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Type the correct answer in each box.
The equation of a hyperbola is x2 − 4y2 − 2x − 15 = 0.
The width the asymptote rectangle is units, and its height is units.
Answers
Answer:
The width the asymptote rectangle is 8 units
The height the asymptote rectangle is 4 units
Step-by-step explanation:
* Lets explain how to solve this problem
- The equation of the hyperbola is x² - 4y² - 2x + 16y - 31 = 0
- The standard form of the equation of hyperbola is
(x - h)²/a² - (y - k)²/b² = 1 where a > b
- The length of the transverse axis is 2a (the width of the rectangle)
- The length of the conjugate axis is 2b (the height of the rectangle)
- So lets collect x in a bracket and make it a completing square and
also collect y in a bracket and make it a completing square
∵ x² - 4y² - 2x - 15 = 0
∴ (x² - 2x) + (-4y²) - 15 = 0
- Take from the second bracket -4 as a common factor
∴ (x² - 2x) + -4(y²) - 15 = 0
∴ (x² - 2x) - 4(y²) - 15 = 0
- Lets make (x² - 2x) completing square
∵ √x² = x
∴ The 1st term in the bracket is x
∵ 2x ÷ 2 = x
∴ The product of the 1st term and the 2nd term is x
∵ The 1st term is x
∴ the second term = x ÷ x = 1
∴ The bracket is (x - 1)²
∵ (x - 1)² = (x² - 2x + 1)
∴ To complete the square add 1 to the bracket and subtract 1 out
the bracket to keep the equation as it
∴ (x² - 2x + 1) - 1 = (x - 1)² - 1
- Lets put the equation after making the completing square
∴ (x - 1)² - 1 - 4(y²) - 15 = 0 ⇒ simplify
∴ (x - 1)² - 4(y)² - 16 = 0 ⇒ add the two side by 16
∴ (x - 1)² - 4(y)² = 16 ⇒ divide both sides by 16
∴ (x - 1)²/16 - y²/4 = 1
∴ (x - 1)²/16 - y²/4 = 1
∴ The standard form of the equation of the hyperbola is
(x - 1)²/16 - y²/4 = 1
∵ The standard form of the equation of hyperbola is
(x - h)²/a² - (y - k)²/b² = 1
∴ a² = 16 and b² = 4
∴ a = 4 , b = 2
∵ The width the asymptote rectangle is 2a
∴ The width the asymptote rectangle = 2 × 4 = 8 units
∵ The height the asymptote rectangle is 2b
∴ The height the asymptote rectangle = 2 × 2 = 4 units
Answer:
[tex]w=8\\h=4[/tex]
Step-by-step explanation:
The given equation is
[tex]x^{2} -4y^{2}-2x-15=0[/tex]
First, we complete squares for each variable to find the explicit form of the hyperbola.
[tex]x^{2} -2x-4y^{2}=15\\x^{2} -2x+(\frac{2}{2} )^{2} -4y^{2} =15+1\\ (x-1)^{2}-4y^{2}=16\\\frac{(x-1)^{2} }{16} -\frac{4y^{2} }{16} =\frac{16}{16}\\\frac{(x-1)^{2} }{16} -\frac{y^{2} }{4}=1[/tex]
Now that we have the explicit form, you can observe that [tex]a^{2}=16 \implies a=4[/tex] and [tex]b^{2}=4 \implies b=2[/tex].
On the other hand, the width of the asymptote rectangle is [tex]2a[/tex] and the height is [tex]2b[/tex].
Therefore, the dimensions are 8 by 4.
[tex]2(4)=8\\2(2)=4[/tex]
Related Questions
The annual salary of each employee at an automobile plant was increased by 6% cost of living raise and then $2000 productivity raise. A) Write a function that transforms old annual salary, S, into the new one, N. B) state any transformations done on the old salary to get to new one.
Answers
Answer:
a) N = 1.06S +2000
b) the old salary is scaled by a factor of 1.06 and translated upward by 2000.
Step-by-step explanation:
a) a 6% raise means the new salary is 100% + 6% = 106% of the old one. A raise of an additional dollar amount simply adds to the scaled salary.
__
b) The translations are "math speak" for the English description of "increased by 6% and then raised by 2000". "Increased by 6%" means that .06 of the amount is added to the amount, effectively multiplying it by 1.06. "Raised by 2000" means 2000 is added.
The edge of a cube was found to be 15 cm with a possible error in measurement of 0.4 cm. Use differentials to estimate the maximum possible error, relative error, and percentage error in computing the volume of the cube and the surface area of the cube. (Round your answers to four decimal places.)
Answers
The maximum error on volume = 270cm³
The relative error on the volume =0.08
The percentage error on volume = 8%.
How to calculate the volume of a given cube?
