Answers
=================================================
Explanation for problem 14
There are 4 aces and 4 copies of "9".
4+4 = 8 cards we want to select out of 52 total
8/52 = (4*2)/(4*13) = 2/13 which points to choice D as the final answer
---------------
Explanation for problem 15
4 kings gives a probability of 4/52 = 1/13.
After the king is chosen, there are 52-1 = 51 cards left since we aren't replacing it. The probability of getting a queen on the second selection is 4/51
(1/13)*(4/51) = 4/663 is the probability of a king, then a queen, with no replacement. We arrive at choice C as the final answer.
----------------
Explanation for problem 16
face card = picture card
face card = Jack, Queen, King
aces are not considered a face card
3*4 = 12 face cards in total
A = 12/52 = (4*3)/(4*13) = 3/13 is the probability of selecting a face card.
That card is then put back (aka replacement) and we still have 52 cards in the deck. We don't drop down to 51.
13 hearts ---> B = 13/52 = 1/4 is the probability of getting a heart
A*B = (3/13)*(1/4) = 3/52 is the probability of getting a face card, then a heart, when replacement is done. The final answer is choice B
Related Questions
A group of 15 athletes participated in a golf competition. Their scores are below:
Score (points) 1 2 3 4 5
Number of Athletes 1 2 3 4 5
Would a dot plot or a histogram best represent the data presented here? Why?
Histogram, because a large number of scores are reported as ranges
Histogram, because a small number of scores are reported individually
Dot plot, because a large number of scores are reported as ranges
Dot plot, because a small number of scores are reported individually
Answers
Answer:
D
Step-by-step explanation:
A dot plot would be the best representation for this data, because the number of scores reported individually is small and a dot plot is ideal for displaying individual data points.
Find the area and perimeter of the triangle.
8m
3m
9m
4m
Answers
Answer: Broken Math Problem
Step-by-step explanation:
I can prove that 2=1, where is the error?
X = 1
X+X = 1+X
2x = 1+X
2x = X+1
2X-2 = X+1-2
2x-2 = X-1
2 (x-1)/(x-1) = X-1/X-1
2 times 1 = 1 1-1 / 1-1
2 = 1
I subtracted -2
because thats the # I chose to subtract with.
Answers
There is no error in steps while proving 2=1
We have to prove 2=1
Let x=1
Add X on both sides
X+X = 1+X
2x = 1+X
2x = X+1
Subtract 2 from both sides
2X-2 = X+1-2
2x-2 = X-1
2(X-1)=x-1
Divide both sides by x-1
2 (x-1)/(x-1) = X-1/X-1
We get 2=1
Hence, while proving 2=1 there is no error in steps
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someone help me with my math problems these are my last questions
Answers
Answer:translation, 12^2+35^2, squre root the answer, thgen add it to 12
Step-by-step explanation:
Write the mixed number as a percent.
1 9/10
Answers
190 percent. you can find this by dividing the numerator by denominater
Answer:
190%
Step-by-step explanation:
19/10 = 19 ÷ 10 = 1.9
1.9 x 100 = 190%
Find the amount in the account for the given principal, interest rate, time, and compounding period. P = $3,830, r= 5.5%, t = 20 years; compounded monthly
Answers
The amount on compound interest is $ 96,685,319.07
What is amount on compound interest?The amount on compound interest is given by A = P(1 + r)ⁿ where P = principal amount, r = interest rate and t = time
To find the amount on compound interest given that
P = $3830r = 5.5 % compounded monthly, so r = 5.5 % × 12 = 66% = 0.66n = 20 yearsSo, substituting the values of the variables into the equation, we have that
A = P(1 + r)ⁿ
A = $3830(1 + 0.66)²⁰
A = $3830(1.66)²⁰
A = $3830(25244.21)
A = $ 96,685,319.07
The amount is $ 96,685,319.07
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Multiply 16, 3 and 29 and then subtract 17
Answers
Answer:
1375
Step-by-step explanation:
16*3*29 = 1392
1392-17 =1375
math hw for tonight
help solve this problem! Thank you!
ap cal bc
Answers
The position of an object is determined as (¹/₃ ln |1 + t³| + 2)i + (- ¹/₂cos(2t) + 3 )j.
