Write the equation in standard form for the hyperbola with vertices (-9,0) and (9,0) and a conjugate axis of length 16

Answers

Answer 1

The given vertices are (-9,0) and (9,0).

Notice that they lie on the x-axis since they have 0 as their y-coordinate.

Hence, the hyperbola is a horizontal hyperbola.

Recall that the equation of a horizontal hyperbola is given as:

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

Where (h,k) is the center and a>b.

As both vertices are equidistant from the origin, the center of the hyperbola is (0,0), and the equation becomes:

[tex]\frac{x^2}{a^2}-\frac{y^2}{b^2}=1[/tex]

Note that the vertices are at (-a,0) and (a,0).

Compare with the given vertices (-9,0) and (9,0). It follows that a=9.

Substitute this into the equation:

[tex]\frac{x^2}{9^2}-\frac{y^2}{b^2}=1[/tex]

Recall that the length of the conjugate axis is given as 2b, it follows that:

[tex]\begin{gathered} 2b=16 \\ \Rightarrow b=\frac{16}{2}=8 \end{gathered}[/tex]

Substitute b=8 into the equation:

[tex]\begin{gathered} \frac{x^2}{9^2}-\frac{y^2}{8^2}=1 \\ \Rightarrow\frac{x^2}{81}-\frac{y^2}{64}=1 \end{gathered}[/tex]

The required equation in standard form is:

[tex]\frac{x^2}{81}-\frac{y^2}{64}=1[/tex]


Related Questions

If a and b are the measure of two first quadrant angles, find the exact value of the functioncsc a =5/3 and tan 5/12 find the cod (a+b)

Answers

Input data

[tex]\begin{gathered} \cos a=\frac{5}{3} \\ \tan b=\frac{5}{12} \end{gathered}[/tex]

Now for cos(a+b)

[tex]\begin{gathered} a=\csc ^{-1}(\frac{5}{3})^{} \\ a=36.87 \end{gathered}[/tex][tex]\begin{gathered} b=\tan ^{-1}(\frac{5}{12}) \\ b=22.62 \end{gathered}[/tex][tex]\begin{gathered} \cos (a+b) \\ \cos (36.87+22.62) \\ \cos 59.5 \\ \frac{33}{65}=0.507 \end{gathered}[/tex]

Raphael has an odd-shaped field shown in Figure 13-2. He wants to put a four-strand barbed wire fence around it for his cattle.A. What is the perimeter of the field?b. How many 80-rod rolls of barbed wire does he need topurchase?c. How many acres will be fenced?

Answers

Answer: Total perimeter = 9, 962.01 feet

The figure is a composite structure

It contains a rectangle and triangle

The perimeter of a rectangle is given as

Perimeter = 2( length + width)

length of the rectangle = 1500ft

Width of the rectangle = 1390 ft

Perimeter = 2( 1500 + 1390)

Perimeter = 2(2890)

Perimeter = 5780 ft

To calculate the perimeter of a triangle

[tex]\begin{gathered} \text{Perimeter = a + b + }\sqrt[]{a^2+b^2} \\ a\text{ = 1050ft and b = 1390 ft} \\ \text{Perimeter = 1050 + 1390 + }\sqrt[]{1050^2+1390^2} \\ \text{Perimeter = 2440 + }\sqrt[]{1,102,\text{ 500 + 1, 932, 100}} \\ \text{Perimeter = 2400 + }\sqrt[]{3,034,600} \\ \text{Perimeter = 2440 + 1,742,01} \\ \text{Perimeter = }4182.01\text{ f}eet \end{gathered}[/tex]

The total perimeter of the field = Perimeter of the rectangle + perimeter of the right triangle

Total perimeter = 5780 + 4182.01

Total perimeter = 9, 962.01 feet

The profit of a cell-phone manufacturer is found by the function y= -2x2 + 108x + 75 , where x is the cost of the cell phone. At what price should the manufacturer sell the phone tomaximize its profits? What will the maximum profit be?

Answers

Hello!

First, let's rewrite the function:

[tex]y=-2x^2+108x+75[/tex]

Now, let's find each coefficient of it:

• a = -2

,

• b = 108

,

• c = 75

As we have a < 0, the concavity of the parabola will face downwards.