To calculate the volume of a given cube, the following steps should be taken as follows:
Formula for volume of a cube = a³
where;
a = 15 cm
Volume(V) = 15³ = 3375cm³
The maximum error on volume(dV);
= 3×side²×dx
= 3×15²×0.4cm
= 270cm³
The relative error on the volume;
= dV/V
= 270/3375
= 0.08
The percentage error on volume;
=Relative error × 100
= 0.08× 100
= 8%
Of 1232 people who came into a blood bank to give blood, 397 people had high blood pressure. Estimate the probability that the next person who comes in to give blood will have high blood pressure. Round to three decimal places as needed.
A. 0.241
B. 0.29
C. 0.373
D. 0.322
Answers
Answer: D. 0.322
Step-by-step explanation:
Given : Of 1232 people who came into a blood bank to give blood, 397 people had high blood pressure.
Then, the probability that the next person who comes in to give blood will have high blood pressure will be :_
[tex]\dfrac{\text{People had high blood pressure}}{\text{Total people}}\\\\=\dfrac{397}{1232}\\\\=0.32224025974\approx0.322[/tex]
Hence, the estimated probability that the next person who comes in to give blood will have high blood pressure = 0.322
The probability that the next person who comes in to give blood will also have high blood pressure is 0.322, calculated by dividing the number of people with high blood pressure (397) by the total number of people (1232).
Explanation:The question is asking for the probability that the next person to come in and give blood will have high blood pressure based on previous data. We calculate probabilities by dividing the number of successful outcomes by the total number of outcomes. In this case, the successful outcome is a person having high blood pressure, and the total number of outcomes is the total number of people.
Simply divide the number of people with high blood pressure (397) by the total number of people (1232). Doing this gives: 397 / 1232 = approximately 0.322.
Therefore, the estimate of the probability that the next person who comes in to give blood will also have high blood pressure is 0.322.
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What is the solution to the system of equations shown below?
2x + y - 3z = 11
-x + 2y + 4z = -3
x - 5y + 2z = - 18
a. (-1, 1, -4)
b, (1, 3, -2)
c.(3, 8, 1)
d. (2, -3, 0)
Answers
Answer:
b. (1, 3, -2)
Step-by-step explanation:
A graphing calculator or scientific calculator can solve this system of equations for you, or you can use any of the usual methods: elimination, substitution, matrix methods, Cramer's rule.
It can also work well to try the offered choices in the given equations. Sometimes, it can work best to choose an equation other than the first one for this. The last equation here seems a good one for eliminating bad answers:
a: -1 -5(1) +2(-4) = -14 ≠ -18
b: 1 -5(3) +2(-2) = -18 . . . . potential choice
c: 3 -5(8) +2(1) = -35 ≠ -18
d: 2 -5(-3) +2(0) = 17 ≠ -18
This shows choice B as the only viable option. Further checking can be done to make sure that solution works in the other equations:
2(1) +(3) -3(-2) = 11 . . . . choice B works in equation 1
-(1) +2(3) +4(-2) = -3 . . . choice B works in equation 2
After substituting the given values from each option into the system of equations, (1, 3, -2) is found to satisfy all three equations. Hence, the correct option is b.
The solution to the system of equations:
2x + y - 3z = 11
-x + 2y + 4z = -3
x - 5y + 2z = -18
can be found using methods such as substitution, elimination, or matrix inversion. In this case, we can find the solution by either of these methods. To check which option is the solution, we can substitute the given (x, y, z) values from each option into the system of equations and see which one satisfies all three equations.
For option a: (-1, 1, -4)
2(-1) + 1 - 3(-4) = -2 + 1 + 12 = 11
-(-1) + 2(1) + 4(-4) = 1 + 2 - 16 = -13
(-1) - 5(1) + 2(-4) = -1 - 5 - 8 = -14
This option does not satisfy the second and third equations.
For option b: (1, 3, -2)
2(1) + 3 - 3(-2) = 2 + 3 + 6 = 11
-(1) + 2(3) + 4(-2) = -1 + 6 - 8 = -3
(1) - 5(3) + 2(-2) = 1 - 15 - 4 = -18
This option satisfies all three equations, hence, it is the correct solution.
The state education commission wants to estimate the fraction of tenth grade students that have reading skills at or below the eighth grade level. in an earlier study, the population proportion was estimated to be 0.23. how large a sample would be required in order to estimate the fraction of tenth graders reading at or below the eighth grade level at the 95% confidence level with an error of at most 0.03? round your answer up to the next integer.
