A new cruise ship line just launched 3 new ships: the Pacific Paradise, the Caribbean Paradise, and the Mediterranean Paradise.
The Caribbean Paradise has 29 more than the Pacific Paradise. The Mediterranean Paradise has 30 fewer deluxe staterooms than twice the number of deluxe staterooms on the Pacific Paradise. Find the number of deluxe staterooms for each of the ships if the total of deluxe staterooms for the total of the three ships is 719.

Answers

Answer 1

c = caribbean, p = pacific, m = mediterranean

c = p + 29
m = 2p - 30
c + m + p = 719

(p + 29) + (2p - 30) + p = 719
4p - 1 = 719
4p = 719 + 1
4p = 720
p = 720/4
p = 180 <=== pacific paradise deluxe staterooms

c = p + 29
c = 180 + 29
c = 209 <=== caribbean paradise deluxe staterooms

m = 2p - 30
m = 2(180) - 30
m = 360 - 30
m = 330 <=== mediterranean paradise deluxe staterooms

Answer 2

Final answer:

To find the number of deluxe staterooms for each of the three ships, we can set up and solve an equation based on the given information.

Explanation:

Let's represent the number of deluxe staterooms on the Pacific Paradise ship as x.

The number of deluxe staterooms on the Caribbean Paradise is 29 more than the Pacific Paradise, so it can be represented as x + 29.

The number of deluxe staterooms on the Mediterranean Paradise is 30 fewer than twice the number of deluxe staterooms on the Pacific Paradise, so it can be represented as 2x - 30.

We can form an equation with the sum of the deluxe staterooms on all three ships: x + (x + 29) + (2x - 30) = 719.

Simplifying the equation gives us 4x - 1 = 719.

By adding 1 to both sides and dividing by 4, we can find the value of x which represents the number of deluxe staterooms on the Pacific Paradise ship.

Finally, we can substitute the value of x into the expressions for the other two ships to find their number of deluxe staterooms.

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Related Questions

If 52/x is a positive integer, how many integer values are possible for x?

Answers

Since there are 6 factors of 52 (1, 2, 4, 13, 26, 52), the answer is 6

There are a total of 6 integer values that are possible for x: 1, 2, 4, 13, 26, and 52.

What is the quotient?

If 52/x is a positive integer, it means that 52 is divisible by x, and x is a factor of 52. In other words, x must be a positive divisor of 52.

The positive divisors of 52 are: 1, 2, 4, 13, 26, and 52.

Thus;

52/1 = 52 (a positive integer)

52/2 = 26 (a positive integer)

52/4 = 13 (a positive integer)

52/13 = 4 (a positive integer)

52/26 = 2 (a positive integer)

52/52 = 1 (a positive integer)

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Factor of x^3-x^2-24x-36

Answers

The trial and error method is used to find an initial factor:
If we let f(x) = x³ - x² - 24x - 36 and all we have to do is sub' in values of x until
f(x) = 0, we can use this to find an initial factor by the factor theorem:
f(1) = (1)³ - (1)² - 24(1) - 36 = -60
f(2) = (2)³ - (2)² - 24(2) - 36 = -80
f(5) = (5)³ - (5)² - 24(5) - 36 = -56
*** f(6) = (6)³ - (6)² - 24(6) - 36 = 0 ***

f(6) = 0 so (x - 6) is a factor of f(x).
This means that: f(x) = x³ - x² - 24x - 36 = (x - 6)(ax² + bx + c).
To find a,b and c, use long division (or inspection) to divide x³ - x² - 24x - 36 by x - 6.
The other 2 factors of f(x) can then be found by factorizing the
ax² + bx + c quadratic the way you would with any other quadratic (i.e. by quadratic formula, CTS or inspection).

Which number is greater? 0 -1 -25 -50?

Answers

The greatest number in this group is 0.
Remember that with positive numbers, 50 is greater than 1, but with negative numbers, -1 is greater than -50.
If that confuses you, think of it like this.
If you owe someone 1 dollar, you yourself have more than if you owe someone 50 dollars. You yourself have even more if you don't owe any money.