So, it will have a maximum point.

To find this maximum point, we must obtain the coordinates of the vertex, using the formulas below:

[tex]\begin{gathered} X_V=-\frac{b}{2\cdot a} \\ \\ Y_V=-\frac{\Delta}{4\cdot a} \end{gathered}[/tex]First, let's calculate the coordinate X by replacing the values of the coefficients:[tex]\begin{gathered} X_V=-\frac{b}{2\cdot a} \\ \\ X_V=-\frac{108}{2\cdot(-2)}=-\frac{108}{-4}=\frac{108}{4}=\frac{54}{2}=27 \end{gathered}[/tex]

So, the coordinate x = 27.

Now, let's find the y coordinate:[tex]\begin{gathered} Y_V=-\frac{\Delta}{4\cdot a} \\ \\ Y_V=-\frac{b^2-4\cdot a\cdot c}{4\cdot a} \\ \\ Y_V=-\frac{108^2-4\cdot(-2)\cdot75}{4\cdot(-2)} \\ \\ Y_V=-\frac{11664+600}{-8}=\frac{12264}{8}=1533 \end{gathered}[/tex]

The coordinate y = 1533.

Answer:

The maximum profit will be 1533 (value of y) when x = 27.

Jina opened a savings account with $600 and was paid simple interest at an annual rate of 3%. When Jina closed the account, she was paid $54 in interest. How long was the account open for, in years?

Answers

Answer: The account has been open for 3 years

Step-by-step explanation:

3% of $600 is 18

18*3 = 54

Answer:

3 years

have $600

interest $54

annual rate 3%

600 - 3% = 582 that is the money she has in bank without interest

600-582 = 18

54÷ 18= 3 years

Given the conversion factor which cube has the larger surface area?

Answers

Given the surface area of a cube as

[tex]\begin{gathered} SA=6l^2 \\ \text{where l is the length} \end{gathered}[/tex]

Given Cubes A and B

[tex]\begin{gathered} \text{Cube A} \\ l=19.5ft \end{gathered}[/tex][tex]\begin{gathered} \text{Cube B } \\ l=6m\text{ } \\ \text{ in ft}\Rightarrow\text{ 1m =3.28ft} \\ l=6\times3.28ft=19.68ft \end{gathered}[/tex]

Find the surface area of the cubes and compare them to know which one is larger

[tex]\begin{gathered} \text{Cube A} \\ SA=6\times19.5^2=6\times380.25=2281.5ft^2 \end{gathered}[/tex][tex]\begin{gathered} \text{Cube B} \\ SA=6\times19.68^2=6\times387.3024=2323.8144ft^2 \end{gathered}[/tex]

Hence, from the surface area gotten above, Cube B has a larger surface area than Cube A

i need help, plotting the ordered pair (0, 0.5) and I need to state in which quadrant or on which axis the point lies.

Answers

The ordered pair:

[tex](x,y)=(0,0.5)[/tex]

it is located at:

Since the point lies on the y-axis it doesn't not lie in any quadrant

solving systems by graphing and tables : equations and inequalities

Answers

Given,

The system of inequalitites are,

[tex]\begin{gathered} 2x+3y>0 \\ x-y\leq5 \end{gathered}[/tex]

The graph of the inequalities is,

The are three possible solution for the inequality.

For (0, 0),

[tex]\begin{gathered} 2x+3y>0 \\ 2(0)+3(0)>0 \\ 0>0 \\ \text{Similarly,} \\ x-y\leq5 \\ 0-0\leq5 \\ 0\leq5 \end{gathered}[/tex]

For (3, -2),

[tex]\begin{gathered} 2x+3y>0 \\ 2(3)+3(-2)>0 \\ 0>0 \\ \text{Similarly,} \\ x-y\leq5 \\ 3-(-2)\leq5 \\ 5=5 \end{gathered}[/tex]

For (5, 0),

[tex]\begin{gathered} 2x+3y>0 \\ 2(5)+3(0)>0 \\ 5>0 \\ \text{Similarly,} \\ x-y\leq5 \\ 5-(0)\leq5 \\ 5=5 \end{gathered}[/tex]

Hence, the solution of the inequalities is (5, 0).

