Write the equation of a hyperbola with vertices (0, -4) and (0, 4) and foci (0, -5) and (0, 5).
Answers
notice, the center is clearly at the origin, and notice how long the "a" component is, also, bear in mind that, is opening towards the y-axis, that means the fraction with the "y" variable is the positive one.
Also notice, the "c" distance from the center to either foci, is just 5 units.
[tex]\bf \textit{hyperbolas, vertical traverse axis }\\\\ \cfrac{(y-{{ k}})^2}{{{ a}}^2}-\cfrac{(x-{{ h}})^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} \end{cases}\\\\ -------------------------------\\\\[/tex]
[tex]\bf \begin{cases} h=0\\ k=0\\ a=4\\ c=5 \end{cases}\implies \cfrac{(y-{{ 0}})^2}{{{ 4}}^2}-\cfrac{(x-{{ 0}})^2}{{{ b}}^2}=1\implies \cfrac{y^2}{16}-\cfrac{x^2}{b^2}=1 \\\\\\ c=\sqrt{a^2+b^2}\implies c^2=a^2+b^2\implies \sqrt{c^2-a^2}=b \\\\\\ \sqrt{5^2-4^2}=b\implies \boxed{3=b} \\\\\\ \cfrac{y^2}{16}-\cfrac{x^2}{3^2}=1\implies \boxed{\cfrac{y^2}{16}-\cfrac{x^2}{9}=1}[/tex]
Related Questions
A rectangular prism with a square base has a lateral area of 192 square yards and a surface area of 264 square yards. What is the length of a base edge?
Answers
TSA = LSA + 2s²
Since the base is said to be the square, its area is equal to s² where s is the base edge. Because there are 2 bases, that's why it's 2s².
264 = 192 + 2s²
s² = (264-192)/2
s = √36
s = 6 yards
(02.02 MC)
Line segment RS is shown on a coordinate grid:
The line segment is rotated 270 degrees counterclockwise about the origin to form R'S'. Which statement describes R'S'?
R'S' is parallel to RS.
R'S' is half the length of RS
. R'S' is twice the length of RS.
R'S' is equal in length to RS.
Answers
length doesnt change, it was just rotated.
SOMEBODY HELP ME WITH THESE PLEASE! I really need the help like NOW PLS!
Answers
[tex]f(n) = 2n^2 - 5n[/tex]
then [tex]g(n) +f(n) = (n-4) + (2n^2 - 5n)[/tex]
[tex]= n - 4 + 2n^2 - 5n[/tex]
[tex]= 2n^2 -5n + n - 4[/tex]
[tex]= 2n^2 - 4n -4[/tex]
95. [tex]f(n) = -n+3[/tex]
[tex]g(n) = n^3 + 3n[/tex]
Then [tex]f(n) . g(n) = (-n+3) . (n^3 + 3n)[/tex]
[tex]= -n^4 + 3n^3 - 3n^2 + 9n [/tex]
97. [tex]f(x) = 3x+1[/tex]
[tex]g(x) = 2x[/tex]
For finding f(g(x)) we will plugin value of g(x) in place of x in f(x).
[tex]f(g(x)) = 3*(2x) + 1[/tex]
[tex]= 6x + 1[/tex]
99. [tex]f(n) = n - 3[/tex]
[tex]g(n) = 2n^2 - 3n[/tex]
[tex]g(-7) = 2*(-7)^2 - 3 * (-7) = 2* 49 + 21 = 98 + 21 = 119[/tex]
[tex]f(g(-7)) = 119 - 3 = 116[/tex]
Answer:
97.
For finding f(g(x)) we will plugin value of g(x) in place of x in f(x).
99.
Step-by-step explanation:
What is the length of a diagonal of a cube with a side length of 10 cm? 200cm, 210cm, 300cm, 320cm
Answers
so the length of diagonal of the cube(y) is
y² = (10)² + (10√2)² =300 so y =10√3cm
The solution is, 10√3cm is the length of a diagonal of a cube with a side length of 10 cm.
What is Pythagorean theorem?Pythagorean theorem, the well-known geometric theorem that the sum of the squares on the legs of a right triangle is equal to the square on the hypotenuse (the side opposite the right angle)—or, in familiar algebraic notation, a2 + b2 = c2.
here, we have,
given that,
the side length is 10cm ,
then the length of diagonal of the base of the cube(x) is
X² =(10)²+(10)²
=200cm
so x = 10√2cm
so the length of diagonal of the cube(y) is
y² = (10)² + (10√2)²
=300
so y =10√3cm
Hence, The solution is, 10√3cm is the length of a diagonal of a cube with a side length of 10 cm.
