Answer:
Step-by-step explanation:
To find the values of the given expressions using the function g(x) = 2x^2 - 2x + 9, we substitute the given values into the function and simplify the expression. Let's calculate each of the following:
a) g(0)
To find g(0), substitute x = 0 into the function:
g(0) = 2(0)^2 - 2(0) + 9
g(0) = 0 - 0 + 9
g(0) = 9
b) g(-1)
To find g(-1), substitute x = -1 into the function:
g(-1) = 2(-1)^2 - 2(-1) + 9
g(-1) = 2(1) + 2 + 9
g(-1) = 2 + 2 + 9
g(-1) = 13
c) g(2)
To find g(2), substitute x = 2 into the function:
g(2) = 2(2)^2 - 2(2) + 9
g(2) = 2(4) - 4 + 9
g(2) = 8 - 4 + 9
g(2) = 13
d) g(-x)
To find g(-x), substitute x = -x into the function:
g(-x) = 2(-x)^2 - 2(-x) + 9
g(-x) = 2x^2 + 2x + 9
e) g(1 - t)
To find g(1 - t), substitute x = 1 - t into the function:
g(1 - t) = 2(1 - t)^2 - 2(1 - t) + 9
g(1 - t) = 2(1 - 2t + t^2) - 2 + 2t + 9
g(1 - t) = 2 - 4t + 2t^2 - 2 + 2t + 9
g(1 - t) = 2t^2 - 2t + 9
Therefore:
a) g(0) = 9
b) g(-1) = 13
c) g(2) = 13
d) g(-x) = 2x^2 + 2x + 9
e) g(1 - t) = 2t^2 - 2t + 9
If is an acute angle and =35 then =?
Answer:
A acute angle is a angle that is less than 90*
Find the center of the ellipse defined by the equation shown below. If necessary, round to the nearest tenth. 100pts
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).
Please help me with this question
An estimate for the mean is 47.6 kg.
How to calculate the mean for the set of data?In Mathematics and Geometry, the mean for this set of data can be calculated by using the following formula:
Mean = [F(x)]/n
Cumulative frequency = 10 + 7 + 2 + 8 + 3
Cumulative frequency = 30
For the total number of data based on the frequency, we have;
Total weight, F(x) = 10(40) + 7(52.5) + 2(65) + 8(77.5) + 3(90)
Total weight, F(x) = 40 + 367.5 + 130 + 620 + 270
Total weight, F(x) = 1427.5
Now, we can calculate the mean weight as follows;
Mean = 1427.5/30
Mean = 47.6 kg.
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A cargo truck traveled 261 miles in 4 hours. About what speed was the truck averaging on this trip?
a.
65 mph
c.
55 mph
b.
1044 mph
Answer:
To find the average speed of the truck, we can divide the total distance travelled by the total time taken.
Average speed = Total distance / Total time
In this case, the truck travelled 261 miles in 4 hours.
Average speed = 261 miles / 4 hours
Average speed = 65.25 mph (rounded to two decimal places)
Therefore, the truck was averaging approximately 65 mph on this trip.
The correct option is (a) 65 mph.
Es el conjunto de deshielo determinante de la matriz x 2
5 7
Es igual a 4 cual es el valor de x
con un conjunto de deshielo determinante igual a 4, es x = 2.
Para determinar el valor de x en la matriz x 2
5 7
dado que el conjunto de deshielo determinante es igual a 4, necesitamos utilizar la propiedad de que el determinante de una matriz 2x2 se puede calcular utilizando la siguiente formula:
determinante = (a × d) - (b × c)
Donde a, b, c, y d son los elementos de la matriz.
En este caso, tenemos la matriz:
x 2
5 7
Aplicando la formula del determinante, podemos establecer la siguiente ecuacion:
( x × 7 ) - ( 2 × 5 ) = 4
Simplificando la ecuacion, obtenemos:
7x - 10 = 4
A continuacion, vamos a resolver la ecuacion para encontrar el valor de x:
7x = 4 + 10
7x = 14
Dividiendo ambos lados de la ecuacion por 7, obtenemos:
x = 2
Por lo tanto, el valor de x en la matriz x 2
5 7
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Suppose that ƒ is a function given as f(x) = 4x² + 5x + 3.