Find the area of the circle. Use 3.14 or 227for π . thxQuestion 2

Answers

Step 1

State the area of a circle using the diameter

[tex]\frac{\pi d^2}{4}[/tex]

Where d=diameter=28in

[tex]\pi=\frac{22}{7}[/tex]

Step 2

Find the area

[tex]A=\frac{22}{7}\times\frac{28^2}{4}=616in^2[/tex]

Answer;

[tex]Area\text{ = }616in^2\text{ when }\pi\text{ =}\frac{22}{7}[/tex]

An airplane is taking off at angle of 9 degrees and traveling at a speed of 200 feet per second in relation to the ground. If the clouds begin at an altitude of 4,000 feet, how many seconds will it take for the airplane to be in the clouds?

Answers

ANSWER

[tex]\begin{equation*} 127.85\text{ }seconds \end{equation*}[/tex]

EXPLANATION

First, let us make a sketch of the problem:

To find the time it will take the airplane to be in the clouds, we first have to find the distance flown by the airplane in attaining that height, x.

To do this, apply trigonometric ratios SOHCAHTOA for right triangles:

[tex]\sin9=\frac{4000}{x}[/tex]

Solve for x:

[tex]\begin{gathered} x=\frac{4000}{\sin9} \\ x=25,569.81\text{ }ft \end{gathered}[/tex]

Now, that we have the distance, we can solve for the time by applying the relationship between speed and distance:

[tex]\begin{gathered} speed=\frac{distance}{time} \\ \Rightarrow time=\frac{distance}{speed} \end{gathered}[/tex]

Substitute the given values into the formula above and solve for time:

[tex]\begin{gathered} time=\frac{25569.81}{200} \\ time=127.85\text{ }seconds \end{gathered}[/tex]

That is the number of seconds that it will take.

Question 13 of 18Graph the solution to the following inequality on the number line.x² - 4x ≥ 12

Answers

Step 1

Given; Graph the solution to the following inequality on the number line.

x² - 4x ≥ 12

Step 2

[tex]\begin{gathered} x^2-4x\ge \:12 \\ Rewrite\text{ in standard form} \\ x^2-4x-12\ge \:0 \\ Factor\text{ the inequality} \\ \left(x+2\right)\left(x-6\right)\ge \:0 \end{gathered}[/tex][tex]\begin{gathered} Identify\text{ the intervals} \\ x\le \:-2\quad \mathrm{or}\quad \:x\ge \:6 \end{gathered}[/tex]

Thus, the number line will look like

Answer; The solution to the inequality graphed on a number line is seen below

[tex]x\le \:-2\quad \mathrm{or}\quad \:x\ge \:6[/tex]

Calculate Sse for the arithmetic sequence {a,}5sequence {1,3 ={}+}=Ο Α. 1463OB. 91220 C. 8,6716D. 9,26767

Answers

Answer:

[tex]\frac{8,671}{6}[/tex]

Explanation:

Here, we want to get the sum of the 58 terms in series

Mathematically, we have the formula to use as:

[tex]S_n\text{ = }\frac{n}{2}(a\text{ + L)}[/tex]

where a is the first term and L is the last term

The first term is when n is 1

We have this calculated as:

[tex]\text{ a}_{}\text{ = }\frac{5}{6}+\frac{1}{3}\text{ = }\frac{5+2\text{ }}{6}\text{ = }\frac{7}{6}[/tex]

The last term is the 58th term which is:

[tex]\text{ a}_{58}\text{ = }\frac{290}{6}\text{ + }\frac{1}{3}\text{ = }\frac{292}{6}[/tex]

We finally substitute these values into the initial equation

Thus, we have it that:

[tex]S_{58}\text{ = }\frac{58}{2}(\frac{292}{6}+\frac{7}{6})\text{ = 29(}\frac{299}{6})\text{ = }\frac{8671}{6}[/tex]

Help me with my schoolwork what is the slope of line /

Answers

The two points given on the line are

[tex]\begin{gathered} (x_1,y_1)\Rightarrow(-2,9) \\ (x_2,y_2)\Rightarrow(6,1) \end{gathered}[/tex]

