Write the domain using interval notation.
Answer:
[tex](f \circ g)(\text{x}) = \frac{13}{13-\text{x}}[/tex]
Domain: [tex](-\infty,0) \cup (0,13) \cup (13,\infty)[/tex]
=================================================
Explanation:
Let's find the function composition.
The notation [tex](f \circ g)(\text{x})[/tex] is the same as [tex]f(g(\text{x}))[/tex]
[tex]f(\text{x}) = \frac{\text{x}}{\text{x}-1}\\\\\\f(g(\text{x})) = \frac{g(\text{x})}{g(\text{x})-1}\\\\\\f(g(\text{x})) = g(\text{x}) \div \Big( g(\text{x}) - 1\Big)\\\\\\[/tex]
Then,
[tex]f(g(\text{x})) = \frac{13}{\text{x}} \div \left(\frac{13}{\text{x}}-1}\right)\\\\\\f(g(\text{x})) = \frac{13}{\text{x}} \div \left(\frac{13}{\text{x}}-\frac{\text{x}}{\text{x}}\right)\\\\\\f(g(\text{x})) = \frac{13}{\text{x}} \div \frac{13-\text{x}}{\text{x}}\\\\\\f(g(\text{x})) = \frac{13}{\text{x}} * \frac{\text{x}}{13-\text{x}}\\\\\\f(g(\text{x})) = \frac{13}{13-\text{x}}\\\\\\[/tex]
-----------------
Now let's find the domain.
If we plugged x = 0 into g(x), then we get a division by zero error.
This means we must exclude this value from the domain.
For similar reasoning, we must exclude x = 13 because we get a division by zero error in [tex]f(g(\text{x})) = \frac{13}{13-\text{x}}[/tex]
We could have any other real number to be plugged in for x.
Here's what the domain looks like in interval notation.
[tex](-\infty,0) \cup (0,13) \cup (13,\infty)[/tex]
We effectively poke holes at 0 and 13 on the number line.
James deposited $10,000 into an account that earns 5.5% compound interest, compounded semiannually. How much interest will James earn after 10 years?
Answer:
$17,204.28
Step-by-step explanation:
So first of all you want to start by finding what P, r and t would be.
P = Principal amount ($$)r = interest rate (%)t = time (years)Once I found all of those I put them into the equation (l = Prt) and solved (by putting it into a calculator obviously). That is how I came up with my answer. Check the screenshot provided to see what P, r and t would be and to see all my work! :)
Hope this helps! :)
Have a great day!
The numbers of polio cases in the world are shown in the table for various years.
Year Number of polio cases (thousands)
1988 350
1992 138
1996 33
2000 4
2005 3.2
2007 1.3
PREDICT THE NUMBER OF POLIO CASES IN 2014
hint: The function F gives the number of thousands of polio cases
The value of t is given as 2.3 years.
How to solvewe get the table for function as
t f(t)
8 350
12 138
16 36
20 4
25 3.2
27 1.3
So it is better to model the data by using an exponential model
using a graphing calculator we get our model as
f(t)=a(b)^t
[tex]f(t)=\boldsymbol{3648.6915(0.7380)^t}[/tex]
we have an exponential decay, so we get a rate of decrease as
b=1-r
r=1-b=1-0.738=0.262=26.2\\%
So the number of polio cases is decreasing by 26.2% per year
The number of polio cases in 2014 is
t=2014-1980=34
f(34)=3648.6915(0.7380)^{34}=0.1991389
this is in thousands so we get the number of cases in 2014 approximately as
[tex]0.1191389*1000 \approx \boldsymbol{119} \textup{ cases}[/tex]
for 1 case of polio :
f(t)=0.001
3648.6915(0.738)^t=0.001
we get
[tex]t=\frac{\ln (\frac{0.001}{3648.6915})}{\ln(0.738)}\approx 50[/tex]
so we get year as
1980+50={2030},
to find half life
[tex]\frac{a}{2}=a(b)^t[/tex]
[tex]b^t=\frac{1}{2}[/tex]
[tex]0.738^t=\frac{1}{2}[/tex]
[tex]t=\frac{\ln (1/2)}{\ln (0.738)}\approx\boldsymbol{2.3} \textup{ years}[/tex]
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a student score 80 on a test last week and 90 on thesame test this week. what is the percent increase in score
Answer:
12.5%
Step-by-step explanation:
We Know
A student scored 80 on a test last week and 90 on the same test this week.
What is the percent increase in the score?