Simplify the expression f(x + h).
f(x + h)
Simplify the difference quotient,
ƒ(x + h) − ƒ(x)
h
=
Submit Question
The derivative of the function at x is the limit of the difference quotient as h approaches zero.
f(x+h)-f(x)
f'(x) =lim
h→0
h
ƒ(x + h) − f(x)
h
=
Answer:
f(x +h) = 4x² +4h² +8xh +5x +5h +3
(f(x+h) -f(x))/h = 4h +8x +5
f'(x) = 8x +5
Step-by-step explanation:
For f(x) = 4x² +5x +3, you want the simplified expression f(x+h), the difference quotient (f(x+h) -f(x))/h, and the value of that at h=0.
F(x+h)Put (x+h) where h is in the function, and simplify:
f(x+h) = 4(x+h)² +5(x+h) +3
= 4(x² +2xh +h²) +5x +5h +3
f(x +h) = 4x² +4h² +8xh +5x +5h +3
Difference quotientThe difference quotient is ...
(f(x+h) -f(x))/h = ((4x² +4h² +8xh +5x +5h +3) - (4x² +5x +3))/h
= (4h² +8xh +5h)/h
(f(x+h) -f(x))/h = 4h +8x +5
LimitWhen h=0, the value of this is ...
f'(x) = 4·0 +8x +5
f'(x) = 8x +5
__
Additional comment
Technically, the difference quotient is undefined at h=0, because h is in the denominator, and we cannot divide by 0. The limit as h→0 will be the value of the simplified rational expression that has h canceled from every term of the difference. This will always be the case for difference quotients for polynomial functions.
<95141404393>
what is the amplitude of the sinusoids graph?
y=2sin3x
Step-by-step explanation:
Y = 2 sin 3x '2' is the amplitude
( 'sin x' usually has amplitude of '1'...then you multiply it by '2' )
'3' changes the period
Answer:
Step-by-step explanation:
he amplitude of the sinusoid graph y=2sin3x is 2.
When a constant force is applied to an object, the acceleration of the object varies inversely with its mass. When a certain constant force acts upon an object
with mass 4 kg, the acceleration of the object is 9 m/s². When the same force acts upon another object, its acceleration is 6 m/s². What is the mass of this
object?
Step-by-step explanation:
a = k/m or ma = k
using 4 and 9 4* 9 = k = 36
then the equation becomes:
ma = 36
using a = 6
6 * m = 36 shows m = 6 kg
Francine currently has $55,000 in her 401k account at work, and plans to contribute $8,000 each year for the next 10 years. How much will she have in the account in 10 years, if the account averages a 4% annual return?
Answer:
Step-by-step explanation:
To calculate the future value of Francine's 401k account in 10 years, considering an annual contribution of $8,000 and an average annual return of 4%, we can use the formula for the future value of a series of regular payments, also known as an annuity.
The formula for the future value of an annuity is:
FV = P * [(1 + r)^n - 1] / r
Where:
FV is the future value
P is the payment amount
r is the interest rate per period
n is the number of periods
In this case:
P = $8,000 (annual contribution)
r = 4% or 0.04 (annual interest rate)
n = 10 (number of years)
Calculating the future value:
FV = $8,000 * [(1 + 0.04)^10 - 1] / 0.04
FV = $8,000 * (1.04^10 - 1) / 0.04
FV ≈ $8,000 * (1.480244 - 1) / 0.04
FV ≈ $8,000 * 0.480244 / 0.04
FV ≈ $8,000 * 12.0061
FV ≈ $96,048.80
Therefore, Francine will have approximately $96,048.80 in her 401k account in 10 years if the account averages a 4% annual return and she contributes $8,000 each year.
What is the third term of the sequence defined by the recursive rule f(1)=0.2,
f(n)=2fn(n-1)/5+1?