None of above is the correct answer.
What is the particle's position?The position of an object is defined as the product of velocity of the object and the time of motion of the object.
So to obtain the position of the particle, we will integrate the velocity of the particle as follows;
The velocity is given as;
v(t) = t²i/(1 + t³) + sin 2t j
Integrate i component as;
Let u = 1 + t³
du/dt = 3t².
dt = du / (3t²)
∫ t²/(1 + t³) dt =∫ (1/u) x (t²/3t²) du
= (1/3) ∫ (1/u) du
= (1/3) ln |u| + C
= (1/3) ln|1 + t³| + C
Integrate j component as follows;
∫ sin(2t) dt = - ¹/₂cos(2t) + C
∫ v(t) dt = (1/3) ln|1 + t³|)i - ¹/₂cos(2t)j
So finally, we add the initial position of 2i + 3j, as follows;
x = (¹/₃ ln |1 + t³| + 2)i + (- ¹/₂cos(2t) + 3 )j
So none of the options matches this solution, the closet is B.
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Exercise C-7 (Algo) Calculate the present value of a single amount (LO C-3)
You have entered into an agreement for the purchase of land. The agreement specifies that you will take ownership of the land
immediately. You have agreed to pay $48,000 today and another $48,000 in three years. Calculate the total cost of the land today.
assuming a discount rate of (a) 4%, (b) 6%, or (c) 8%. (FV of $1. PV of $1. EVA of $1, and PVA of $1) (Use tables, Excel, or a financial
calculator. Round your answers to 2 decimal places.)
a.
b.
C.
Payment
Amount
$ 48,000
48,000
48,000
Interest
Rate
6%
8%
Compounding
Annually
Annually
Annually
Period Due
3 years
3 years
3 years
Total Cost of Land
Today
Answers
According to the compound interest concept the total cost of the land today is $89,306.64, $86,249.94, $83,693.44 respective to the discount percentage.
To calculate the PV of the second payment of $48,000 due in three years, we need to use the formula for the present value of a single sum:
PV = FV / (1 + r)ⁿ
where PV is the present value, FV is the future value, r is the discount rate, and n is the number of periods.
Using the given information, the FV is $48,000, the discount rate is 4%, 6%, or 8%, depending on the question, and the number of periods is three. Using a financial calculator, Excel, or present value tables, we can calculate the PV of the second payment:
For a discount rate of 4%:
PV = $48,000 / (1 + 0.04)³ = $41,306.64
For a discount rate of 6%:
PV = $48,000 / (1 + 0.06)³ = $38,249.94
For a discount rate of 8%:
PV = $48,000 / (1 + 0.08)³ = $35,693.44
Next, we need to add the PV of the second payment to the initial payment of $48,000 to get the total cost of the land today:
For a discount rate of 4%:
Total cost = $48,000 + $41,306.64 = $89,306.64
For a discount rate of 6%:
Total cost = $48,000 + $38,249.94 = $86,249.94
For a discount rate of 8%:
Total cost = $48,000 + $35,693.44 = $83,693.44
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Find the line parallel to y = -9x-1 that includes the point (2, -3). y- [?] = [?] ( x - [?])
Answers
Answer:
y + 3 = - 9(x - 2)
Step-by-step explanation:
the equation of a line in slope- intercept form is
y = mx + c ( m is the slope and c the y- intercept )
y = - 9x - 1 ← is in slope- intercept form
with slope m = - 9
• Parallel lines have equal slopes
then slope of parallel line is m = - 9
the equation of a line in point- slope form is
y - b = m(x - a)
where m is the slope and (a, b ) a point on the line
m = - 9 and (a, b ) = (2, - 3 ) , then
y - (- 3) = - 9(x - 2) , that is
y + 3 = - 9(x - 2)
A person ordering a certain model of car can choose any of 4 colors, either manual or automatic transmission, and
any of 6 audio systems, How many ways are there to order this model of car?