Answers
Answer:
The sample size must be greater than or equal to 756
Step-by-step explanation:
The formula to calculate the error of the proportion is the following
[tex]E=z_{\alpha/2}*\sqrt{\frac{p(1-p)}{n}}[/tex]
where p is the proportion, n the sample size, E is the error and z is the z-score for a confidence level of 95%
For a confidence level of 95% [tex]z_{\alpha/2}=1.96[/tex]
We know that for this case [tex]p=0.23[/tex]
We require that the error be 0.03 as maximum
Therefore we solve for the variable n
[tex]z_{\alpha/2}*\sqrt{\frac{p(1-p)}{n}}\leq0.03\\\\1.96*\sqrt{\frac{0.23(1-0.23)}{n}}\leq0.03\\\\\sqrt{\frac{0.23(1-0.23)}{n}}\leq \frac{0.03}{1.96}\\\\(\sqrt{\frac{0.23(1-0.23)}{n}})^2\leq (\frac{0.03}{1.96})^2\\\\\frac{0.23(1-0.23)}{n}\leq (\frac{0.03}{1.96})^2\\\\\frac{0.23(1-0.23)}{(\frac{0.03}{1.96})^2}\leq n\\\\n\geq\frac{0.23(1-0.23)}{(\frac{0.03}{1.96})^2}\\\\n\geq756[/tex]
Answer:
1305
Step-by-step explanation:
Two long conducting cylindrical shells are coaxial and have radii of 20 mm and 80 mm. The electric potential of the inner conductor, with respect to the outer conductor, is +600 V. An electron is released from rest at the surface of the outer conductor. What is the speed of the electron as it reaches the inner conductor?
Answers
Answer:
v = 1.45 × 10⁷ m/s
Step-by-step explanation:
Given:
Inner radius of the cylinder, r₁ = 20 mm = 0.2 m
outer radius of the cylinder, r₂ = 80 mm = 0.8 m
Potential difference, ΔV = 600V
Now, the work done (W) in bringing the charge in to the inner conductor
W = [tex]\frac{1}{2}mv^2[/tex]
where, m is the mass of the electron = 9.1 × 10⁻³¹ kg
v is the velocity of the electron
also,
W = qΔV
where,
q is the charge of the electron = 1.6 × 10⁻¹⁹ C
equating the values of work done and substituting the respective values
we get,
qΔV = [tex]\frac{1}{2}mv^2[/tex]
or
1.6 × 10⁻¹⁹ × 600 = [tex]\frac{1}{2}\times 9.1\times 10^{-31}v^2[/tex]
or
[tex]v = \sqrt\frac{2\times 600\times 1.6\times 10^{-19}}{9.1\times 10^{-31}}[/tex]
or
v = 14525460.78 m/s
or
v = 1.45 × 10⁷ m/s
Final answer:
The speed of the electron as it reaches the inner conductor is calculated using conservation of energy, giving a final speed of approximately 1.46 × 107 m/s.
Explanation:
Electron Velocity in Coaxial Conductors
An electron released from the outer conductor will be accelerated towards the higher potential inner conductor due to the electric field between them. To calculate the speed of the electron as it reaches the inner conductor, we use the concept of conservation of energy. The electrical potential energy lost by the electron as it moves from the outer to the inner conductor is converted into kinetic energy.
The initial potential energy (Ui) of the electron can be given by:
Ui = qVwhere q is the charge of the electron (q = -1.6 × 10-19 C) and V is the potential difference (V = 600 V).
The final kinetic energy (Kf) when the electron reaches the inner conductor is:
Kf = ½ [tex]mv^2[/tex]where m is the mass of the electron (m = 9.11 × 10-31 kg) and v is the final speed we want to find.
Using conservation of energy (Ui + Ki = Kf + Uf and noting that both the initial kinetic energy Ki and the final potential energy Uf are zero), we get:
qV = ½ [tex]mv^2[/tex](-1.6 × 10-19 C)(600 V) = ½ (9.11 × 10-31kg)[tex]v^2[/tex]Solving for v gives us the final speed of the electron:
v = √[(2qV)/m]v = √[(2(-1.6 × 10-19 C)(600 V))/(9.11 × 10-31kg)]v = 1.46 × 107 m/sThis is the speed of the electron when it reaches the inner conductor.
Choose the system of inequalities whose solution is represented by the graph.
Answers
Answer:
-x + y > -4; x + y < 3 . . . . . last choice
Step-by-step explanation:
The boundary lines are both dashed, so there will be no "or equal to" as part of the inequality symbols (eliminates the second choice).
The downward sloping line has x- and y-intercepts that are both 3, so it will have the equation in intercept form ...
x/3 + y/3 = 1
Multiplying by 3 gives x+y=3. The shading is below it, so the inequality with that line as the boundary is ...
x + y < 3
This inequality is only part of the last choice.
__
The upward sloping line has x- and y- intercepts of 4 and -4, so its equation in intercept form is ...
x/4 + y/-4 = 1
Multiplying by -4 gives -x+y=-4. The shading is above it, so the inequality with that boundary line is ...
-x + y > -4
This inequality is included in the last choice.
The graph represents a system of inequalities that define a region satisfying all inequalities simultaneously. The relationship between variables on the axes gives the inequalities' data points, and quadratic equations' solutions provide boundaries if they're relevant and positive.
Explanation:To identify the system of inequalities that the graph represents, you need to consider the relationships between the variables represented on the x-axis and the y-axis. This is an exercise in two-dimensional graphing. The values on the x-axis (independent variable) and the y-axis (dependent variable) provide the data points for the inequalities.
It's essential to note that in a system of inequalities, the solution is the region that satisfies all of the inequalities simultaneously. Depending on the inequality, the graphical representation could either be above or below a certain line, or within a particular region of the graph.