Answer:
f(3) = 1.432
Step-by-step explanation:
You want to know the third term of the sequence defined by ...
f(1) = 0.2f(n) = (2/5)f(n-1) +1SequenceThe terms of the sequence can be found one at a time by evaluating the recursive relation. The attached calculator output shows the first three terms are ...
f(1) = 0.2 . . . . . . . given
f(2) = 0.4(0.2) +1 = 1.08
f(3) = 0.4(1.08) +1 = 1.432
The third term of the sequence is 1.432.
__
Additional comment
The explicit form of the function is ...
f(n) = 5/3 -11/3(2/5)^n
Terms will asymptotically approach a value of 5/3.
<95141404393>
D
Drag the expressions to the correct locations on the image. Not all expressions will be used.
Consider this quotient.
(2³ - 8z + 6) ÷ (2² - 2x + 1)
Use long division to rewrite the quotient in an equivalent form as q(z) +
2³ - 8z + 6
2² - 2x + 1
where g(z) is the quotient, r(2) is the remainder, and b(z) is the divisor.
-5z + 4
Reset
H
Next
-11x + 12
x + 2
The solution to the polynomial division in quotient and remainder form is: (x + 2) + (-5x + 4)/(x² - 2x + 1)
How to carry out Long Division of Polynomials?The polynomials we want to divide are:
x³ - 8x + 6 by x² - 2x + 1 and as such we can write it as:
x + 2
x² - 2x + 1|x³ - 8x + 6
- x³ - 2x² + x
2x² - 9x + 6
- 2x² - 4x + 2
-5x + 4
Thus, the solution expressed in quotient and remainder form is:
(x + 2) + (-5x + 4)/(x² - 2x + 1)
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Seafloor rocks from a secret area above the Arctic Circle will recently analyzed by the globe. The company had a contract with spacegov.bids to test the rocks for nickel. Zieglow found that setting samples from the first location were composed of an average of 8.43% nickel. Six samples from the second location yielded an average of 7.81% nickel. What was the overall average nickel content of the rock samples
Answer:
The overall average nickel content of the rock samples is approximately 7.97%.
Step-by-step explanation:
To find the overall average nickel content of the rock samples, we need to take into account the number of samples from each location. Since we know the average nickel content of each set of samples, we can use a weighted average formula:
overall average nickel content = (total nickel content from first location + total nickel content from second location) / (total weight of samples from both locations)
To calculate the total nickel content from each location, we need to multiply the average nickel content by the number of samples:
total nickel content from first location = 8.43% x 1 sample = 8.43%
total nickel content from second location = 7.81% x 6 samples = 46.86%
To calculate the total weight of the samples from both locations, we need to add up the number of samples:
total weight of samples from both locations = 1 + 6 = 7
Now we can substitute these values into the formula and calculate the overall average nickel content:
overall average nickel content = (8.43% + 46.86%) / 7 ≈ 7.97%
Therefore, the overall average nickel content of the rock samples is approximately 7.97%.
I need help with 53 please
Answer:
53)
[tex]f(x) = \frac{7(x - 4)(x + 6)}{(x + 4)(x + 5)} [/tex]
Here is a unit circle with point P at (1, 0) Find the coordinates of P after the circle rotates the given amount counter clockwise around its center
1. 1/3 of a full rotation: ?
2 1/2 of a full rotation: ?
3. 2/3 of a full rotation: ?
1. 1/3 of a full rotation: The coordinates of point P after rotating 1/3 of a full rotation counterclockwise are approximately (0.5, 0.866).
2. 1/2 of a full rotation: The coordinates of point P after rotating 1/2 of a full rotation counterclockwise are (-1, 0).
3. 2/3 of a full rotation: The coordinates of point P after rotating 2/3 of a full rotation counterclockwise are approximately (-0.5, -0.866).
1/3 of a full rotation:
To find the coordinates of point P after rotating 1/3 of a full rotation counter clockwise, we need to determine the angle of rotation.
A full rotation around the unit circle is 360 degrees or 2π radians.
Since 1/3 of a full rotation is (1/3) [tex]\times[/tex] 360 degrees or (1/3) [tex]\times[/tex] 2π radians, we have:
Angle of rotation = (1/3) [tex]\times[/tex] 2π radians
Now, let's use the properties of the unit circle to find the new coordinates.
At the initial position, point P is located at (1, 0).