A) 48 ways
B) 44 ways
C 58 ways
D) 56 ways
Answers
Answer:
Step-by-step explanation:
4*2*6=48a
A cargo container is 25 ft long, 10 ft tall, and 12 ft wide. Find its volume in cubic yards. Round to
the nearest hundredth.
Answer: ________yd3
Answers
Answer:
3,000
Step-by-step explanation:
v = lwh
v = (25)(10)(12)
v = 3000
Helping in the name of Jesus.
Let A={a,b,k,s,u} and B={y,t,p,l,m,a}, find the least possible universal set for A and B
Answers
Answer:
The universal set is the set of all elements that can possibly belong to either set A or set B. To find the least possible universal set for A and B, we need to combine all the elements from both sets while avoiding duplicates.
The combined set of A and B is {a, b, k, s, u, y, t, p, l, m}. Therefore, this is the least possible universal set for A and B.
Step-by-step explanation:
The universal set is the set of all elements that can possibly belong to either set A or set B. To find the least possible universal set for A and B, we need to combine all the elements from both sets while avoiding duplicates.
The combined set of A and B is {a, b, k, s, u, y, t, p, l, m}. Therefore, this is the least possible universal set for A and B.
In which quadrant is point B located?
Answers
Answer:
Step-by-step explanation:
Quadrant II
A is in Quadrant I then you count counterclockwise
B quadrant II (2)
C Quadrant III (3)
D Quadrant VI (4)
Roman numerals are used to for quadrant number
At a large university, 20% of students are enrolled in the nursing program. The dean of students selects a random sample of 20 students and records n the number of students enrolled in the nursing program. What is an appropriate assignment of digits to the outcomes for a simulation of this random process? = O Let 1 = the student is enrolled in the nursing program. Let 2-5 = the student is not enrolled in the nursing program. Skip the digits 6-9, and 0. ○ Let 0 and 1 = the student is enrolled in the nursing program. Let 2-5 = the student is not enrolled in the nursing program. Skip the digits 6, 7, 8, 9, and 0. O Let 1 = the student is enrolled in the nursing program. Let 2-9 = the student is not enrolled in the nursing program. Skip the digit 0. O Let 1 and 2 = the student is enrolled in the nursing program. Let 3-9 = the student is not enrolled in the nursing program. Skip the digit 0.
Answers
An appropriate assignment of digits to the outcomes for a simulation of this random process would be:
Option 1: Let 0 and 1 = the student is enrolled in the nursing program. Let 2-5 = the student is not enrolled in the nursing program. Skip the digits 6, 7, 8, 9, and 0.
This option assigns two digits, 0 and 1, to represent the two possible outcomes: a student is either enrolled in the nursing program (outcome 1) or not enrolled (outcome 2-5). The digits 2-5 represent the non-nursing program outcomes, and the digits 6-9 and 0 are skipped.
Option 1 is appropriate because it assigns a unique digit to each possible outcome, and the skipped digits are not relevant to the simulation. Additionally, the proportion of digits assigned to each outcome (2 out of 10) corresponds to the proportion of students in the nursing program (20%).
Determine whether the graph of the equation is symmetric with respect to the x axis, y axis and or the origin.
X^2 + Y -36 =0
Answers
The graph of the equation x² + y - 36 = 0 is symmetric with respect to the y-axis and the origin, but not with respect to the x-axis.
x² + y - 36 = 0
Symmetry with respect to the x-axis:
Replace y with -y: x² - y - 36 = 0
This equation is not equivalent to the original equation, so the graph is not symmetric with respect to the x-axis.
Symmetry with respect to the y-axis:
Replace x with -x: (-x)² + y - 36 = 0
Simplifying, we get x² + y - 36 = 0, which is equivalent to the original equation.
Therefore, the graph is symmetric with respect to the y-axis.