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HELP !! HELP ME PLEASE WITH THESE 3 PROBLEMS !!!!25 POINTS !!!
Answers
[tex]\bf \cfrac{model}{tower}\qquad \cfrac{2}{25}=\cfrac{m}{160}[/tex]
solve for "m"
5)
[tex]\bf \textit{sprinkle park to parking lot}\quad \cfrac{drawing}{park}\qquad \cfrac{15}{60}\implies \cfrac{1}{4} \\\\\\ \textit{playground to sprinkle park}\qquad \cfrac{drawing}{park}\qquad \cfrac{1}{4}=\cfrac{12}{p}[/tex]
solve for "p"
6)
[tex]\bf \begin{array}{ccllll} cake&people\\ \text{\textemdash\textemdash\textemdash}&\text{\textemdash\textemdash\textemdash}\\ 1&18\\ x&108 \end{array}\implies \cfrac{1}{x}=\cfrac{18 }{108}[/tex]
solve for "x".
If measure of angle p=(10x-2),measure of angle o=(3x+9) , and measure of angle q=(3x-3) , list the lengths of the sides of triangle OPQ from longest to shortest.
Answers
So we can find the value of x...
10x-2+3x+9+3x-3=180
16x+4=180
16x=176
x=11
So the angles are 108°, 42°, 30°
Now using the Law of Sines we can solve for all sides.
(sina/A=sinb/B=sinc/C)
a/sin30=b/sin42=c/sin108
However, we would need to know at least one side length to solve for sides a,b, and c numerically, otherwise it can be any multiple of an arbitrary choice for the smallest side. Let a=1 for example:
b=sin42/sin30, c=sin108/sin30
b≈1.34, c≈1.90
a=1, b=1.34, c=1.9
Anyway...
If you are given any side length:
a/sin30=b/sin42=c/sin108 still holds true and you can solve for the other side lengths...
Answer:
OQ > PQ > OP
Step-by-step explanation:
In this question measure of all angles in a triangle has been given.
m∠ p = (10x - 2)
m∠ q = (3x - 3)
m∠ o = (3x + 9)
Since total of all angles is 180°, so we will equate the total of all angles to 180°
∠p + ∠q + ∠o = 180°
(10x - 2) + (3x - 3) + (3x + 9) = 180°
(10x + 3x + 3x) -2 - 3 + 9 = 180
16x - (2 + 3 - 9) = 180°
16x + 4 = 180
16x = 180 - 4
16x = 176
x = [tex]\frac{176}{16}=11[/tex]
Now we will find the measure of each angle.
∠p = (10x - 2) = 10×11 - 2 = (110 - 2) = 108°
∠q = (3x - 3) = 3×11 - 3 = (33 - 3) = 30°
∠o = (3x + 9) = 3×11 + 9 = (33 + 9) = 42°
As we know in a triangle, angle opposite to the largest side is largest and angle opposite side is the smallest.
Since ∠p > ∠o > ∠q
Therefore, in Δ OPQ opposite sides to these angles will be in the same ratio.
OQ > PQ > OP
1. is acute. 2. is isosceles. 3. is right. Which two statements contradict each other?
Answers
Answer:
the answer is 2 and 3
Step-by-step explanation:
I took the test
A basket contains 11 pieces of fruit: 4 apples, 5 oranges, and 2 bananas. Jonas takes a piece of fruit at random from the basket, and then Beth takes a piece at random. What is the probability that Jonas will get an orange and Beth will get an apple?
Answers
The probability that Jonas will get an orange and Beth will get an apple is 20/121.
Explanation:To find the probability that Jonas will get an orange and Beth will get an apple, we need to find the probability of Jonas getting an orange and then multiply it by the probability of Beth getting an apple.
The probability of Jonas getting an orange is the number of oranges in the basket (5) divided by the total number of fruits (11):
P(orange for Jonas) = 5/11
The probability of Beth getting an apple is the number of apples in the basket (4) divided by the total number of fruits (11):
P(apple for Beth) = 4/11
To find the combined probability, we multiply these two probabilities together:
P(orange and apple) = P(orange for Jonas) * P(apple for Beth) = (5/11) * (4/11) = 20/121
A pizza delivery shop averages 20 minutes per delivery with a standard deviation of 4 minutes. What is the probability that a pizza takes less than 15 minutes to be delivered?