Quadratic equations sometimes provide a boundary for these inequalities, particularly when we are only interested in the real and positive root solutions. So, considering these aspects, it's possible to define a system of inequalities that match the graph provided.
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Use the system of equations to answer the questions. 2x + 3y = 3 y = 8 – 3x The value of y from the second equation is substituted back into the first equation. What is the resulting equation? What is the value of x? What is the value of y?
Answers
Answer:
2x +3(8 -3x) = 3x = 3y = -1Step-by-step explanation:
The second equation tells you ...
y = 8 -3x
Using this expression in the first equation gives you ...
2x +3(8 -3x) = 3
2x +24 -9x = 3 . . . . . eliminate parentheses
21 = 7x . . . . . . . . . . . add 7x -3
3 = x . . . . . . . . . . . . . . divide by 7
y = 8 -3×3 = -1 . . . . . . use the second equation to find y
The solution is (x, y) = (3, -1).
Answer:
the correct answers for edu are 2x+3(8-3x)=3 than 3 and last -1
Step-by-step explanation:
A mixture contains forty ounces of glycol and water and is ten percent glycol. If the mixture is to be strengthened to twenty-five percent, how much glycol is to be added?
Answers
Answer:
40oz glycol and water
10% is glycol
4oz is glycol
36oz is water
if glycol is to be added to make glycol 25% of all then
note: water does not change
100-25=75
water does not change so
36oz=75%
12=25%
there should be 12 oz of glycol total
4 now
12-4=8
8 oz should be added
sorry, I just wrote what I was thinking
answer is 8oz
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Step-by-step explanation:
i rlly hate math, pweaseeee help me asap!!!
Answers
Answer:
See explanation.
Step-by-step explanation:
Part C)
[tex]5+(-10)[/tex]
When adding a positive to a negative, the result will be the sign of the larger absolute value. In this case, 10 is greater than 5, so the result will be negative.
Adding a negative is also the same thing as subtracting a positive.
[tex]5+(-10)\\5-10\\-5[/tex]
For your second problem, 5 is larger than 3, so the result will be positive.
[tex]5+(-3)\\5-3\\2[/tex]
-----
When adding a positive to another positive, the result will be positive.
[tex]5+3\\8[/tex]
When adding a number to zero, the answer will always be the number.
[tex]5+0\\5[/tex]
Part D)
When multiplying a positive times a negative, the result will always be negative.
[tex]5*(-10)\\-50[/tex]
[tex]5*(-3)\\-15[/tex]
-----
When multiplying a positive times a positive, the result will always be positive.
[tex]5*10\\50[/tex]
When multiplying a positive or a negative times zero, the result will always be zero.
[tex]5*0\\0[/tex]
Which expression is a sum of cubes?
Answers
Answer:
-27a³b⁶+ 8a⁹b¹²
Step-by-step explanation:
In the expression above -27 is a perfect cube of -3, 8 is a perfect cube of 2.
The exponents of a and b in both terms in the expression are divisible by 3.
The cube root of x, that is ∛xⁿ=x∧(n/3) where n is an integer.
ANSWER
[tex]- 27 {a}^{3} {b}^{6} + 8 {a}^{9} {b}^{12} [/tex]
EXPLANATION
When we can write an expression in the form
[tex] {(x)}^{3} + {(y)}^{3} [/tex]
then it is a sum of cubes.
To write a given sum as sum of cubes, then the coefficients of the terms should cube be numbers and the exponents of any power should be a multiple of 3.
This tells us that the first option will be the best choice.
[tex] - 27 {a}^{3} {b}^{6} + 8 {a}^{9} {b}^{12} [/tex]
We can rewrite this as:
[tex] { (- 3)}^{3} {a}^{3} {b}^{2 \times 3} + {2}^{3} {a}^{3 \times 3} {b}^{4 \times 3} [/tex]
We apply this property of exponents:
[tex] ({a}^{m} )^{n} = {a}^{mn} [/tex]
This gives us
[tex] {( - 3a {b}^{2}) }^{3} + {(2{a}^{3} {b}^{4} })^{3} [/tex]
Therefore the correct option is A
Can someone help me with this math question
Answers
Answer:
The coordinates of D' are (1,-1)
Step-by-step explanation:
The point D in the figure has co-ordinates (2,-2) as shown in the figure.
The figure is dilated by a factor of 1/2
So, multiply the coordinates of D (2,-2) by 1/2
D' = (1/2*2, 1/2*-2)
D' = (1,-1)
So, the coordinates of D' are (1,-1)
Louis kicked a football during the opening play of a high school football game. Which type of function could model the height of the football after the kick?
Answers
Answer:
Ballistic motion is usually modeled by a quadratic function.
Step-by-step explanation:
The usual assumption is that the only force acting on the object is that due to gravity, and that it is constant and directed downward. With this assumption, along with the assumption of a flat Earth, the resulting model is a downward-opening quadratic function.
Answer:
quadratic function.
Step-by-step explanation: "
ballistic motion is modeled with a quadratic function"
A boater travels 532 miles. Assuming the boat averages 6.3 miles per gallon, how many gallons of gasoline(to the nearest then of gallon) were used? plz show work
Answers
Answer:
84.4 gallons to the nearest tenth.