Symmetry with respect to the origin:
Replace x with -x and y with -y: (-x)² + (-y) - 36 = 0
Simplifying, we get x² + y - 36 = 0, which is equivalent to the original equation.
Therefore, the graph is symmetric with respect to the origin.
Hence, the graph of the equation x² + y - 36 = 0 is symmetric with respect to the y-axis and the origin, but not with respect to the x-axis.
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The relative locations of Marilyn's house, Bobby's house, and Kimberly's house are shown in the figure.
What is the distance from Kimberly's house to Marilyn's house?
Enter your answer in the box. Round your final answer to the nearest whole number.
Answers
The distance from Kimberly's house to Marilyn's house = 12.04 mi
Let us assume that A be the angle at Kimberly's house, B represents the angle at Bobby's house and S represents the angle at Marilyn's house.
Let a, b, c represents the sides(distance between two houses) opposite to angles A, B and C.
Using sine rule to triangle ABC,
sin A/a = sin B/b = sin C/c
Consider equation,
sin A/a = sin B/b
sin(63°) / 14 = sin(50°) / b
b = (0.766 × 14) / 0.891
b = 12.04 mi
Thus, the required distance between Kimberly's house and Marilyn's house = 12.04 mi
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(06.07 LC)
The graph below plots the values of y for different values of x
20
15
10
Which correlation coefficient best matches the data plotted on the graph? (1 point)
-0.5
0
0.25
0.99
Answers
The correlation coefficient which best matches the data plotted on the graph is 0.25, the correct option is C.
We are given that;
The graph of different values of x
Now,
we can see that there is a weak positive linear relationship between x and y, as the points are scattered around a slightly upward-sloping line. The closer the points are to the line, the stronger the correlation. The farther the points are from the line, the weaker the correlation. Hence, we can expect the correlation coefficient to be a small positive number, less than 1.
Therefore, by correlation coefficient the answer will be 0.25.
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The table below show the data to find an exponential model.
x 1 4 7 8 10
y. 798 1078 1519 2075 3102
1 goes with 798, 4 goes with 1078, 7 goes with 1519, 8 goes with 2075, and 10 goes with 3102
a. Write the data set to be used in order to linearize the data.
b. Write the system of equations
c. Write the matrices.
d. Indicate the c values
e. Write the exponential model in the form Y=C* 10^bx
Answers
The data set to be used to linearize the data is:
x ln(y)
1 ln(798)
4 ln(1078)
7 ln(1519)
8 ln(2075)
10 ln(3102)
b. The system of equations is:
$\begin{pmatrix}5 & \sum_{i=1}^{5} x_i \\ \sum_{i=1}^{5} x_i & \sum_{i=1}^{5} x_i^2 \\\end{pmatrix}\begin{pmatrix}b \\ ln(C) \\\end{pmatrix}=\begin{pmatrix}\sum_{i=1}^{5} ln(y_i) \\ \sum_{i=1}^{5} x_i ln(y_i) \\\end{pmatrix}$
c. The matrices are:
$\begin{pmatrix}5 & \sum_{i=1}^{5} x_i \\ \sum_{i=1}^{5} x_i & \sum_{i=1}^{5} x_i^2 \\\end{pmatrix}\begin{pmatrix}b \\ ln(C) \\\end{pmatrix}=\begin{pmatrix}\sum_{i=1}^{5} ln(y_i) \\ \sum_{i=1}^{5} x_i ln(y_i) \\\end{pmatrix}$
$\begin{pmatrix}5 & 30 \\ 30 & 266 \\\end{pmatrix}\begin{pmatrix}b \\ ln(C) \\\end{pmatrix}=\begin{pmatrix}7.986 \\ 27.214 \\\end{pmatrix}$
d. Solving for b and ln(C) using a calculator or a matrix-solving tool, we get:
b = 0.2695
ln(C) = 6.6195
Taking exponential of ln(C) we get C = 760.9305.
Therefore, the values of c are: C = 760.9305.
e. The exponential model in the form Y=C * 10^(bx) is:
Y = 760.9305 * 10^(0.2695x).