Answers
X ≈ N(20, 4²)
We want P(X<15)
Standardising gives
P(X<15) = P(Z<[tex] \frac{15-20}{4} [/tex])
P(X<15) = P(Z< -1.25) ⇒ When the value of the z-score is negative, we will read the value of z<1.25 on the z-table then subtract the answer from 1
P(X<15) = 1 - P(Z<1.25)
P(X<15) = 1 - 0.8944
P(X<15) = 0.1056
Hence, the probability that pizza takes less than 15 minutes to be delivered is 0.1056 = 10.56%
(25 Points)Enter numbers to write 4.23×10^3 in standard notation.
Answers
So the answer will be 4230
Find the total area of the prism in ( _+_√_ )
Answers
a^2 + b^2 = c^2
4^2 + 6^2 = c^2
16 + 36 = c^2
52 = c^2
2√13 = c
Triangle bases:
A = (1/2) * b * h * 2
A = (1/2) * 4 * 6 * 2
A = 24 ft^2
Front Rectangle: Left side Rectangle Back Side Rectangle
A = 8 * 2√13 = 16√13 ft^2 A = 4 * 4 = 32 ft^2 A = 6 * 8 = 48 ft^2
24 + 32 + 48 + 16√13 = Total area = 104 + 16√13 ft^2
Solve the equation ,and check the solution.
r - 3 r
____ = __
10 13 Check:
Answers
I'm assuming the equation should be (r-3)/10 = r/13.
Use parenthesis to indicate what is being divided. In this case all of "r-3" is over 10.
If my assumption above holds up, then we can solve for r like so
(r-3)/10 = r/13
13(r-3) = 10r ... cross multiply
13r-39 = 10r
13r-39-13r = 10r-13r ... subtract 13r from both sides
-39 = -3r
-3r = -39
-3r/(-3) = -39/(-3) ... divide both sides by -3
r = 13
The value of r is r = 13
-----------------------
Check:
Replace every copy of "r" with "13" to get
(r-3)/10 = r/13
(13-3)/10 = 13/13
10/10 = 13/13
1 = 1 ... equation is true; answer has been confirmed
To solve the equation, we cross multiply to eliminate the fractions and solve for r. The solution is r = 13. We can check the solution by substituting it back into the original equation.
Explanation:To solve the equation (r - 3) / 10 = r / 13, we can cross multiply to eliminate the fractions. This gives us 13(r - 3) = 10r. Distributing, we get 13r - 39 = 10r. We can then solve for r by subtracting 10r from both sides and adding 39 to both sides. This gives us 3r = 39. Dividing both sides by 3, we find that r = 13.
To check the solution, we substitute r = 13 back into the original equation. The left side becomes (13 - 3) / 10 = 10 / 10 = 1, and the right side becomes 13 / 13 = 1. Since both sides are equal to 1, we can confirm that the solution r = 13 is correct.
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The sum of four consecutive whole numbers is 54, what are the four numbers
Answers
The sum of the four numbers is 54, therefore
x + (x+1) + (x+2) + (x+3) = 54
4x + 6 = 54
4x = 54 - 6 = 48
x = 12
Answer:
The four numbers are 12, 13, 14, and 15
Use the functions f(x) = 4x − 5 and g(x) = 3x + 9 to complete the function operations listed below.
Part A: Find (f + g)(x). Show your work. (3 points)
Part B: Find (f ⋅ g)(x). Show your work. (3 points)
Part C: Find f[g(x)]. Show your work. (4 points)
Answers
(f+g)(x)=f(x)+g(x)
(f+g)(x)=4x-5+3x+9
(f+g)(x)=7x+4
B:
(f•g)(x)=f(x)•g(x)
(f•g)(x)=(4x-5)(3x+9)
(f•g)(x)=12x^2-15x+36x-45
(f•g)(x)=12x^2+21x-45
C:
(f○g)(x)=f(g(x))
(f○g)(x)=4(3x+9)-5
(f○g)(x)=12x+36-5
(f○g)(x)=12x+31
A rectangular garden has length twice as great as its width. A second rectangular garden has the same width as the first garden and length that is 4 meters greater than the length of the first garden. The second garden has area of 70 square meters. What is the width of the two gardens?
Answers
A rectangle is 42 square feet. what percent of the area of the rectangle is a square with side lengths of 6 feet?