Step-by-step explanation:
Average usage = miles travelled / gallons used so:
6.3 = 532 / gallons used
Gallons used = 532 / 6.3
= 84.44.
Final answer:
To find the gallons of gasoline used, divide the total miles (532) by the average miles per gallon (6.3). This calculation results in approximately 84.4444 gallons, which can be rounded to 84.4 gallons of gasoline used.
Explanation:
To calculate the amount of gasoline used by the boater who traveled 532 miles averaging 6.3 miles per gallon, you need to divide the total miles traveled by the average miles per gallon. The formula to use is:
Gallons used = Total miles traveled ÷ Average miles per gallon
Plugging in the values given:
Gallons used = 532 miles ÷ 6.3 miles/gallon
This gives us:
Gallons used = 84.4444... gallons
To round to the nearest tenth of a gallon, we would round 84.4444... to 84.4 gallons. Thus, the boater used approximately 84.4 gallons of gasoline.
PLEASE HELP ME WITH THIS MATH PROBLEM
Answers
Answer:
22π
Step-by-step explanation:
The area formula for a sector is
[tex]A=\frac{\theta }{360}*\pi r^2[/tex]
The angle theta is to be the angle that is a part of the sector for which we are trying to find the area. If we are looking for the area of the larger sector, we are not using 140 as our angle theta, we are using 360 - 140 = 220 as our angle theta since that is the angle for the larger of the 2 sectors. Filling in our formula using r = 6:
[tex]A=\frac{220}{360}*36\pi[/tex]
The easiest way to handle this math is to multiply the 220 by the 36, hit enter on your calculator, then divide that product by 360. When you do that your answer, in terms of pi, is 22π
19. Solve sin O+ 1 = cos 20 on the interval 0≤x < 2xpi
Answers
Answer:
[tex]\theta=\frac{\pi}{2},\frac{3\pi}{2},\frac{2\pi}{3},\frac{4\pi}{3}[/tex]
Step-by-step explanation:
If I'm interpreting that correctly, you are trying to solve this equation:
[tex]sin(\theta )+1=cos(2\theta)[/tex]
for theta. To do this, you will need a trig identity sheet (I'm assuming you got one from class) and a unit circle (ditto on the class thing).
We need to solve for theta. If I look to my trig identities, I will see a double angle one there that says:
[tex]cos(2\theta)=1-2sin^2(\theta)[/tex]
We will make that replacement, then we will have everything in terms of sin.
[tex]sin(\theta)+1=1-2sin^2(\theta)[/tex]
Now get everything on one side of the equals sign to solve for theta:
[tex]2sin^2(\theta)+sin(\theta)=0[/tex]
We can factor out the common sin(theta):
[tex]sin\theta(2sin\theta+1)=0[/tex]
By the Zero Product Property, either
[tex]sin\theta=0[/tex] or
[tex]2sin\theta+1=0[/tex]
Now look at your unit circle and find that the values of theta where the sin is 0 are located at:
[tex]\theta=\frac{\pi }{2},\frac{3\pi}{2}[/tex]
The next one we have to solve for theta:
[tex]2sin\theta+1=0[/tex] simplifies to
[tex]2sin\theta=-1[/tex] and
[tex]sin\theta=-\frac{1}{2}[/tex]
Look at the unit circle again to find the values of theta where the sin is -1/2:
[tex]\theta=\frac{2\pi}{3},\frac{4\pi}{3}[/tex]
Those ar your values of theta!
Write an equation that could be used to find the measure of angle A
Answers
Answer: C
Step-by-step explanation: The numerator is the angle measure, and the denominator is the side length. For angle B, the angle is 47 degrees. The opposite side is b, which is 85. We are finding the angle A, which is the numerator. The side 94 is opposite of angle A.
You are hiking and are trying to determine how far away the nearest cabin is, which happens to be due north from your current position. Your friend walks 200 yards due west from your position and takes a bearing on the cabin of N 30.7°E. How far are you from the cabin?
Answers
Answer:
336.7 yards away from the cabin....
Step-by-step explanation:
The angle 30.7° is also the angle of the upper interior angle of the triangle (near the cabin)
Use the tan function:
opposite = 200 yards
adjacent = x
tan(30.7°) = (opposite / adjacent)
tan(30.7°) = 200 yards/x
x * tan(30.7°) = 200 yards
x = 200 yards/ tan(30.7°)
x= 200/ 0.594
x = 336.7 yards.
336.7 yards away from the cabin....
Answer: 337
Step-by-step explanation: you have to round up
Determine the horizontal, vertical, and slant asymptotes: y=x2+2x-3/x-7
Answers
Answer:
vertical asymptote: x = 7slant asymptote: y = x+9Step-by-step explanation:
The vertical asymptotes are found where a denominator factor is zero (and there is no corresponding numerator factor to cancel it). Here, that is at x = 7.
There is no horizontal asymptote because the numerator is of higher degree than the denominator.