We Tale
(90 ÷ 80) x 100 = 112.5%
Then we take
112.5 - 100 = 12.5%
So, the percent increase in score is 12.5%
7. Find (i) the length of an arc and (ii) the area of a sector in a circle of radius 7 meters subtended by the central angle of 85°.
i) The length of the arc is approximately 73.148 meters.
ii) The area of the sector is approximately 85.584 square meters.
(i) The length of an arc can be calculated using the formula:
Arc Length = (θ/360) × 2πr
where;
θ = central angle in degrees
r = radius of the circle
π = mathematical constant approximately equal to 3.14159.
Given that the central angle is 85° and the radius is 7 meters, we can substitute these values into the formula:
Arc Length = (85/360) × 2π × 7
= (17/72) × 2 × 3.14159 × 7
≈ 1.7454 × 6.28318 × 7
≈ 73.148 meters (rounded to three decimal places.
(ii) The area of a sector can be calculated using the formula:
Sector Area = (θ/360) × πr²
Given that the central angle is 85° and the radius is 7 meters, we can substitute these values into the formula:
Sector Area = (85/360) × π × 7²
= (17/72) × 3.14159 × 7²
≈ 1.7454 × 3.14159 × 49
≈ 85.584 square meters (rounded to three decimal places)
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The correct question is:
Find:
(i) the length of an arc
(ii) the area of a sector in a circle of radius 7 meters subtended by the central angle of 85°.
5. Kamal said that he can measure
area using squares that are 2 units
long and 1 unit wide. What mistake
did Kamal make?
Answer:
A square's length & width are equal
Step-by-step explanation:
Kamal's shape is not a square, because a square is equilateral (equal length in all sides), but his square is 2:1,
. The students in a furniture-making class make a tabletop shaped like the figure shown. The tabletop has squares cut out of the corners. a. What is the area of the tabletop?
The area of the tabletop made by the students in the furniture - making class would be 26 ft ²
How to find the area ?We can see that this tabletop made by students in the furniture - making class takes the form of a composite shape with two smaller rectangles and one large rectangle.
The area of the large rectangle is:
= 4 ft x ( 1 + 3 + 1 )
= 4 ft x 5 ft
= 20 ft ²
The area of the two smaller rectangles is:
= 2 x ( 3 ft x 1 ft )
= 2 x 3
= 6 ft ²
The total area of the tabletop is:
= 20 + 6
= 26 ft ²
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find the inverse of f(x)=2^x+2
us the function odd, even, or neither?
I do not understand how to round to whole number
The given fraction can be converted into whole number as
16/5 +52/9 =9
99/7 -15/2=7
13/8+ 27/5=7
41/4-26/15=9
27/10+44/9=8
How can the fraction be converted to whole number?We can see that all the expression was given as a fraction the to convert to whole number we will need to solve them by performing the neccessary addition as well as substraction operations then it will converted to whole number.
16/5 +52/9
=404/45.
=8.977
=9
99/7 -15/2=
= 93/14
=6.643
7
13/8+ 27/5
= 281/40
=7.025
7
41/4-26/15
=511/60
=8.52
=9
27/10+44/9
= 683/90
=7.59
=8
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The population of a small town of 29,000 people is expected to grow exponentially at a rate of 1.9% per year. Estimate the population in 2 years' time.
Answer:
Step-by-step explanation:
The trinomial: x² - 5x +11 is not factorable. Explain why this is.
Answer:
Step-by-step explanation:
I cannot find 2 numbers that multiply to the last term (+11) and add to middle term (-5)
So it can't be factored that way.
After further inspection. i try to plug it into b²-4ac from the quadratic formula.
b=-5
c=11
(-5)²-4(1)(11)
= -19 you can never get a negative under the square root. so this quadratic will produce 2 imaginary solutions. so it is not factorable.
Me González designed a ramp for his caras shown in the picture below
what is the volume of the ramp in cubic feet
The ramp has a volume capacity of around 37.5 cubic feet.
How to solveCalculating the volume of a triangular prism shaped ramp follows a specific formula. First, identify the shape and refer to the equation:
Volume = (1/2) * Base Area * Height
To begin, the base is always in a right-angled triangle form, where its area depends on its width denoted as W, and height noted specifically as H.
Based on these computations, knowing that its triangular base measures 5 feet at its length while taking its height equates 3 feet therefore assessing how much it covers can be found using:
Triangle Area = (1/2) * Base * Height
Substituting values into solving areas leads us to have an answer of over 7 square feet.