A company's survey of 150 employees find that on average an employee drinks 10.2 cups of coffee during the workweek with a margin of error of #0.6. Using this data, it is estimated that a maximum of 6, 048 cups of coffee will be consumed during a workweek. What is the total number of employees in the company?
• 403
• 560
• 593
• 630
Answers
The total number of employees in the company is 560.
What is total number of employees?
The total number of employees in the company is calculated as follows;
the true average number of cups of coffee consumed per employee during the workweek is 10.2 ± 0.6 cups.
maximum = 10.8 cups/employee
minimum = 9.6 cups/employee
The minimum number of employees is calculated as;
n_m = 6,048 cups / 10.8
n_m = 560
The maximum number is calculated as;
n_max = 6,048 cups/9.6
n_max = 630
Based on the error margin, we take the minimum number.
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A committee of four people is formed by selecting members from a list of 12 people.
How many different committees can be formed?
__ committees
Answers
The number of different committees that can be formed is 495
How many different committees can be formed?From the question, we have the following parameters that can be used in our computation:
People = 12
Committee = 4
These can be represented as
n = 12 and r = 4
The number of different committees that can be formed is
Number = nCr
So, we have
Number = 12C4
Evaluate
Number = 495
Hence, the committee are 495
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What is the equation of the line in slope-intercept form?
Answers
Answer:
The answer is the second choice.
Step-by-step explanation:
The slope is rise over run, which is 3/-1, which is -3, which goes before the x. The line hits the y axis at -1, so it is -3x–1
Using the simple interest formula, find the amount earned after 4 years with a present value of $600 and an APR of 5%. Verify that you get the same answer as in the table.
Answers
The simple interest after 4 years with the given principal and rate is $120.
Given that, principal =$600, rate=5% and time=4 years.
We know that, simple interest=(Principal×Rate×Time)/100.
Simple interest = (600×5×4)/100
= $120
Therefore, the simple interest after 4 years with the given principal and rate is $120.
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A ladder is leaning against a house at a 50 degree angle. If the ladder is 80 feet long, how high up is it from the ground at the top of the ladder?
We are missing a side or and angle?
Regular or Inverse Trig?
Answers
The height from the ground to the top of the ladder is 61. 28 feet
How to determine the heightFirst, we need to know that their are six different trigonometric identities.
These identities includes;
secantcotangenttangentcosinesinecosecantFrom the information given, we have that;
the height of the ladder that is leaning against the house = 80 feet
This the hypotenuse side
The angle of elevation is 50 degrees
Then, the opposite side is the height from the ladder = x
Using the sine identity, we have;
sin 50 = x/80
cross multiply the values
x = 61. 28 feet
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Which prism has a volume between 38 and 48 cubic inches? Four prisms named A, B, C, and D. All the prisms are measured in cubic inches. Prism A has four rows, five columns, and two layers. Prism B has three rows, four columns, and three layers. Prism C has four rows, four columns, and one layer. Prism D has four rows, four columns, and six layers. A B C D
Answers
Answer:
To find the prism with a volume between 38 and 48 cubic inches, we need to calculate the volume of each prism and then compare it to the given range.
Prism A:
Number of cubes in prism A = 4 x 5 x 2 = 40
Volume of each cube = 1 cubic inch
Volume of prism A = Number of cubes x Volume of each cube = 40 x 1 = 40 cubic inches
Prism B:
Number of cubes in prism B = 3 x 4 x 3 = 36
Volume of each cube = 1 cubic inch
Volume of prism B = Number of cubes x Volume of each cube = 36 x 1 = 36 cubic inches
Prism C:
Number of cubes in prism C = 4 x 4 x 1 = 16
Volume of each cube = 1 cubic inch
Volume of prism C = Number of cubes x Volume of each cube = 16 x 1 = 16 cubic inches
Prism D:
Number of cubes in prism D = 4 x 4 x 6 = 96
Volume of each cube = 1 cubic inch
Volume of prism D = Number of cubes x Volume of each cube = 96 x 1 = 96 cubic inches
Therefore, the prism with a volume between 38 and 48 cubic inches is prism A, since its volume is 40 cubic inches which falls within the given range.