The slope of line that passes through (x1,y1) and (x2,y2) is gotten using the formula below

[tex]\begin{gathered} m=\frac{\text{change in y}}{\text{change in x}} \\ m=\frac{y_2-y_1}{x_2-x_1} \end{gathered}[/tex]

By substituting the values, we will have

[tex]\begin{gathered} m=\frac{y_2-y_1}{x_2-x_1} \\ m=\frac{1-9}{6-(-2)} \\ m=-\frac{8}{6+2} \\ m=-\frac{8}{8} \\ m=-1 \end{gathered}[/tex]

Therefore,

The slope of the line = -1

match the system of equations with the solution set.hint: solve algebraically using substitution method.A. no solutionB. infinite solutionsC. (-8/3, 5)D. (2, 1)

Answers

We will solve all the systems by substitution method .

System 1.

By substituting the second equation into the first one, we get

[tex]x-3(\frac{1}{3}x-2)=6[/tex]

which gives

[tex]\begin{gathered} x-x+6=6 \\ 6=6 \end{gathered}[/tex]

this means that the given equations are the same. Then, the answer is B: infinite solutions.

System 2.

By substituting the first equation into the second one, we have

[tex]6x+3(-2x+3)=-5[/tex]

which gives

[tex]\begin{gathered} 6x-6x+9=-5 \\ 9=-5 \end{gathered}[/tex]

but this result is an absurd. This means that the equations represent parallel lines. Then, the answer is option A: no solution.

System 3.

By substituting the first equation into the second one, we obtain

[tex]-\frac{3}{2}x+1=-\frac{3}{4}x+3[/tex]

by moving -3/4x to the left hand side and +1 to the right hand side, we get

[tex]-\frac{3}{2}x+\frac{3}{4}x=3-1[/tex]

By combining similar terms, we have

[tex]-\frac{3}{4}x=2[/tex]

this leads to

[tex]x=-\frac{4\times2}{3}[/tex]

then, x is given by

[tex]x=-\frac{8}{3}[/tex]

Now, we can substitute this result into the first equation and get

[tex]y=-\frac{3}{2}(-\frac{8}{3})+1[/tex]

which leads to

[tex]\begin{gathered} y=4+1 \\ y=5 \end{gathered}[/tex]

then, the answer is option C: (-8/3, 5)

System 4.

By substituting the second equation into the first one, we get

[tex]-5x+(2x-3)=-9[/tex]

By combing similar terms, we have

[tex]\begin{gathered} -3x-3=-9 \\ -3x=-9+3 \\ -3x=-6 \\ x=\frac{-6}{-3} \\ x=2 \end{gathered}[/tex]

By substituting this result into the second equation, we have

[tex]\begin{gathered} y=2(2)-3 \\ y=4-3 \\ y=1 \end{gathered}[/tex]

then, the answer is option D

w=3? What is the value of the expression below when w = 5w+ 2

Answers

Answer:

The value of the expression at w=3 is;

[tex]17[/tex]

Explanation:

Given the expression;

[tex]5w+2[/tex]

Then when w=3, the value of the expression is;

[tex]\begin{gathered} 5w+2 \\ =5(3)+2 \\ =15+2 \\ =17 \end{gathered}[/tex]

The value is gotten by replacing/substituting w with 3 in the expression;

Therefore, the value of the expression at w=3 is;

[tex]17[/tex]

Find the sum of the arithmetic series given a1 =2, an =35 an n = 12

Answers

Given:

[tex]a_1=2,a_n=35,n=12[/tex]

Required:

Find the sum of the arithmetic series.

Explanation:

The sum of the arithmetic series when the first and the last term is given by the formula.

[tex]S_n=\frac{n}{2}(a_1+a_n)[/tex]

Substitute the given values in the formula.

[tex]\begin{gathered} S_n=\frac{12}{2}(2+35) \\ =6(37) \\ =222 \end{gathered}[/tex]

Final Answer:

Option D is the correct answer.