Guess the value of the limit (correct to six decimal places). (if an answer does not exist, enter dne.) lim hâ0 (4 + h)5 â 1024 h

Answers

[tex]\displaystyle\lim_{h\to0}\frac{(4+h)^5-1024}h=\lim_{h\to0}\frac{(4+h)^5-4^5}h[/tex]

Recall the definition of the derivative of a function [tex]f(x)[/tex] at a point [tex]x=c[/tex]:

[tex]f'(c)=\displaystyle\lim_{h\to 0}\frac{f(c+h)-f(c)}h[/tex]

We can then see that [tex]f(c)=c^5[/tex], and by the power rule we have [tex]f'(c)=5c^4[/tex]. Then replacing [tex]c=4[/tex], we arrive at

[tex]\displaystyle\lim_{h\to0}\frac{(4+h)^5-4^5}h=5\times4^4=1280[/tex]

Alternatively, we could have expanded the binomial, giving

[tex]\dfrac{(4+h)^5-4^5}h=\dfrac{(4^5+5\times4^4h+10\times4^3h^2+10\times4^2h^3+5\times4h^4+h^5)-4^5}h[/tex]
[tex]=\dfrac{1280h+640h^2+160h^3+20h^4+h^5}h[/tex]
[tex]=1280+640h+160h^2+20h^3+h^4[/tex]

and so as [tex]h\to0[/tex] we're left with 1280, as expected.

An isosceles triangle has a base with length 15 inches and two congruent sides with lengths of 15 inches each. Find the height of the triangle. (Round your answer to the nearest tenth of an inch.)

Answers

An isosceles triangle is a special triangle having at least 2 equal sides equal. But in this case, since all sides are equal having a length of 15 inches, this is actually an equilateral triangle.

When you create a perpendicular bisector from the top vertex to the midpoint of the base, you get a right triangle with the hypotenuse equal to 15 inches, a base equal to half of 15 because it was bisected, and the height. Therefore, we use the pythagorean theorem. The solution would be

h = √(15)^2 +(15/2)^
h = 13.0 meters

Final answer:

By using the Pythagorean theorem on one of the right triangles formed by dividing the isosceles triangle, we find the height to be approximately 13.0 inches when rounded to the nearest tenth.

Explanation:

To find the height of an isosceles triangle with a base of 15 inches and congruent sides each 15 inches, we can use the Pythagorean theorem or note that the triangle can be split into two right triangles by a height dropping from the vertex opposite the base to the midpoint of the base.

The midpoint will divide the 15-inch base into two segments, each 7.5 inches. The height of the isosceles triangle is the same as the height of the right triangle.

Using the Pythagorean theorem (a² + b² = c²), we can find the height (h) knowing that the hypotenuse (c) is one of the congruent sides (15 inches) and one leg (a, half of the base) is 7.5 inches:

h² + (7.5 inches)² = (15 inches)²

h² + 56.25 inches sup>2 = 225 inches²

h² = 225 inches² - 56.25 inches²

h² = 168.75 inches²

h = sqrt{168.75 inches²}

h approx 12.99 inches, which rounds to 13.0 inches to the nearest tenth.

Therefore, the height of the isosceles triangle rounds to 13.0 inches.

what is e/4 + 2f-3 when e =12 andf =1/2

Answers

Hey!

When putting in a number for a variable, we want to put it on parenthesis. So, let's write the problem with the variables substituted.
[tex](12)/4+2(1/2)-3[/tex]
We could also write it like this:
[tex]\frac{\left(12\right)}{4}+2\left(\frac{1}{2}\right)-3=1[/tex]
To start, we have to remove the parenthesis.
[tex]=\frac{12}{4}+2\cdot \frac{1}{2}-3[/tex]
Let's simplify the first fraction.
[tex]\frac{12}{4}=3[/tex]
Let's simplify the second part:
[tex]2\cdot \frac{1}{2}[/tex]
[tex]=\frac{1\cdot \:2}{2}[/tex]
[tex]=\frac{2}{2}[/tex]
[tex]=1[/tex]

We would be left with,
[tex]=3+1-3[/tex]
Simplify.
[tex]=1[/tex]

Thanks!
-TetraFish

Hello there! Thank you for asking your question here at Brainly. I will be assisting you with this problem and will teach you how to handle it on your own in the future.