Step-by-step explanation:
Answer:
Prism A:
Number of cubes in prism A = 4 x 5 x 2 = 40
Volume of each cube = 1 cubic inch
Volume of prism A = Number of cubes x Volume of each cube = 40 x 1 = 40 cubic inches
Prism B:
Number of cubes in prism B = 3 x 4 x 3 = 36
Volume of each cube = 1 cubic inch
Volume of prism B = Number of cubes x Volume of each cube = 36 x 1 = 36 cubic inches
Prism C:
Number of cubes in prism C = 4 x 4 x 1 = 16
Volume of each cube = 1 cubic inch
Volume of prism C = Number of cubes x Volume of each cube = 16 x 1 = 16 cubic inches
Prism D:
Number of cubes in prism D = 4 x 4 x 6 = 96
Volume of each cube = 1 cubic inch
Volume of prism D = Number of cubes x Volume of each cube = 96 x 1 = 96 cubic inches
Therefore, the prism with a volume between 38 and 48 cubic inches is prism A, since its volume is 40 cubic inches which falls within the given range.
Step-by-step explanation:
Write your answer as an integer or as a decimal rounded to the nearest tenth.
Answers
The measure of WY using trigonometric identities is 5.4405.
We have,
Using trigonometric identities,
tan 33 = YW/√70
Now,
√70 = 8.37
tan 33 = 0.65
Substituting,
0.65 = YW/8.37
YW = 0.65 x 8.37
YW = 5.4405
Thus,
The measure of WY is 5.4405.
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The length of the arc intercepted by a central angle of 5 radians in a circle of radius
73 is
The length of the arc intercepted by a central angle of 86" in a circle of radius 15 is
Answers
Answer:
The length of the arc intercepted by a central angle of 3 radians in a circle of radius 76 is 228 and the length of the arc intercepted by a central angle of 2 radians in a circle of radius 15 is 30.
Explain:
The following formula determines the length of an arc that a circle's central angle intercepts:
Arc length is equal to (central angle / 2) 2r r r
where r denotes the circle's radius. We can determine the length of the two arcs using the following formula:
For the first circle, with a radius of 76 and a center angle of 3 radians:
Arc length = 3 x 76 = 228
Therefore, in a circle with a radius of 76, the length of the arc that is intercepted by a central angle of 3 radians is 228.
For the second circle, whose radius is 15, and whose center angle is 2 radians:
Arc length = 2 x 15 = 30
As a result, the length of the arc in a circle with a radius of 15 is 30 when it is intercepted by a central angle of 2 radians.
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Write the quadratic equation in standard form:-4x – 19 = x
Answers
Answer: x^2 + 0x + 19/5 = 0
Step-by-step explanation: The initial step involves the equation -4x - 19 = x, which can be simplified by performing the operation of subtracting x from both sides.
The given equation can be represented as -5x - 19 = 0 in standard algebraic form, where 'x' denotes the unknown variable.
The process of combining similar or like terms leads to the consolidation of terms that share the same variable and corresponding exponents.
The given expression, 5x - 19 = 0, can be rewritten in an academic manner as a linear equation. Specifically, this expression represents a linear equation in one variable, where x is the unknown. The equation can be solved to find the value of x that satisfies it. This can be done through various methods, such as isolation of x, substitution, or using algebraic properties. The solution of this equation is x = 19/5, which can be verified by plugging it back into the original equation and observing its validity.
In order to render this equation in a standard format, it is imperative to eliminate the fixed element present on the left-hand side. The aforementioned task can be accomplished by simultaneously augmenting 19 to each side of the equation.
The equation 5x = 19 can be expressed in an academic manner as a linear equation with one variable. Specifically, it states that there is a value, x, that when multiplied by the constant factor of 5, results in a final product of 19. This equation could be used as a starting point for further mathematical analysis or application, such as solving for the specific value of x or incorporating it into a larger system of equations.