Answers
area = L x w
42 = 6 x w
w=42/6 = 7
rectangle is 6 x 7
square would be 6 x 6 = 36 square feet
36/42 = 0.857 = 85.7% ( Round answer if needed)
A) side-side-side triangle similarity postulate
B) angle-angle triangle similarity postulate
C) angle-side-angle triangle similarity postulate
D) hypotenuse-lag triangle similarity postulate
Answers
For similarity you only have to know that 2 corresponding angles are equal.
What are the roots of the function y = 4x2 + 2x – 30? To find the roots of the function, set y = 0. The equation is 0 = 4x2 + 2x – 30.
Factor out the GCF of .
Next, factor the trinomial completely. The equation becomes .
Use the zero product property and set each factor equal to zero and solve. The roots of the function are .
Answers
factor out the GCF:
2(2x² + x - 15) = 0
factor the trinomial completely:
2x² + x - 15 = 0
2x² + 6x - 5x - 15 = 0
2x(x + 3) - 5(x + 3) = 0
(2x - 5)(x + 3) = 0
use the zero product property and set each factor equal to zero and solve:
2x - 5 = 0 or x + 3 = 0
2x = 5 x = -3
x = 2.5
The roots of the function are x=-3, x=2.5
The roots of the equation [tex]y=4x^2+2x-30[/tex] are x = -3 and x = 5/2
Roots of a quadratic equationThe given quadratic equation is:
[tex]y=4x^2+2x-30[/tex]
Set y = 0
[tex]4x^2+2x-30=0[/tex]
Factor the trinomial completely
[tex]4x^2-10x+12x-30=0\\\\2x(2x-5)+6(2x-5)=0\\\\(2x-5)(2x+6)=0[/tex]
Set each factor to zero and solve
2x - 5 = 0
2x = 5
x = 5/2
2x + 6 = 0
2x = -6
x = -6/2
x = -3
The roots of the equation [tex]y=4x^2+2x-30[/tex] are x = -3 and x = 5/2
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The transformation from f to g represents a __________ stretch. f(x) = Square root of x. and g(x) = 6Square root of x.
Answers
The transformation from f to g represents a Vertical stretch.
Step-by-step explanation:We are given a parent function f(x) as:
[tex]f(x)=\sqrt{x}[/tex]
and the transformed function g(x) as:
[tex]g(x)=6\sqrt{x}[/tex]
We know that any function transformation of the type:
f(x) → a f(x)
represents either a vertical stretch stretch or compression depending on the value of a.
If 0<a<1 then the transformation is a vertical compression and if a>1 then the transformation is a vertical stretch.
Here we have a=6>1
Hence, the transformation is a VERTICAL STRETCH.
How to write 100000 in scientific notation?
Answers
1×10^5
Answer:
100000 in scientific notation:[tex]100,000=1.0\times 10^{5}m/s[/tex]
Step-by-step explanation:
Scientific notation is defined as the notation in which a number is expressed in the decimal form. This means that the number is always written in the power of 10 form. The numerical digit lies between 0.1.... to 9.9.....
If the decimal is shifting to right side, the power of 10 is negative and if the decimal is shifting to left side, the power of 10 is positive.
We are given: 100,000
Converting this into scientific notation, we get:
[tex]\Rightarrow 100,000=1.0\times 10^{5}m/s[/tex]
As, the decimal point is shifting to left side, thus the power of 5 is positive.
(07.05 MC)
An equation is shown below:
4x + 2(x – 3) = 4x + 2x – 11
Part A: Solve the equation and write the number of solutions. Show all the steps. (6 points)
Part B: Name one property you used to solve this equation. (4 points
Answers
4x + 2x - 6 = 4x + 2x - 11
6x - 6 = 6x - 11
6x - 6x = -11 + 6
0 = -5...incorrect......this has no solution...0 solutions
one property used was the distributive property :2(x - 3) = 2x - 6
Answer:
-6 = -11
Distributive Property
Step-by-step explanation:
A.
4x + 2(x – 3) = 4x + 2x – 11
4x + 2x - 6 = 4x + 2x - 11
6x - 6 = 6x - 11
-6 = -11
since -6 is not equal to -11, so there's no solution
B. Distributive property
The length of LN is 28 centimeters. What is the length of LP?