Let's take a look at our question.
"What is e/4 + 2f -3 when e = 12 and f = 1/2?"

To solve for this, we need to plug in our values to their proper values.

Let's replace "e" with "12" and "f" with "0.5" (0.5 is equal to 1/2).

Our expression should look like this now:
12/4 + 2(0.5) -3

Well, we can simplify two parts of our expression: 12/4 and 2(0.5).

We can simplify 12/4 into 3, as 12 divided by 4 is 3.
We can also simplify 2(0.5) into 1, as multiplying by 0.5 is the same as dividing by 2.

We now have:
3 + 1 - 3
We can cancel out 3 and -3 to make 0.
We are now left with 1.

Your answer remains as "1".

I hope this helps! :)

What is the greatest number of triangular sections, each with a base of 5 inches and a height of 8 inches, that can be cut from a rectangular piece of paper measuring 30 inches by 40 inches?

Answers

12 triangular sections- you only need the base for this question- in an area of paper you can see that there is a square -you can make 2 triangles out of this square. Use 30 divided by 5 and get 6. Multiply that by 2 triangles from the area we just talked about and get 12 triangles.

Jane is one of 50 students to take a standardized math test that includes 100 multiple choice questions. If she has the highest score of any student with a raw score of 87, what is her percentile score?

Answers

Answer: 87%
Solution:
87 correct questions over 100 = 87/100 = .87 = (change to percent by moving the decimal 2 places to the right) 87%

"let v = r 2 with the usual addition and scalar multiplication defined by k(u1, u2) = (ku1, 0). determine which of the five axioms of vector spaces involving scalar multiplication v satisfies and which fail. for the ones it satisfies, prove that it satisfies the axiom. for those that fail, show that it fails with a counterexample."

Answers

Let [tex]\mathbf u\in\mathbb R^2[/tex], where

[tex]\mathbf u=(u_1,u_2)[/tex]

and let [tex]k\in\mathbb R[/tex] be any real constant.

Given this definition of scalar multiplication, we can see right away that there is no identity element [tex]e[/tex] such that

[tex]e\mathbf u=\mathbf u[/tex]

because

[tex]e\mathbf u=e(u_1,u_2)=(eu_1,0)\neq(u_1,u_2)=\mathbf u[/tex]

Two cars that are 150 miles apart start driving toward each other on parallel roads. The average speed of the first car is 60 miles per hour. The average speed of the second car is 55 miles per hour. Which equation can be used to determine t, the time it takes for the two cars to pass each other?

60t – 55t = 0
60t + 55t = 1
60t + 55t = 150
60t – 55t = 150

Answers

Answer: 60t + 55t = 150

The equation that can be used to determine t, the time it takes for the two cars to pass each other is:

60t + 55t = 150

Option C is the correct answer.

What is an equation?

An equation contains one or more terms with variables connected by an equal sign.

Example:

2x + 4y = 9 is an equation.

2x = 8 is an equation.

We have,

The equation that can be used to determine t, the time it takes for the two cars to pass each other is:

60t + 55t = 150

Here,

60t represents the distance covered by the first car in time t, and 55t represents the distance covered by the second car in time t.

When they meet, the sum of their distances would be equal to the total distance between them, which is 150 miles.

Therefore,

We can add their distances and set them equal to 150 miles to solve for t.

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Using 8(-20) in a real world situation

Answers

Maybe, the price went down $20 eight times?

what is the answer to 11 over 7 in improper fractions

Answers

if its improper fraction is 11/7 if you want the mixed number its 1 4/7

if u change it to a proper fraction it would be a mixed number and the mixed number is 1 4/7

Let (-7, -4) be a point on the terminal side of theta. Find the exact values of Sin, CSC and COT

Answers

First you need to get the radius of the circle. We do that by using the (x,y) point. So we have (-7, -4).