Which representation does not show y as a function of x?1.II.€9> 10III.x 1 3 5 7y -6 -18 -30 -42IV. {(-2,3), (-1,4), (0,4), (3, 2)}a) I and IIb) I, II, and IIIc) I and IVd) All of the above are functions

Answers

We can say that I is not a function because inputs can only have one output.

II it's not a function since if you draw an horizontal line through the function intersect in two points, then it's not a function.

The answer is A.

Which of the following statements about the table is true?
Select all that apply.
The table shows a proportional relationship.
All the ratios for related pairs of x and y are equivalent to 7.5.
When x is 13.5, y is 4.5.
When y is 12, x is 4.
The unit rate of for related pairs of x and y is .
26
22 Undertond Proportional Relationships: Fouivalent Ratios
C
C
y
10.5 3.5
15.9 5.3
22.5 7.5
27
9
3

Answers

Answer:

there is a lot of ratios here, but I will try my best. A proportional relationship is the relationship that is proportional obviously. and if the ratio is related, pairs are equivalent to 7.5 then that must mean that the proportional relationship is fuevalent

“Use the properties to rewrite this expression with the fewest terms possible:3+7(x - y) + 2x - 5y”

Answers

[tex]-5y+2x+7(x-y)+3[/tex]

Expanding 7(x - y) in the above expression gives

[tex]-5y^{}+2x+7x-7y+3[/tex]

adding the like terms (2x+ 7x) and (-5y-7y) gives

[tex](-5y-7y)+(2x+7x)+3[/tex][tex]\rightarrow\textcolor{#FF7968}{-12y+8x+3.}[/tex]

The last expression is the simplest form we can convert our expression into.

Can you please help me because I don’t understand this and I would like to really understand it

Answers

Answer:

Explanation:

Given the expression:

[tex]\sqrt{12(x-1)}\div\sqrt{2(x-1)^{2}}[/tex]

By the division law of surds:

[tex]\sqrt[]{x}\div\sqrt[]{y}=\sqrt[]{\frac{x}{y}}[/tex]

Therefore:

[tex]\sqrt[]{12(x-1)}\div\sqrt[]{2(x-1)^2}=\sqrt[]{\frac{12(x-1)}{2(x-1)^2}}[/tex]

The result obtained can be rewritten in the form below:

[tex]=\sqrt[]{\frac{2\times6(x-1)}{2(x-1)(x-1)^{}}}[/tex]

Canceling out the common factors, we have:

[tex]=\sqrt[]{\frac{6}{(x-1)^{}}}[/tex]

An equivalent expression is Opt

why does a cubic graph have both an x intercept and a y intercept

Answers

Answer:

All cubic function has domain (-∞,∞) and range (-∞,∞)

Step-by-step explanation:

3/5 ÷ 1/3 = ?????????

Answers

[tex]\frac{3}{5}\text{divide by}\frac{1}{3}[/tex]

Change the division sign to multiplication and then invert 1/3

That is;

[tex]\frac{3}{5}\times3[/tex][tex]=\frac{9}{5}\text{ =1}\frac{4}{5}[/tex]

Determine the minimum and maximum value for f(x) = -5x²-3x+7 over interval [-1, 3].

Answers

The maximum and minimum value of the equation "f(x) = -5x²-3x+7" over the interval [-1, 3] is 5 and -47.

What are equations?A mathematical statement that has an "equal to" symbol between two expressions with equal values is called an equation. A number that can be entered for the variable to produce a true number statement is the solution to an equation. 3(2)+5=11, which states that 6+5=11, is accurate. The answer is 2, then. The point-slope form, standard form, and slope-intercept form are the three main types of linear equations.

So, the minimum and maximum values when x are -1 and 3:

(1) When x = -1:

f(x) = -5x²-3x+7f(x) = -5(-1)²-3(-1) +7f(x) = -5(1) + 3 +7f(x) = -5 + 10f(x) = 5

(2) When x = 3:

f(x) = -5x²-3x+7f(x) = -5(3)² -3(3)+7f(x) = -5(9) -9 +7f(x) = -45 -9 +7f(x) = - 54 + 7f(x) = - 47

Therefore, the maximum and minimum value of the equation "f(x) = -5x²-3x+7" over the interval [-1, 3] is 5 and -47.