When you divide the numerator by the denominator, you get ...
y = (x +9) +60/(x -7)
Then when x gets large, the behavior is governed by the terms not having a denominator: y = x +9. This is the equation of the slant asymptote.
The function y = (x^2 + 2x - 3) / (x - 7) has a vertical asymptote at x = 7 and a slant asymptote at y = x. It does not have a horizontal asymptote.
Explanation:
The function given is y = (x^2 + 2x - 3) / (x - 7). When identifying the asymptotes, we need to consider three types: horizontal, vertical, and slant.
For vertical asymptotes, we look for values of x that make the denominator of the function zero. In this case, x = 7 is a vertical asymptote.
For horizontal or slant asymptotes, we compare the degree of the polynomial in the numerator with the degree of the polynomial in the denominator. Since the degree of the numerator (which is 2) is larger than the degree of the denominator (which is 1), there's no horizontal asymptote. However, we have a slant asymptote which is determined by long division of the polynomials or using synthetic division.
The slant asymptote is therefore y = x.
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Help please!
What is the magnitude of the position vector whose terminal point is (-2, 4)?
Answers
sqrt(4+16)=sqrt 20 = 2*sqrt5
hat is the value of the car two years after Fred buys it? Explain how you got your answer or show your work.
When will the car be worth less than $500,000? Explain or show your work
Answers
Answer:
Step-by-step explanation:
Since we don't have the information we need to fit this into an exponential equation, we will do it the simple way, using recursion. We will take the value of the car and subtract from it the depreciation, using the value at the end of one calculation as the initial value of the one following. Here's where it gets tricky, though. If the car's value at the end of a calculation is 80% of its initial value, then it depreciates 20%. Here is what the equation looks like after one year:
2,700,000 - .2(2,700,000) = 2,160,000
We subtracted away 20% of the initial to get the new initial. Now we use that value in the next recursion.
2,160,000 - .2(2,160,000) = 1,728,000
This is the value of the car after 2 years.
If we continue this process, we would find that after the 8th year, the car's value drops below 500,000 (namely, $452,984.83)
Find the value of x. Then find the measure of each labeled angle.
x = 110; the labeled angles are 110º and 70º.
x = 70; the labeled angles are 110º and 70º.
x = 20; the labeled angles are 160º and 20º.
x = 20; the labeled angles are 60º and 20º.
Answers
Answer:
x = 70; the labeled angles are 110º and 70º.
Step-by-step explanation:
The full degrees will add up to 180º. There are 2x and +40.
Equation: [tex]2x+40=180[/tex]
Solve: [tex]2x+40=180\\\\180-40=140\\\\2x=140\\\\x=70[/tex]
Since x=70, 70+40 will equal to 110.
Therefore, "x = 70; the labeled angles are 110º and 70º." is correct.
A rectangular prism with a volume of 666 cubic units is filled with cubes with side lengths of \dfrac12 2 1 ? start fraction, 1, divided by, 2, end fraction unit. How many \dfrac12 2 1 ? start fraction, 1, divided by, 2, end fraction unit cubes does it take to fill the prism?
Answers
Answer:
[tex]48\ cubes[/tex]
Step-by-step explanation:
we know that
The volume of the rectangular prism is equal to
[tex]V=6\ unit^{3}[/tex]
step 1
Find the volume of one cube
The volume of the cube is equal to
[tex]V=b^{3}[/tex]
where
b is the side length of the cube
we have
[tex]b=\frac{1}{2}\ unit[/tex]
substitute
[tex]V=(\frac{1}{2})^{3}[/tex]
[tex]V=\frac{1}{8}\ unit^{3}[/tex]
step 2
To find out the number of cubes needed to fill the prism, divide the volume of the rectangular prism by the volume of one cube
so
[tex]6/(1/8)=48\ cubes[/tex]
Find the coordinates of the orthocenter of ΔYAB that has vertices at Y(3, –2), A(3, 5), and B(9, 1). (JUSTIFY)
Answers
Answer:
So (5,1) is the orthocenter.
Step-by-step explanation:
So we have to find the slopes of all three lines in burgundy (the line segments of the triangle). We also need to find the equations for the altitudes with respect from all sides of the triangle (we are looking for perpendicular lines).
The vertical line there is just going to be x=a number so that line is x=3 because all the points on that line are of the form (3,y). x=3 says we don't care what y is but x will always be 3. So the line for AY is x=3.
So the altitude of the triangle with respect to that side (that line segment) would be a line that is perpendicular to is which would be a horizontal line y=1. I got y=1 because it goes through vertex B(9,1) and y=1 is perpendicular to x=3.
So we now need to find the equations of the other 2 lines.
One line has points A(3,5) and B(9,1).
To find the slope, you may use [tex]\frac{y_2-y_1}{x_2-x_1}[/tex].
Or you could just line up the points vertically and subtract then put 2nd difference over 1st difference.
Like this:
(9 , 1)
-(3 ,5)
----------
6 -4
So the slope is -4/6=-2/3.