Rotating counterclockwise by an angle of (1/3) [tex]\times[/tex] 2π radians, we move along the circumference of the unit circle.
The new coordinates of point P after the rotation will be (cos(angle), sin(angle)).
Substituting the angle of rotation into the cosine and sine functions, we get:
New coordinates of P = (cos((1/3) [tex]\times[/tex] 2π), sin((1/3) [tex]\times[/tex] 2π))
Calculating the values:
cos((1/3) [tex]\times[/tex] 2π) ≈ 0.5
sin((1/3) [tex]\times[/tex] 2π) ≈ 0.866
Therefore, the coordinates of point P after rotating 1/3 of a full rotation counterclockwise are approximately (0.5, 0.866).
1/2 of a full rotation:
Following a similar process, when rotating 1/2 of a full rotation counterclockwise, we have an angle of (1/2) [tex]\times[/tex] 2π radians.
New coordinates of P = (cos((1/2) [tex]\times[/tex] 2π), sin((1/2) [tex]\times[/tex] 2π))
Calculating the values:
cos((1/2) [tex]\times[/tex] 2π) = cos(π) = -1
sin((1/2) [tex]\times[/tex] 2π) = sin(π) = 0
Therefore, the coordinates of point P after rotating 1/2 of a full rotation counterclockwise are (-1, 0).
2/3 of a full rotation:
For a rotation of 2/3 of a full rotation counterclockwise, the angle is (2/3) [tex]\times[/tex] 2π radians.
New coordinates of P = (cos((2/3) [tex]\times[/tex] 2π), sin((2/3) [tex]\times[/tex] 2π))
Calculating the values:
cos((2/3) [tex]\times[/tex] 2π) ≈ -0.5
sin((2/3) [tex]\times[/tex] 2π) ≈ -0.866
Therefore, the coordinates of point P after rotating 2/3 of a full rotation counterclockwise are approximately (-0.5, -0.866).
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Find the measure of the indicated arc.
T
56°
S
?
U
The measure of the intercepted arc SU in the circle is 112 degrees.
What is the measure of the arc length?An inscribed angle is simply an angle with its vertex on the circle and whose sides are chords.
The relationhip between an an inscribed angle and intercepted arc is expressed as:
Inscribed angle = 1/2 × intercepted arc.
From the diagram:
Inscribed angle = 56 degrees
Intercepted arc SU= ?
Plug the given value into the above formula and solve for the intercepted arc.
Inscribed angle = 1/2 × intercepted arc
56 = 1/2 × arc SU
Multiply both sides by 2:
56 × 2 = 1/2 × 2 × arc SU
112 = arc SU
Arc SU = 112°
Therefore, the intercepted arc measure 112 degrees.
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subtract (9x^2+4x) from ( -4x^2-7)
Answer: -13x² - 4x - 7
Step-by-step explanation:
We will subtract (9x² + 4x) from (-4x² - 7).
Given:
-4x² - 7 - (9x² + 4x)
Distribute the negative:
-4x² - 7 - 9x² - 4x
Reorder terms by degree:
-4x² - 9x² - 4x - 7
Combine like terms:
-13x² - 4x - 7
I need help with a question
The function for which f(x) is equal to f⁻¹(x) is: C. [tex]f(x) = \frac{x+1}{x-1}[/tex]
What is an inverse function?In this exercise, you are required to determine the inverse of the function f(x) with an equivalent inverse function f⁻¹(x). This ultimately implies that, we would have to swap (interchange) both the independent value (x-value) and dependent value (y-value) as follows;
[tex]f(x) = y = \frac{x+6}{x-6} \\\\x=\frac{y+6}{y-6}[/tex]
x(y - 6) = y + 6
y = xy - 6x - 6
f⁻¹(x) = (-6x - 6)/(x - 1) ⇒ Not equal.
Option B.
[tex]f(x) = y = \frac{x+2}{x-2} \\\\x=\frac{y+2}{y-2}[/tex]
x(y - 2) = y + 2
y = xy - 2x - 2
f⁻¹(x) = (-2x - 2)/(x - 1) ⇒ Not equal.
Option C.