Know more about equations here:

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If TW =6, WV =2, and UV =25, find XV to the nearest hundredth.

Answers

TW = 6

WV = 2

UV = 25

XV = ?

XV/UV = WV/TV

XV/25 = 2 /(6 + 2)

XV = 2(25)/7

XV = 50/7

XV = 7.1428

Rounded to the nearest hundredth

XV = 7.14

hannah paid 15.79 for a dress that was originally marked 24.99 what js the percent of discount

Answers

The percentage of discount is 37%

Here, we want to calculate the percentage of discount

The first thing we need to do here is to calculate the discount amount

Mathematically, we have this as;

[tex]24.99-15.79\text{ = 9.2}[/tex]

Now, we find the percentage of 24.99 is this discount

We have this as;

[tex]\frac{9.2}{24.99}\text{ }\times100\text{ \% = 36.8\%}[/tex]

The percentage of discount is approximately 37%

A psychology test has personality questions numbered 1, 2, 3, intelligence questions numbered 1, 2, 3, 4, and attitudequestions numbered 1,2. If a single question is picked at random, what is the probability that the question is an intelligence question OR has an odd number?

Answers

Answer:

7/9.

Step-by-step explanation?

Total number of questions: 3 + 4 + 2 = 9.

Number of Intelligence questions: 4

Number of questions that have an odd number: 5

The probability of a question is Intelligence questions = 4/9

The probability a question has an odd number = 5/9

The probability a question is Intelligence questions and has an odd number = 2/9

The probability a question is Intelligence question OR has an odd number is:

4/9 + 5/9 - 2/9 = 7/9.

A half-marathon has 53 runners. A first-, second-, and third-place trophy will be awarded. Howmany different ways can the trophies be awarded?

Answers

Let's use the combination formula:

[tex]\begin{gathered} C(n,k)=nCk=\frac{n!}{k!(n-k)!} \\ n=53 \\ k=3 \\ C(53,3)=53C3=\frac{53!}{3!(50)!}=23426 \end{gathered}[/tex]

The function is defined by h(x) = x - 2 . Find h(n + 1) .

Answers

SOLUTION:

Case: Functions

Method:

The function

[tex]\begin{gathered} h(x)=x-2 \\ Hence \\ h(n+1)=(n+1)-2 \\ h(n+1)=n+1-2 \\ h(n+1)=n-1 \end{gathered}[/tex]

Final answer:

[tex]h(n+1)=n-1[/tex]

3 166.40 266.24 3. Consider the following functions which all have an or decay? By what percent? Rewrite as (1+r) or (1-r) f(t) = 30(1.04) p(x) = 30(0.65)Solve f(t)

Answers

ANSWER

Function f(t) represents a growth by 4%

EXPLANATION

If the function represents a decay it is written as:

[tex]f(t)=a(1-r)^t[/tex]

and if it represents a growth it's:

[tex]f(t)=a(1+r)^t[/tex]

We can see if it's a growth or decay by looking at the number we have between parenthesis: if it's greater than 1, then it's a growth and if it's less than 1 then it's a decay.

For function f(t) we have

[tex]1+r=1.04[/tex]

Therefore, r = 0.04 which, expressed as a percent is 4%

Some airlines charge a fee for each checked luggage item that weighs more than 21,000 grams. How many kilograms is​ this?

Answers

The value of 21,000 grams to kilograms is 21 kilograms

How to convert kilograms to grams ?

1000 grams = 1kg

The first step is to convert 21,000 grams to kilograms

It can be calculated as follows;

= 21000/1000

= 21

Hence the value of 21,000 grams in kilograms is 21 kilograms

Read more on grams here

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What is the measure of EDH?EHFO 10°O 40°50°90

Answers

To find the measure of angle EDH we must solve for x first. Formulating an equation to find x, we have:

5x + 4x= 90 (Given that the sum of the angles EDH and HDG is equal to 90°)

9x = 90 (Adding like terms)

x= 90/9 (Dividing on both sides of the equation by 9)

x= 10

Replacing in the expression for angle EDH, we have:

m∠EDH = 5*(x) = 5*(10) = 50° (Multiplying)

The answer is m∠EDH =50°.

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