So a line that is perpendicular will have opposite reciprocal slope. That means we are looking for a line with 3/2 as the slope. We want this line from segment AB going to opposite point Y so this line contains point (3,-2).
Point slope-form is
[tex]y-y_1=m(x-x_1)[/tex] where [tex](x_1,y_1)[/tex] is a point on the line and [tex]m[/tex] is the slope.
So the line is:
[tex]y-(-2)=\frac{3}{2}(x-3)[/tex]
[tex]y+2=\fac{3}{2}x-\frac{9}{2}[/tex]
Subtract 2 on both sides:
[tex]y=\frac{3}{2}x-\frac{9}{2}-2[/tex]
Simplify:
[tex]y=\frac{3}{2}x-\frac{13}{2}[/tex].
Let's find the the third line but two lines is plenty, really. The othorcenter is where that perpendicular lines will intersect.
Now time for the third line.
BY has points (9,1) and (3,-2).
The slope can be found by lining up the points vertically and subtracting, then put 2nd difference over 1st difference:
(9 ,1)
-(3,-2)
---------
6 3
So the slope is 3/6=1/2.
A perpendicular line will have opposite reciprocal slope. So the perpendicular line will have a slope of -2.
We want this line segment to go through A(3,5).
We are going to use point-slope form:
[tex]y-5=-2(x-3)[/tex]
Add 5 on both sides:
[tex]y=-2(x-3)+5[/tex]
Distribute:
[tex]y=-2x+6+5[/tex]
Combine like terms:
[tex]y=-2x+11[/tex]
So the equation of the 3rd altitude line is y=-2x+11.
So the equations we want to find the intersection to is:
y=(3/2)x-(13/2)
y=1
y=-2x+11
I like the bottom two equations so I'm going to start there and then use my third line to check some of my work.
y=1
y=-2x+11
Replacing 2nd y with 1 since y=1:
1=-2x+11
Subtract 11 on both sides:
1-11=-2x
Simplify:
-10=-2x
Divide both sides by -2:
5=x
The point of intersection between y=1 and y=-2x+11 is (5,1).
Let's see if (5,1) is on that third line.
y=(3/2)x-(13/2)
1=(3/2)(5)-(13/2)
1=(15/2)-(13/2)
1=(2/2)
1=1
So (5,1) is the intersection of all three lines.
So (5,1) is the orthocenter.
The product shown is a difference of squares. What is the missing constant term in the second factor?(–5x – 3)(–5x + )
Answers
Answer:
3
Step-by-step explanation:
the missing no is 3
I have answered ur question
Answer:
3
Step-by-step explanation:
The regular octagon has a perimeter of 122.4 cm. Which statements about the octagon are true? Check all that apply.
Answers
Answer:
The length of segment XY can be found by solving for a in
[tex]20^2-7.65^2=a^2[/tex]
The measure of the central angle [tex]\angle ZXW[/tex] is [tex]45\degree[/tex].
Step-by-step explanation:
If the regular octagon has a perimeter of 122.4cm, then each side is [tex]\frac{122.4}{8}=15.3cm[/tex]
The measure of each central angle is [tex]\frac{360\degree}{8}=45\degree[/tex]
The angle between the apothem and the radius is [tex]\frac{45}{2}=22.5\degree[/tex]
The segment XY=a is the height of the right isosceles triangle.
We can use the Pythagoras Theorem with right triangle XYZ to get:
[tex]a^2+7.65^2=20^2[/tex]
[tex]a^2=20^2-7.65^2[/tex]
Therefore, the correct options are:
The length of segment XY can be found by solving for a in
[tex]20^2-7.65^2=a^2[/tex]
The measure of the central angle [tex]\angle ZXW[/tex] is [tex]45\degree[/tex].
Answer:3rd and 4th anwser
What’s the length of BD ?
Answers
Answer:
BD = 4.99
Step-by-step explanation:
You can simply use the trigonometric identity tangent to solve for length BD.
Tan = opposite/adjacent
In this case we have,
Tan 31 = 3/BD
BD = 3/Tan 31
BD = 4.99
Since tan = oposite/adjacent we can plug in the values we know.
Solve 3x^2 + x + 10 = 0 round solutions to the nearest hundredth
A. X= -2.83 and x=0.83
B. No real solutions
C. X= -2.01 and x= 1.67
D. X= -1.67 and x=2.01
Answers
Answer:
C. X= -2.01 and x= 1.67
Step-by-step explanation:
[tex]3x {}^{2} + x + 10 = 0 \\ 3x {}^{2} + 6x - 5x + 10 = 0 \\ 3x(x + 2) - 5(x + 2) \\ (x + 2)(3x - 5 )\\ x + 2 = 0 \: \: or \: \: 3x - 5 = 0 \\ x = - 2 \: \: or \: \: x = \frac{5}{3} [/tex]
ANSWER
B. No real solutions
EXPLANATION
The given equation is
[tex]3 {x}^{2} + x + 10 = 0[/tex]
By comparing to
[tex]a {x}^{2} + bx + c= 0[/tex]
We have a=3,b=1 and c=10.