Answers
so LP = 1/2 * 28 = 14
Answer:
we need a diagram or somthing
Step-by-step explanation:
find the measure of a base angle of a isosceles triangle if the measure of the vertex angle is 100
Answers
Isosceles, two of them are equal: 100 + 2x =180, x = 40 degrees! <- That's the answer ;)
(Solve for r) 0.5r − 3.8 = 5.66
Answers
0.5r = 5.66 + 3.8
0.5r = 9.46...divide both sides by 0.5
r = 9.46 / 0.5
r = 18.92 <==
0.5r = 9.46
r = 9.46 ÷ 0.5
r = 18.92
r = 18/23/25
Write an equation for the line that is parallel to the given line and that passes through the given point.
y = 1/2 – 8; (–6, –17)
A.) y = 2x – 14
B.) y = 1/5x + 5/2
C.) y = -2x + 14
D.) y = 1/2x – 14
Answers
Evaluate S5 for 400 + 200 + 100 + … and select the correct answer below.
25
775
1,125
500
Answers
The sum of first five terms for the geometric series is 775. The correct answer is (B).
What is a geometric sequence?In a geometric sequence, the ratio of two consecutive terms are always equal.
The general expression for the nth term is given as aₙ = a₁ × rⁿ⁻¹ .
The sum of n terms for this sequence is given as Sₙ = (rⁿ - 1)/(r - 1)
The given series is 400 + 200 + 100 + …
It is a geometric series.
Its first term is a₁ = 400.
And, common ratio is r = 200/400 = 0.5.
Now, the expression for the sum of n terms is given as below,
Sₙ = a(1 - rⁿ)/(1 - r)
Then, substitute the value of a₁, a₅ and n = 5 to obtain,
S₅ = 400(1 - 0.5⁵)/(1 - 0.5)
⇒ S₅ = 775.
Hence, the sum of first five terms of the given series is 775.
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let n be the first 3 of consecutive even integers. what is the sum of those integers?
Answers
even integers are 2 apart
so the 3 integers are n,n+2,n+4
the sum is n+n+2+n+4=3n+6
the sum is 3n+6
I don't get it I got a different answer then these
Answers
So you multiply the first fraction of both numerator and denominator by 3, and the second by 2:
3(g-2) / 18 - 2(g+3) / 18 = (3g - 6 - 2g - 6) / 18 = (g-12)/18
= ---------------------
18
3g - 6 -2g - 6
= ---------------------
18
g - 12
= -----------
18
this is what i got
What is the value of the fourth term in a geometric sequence for which a1 = 30 and r = 1/2?
Answers
The value of the fourth term in the geometric sequence is 3.75.
Explanation:The value of the fourth term in a geometric sequence can be found using the formula:
an = a1 * r(n-1)
Given that a1 = 30 and r = 1/2, we can substitute these values into the formula and solve for a4:
a4 = 30 * (1/2)(4-1) = 30 * (1/2)3 = 30 * (1/8) = 3.75
Therefore, the value of the fourth term in the geometric sequence is 3.75.
Final answer:
The value of the fourth term in the geometric sequence is 3.75.
Explanation:
A geometric sequence is a sequence of numbers where each term is found by multiplying the previous term by a constant ratio. In this case, the first term (a1) is 30 and the common ratio (r) is 1/2. To find the fourth term, we can use the formula:
an = a1 * r(n-1)
Substituting the values given, we have:
a4 = 30 * (1/2)(4-1)
Simplifying this expression, we get:
a4 = 30 * (1/8) = 3.75
Therefore, the value of the fourth term in this geometric sequence is 3.75.
How to go from sum of products to product of sums?
Answers
Multiplication gives us distribution over the products, so
(a′+b+d′) (a′+b+c′+f′) = a′ (a′+b+c′+f′) + b (a′+b+c′+f′) + d′ (a′+b+c′+f′)
And then you can then distribute again each of the factors on the right.
Then you should simplify in any given number of ways. To take as an example, you have a′b and ba′, and since a′b + a′b = a′b + a′b = a′b, you can just drop one of them. Since bb = b, you can rewrite bb as b and etc.
So in the end part we should arrive at a sum of products. Then you can just invert. For example, if at the end you had:
p′ = a′b + bc′ + d′f ′+ a′f′
Then we would have
p = p′′ = (a′b + bc′ + d′f′ + a′f′)′ = (a′b)′⋅(bc′)′⋅(d′f′)′⋅(a′f′)′
Then applying De Morgan's laws to each of the factors, e.g., (a′b)′ = a+b′, so we would have
p = (a+b′)⋅(b′+c)⋅(d+f)⋅(a+f)
which is a product of sums.