The equation of the hyperbola with directrices at x = ±2 and foci at (5, 0) and (−5, 0) is [tex]\frac{x^2}{10} + \frac{y^2}{15} = 1[/tex]
How to determine the equation of the hyperbola?The given parameters are:
Directrices at x = ±2 Foci at (5, 0) and (−5, 0)The foci of a hyperbola are represented as:
Foci = (k ± c, h)
The center is:
Center = (h,k)
And the directrix is:
Directrix, x = h ± a²/c
By comparison, we have:
k ± c = ±5
h = 0
h ± a²/c = ±2
Substitute h = 0 in h ± a²/c = ±2
0 ± a²/c = ±2
This gives
a²/c = 2
Multiply both sides by c
a² = 2c
k ± c = ±5 means that:
k ± c = 0 ± 5
By comparison, we have:
k = 0 and c = 5
Substitute c = 5 in a² = 2c
a² = 2 * 5
a² = 10
Next, we calculate b using:
b² = c² - a²
This gives
b² = 5² - 10
Evaluate
b² = 15
The hyperbola is represented as:
[tex]\frac{(x - k)^2}{a^2} + \frac{(y - h)^2}{b^2} = 1[/tex]
So, we have:
[tex]\frac{(x - 0)^2}{10} + \frac{(y - 0)^2}{15} = 1[/tex]
Evaluate
[tex]\frac{x^2}{10} + \frac{y^2}{15} = 1[/tex]
Hence, the equation of the hyperbola is [tex]\frac{x^2}{10} + \frac{y^2}{15} = 1[/tex]
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The Chang family is on their way home from a cross-country road trip. During the trip, the function D(t)=3260−55t can be used to model their distance, in miles, from home after t hours of driving.
part A:Find D(12) and interpret the meaning in the context of the problem.
part B: If D(t)=2490, find the value of t and interpret its meaning in the context of the problem.
Considering the given function, we have that:
a) D(12) = 2600, which means that after 12 hours of driving, the family is 2600 miles from home.
b) D(t) = 2490 when t = 14, which means that the family is 2490 miles from home after 14 hours of driving.
What is the function in this problem?The distance that the family is from home, after t hours of driving, is given by:
D(t) = 3260 - 55t.
Then, after 12 hours, the distance is given by D(12), that is:
D(12) = 3260 - 55 x 12 = 2600.
D(12) = 2600, which means that after 12 hours of driving, the family is 2600 miles from home.
D(t)=2490, hence:
3260 - 55t = 2490
55t = 770
t = 770/55
t = 14
D(t) = 2490 when t = 14, which means that the family is 2490 miles from home after 14 hours of driving.
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Find Median of 10,12,14 and 16
Step-by-step explanation:
the median is the value, where half of the other values is smaller (or equal), and the other half bigger (or equal).
with an even number of data points that is not directly possible to find.
in this case the 2 middle values are taken and their mean value is the median.
in our case, the 2 middle values are 12 and 14.
their mean value is
(12+14)/2 = 26/2 = 13
so,
13 is the median (even though if is not a data point itself).
Please help me with this
Answer:
10 percent
Step-by-step explanation:
look closl
it's 10 percent ...................... 10%
What is the sum of -26 and -37?
Answer:
-66
Step-by-step explanation:
add the numbers
-26 -37 = -66
[tex] - 26 - 37 = - 63[/tex]
Whenever the sign of the numbers is same , add them but don't forget the negative sign.
Please give brainliest
Determine if the following polygons are similar.
Problem 1
Triangle ABE is similar to triangle DCE. This is because we have two pairs of congruent corresponding angles.
angle AEB = angle DEC (vertical angles)angle ABE = angle DCE (alternate interior angles)Refer to the angle angle (AA) similarity theorem for more info.
-------------------
For the other pair of triangles, we don't have enough info to determine if they are similar or not. We have info about one pair of sides, and a pair of angles, but that's about it.
We need to know if JL is proportional to MO, and in the same ratio as KL & NO are, if we wanted to use the side angle side (SAS) similarity theorem.
-------------------
Summary:Triangle ABE is similar to triangle DCE by the AA (angle angle) similarity theoremWe don't have enough info to know if triangles JKL and MNO are similar or not.============================================================
Problem 2
Divide the corresponding sides
LN/PO = 30/15 = 2LN/OQ= 26/13 = 2ML/QP = 26/13 = 2We see the three ratios are equal to the same value (2) and that proves the triangles LNM and POQ are similar by the SSS (side side side) similarity theorem.
-------------------
Like before, we'll divide the corresponding sides
1st Rectangle: horizontal/vertical = 10/4 = 2.52ndRectangle: horizontal/vertical = 3/8 = 0.375We don't have a match. Let's try rotating the second rectangle so that the side '8' is horizontal, and the side '3' is vertical
1st Rectangle: horizontal/vertical = 10/4 = 2.52nd Rectangle: horizontal/vertical = 8/3 = 2.67 approximatelyWe don't have a match here either. There are other approaches we could take, but basically the two rectangles are not similar.
-------------------
Summary:Triangles LNM and POQ are similar by the SSS (side side side) similarity theorem.The rectangles are not similar.