To get the radius, we us the following formula.

r = √(-7)² + (-4)²

This reads:
radius = square root of -7 to the power of 2 plus -4 to the power of 2.

I am new here and I am not sure if they support latex. Other wise I would of used latex.

Ok so r (radius) = r = √(-7)² + (-4)² = √65

r = √65

Now we have our radius we can find the values for sin csc and cot

sin = y / r
csc = r / y
cot = x / y

I will do the first one for you which is sin.

sin = y / r = -4 / √65

The above is saying sin = y divided by r = -4 divided by the square root of 65.

If you need help with the others let me know.

The above is saying sin = y divided by r = -4 divided by the square root of 65.

The amount of carbon-14 present in animal bones t years after the animal's death is given by P(t)= -0.00012097t. How old is an ivory tusk that has lost 34% of its carbon-14?

Answers

I believe the equation you gave is wrong because the standard form of equation for C-14 decay is in the form of:

A = Ao e^-kt

So I think the right form of equation is (correct me if I’m wrong):

P(t) = Po e^(-0.00012097t)

Where,

Po = initial value of C-14 at t = 0

t = time elapsed

Since it is given that:

P = (1 – 0.34) Po

P = 0.66 Po

Therefore, t is:

0.66 Po = Po e^(-0.00012097 t)

0.66 = e^(-0.00012097 t)

taking ln of both side:

ln 0. 66 = -0.00012097 t

t = - ln 0. 66 / 0.00012097

t = 3,434.86 years

Therefore the ivory tusk is about 3,435 years old.

What is the next number in the series? 83 79 75 71 67

Answers

This is an arithmetic sequence with a common difference of -4, meaning that each term is four less than the previous term. So the next term in that sequence (technically the term "series" should be reserve for infinite sequences) will be 67-4=63.

83-79 =4

79-75 = 4

the numbers are decreasing by 4

so the next number would be 67-4 = 63

If 4 1/2 = 2, 8 1/3 = 2, and 16 1/4 = 2, then for what value of x would x 1/5 = 2?

2
4
24
32

Answers

D. 32
By paying attention to the hole number and ignoring the fraction. You can see the pattern is doubling each time. 4, 8, 16, 32.

Find the length of the third Angelou a triangle given that the first two angels are 35 and 70 show your work

Answers

angles in a triangle = 180 degrees

70 +35 = 105 degrees

180-105 = 75 degrees

3rd angle = 75 degrees

What's 10⁄12 written as a fraction in simplest form?

A. 5⁄6
B. 10⁄6
C. 3⁄5
D. 5⁄12

Answers

That would be
A. 5/6

A. 5/6 because 10/12 divided by two would = 10 divided by 2=5 and 12 divided by 2=6

Inez waters her plants every two days. She trims them every 15 days. She did both today. When will she do them both again?

Answers

The answer is 30 days.

The solution for this is to find the least common multiple. By getting the multiple of both numbers.

Multiples of 2: 2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40…..

Multiples of 15: 15,30,45,60,75……..

The Least Common Multiples of 2 and 15 is 30. So, Inez will do them both again in 30 days.

The least common multiple (LCM) of two numbers is the smallest number that is multiple by the both number.

Which pair of non-congruent figures must be similar? two squares two parallelograms (not rectangles) two right triangles two isosceles triangles (not equilateral)

Answers

Similar figures are figures which has the same shape however they do not always have the same size. It should not be used interchangeably with congruent figures since the latter would refer to figures which has the same size and shape. Thus, congruent figures are always similar figures but similar figures are not always congruent figures. From the choices given, I think the best choice would be the first option. The two squares would be the pair of non-congruent figures that are always similar. This is because no matter what are the side lengths of the squares, it will always have the form of the square.

