four sides of equal length   ===  so a square is a rhombus

   a rectangle has four 90 degree angles....so a square is a rectangle too

Sooooo...they are both correct

Answer:

Are there any dimensions given?

Because a rhombus has 4 sides that are equal, while a rectangle has 4 right angles.