What is the value of the expression 9+4n-1 when n = 3?
Make this into an expression. the product of four and a number, divided by seven
Answer:
4x ÷ 7
Step-by-step explanation:
the product of 4 and a number (x)
would be 4x. Then you would divide by 7.
What is the distance in units between (15,-17) and (-20,-5)
Answer:
41.34
Step-by-step explanation:
We can use the Pythagorean theorem to solve for the distance. First we will create an imaginary triangle with the leg lengths being the x distance and y distance between the 2 points and the hypotenuse being the distance between the 2 points.
x distance =
[tex]15 - (-20) = 35[/tex]
y distance =
[tex] |-17-5| = | - 22| = 22[/tex]
We can now use the Pythagorean theorem (a²+b²=c²) with our imaginary triangle
xdistance²+ydistance²=distance²
35²+22²=distance²
1709=distance²
[tex] \sqrt{1709} = distance[/tex]
In decimal form the distance would be around 41.34.
That is the answer
- Kan Academy Advance
Translate the shape A three squares right and three squares up.What are the coordinates of the vertices of the image?
Answer:
see explanation
Step-by-step explanation:
a translation of 3 squares right means add 3 to the x- coordinate
a translation of 3 squares up means add 3 to the y- coordinate
then
(1, 1 ) → (1 + 3, 1 + 3 ) → (4, 4 )
(3, 1 ) → (3 + 3, 1 + 3 ) → (6, 4 )
(1, 4 ) → (1 + 3, 4 + 3 ) → (4, 7 )
Considering the probability you found in part D, which option makes more sense for Jake to choose? Explain your reasoning.
The probability that can be chosen by Jake will be 33.5%.
How to calculate the probability?The area of the whole dartboard will be:
= πr²
where r will be:
= 3 + (4 × 4)
= 3 + 16
= 19
Area = π(19)² = 361π²
The area of 30, 40, and 50 points will be:
= π × [3 + (2 × 4)]²
= 121π²
The probability will be:
= (361π - 121π) × 100
= 33.5%
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three consecutive odd integers have a sum of 51. what is the smallest integer
Answer:
15
Step-by-step explanation:
Let 2n+1 be the first odd integer.
Therefore, the consecutive odd integers will be 2n+3 and 2n+5
If the sum of the consecutive odd integers is 51 then:
⇒ 2n + 1 + 2n + 3 + 2n + 5 = 51
⇒ 6n + 9 = 51
⇒ 6n = 42
⇒ n = 7
To find the values of the integers, substitute the found value of n into the expressions:
First integer: 2(7) + 1 = 15Second integer: 2(7) + 3 = 17Third integer: 2(7) + 5 = 19Therefore, the three consecutive odd integers that sum to 51 are 15, 17 and 19. The smallest of which is 15.
Find the value of x and y from the following figure.
Answer:
X=70
Step-by-step explanation:
aw we can for a triangle, where the 2 angles are complementary from the outer ones, so the 2 angles are 50, and 60
so the third one will be 70
Please help i dont know what to do :(
In AGHI, ZG=590 Zl=900, and h=67m.
What is the length off-to the nearest metre?t9i& t)
what are the least common denominators for 4 and 6
2y = x + 10
3y = 3x + 15
2y=x+10
3y=3x+15
2y-10=x
3y=3(2y-10)+15
2y-10=x
3y=6y-30+15
2y-10=x
0=3y+15
2y-10=x
3y=15
x=2y-10
y=5
2y=x+10
2×5=x+10
x=0
x=0, y=5
Which of the following is a solution made up by x + y=27 and y=2x?
Answer:
4. x = 9 and y = 18
Step-by-step explanation:
Hello there!
When given such a problem we have to methods to handle this:
substitution methodelimination method.On this problem, let's limit ourself to substitution method;
x + y = 27 ..................... eq.1
y = 2x ........................ eq.2
Eq.2 is already manipulated. Substitute it into eq.1
x + y = 27
=> x + 2x = 27
3x = 27
x = 9
Substitute x into eq.2
y = 2 (9)
= 18
In terms of x and y;
(x, y) => (9, 18)
I hope this helps you.
Have a nice time and nice studies.
Slope of (-4,-7) and (-2,-19)
Hi student, let me help you out! :)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
We are asked to find the slope of (-4, -7) and (-2, -19).
[tex]\triangle~\fbox{\bf{KEY:}}[/tex]
Use the slope formula.Here's the slope formula:
[tex]\star~\boxed{\mathrm{\cfrac{y2-y1}{x2-x1}}}[/tex]
Where
y2 and y1 are y-coordinates
x2 and x1 are x-coordinates
Input the values, which are:
-19: y2
-2: x2
-7: y1
-4: x1
[tex]\star~\mathrm{\cfrac{-19-(-7)}{-2-(-4)}}[/tex]
[tex]\star~\mathrm{\cfrac{-19+7}{-2+4}}[/tex]
Simplify!
[tex]\star~\mathrm{\cfrac{-12}{2}}[/tex]
[tex]\star~\mathrm{-6}[/tex]
Hope it helps you out! :D
Ask in comments if any queries arise.