What is the median of the data set given below? 19, 22, 46, 24, 37, 16, 19, 33

Answers

u have to put tje numbers in order...
16,19,19,22,24,33,37,46....the median is the middle number. If there is an even number of numbers, like this one, there will be 2 middle numbers. Find those 2 middle numbers, divide them by 2, and that is ur median.

start from the ends and count inwards till u come to the 2 middle numbers...
(22 + 24) / 2 = 46/2 = 23

** now if there would have been an odd number of numbers, u would just have 1 middle number, and that number would be the median.

To find the median, first arrange the data in either ascending or descending order;

16,19,19, 22, 24, 33, 37, 46

If the total amount of data [let's take it as n] is odd, then the median is data on [[tex] \frac{n}{2} [/tex]].

But here the total amount of data is even [n=8], so the median is the average of the data on [[tex] \frac{n}{2} [/tex]] and [[tex] \frac{n}{2} +1[/tex]].

So the median is

[tex] \frac{22+24}{2} = \frac{46}{2} = 23[/tex]

M(5, 7) is the midpoint of rs The coordinates of S are (6, 9). What are the coordinates of R?
(5.5, 8)

(7, 11)

(10, 14)

(4, 5)

Answers

the correct answer is D (4,5)

[tex]\text{Let the coordinates of R be (x, y)}[/tex]

[tex]x_M = \frac{x_R + x_S}{2}[/tex]
[tex]5 = \frac{x + 6}{2}[/tex]
[tex]10 = x + 6[/tex]
[tex]x = 4[/tex]

[tex]y_M = \frac{y_R + y_S}{2}[/tex]
[tex]7 = \frac{y + 9}{2}[/tex]
[tex]14 = y + 9[/tex]
[tex]y = 5[/tex]

[tex]\therefore R(4, 5)[/tex]

A psychologist claims that less than 5.8 percent of the population suffers from professional problems due to extreme shyness. Express the null hypothesis and the alternative hypothesis in symbolic form. Use p, the true percentage of the population that suffers from extreme shyness. Select one: a. H0: p = 5.8% H1: p < 5.8% b. H0: p > 5.8% H1: p ≤ 5.8% c. H0: p < 5.8% H1: p ≥ 5.8% d. H0: p = 5.8% H1: p > 5.8%

Answers

In statistics, you may test a hypothesis using different statistical techniques like ANOVA or the F-test. You should have the null hypothesis, denoted as H0 and the alternative hypothesis, denoted as H1. Null hypothesis is the hypothesis you would like to test on. This is called as such because this is the hypothesis statisticians would like to test to nullify. This would be p < 5.8%. On the other hand, the alternative hypothesis is the contrary of the null hypothesis. Hence, the alternative hypothesis is p ≥ 5.8%. The answer is C.

Answer:

Step-by-step explanation:

Solve the equation by graphing. x^2+14x+45=0 First, graph the associated parabola by plotting the vertex and four additional points, two on each side of the vertex. Then, use the graph to give the solution(s) to the equation. If there is more than one solution, separate them with commas.

Points:
Solution(s):

Answers

wheres the image or graph so i can help u?

ok, we know that for
y=ax²+bx+c
the x value of the vertex is -b/(2a)
so
y=1x²+14x+45
vertex is -14/(2*1)=-14/2=-7

sub back to find y value
y=(-7)²+14(-7)+45
y=49-98+45
y=-4
the vertex is (-7,-4)

some other points are
(1,60)
(-1,32)
(-8,-3)
(-9,0)

we see that the solutions to y=0 and y=x^2+14x+45 is at x=-9 and x=-5

How many different committees can be formed from 10 teachers and 41 students if the committee consists of 2 teachers and 2 students?