[tex]f(x) = y = \frac{x+1}{x-1} \\\\x=\frac{y+1}{y-1}[/tex]
x(y - 1) = y + 1
y - xy = x + 1
f⁻¹(x) = (x + 1)/(x - 1) ⇒ equal.
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Which of the following functions is graphed below ?
Answer:
A) [tex]\displaystyle y=\left \{ {{x^3-4,\,x\leq 1} \atop {x^2-3,\,x > 1}} \right.[/tex]
Step-by-step explanation:
The first "piece" of the piecewise function, [tex]y=x^3-4[/tex], contains [tex]x=1[/tex] because of the closed dot there.
The second "piece" of the piecewise function, [tex]y=x^2-3[/tex], doesn't contain [tex]x=1[/tex] because of the open dot there.
What occurs between the two pieces is called a jump discontinuity.
Therefore, A is the correct answer.
Given just the graph what 3 steps are required to write the equation of a line?
Answer:
Step-by-step explanation:
step 1:
determining the values for standard form for the equation of a line,
y = mx + c
Step 2:
calculation of m, where m is the gradient or slope which determines how steep the line is.
step 3:
calculation of c, where c is the height at which the line crosses the y - axis also known as y - intercept
how can you write the expression with a rationalized denominator?
3 sqrt 2 / 3 sqrt 6
see photo attached for answers
The expression (3√2) / (3√6) with a rationalized denominator is 3√9 / 6. Option C is the correct answer.
To rationalize the denominator in the expression (3√2) / (3√6), we can multiply both the numerator and denominator by the conjugate of the denominator. The conjugate of √6 is -√6, so we multiply the expression by (-√6) / (-√6):
(3√2 / 3√6) * (-√6 / -√6)
This simplifies to:
-3√12 / (-3√36)
Further simplifying, we have:
-3√12 / (-3 * 6)
-3√12 / -18
Finally, we can cancel out the common factor of 3:
- 3√9 / - 6.
Simplifying further, we get:
3√9 / 6.
Option C is the correct answer.
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The school cafetteria recently served a new kind of snack to all the senior high school student. They want to know if more than 50% of the student like the newly served snack, thus, the cafeteria conducted a survey for asking 60 random selection of students whether they like (1), or Do not like (0), the new snack. They responses are show as follows
The cafeteria can conclude that a majority of the senior high school students like the newly served snack.
To determine if more than 50% of the students like the newly served snack, we need to analyze the responses of the 60 randomly selected students.
Analyzing the responses:
Out of the 60 students surveyed, we have:
- Number of students who responded with "1" (liking the snack): 32 students.
- Number of students who responded with "0" (not liking the snack): 28 students.
To determine the percentage of students who liked the snack, we divide the number of students who liked it by the total number of students surveyed and multiply by 100: (32/60) * 100 = 53.33%.
Since the percentage of students who liked the newly served snack is 53.33%, which is greater than 50%, we can conclude that more than 50% of the students like the snack based on the given survey results.
Therefore, the cafeteria can conclude that a majority of the senior high school students like the newly served snack.
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Find the equations of the asymptotes of the hyperbola defined by the equation shown below. If necessary, round to the nearest tenth. 100pts
The equations of the asymptotes of the hyperbola are y = (5/9)x - 79/9 and y = -(5/9)x + 79/9.
To find the equations of the asymptotes of the hyperbola defined by the equation:
[tex]-25x^2 + 81y^2 + 100x + 1134y + 1844 = 0[/tex]
We can rewrite the equation in the standard form by isolating the x and y terms:
[tex]-25x^2 + 100x + 81y^2 + 1134y + 1844 = 0[/tex]
Rearranging the terms:
[tex]-25x^2 + 100x + 81y^2 + 1134y = -1844[/tex]
Next, let's complete the square for both the x and y terms:
[tex]-25(x^2 - 4x) + 81(y^2 + 14y) = -1844\\-25(x^2 - 4x + 4 - 4) + 81(y^2 + 14y + 49 - 49) = -1844\\-25((x - 2)^2 - 4) + 81((y + 7)^2 - 49) = -1844[/tex]
Expanding and simplify
[tex]-25(x - 2)^2 + 100 - 81(y + 7)^2 + 3969 = -1844\\-25(x - 2)^2 - 81(y + 7)^2 = -1844 - 100 - 3969\\-25(x - 2)^2 - 81(y + 7)^2 = -4913[/tex]
Dividing both sides by -4913:
[tex](x - 2)^2/(-4913/25) - (y + 7)^2/(-4913/81) = 1[/tex]
Comparing this equation to the standard form of a hyperbola:
[tex](x - h)^2/a^2 - (y - k)^2/b^2 = 1[/tex]
We can determine that the center of the hyperbola is (h, k) = (2, -7). The value of [tex]a^2[/tex] is (-4913/25), and the value of [tex]b^2[/tex] is (-4913/81).