#StudyWithBrainly
~Just a smiley person helping fellow students out :)
jamie is shopping for sports equipment. he purchases equipment worth 202 bout pays only 168. What is the discount amount?
Answer:
Discount amount = 34
Step-by-step explanation:
Discount amount = 202 - 168 = 34
WILL GIVE BRAINLIEST, ANSWER FAST PLEASE
Answer:
[tex]2( \frac{2}{5} x + 2) = \frac{4}{5} x + 4[/tex]
Step-by-step explanation:
Use the distributive property.
(x + 8/3)^2 + y^2 = 1 What is the center of the circle? what is the radius?
(x+8/3)²+y²=1
(x+8/3)²+y²+0²+2×0y=1
(x+8/3)²+(y+0)²=(V1)²
->center: C(8/3,0)
-> radius: V1
garcia is modeling the equation 5x
* BRAINLIEST IT WILL AFFECT MY GRADE!! *
Problem 5.2
Here are three lines that intersect at
one point.
Determine the values of x, y, and z
Answer:
z=98 degrees
x=64 degrees
y=18 degrees
Step-by-step explanation:
y is congruent to 18 since theyre opposite of eachother, same with x and 64. find z by using the straight 180 line. (180=64+18+z, so z=98)
Find the x-intercepts for the parabola defined
by this equation:
y = x² + 3x - 4
Write your answer as two ordered pairs
Answer:
(-4,1)
Step-by-step explanation:
Factoring, Guess and check.
can someone please explain how to factor a polynomial. {will mark as brainliest}
To factorize a polynomial you need to find the zeros, and then you can write the polynomial as a product of bynomials.
How to factor a polynomial?
For a polynomial of the form:
[tex]p(x) = A_n*x^n + A_{n-1}*x^{n-1} + ... + A_1*x + A_0[/tex]
We define the zeros as the values of x such that:
p(x) = 0.
On the above case, the degree of the polynomial is n (the maximum exponent). Then the number of zeros of the polynomial is n (same as the degree).
Once we know the n zeros, we can factorize the polynomial as:
[tex]p(x) = A_n*(x - x_1)*(x - x_2)*...*(x - x_n)[/tex]
For example, for the polynomial:
[tex]p(x) = 2x^2 - 8[/tex]
The two zeros are x = 2 and x = -2, then it can be rewriten as:
[tex]p(x) = 2*(x + 2)*(x - 2)[/tex]
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If f(x) = x² + 2, which interval describes the range of this function?
(1) (-∞, ∞)
(2) [0, ∞)
(3) [2, ∞)
(4) (-∞ ,2]
What is the median of the following data set?
{6, 3, 9, 1, 7}
A. 3
B. 6
C. 8
D. 9
The formula for the volume, V, of a cone having the radius, r, and the height, h, is shown below.
Write the formula to calculate the height, h.
The formula for calculating the height is h = 3V/πr²
Subject of the formulaThe formula given is the formula for calculating the volume of a cone
V = 1/3πr²h
where
h is the height
r is the radius
Make h the subject of the formula
Cross multiply
3V = πr²h
Divide both sides by πr²
3V/πr² = πr²h/πr²
h = 3V/πr²
Hence the formula for calculating the height is h = 3V/πr²
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Triangle DEF, with vertices D(9,-8), E(8,-3), and F(2,-5), is drawn inside a rectangle,
as shown below.
What is the area, in square units, of triangle DEF?
[
Answer: A=?
units²
The area of the triangle DEF is 2 square units
How to determine the area?The given parameters are:
D(9,-8), E(8,-3), and F(2,-5),
The area of the triangle is calculated using:
Area = 0.5|x1 * (y2 -y3) + x2 * (y1 -y3) + x3 * (y2 -y1) |
This gives
Area = 0.5|9 * (-3 + 5) + 8 * (-8 + 5) + 2 * (-3 + 8) |
Evaluate
Area = 0.5|4|
Expand
Area = 2
Hence, the area of the triangle DEF is 2 square units
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Coach palmer makes 9 quarts of iced tea. she fills 12 bottles with 2 cups of iced tea each to share with her team. how many quarts of iced tea are left?
The amount of iced tea left in quartz is 3 quartz
How to find amount of tea left in quartz?
The coach makes 9 quarts of iced tea.
she fills 12 bottles with 2 cups of iced tea each to share with her team.
1 quartz = 4 cups
Therefore, she shared the following amount of tea in cups:
amount shared = 12 × 2 = 24 cups
Then, let's convert to quartz,
4 cups = 1 quartz
24 cups = ?
amount shared in quartz = 24 / 4 = 6 quarts
Therefore,
the amount of ice tea left = 9 - 6 = 3 quartz
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1. You have a fair coin, flipping the coin has a 50% chance of showing a head. What is the probability of getting all heads when flip 10 times.
Answer:
1 / 1024
Step-by-step explanation:
Each flipping has 1/2 chance
In ten flippings:
[tex](\frac{1}{2} )^{10} =\frac{1^{10} }{2^{10} } =\frac{1}{1024}[/tex]
Hope this helps