Answers

If we define "n choose r" as C(n,r)=n!/(r!(n-r)!)
where C(n,r) represents the number of ways (order not important) we can choose r objects out of n, then

Number of ways to choose teachers = 10 choose 2 = C(10,2), and
number of ways to choose students = 41 choose 2 = C(41,2)

So the number of different committees
= C(10,2)*C(41,2)
= 45*820
= 36900

[tex]10C2\cdot41C2=\dfrac{10!}{2!8!}\cdot\dfrac{41!}{2!39!}=\dfrac{9\cdot10}{2}\cdot\dfrac{40\cdot41}{2}=36,900[/tex]

The perimeter of a rectangle is 56 feet. Describe the the possible lengths of a side if the area of the rectangle is not to exceed 132 square ft

Answers

P = 2(L + W)
56 = 2(L + W)
56/2 = L + W
28 = L + W.....L = 28 - W

A = L * W
132 = W(28 - W)
132 = -W^2 + 28W
W^2 - 28W + 132 = 0
(W - 6)(W - 22) = 0

W - 6 = 0 L = 28 - W
W = 6 L = 28 - 6
L = 22

W - 22 = 0 L = 28 - W
W = 22 L = 28 - 22
L = 6

not sure which (length or width)....one of them is 6 ft and the other is 22 ft

The possible lengths of sides of the rectangle will be [tex](6,22)[/tex] or [tex](22,6)[/tex] .

What is area ?

Area is the space which is occupied by any [tex]2D[/tex] surface.

Area of rectangle [tex]=Length\ *\ Breadth[/tex]

Perimeter of rectangle [tex]=2(Length\ +\ Breadth)[/tex]

We have,

Perimeter of rectangle [tex]=56[/tex] feet

Area of rectangle [tex]=132[/tex] squared feet

Let,

Length of rectangle [tex]=x[/tex]

Breadth of rectangle [tex]=y[/tex]

So,

Perimeter of rectangle [tex]=2(Length\ +\ Breadth)[/tex]

[tex]56=2(x+y)[/tex]

[tex]28=(x+y)[/tex]

⇒ [tex]x=28-y[/tex] [tex]........(i)[/tex]

Now,

Area of rectangle [tex]=Length\ *\ Breadth[/tex]

[tex]132=x\ *\ y[/tex] [tex]........(ii)[/tex]

Now substitute value of [tex]x[/tex] in [tex](ii)[/tex],

[tex]132=(28-y)y[/tex]

[tex]132=28y-y^2[/tex]

⇒[tex]y^2-28y+132=0[/tex]

Now using Middle term split method;

[tex]y^2-28y+132=0[/tex]

[tex]y^2-22y-6y+132=0[/tex]

[tex]y(y-22)-6(y-22)=0[/tex]

[tex](y-6)(y-22)=0[/tex]

⇒

[tex](y-6)=0[/tex]

⇒ [tex]y=6[/tex]

[tex](y-22)=0[/tex]

⇒ [tex]y=22[/tex]

So,

Now if we take [tex]y=6[/tex], then

[tex]x=28-6[/tex]

[tex]x=22[/tex]

And,

if we take [tex]y=22[/tex], then

[tex]x=28-22[/tex]

[tex]x=6[/tex]

So, the possible lengths of sides of the rectangle are [tex](6,22)[/tex] or [tex](22,6)[/tex] .

Hence, we can say that he possible lengths of sides of the rectangle are [tex](6,22)[/tex] or [tex](22,6)[/tex] .

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Find an ordered pair (x , y) that is a solution to the equation

Answers

4x+y = 3
if x = 0, y = 3

ordered pair (0,3)

a,(-a)^2=a^2 This is true or false?

Answers

[tex]\bf (-a)^2\implies (-a)(-a)\impliedby \textit{recall \underline{minus} times \underline{minus} is \underline{plus}} \\\\\\ -1(a)\cdot -1(a)\implies (-1\cdot -1)(a)(a)\implies +1(a)(a)\implies a^2[/tex]

Which question is not a good survey question?

a) Don’t you agree that the financial crisis is essentially over?

b) On average, how many hours do you sleep per day?

c) What is your opinion of educational funding this year?

d) Are you happy with the availability of electronic products in your state?

Answers

C, as it asks opinion, and not a specific question, so there may be a versed answer.