The equations of the asymptotes can be found using the formula:
y - k = ±(b/a)(x - h)
Substituting the values we found:
y + 7 = ±(√(-4913/81) / √(-4913/25))(x - 2)
Simplifying:
y + 7 = ±(√(4913) / √(81)) × √(25/4913) × (x - 2)
y + 7 = ±(√(4913) / 9) × √(25/4913) × (x - 2)
Rationalizing the denominators and simplifying:
y + 7 = ±(5/9) ×(x - 2)
Finally, rearranging the equation to isolate y:
y = ±(5/9)x - 10/9 - 7
Simplifying further:
y = ±(5/9)x - 79/9
In light of this, the equations for the hyperbola's asymptotes are y = (5/9)x - 79/9 and y = -(5/9)x + 79/9.
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Answer:
[tex]\boxed{y=\dfrac{5}{9}x-\dfrac{73}{9}}\;\; \textsf{and} \;\;\boxed{ y=-\dfrac{5}{9}x-\dfrac{53}{9}}[/tex]
Step-by-step explanation:
First, rewrite the given equation in the standard form of a hyperbola by completing the square.
Given equation:
[tex]-25x^2+81y^2+100x+1134y+1844=0[/tex]
Arrange the equation so all the terms with variables are on the left side and the constant is on the right side:
[tex]-25x^2+100x+81y^2+1134y=-1844[/tex]
Factor out the coefficient of the x² term and the coefficient of the y² term:
[tex]-25(x^2-4x)+81(y^2+14y)=-1844[/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]-25(x^2-4x+4)+81(y^2+14y+49)=-1844-25(4)+81(49)[/tex]
Factor the two perfect trinomials on the left side and simplify the right side:
[tex]-25(x-2)^2+81(y+7)^2=2025[/tex]
Divide both sides by the number of the right side so the right side equals 1:
[tex]\dfrac{-25(x-2)^2}{2025}+\dfrac{81(y+7)^2}{2025}=\dfrac{2025}{2025}[/tex]
[tex]\dfrac{-(x-2)^2}{81}+\dfrac{(y+7)^2}{25}=1[/tex]
[tex]\dfrac{(y+7)^2}{25}-\dfrac{(x-2)^2}{81}=1[/tex]
As the y²-term is positive, the hyperbola is vertical (opening up and down).
The standard equation of a vertical hyperbola is:
[tex]\boxed{\dfrac{(y-k)^2}{a^2}-\dfrac{(x-h)^2}{b^2}=1}[/tex]
Therefore, comparing this with the rewritten equation:
h = 2k = -7a² = 25 ⇒ a = 5b² = 81 ⇒ b = 9The formula for the equations of the asymptotes of a vertical hyperbola is:
[tex]\boxed{y=\pm \dfrac{a}{b}(x-h)+k}[/tex]
Substitute the values of h, k, a and b into the formula:
[tex]y=\pm \dfrac{5}{9}(x-2)-7[/tex]
Therefore, the equations for the asymptotes are:
[tex]\boxed{y=\dfrac{5}{9}x-\dfrac{73}{9}}\;\; \textsf{and} \;\;\boxed{ y=-\dfrac{5}{9}x-\dfrac{53}{9}}[/tex]
Find the center of the ellipse defined by the equation... 100 points
Answer:
(-4,4)
Step-by-step explanation:
You rewrite the terms:
(x + 4)^2 => [x - (-4)]^2
(y - 4)^2 => [y - (4)]^2
so h = -4 and k = 4
so center of ellipse is (h,k) or (-4,4)
Answer:
Center = (-4, 4)
Step-by-step explanation:
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]
The given equation is:
[tex]\dfrac{(x+4)^2}{25}+\dfrac{(y-4)^2}{9}=1[/tex]
Comparing the given equation with the standard form, we can see that h = -4 and k = 4. Therefore, the center (h, k) of the ellipse is (-4, 4).