We substitute these values into the formula
[tex]D = {b}^{2} - 4ac[/tex]
to determine the nature of the roots.
[tex]D = {1}^{2} - 4(3)(10)[/tex]
[tex]D = 1 - 120[/tex]
[tex]D = - 119[/tex]
The discriminant is negative.
This means that the given quadratic equation has no real roots.
The midpoint of a segment is (−2,−3) and one endpoint is (3,0) . Find the coordinates of the other endpoint.
A. (8, 3)
B. (-7, 3)
C. (8, -6)
D. (-7, -6)
Answers
The midpoint can be defined using formula,
[tex]M(x_m=\dfrac{x_1+x_2}{2},y_m=\dfrac{y_1+y_2}{2})[/tex]
So by knowing [tex]x_m, x_1[/tex] and [tex]y_m, y_1[/tex] we can calculate [tex]x_2, y_2[/tex]
First we must derive two equations,
[tex]x_m=\dfrac{x_1+x_2}{2}\Longrightarrow x_2=2x_m-x_1[/tex]
and
[tex]y_m=\dfrac{y_1+y_2}{2}\Longrightarrow y_2=2y_m-y_1[/tex]
Then just put in the data,
[tex]x_2=2\cdot(-2)-3=-7[/tex]
[tex]y_2=2\cdot(-3)-0=-6[/tex]
So the other endpoint has coordinates [tex](x,y)\Longrightarrow(-7, -6)[/tex] therefore the answer is D.
Hope this helps.
r3t40
To work out the mid point of two points you, add the x coordinates and divide by 2, and you take the y coordinates and divide by two:
So:
[tex]midpoint = \frac{sum.of.x-coords}{2}, \frac{sum.of.y-coords}{2}[/tex]
------------------------------------
So the x-coords of the midpoint is:
[tex]\frac{sum.of.x-coords}{2}[/tex]
and
y -coords of midpoint is:
[tex]\frac{sum.of.y-coords}{2}[/tex]
------------------------------------
However, in this question we are trying to work out one of the endpoints.
First let's say that the coordinates of the missing endpoint is:
(x , y)
_____________________________________________
That means that the x-coords of the midpoint of (x, y) and the other endpoint (3, 0) is :
[tex]\frac{3 + x}{2}[/tex]
However, we already know the x-coord of the midpoint ( it's -2). So we can form an equation to workout x:
[tex]\frac{3 + x}{2} = -2[/tex] (multiply both sides by 2)
[tex]3 + x = -4[/tex] (subtract 3 from both sides)
[tex]x = -7[/tex]
This is the x-coord of the other endpoint
_______________________________________________
Let's do the same for the y coordinates:
We know y coords for the midpoint of (x, y) and (3, 0) is:
[tex]\frac{0 + y}{2}[/tex]
But we also know the ycoord is -3. So we can form an equation and solve for y:
[tex]\frac{0+y}{2} = -3[/tex]
[tex]\frac{0 + y}{2} = -3[/tex] (multiply both sides by 2)
[tex]0 + y = -6[/tex] (simplify)
[tex]y = -6[/tex]
This is the y-coord of the other endpoint
___________________________________
Now we just put these coords together to get the coordinate of the other endpoint:
Endpoint is at:
(x, y) (substitute in values that we worked out)
= (-7, -6)
_________________________________________________
Answer:D. (-7, -6)
________________________________________________
Note:
If there is anything you don't quite understand or was unclear
- please don't hesitate to ask below in the comments.
Please help will give the brainliest
Find the coordinates of the vertices formed by the system of inequalities.
X≤ 3
-x + 3y ≤ 12
4x + 3y ≥ 12
A (0, 3), (4, 0), (5, 3)
B (3, 0), (0, 4), (3, 5)
C (-3, 3), (1, 3), (0, 4)
D (3, -3), (3, 1), (4, 0)
2. At What point is the maximum value found in the system of inequalities graphed below for the function f(x, y) = x - 2y?
A (0, 3)
B (0, 0)
C (5, 0)
D (5, 3)
Answers
Final answer:
To find the vertices of the system of inequalities x ≤ 3, -x + 3y ≤ 12, and 4x + 3y ≥ 12, we can solve each pair of inequalities to find the intersection points. The vertices are the points where the lines intersect. The coordinates of the vertices are (3, 0), (0, 4), and (3, 5). For the function f(x, y) = x - 2y, the point where the maximum value is found in the system of inequalities is (5, 3).
Explanation:
To find the coordinates of the vertices formed by the system of inequalities x ≤ 3, -x + 3y ≤ 12, and 4x + 3y ≥ 12, we can solve each pair of inequalities to find the intersection points. The vertices are the points where the lines intersect.
The solution is (3, 0), (0, 4), and (3, 5), so the correct answer is B.
For the second question, to find the point where the maximum value is found for the function f(x, y) = x - 2y in the system of inequalities graphed below, we need to locate the highest point on the graph. From the given options, (5, 3) is the point where the maximum value is found, so the correct answer is D.