The left-hand side of the expression contains the linear term and the constant term, while the right-hand side is equal to 0. In order to express this equation in a standardized form, it is necessary to perform division by a factor of negative five to isolate the variable x.
The numerical value of the variable x is equivalent to negative nineteen divided by five, expressed as x = -19/5.
The standard form of the quadratic equation can be expressed as follows:
The mathematical expression, x + 19/5 = 0, may be restated in a formal academic style as follows: "The equation is of the form x + 19/5 = 0."
In order to determine the coefficients of the squared and constant terms, it can be observed that the coefficient of the squared term equates to 1 by virtue of the squared exponent of x, while the constant term evaluates to 19/5. This analytical approach yields the appropriate identification of the relevant coefficients in the given equation. Henceforth, the standard format for the quadratic equation is expressed as follows:
The equation in question is x^2 + 0x + 19/5 = 0.
BRAINLIEST!!! Show all work to identify the asymptotes and state the end behavior of the function ..
f(x)= 3x/x-9
SHOW ALL STEPS PLEASE.
Answers
The function f(x) = 3x/(x-9) has a vertical asymptote at x=9, a horizontal asymptote at y=3 as x->±∞, and a horizontal asymptote at y=-3 as x->±∞.
To identify the asymptotes and state the end behavior of the function f(x) = 3x/(x-9), we need to analyze the behavior of the function as x approaches positive and negative infinity.
First, let's look at the denominator of the fraction, which is x-9. This means that the function has a vertical asymptote at x=9, as the denominator approaches zero at that point. This vertical asymptote divides the x-axis into two regions, x<9 and x>9.
Now let's consider the behavior of the function as x approaches positive infinity. In this case, the numerator grows faster than the denominator, and the function approaches a horizontal asymptote at y=3. To see this, we can use the limit definition:
lim (x->∞) f(x) = lim (x->∞) 3x/(x-9) = lim (x->∞) 3(1/(1-9/x)) = 3
Similarly, as x approaches negative infinity, the function also approaches a horizontal asymptote at y=-3. Again, we can use the limit definition to confirm this:
lim (x->-∞) f(x) = lim (x->-∞) 3x/(x-9) = lim (x->-∞) 3(1/(1-9/x)) = -3
Therefore, the function f(x) = 3x/(x-9) has a vertical asymptote at x=9, a horizontal asymptote at y=3 as x->±∞, and a horizontal asymptote at y=-3 as x->±∞.
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Write a function of the form y= A sin (Bx-C)+D that has period 8, phase shift -2, and the range -12 ≤ y ≤-4.
Answers
The general form of the function is:
y = A sin(Bx - C) + D
To determine the specific values of A, B, C, and D that satisfy the given conditions, we can use the following steps:
Period: The period of a sinusoidal function is given by 2π/B, where B is the coefficient of x in the argument of the sine function. In this case, the period is 8, so we have:
2π/B = 8
Solving for B, we get:
B = π/4
Phase shift: The phase shift of a sinusoidal function is given by C/B, where C is the constant inside the argument of the sine function. In this case, the phase shift is -2, so we have:
C/B = -2
Substituting B from step 1, we get:
C/(π/4) = -2
Solving for C, we get:
C = -π/2
Amplitude and vertical shift: The range of the function is given as -12 ≤ y ≤ -4. The amplitude of a sinusoidal function is half the distance between its maximum and minimum values. In this case, the amplitude is:
(amplitude) = (maximum - minimum)/2 = (-4 - (-12))/2 = 4
The vertical shift of the function is given by the constant term D. Since the minimum value of the function is -12, we have:
D + (amplitude) = -12
Substituting the value of the amplitude from above, we get:
D + 4 = -12
Solving for D, we get:
D = -16
Therefore, the function of the form y = A sin(Bx - C) + D that has period 8, phase shift -2, and the range -12 ≤ y ≤ -4 is:
y = 4 sin(π/4 x + π/2) - 16