The numbers on two consecutively numbered gym lockers have a sum of 129,
What are the locker numbers?
Answer:
64 and 65
Step-by-step explanation:
Let the two lockers be x and (x+1) since they are consecutive
The sum is 129 so,
x + (x+1) = 129
2x + 1 = 129
2x = 128
x = 64
x + 1 = 65
The locker numbers are 64 and 65
What is the square root of 184
The square root of 184 is approximately 13.5647. It is a non-repeating, non-terminating decimal.
The square root is obtained by finding the number that, when multiplied by itself, equals 184. In this case, 13.5647 multiplied by itself is approximately equal to 184. To explain the answer further, the square root is a mathematical operation that determines the value which, when multiplied by itself, gives the original number.
In the case of 184, the square root is an irrational number, meaning it cannot be expressed as a fraction or a terminating decimal. The approximate value of 13.5647 is derived using numerical methods or a calculator. This value represents the principal square root of 184, and it is positive since the square of a negative number would yield a positive result.
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Answer:
The square root of 184 is expressed as √184 in the radical form and as (184)½ or (184)0.5 in the exponent form. The square root of 184 rounded up to 5 decimal places is 13.56466. It is the positive solution of the equation x2 = 184. We can express the square root of 184 in its lowest radical form as 2 √46.
Square Root of 184: 13.564659966250536
Square Root of 184 in exponential form: (184)½ or (184)0.5
Square Root of 184 in radical form: √184 or 2 √46
please help- (in need of answer please don't put gibberish this is serious work)
Answer:
W = V/(LH)
Step-by-step explanation:
All we are doing is isolating W. Since V=LWH, then dividing both sides by LH will put W by itself on the right-hand side, you have V/(LH) = W as your equation
Assume a class has 26 members.
a. In how many ways can a president, a vice president, and a secretary be selected?
b. How many committees of 4 people can be chosen?
a. The number of ways to select a president, a vice president, and a secretary is
b. The number of ways to form a 4-person committee is
$0.
a. There are 15,600 ways to select a president, a vice president, and a secretary from a class of 26 members.
b. There are 14,950 ways to form a 4-person committee from a class of 26 members.
a. To select a president, a vice president, and a secretary from a class of 26 members, we can use the concept of permutations.
For the president position, we have 26 choices. After selecting the president, we have 25 choices remaining for the vice president position. Finally, for the secretary position, we have 24 choices left.
The total number of ways to select a president, a vice president, and a secretary is obtained by multiplying the number of choices for each position:
Number of ways = 26 * 25 * 24 = 15,600
Therefore, there are 15,600 ways to select a president, a vice president, and a secretary from a class of 26 members.
b. To form a 4-person committee from a class of 26 members, we can use the concept of combinations.
The number of ways to choose a committee of 4 people can be calculated using the formula for combinations:
Number of ways = C(n, r) = n! / (r!(n-r)!)
where n is the total number of members (26 in this case) and r is the number of people in the committee (4 in this case).
Plugging in the values, we have:
Number of ways = C(26, 4) = 26! / (4!(26-4)!)
Calculating this expression, we get:
Number of ways = 26! / (4! * 22!)
Using factorials, we simplify further:
Number of ways = (26 * 25 * 24 * 23) / (4 * 3 * 2 * 1) = 14,950
Therefore, there are 14,950 ways to form a 4-person committee from a class of 26 members.