Answer:

See image

Step-by-step explanation:

Plato

An ellipse has vertices along the major axis at (0, 8) and (0, –2). The foci of the ellipse are located at (0, 7) and (0, –1). What are the values of a, b, h, and k, given the equation below? (y-k)^2/a^2+(x+h)^2/b^2=1

Answers

check the picture below, so it looks like so.

now hmm, from the provided vertices and focus point, you can pretty much see what "a" is, half of the major axis, is just 5.

now, the center is from either vertex to half-way up, or "a" units up, so say from -2 + 5, is at 3, so the center is at 0, 3.

now, the distance from a focus point to the center, is 4 units, like say from 0, 3 up to 0,7.

[tex]\bf \textit{ellipse, vertical major axis}\\\\ \cfrac{(y-{{ h}})^2}{{{ a}}^2}+\cfrac{(x-{{ k}})^2}{{{ b}}^2}=1 \qquad \begin{cases} center\ ({{ h}},{{ k}})\\ vertices\ ({{ h}}, {{ k}}\pm a)\\ c=\textit{distance from}\\ \qquad \textit{center to foci}\\ \qquad \sqrt{{{ a }}^2-{{ b }}^2}\\ ----------\\ h=0\\ k=3\\ a=5\\c=4 \end{cases} \\\\\\ \cfrac{(y-3)^2}{5^2}+\cfrac{(x-0)^2}{b^2}=1[/tex]

now, let' s find "b".

[tex]\bf c=\sqrt{a^2-b^2}\implies c^2=a^2-b^2\implies b^2=a^2-c^2 \\\\\\ b=\sqrt{a^2-c^2}\implies b=\sqrt{5^2-4^2}\implies b=3[/tex]

so, just plug that in.

Answer: The values for a, b, h, and k are a = 5, b = 3, h = 0, k = -3.

Step-by-step explanation: In this problem, we know ellipse has vertices along the major axis at (0, 8) and (0, -2). The foci of the ellipse are located at (0, 7) and (0, -1). We are asked to determine the values of a, b, h, and k.

We were also then provided with the equation for vertical eclipse:

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

Before we begin, we need to first define our values for a, b, h, and k.

a - distance to vertices from the centerb - distance to co-vertices from the center(h, k) - represents the center of the eclipse

The first step, we need to determine the center of the eclipse. We can use the midpoint formula to determine the midpoint between the vertices along the major axis: (0, 8) and (0, -2).

[tex]M = (\frac{x_{1} +x_{2} }{2} , \frac{y_{1} + y_{2} }{2} )[/tex]

[tex]M = (\frac{0 + 0}{2} , \frac{-2 + 8}{2} )\\M = (0, 3)[/tex]

We now know that our center (h, k) is (0, 3). Which means our values for h and k are 0 and 3. Next, we have to determine our values for a and b. Considering the center of our eclipse is not at the center, we can use one of our vertices to determine our value for a.

[tex]V_{1}[/tex] = (h, k±a)

(0, 8) = (0, 3±a)

3 ± a = 8

±a = 5

Now, we know that a = 5. For us to get b, we need to use this formula: [tex]c^2 = a^2 - b^2[/tex]. Let's rewrite this formula, so we can focus on getting our b-value.

[tex]c^2 - a^2 = -b^2[/tex]

For us to use this formula, we need to determine our c value. To find our c-value, we have use of our foci points: (h, k±c). C is the units away/further from the center towards our foci points.

(0, 3±c) = (0, 7)

3 + c = 7

7 - 3 = c

4 = c

Now, we know that our value for c is 4. Now, let's plug into the formula.

[tex](4)^2 - (5)^2 = -b^2\\16 - 25 = -b^2\\\frac{-9}{-1} = \frac{-b^2}{-1} \\\sqrt{b^2} = \sqrt{9} \\b = 3[/tex]

Our value for b is 3. If we put into our eclipse formula:

[tex]\frac{(x-0)^2}{3^2} + \frac{(y -(-3))^2}{5^2}[/tex]

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