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1.Lim as x approaches 0
(sin3x)/(2x-Sinx)
2. Lim as x approaches infinity
x^-1 lnx
3. Lim x approaches infinity
x/ e^x
Using L’Hospals rule for all
Lim as x approaches 0: (sin3x)/(2x-Sinx)
Taking the derivative of the numerator and denominator separately:
Numerator: d/dx(sin3x) = 3cos3x
Denominator: d/dx(2x - sinx) = 2 - cosx
Now, evaluate the limit using L'Hôpital's Rule:
Lim as x approaches 0: (3cos3x)/(2 - cosx)
Plugging in x = 0:
Lim as x approaches 0: (3cos(0))/(2 - cos(0))
= 3/2
Therefore, the limit as x approaches 0 of (sin3x)/(2x-Sinx) is 3/2.
Lim as x approaches infinity: x^-1 lnx
Taking the derivative of the numerator and denominator separately:
Numerator: d/dx(x^-1 lnx) = (1/x)lnx
Denominator: d/dx(1) = 0
Since the denominator is 0, we cannot apply L'Hôpital's Rule. However, we can still evaluate the limit:
Lim as x approaches infinity: x^-1 lnx
As x approaches infinity, the natural logarithm (lnx) grows without bound, so the overall limit is 0.
Therefore, the limit as x approaches infinity of x^-1 lnx is 0.
Lim x approaches infinity: x/ e^x
Taking the derivative of the numerator and denominator separately:
Numerator: d/dx(x) = 1
Denominator: d/dx(e^x) = e^x
Now, evaluate the limit using L'Hôpital's Rule:
Lim as x approaches infinity: 1/ e^x
As x approaches infinity, the exponential function e^x grows without bound, so the overall limit is 0.
Therefore, the limit as x approaches infinity of x/ e^x is 0.
A triangular pyramid has a base shaped like an equilateral triangle. The legs of the equilateral triangle are all 11 yards long, and the height of the equilateral triangle is 9.5 yards. The pyramid's slant height is 17 yards. What is its surface area?
The surface area of the triangular pyramid is approximately 331.93 square yards.
To find the surface area of the triangular pyramid, we need to calculate the areas of its individual components and then sum them up.
The triangular pyramid has a base shaped like an equilateral triangle. The legs of the equilateral triangle are all 11 yards long, and the height of the equilateral triangle is 9.5 yards. The formula to calculate the area of an equilateral triangle is:
Area = (√3/4) * [tex]side^2[/tex]
Plugging in the values, we get:
Area of the base equilateral triangle = (√3/4) * 11^2 ≈ 52.43 square yards
The triangular pyramid also has three triangular faces. Each face is an isosceles triangle, with two sides measuring 11 yards (same as the sides of the base equilateral triangle) and a slant height of 17 yards. We can use the formula for the area of an isosceles triangle:
Area = (1/2) * base * height
Since the base of the isosceles triangle is 11 yards and the height is 17 yards, the area of each triangular face is:
Area of each triangular face = (1/2) * 11 * 17 = 93.5 square yards
Now, we can calculate the total surface area of the triangular pyramid by summing up the areas of the base and the three triangular faces:
Surface area = Area of the base equilateral triangle + 3 * Area of each triangular face
Surface area = 52.43 + 3 * 93.5
Surface area ≈ 331.93 square yards
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Find the midpoint of WZ of WXYZ with the vertices W(0, 0), X(h, 0), Y(h,b), and Z(0, b).
(0, h/2)
(h/2, b/2)
(0, b/2)
(h/2, 0)
Third option is correct.The midpoint of WZ of WXYZ with the vertices W(0, 0), X(h, 0), Y(h,b), and Z(0, b) is (0, b/2).
To find the midpoint of segment WZ, we need to average the x-coordinates and the y-coordinates of the endpoints.
The coordinates of point W are (0, 0), and the coordinates of point Z are (0, b).
To find the x-coordinate of the midpoint, we average the x-coordinates of W and Z:
(x-coordinate of W + x-coordinate of Z) / 2 = (0 + 0) / 2 = 0 / 2 = 0
To find the y-coordinate of the midpoint, we average the y-coordinates of W and Z:
(y-coordinate of W + y-coordinate of Z) / 2 = (0 + b) / 2 = b / 2
Therefore, the midpoint of segment WZ is (0, b/2).
So, the correct answer is (0, b/2).
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