Next is computing the volume of the said triangular prism. Placing all given data results in the same formula:
Volume = (1/2) * Base Area * Height
With one-half of the product of both areas which resulted in 7.5 square feet and then multiplying the figure by 10 ft, this yields us an approximate volume of 37.5 cubic feet.
In conclusion, we have determined that the ramp has a volume capacity of around 37.5 cubic feet.
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Length (L) = 10 feet
Width (W) = 5 feet
Height (H) = 3 feet
Mr. González designed a ramp for his car with a length of 10 feet, a width of 5 feet, and a height of 3 feet. What is the volume of the ramp in cubic feet?
HELP!!!!!!!! Which system of linear equations can be solved using the information below?
The system of linear equations that can be solved from the matrices is given as follows:
-5x + 4y = 3.-8x + y = -6.How to obtain the system of equations?Considering that the row [3, -6] is common to matrices Ax and Ay, the matrix A is given as follows:
A = [-5 4; -8 1]
Hence the multiplication of matrices representing the system is given as follows:
[-5 4; -8 1][x; y] = [3; -6]
Applying the multiplication of matrices, the system of equations is given as follows:
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Can someone please help with this question and break it down so I can learn a better way to do it.
Using trigonometry and the Pythagorean theorem, the length of the hypotenuse AC is found to be 3.5 cm, and using the Pythagorean theorem again, the length of BC is found to be approximately 1.75 cm.
Using trigonometry and the given angle and side length information, we can solve for the length of side BC (x).
We know that
sin(A) = opposite/hypotenuse
sin(30) = AC/7
AC = 7 × sin(30)
AC = 3.5 cm
Using the Pythagorean theorem, we have
BC² = AC² - AB²
BC² = (3.5)² - (7)² sin²(30)
BC² = 3.0625
BC = √3.0625
BC = 1.75 cm (rounded to two decimal places)
Therefore, the value of x is approximately 1.75 cm.
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Need help in this question, with explanation please.
Answer:
a) (1/2)(5)(CD) = 20 cm^2
5CD = 40, so CD = 8 cm
b) (1/2)(12)(8) = 48 cm^2
some the following equation
x^4+4x^3+3x^2=0
Answer:
x = 0, -1, -3
Step-by-step explanation:
Move all terms to the left side and set equal to zero. Then set each factor to equal zero.
The ordered pairs model an exponential function, where w is the function name and t is the input variable.
{(1, 60), (2, 240), (3, 960), (4, 3840)}
What is the function equation in sequence notation?
ANSWER: 60(4)t−1
just took the test
Answer: w(t) = 60 * 4^(t-1).
Step-by-step explanation:
The given ordered pairs model an exponential function of the form w(t) = ab^t, where w is the function name, t is the input variable, a is the initial value, and b is the growth factor.
From the first ordered pair (1, 60), we can see that the initial value a is 60. From the second and third ordered pairs (2, 240) and (3, 960), we can see that the output value is multiplied by 4 when the input value increases by 1. This means that the growth factor b is 4.
So, the function equation in sequence notation is w(t) = 60 * 4^(t-1).
A small country emits 80,000 kilotons of carbon dioxide per year. In a recent global agreement, the country agreed to cut its carbon emissions by 2.5% per year for the next 7 years. In the first year of the agreement, the country will keep its emissions at 80,000 kilotons and the emissions will decrease 2.5% in each successive year. How many total kilotons of carbon dioxide would the country emit over the course of the 7 year period, to the nearest whole number?
The total carbon dioxide emission over a period of 7 years will be 5,13,183 kilotons.
Considering the carbon emissions follow an exponential rate then
A(t)=A₀(1-r)^t
where A₀ is the initial value, and r is the decay rate as a decimal.
For the given problem A₀ =80000 kilotons and emissions will decrease at the rate of 2.5% per year therefore, r=0.025
The general equation for emission becomes
[tex]A(t)=80000(1-0.025)^t[/tex]
[tex]A(t)=80000(0.975)^t[/tex]
To calculate total emissions for 7 years it is calculated as follows
[tex]I=\int\limits^{7}_0 {A(t)} .dt=\int\limits^{7}_0 {80000(0.975)^tdt}[/tex]
[tex]I=80000(0.975)^t/ln (0.975)\left \{ {{t=7} \atop {t=0}} \right.[/tex]
I=5,13,183Kilotonnes
Hence, the total emission of 5,13,183 kilotons of carbon dioxide would be emitted over the course of the 7-year period.
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A theme park has a ride that is located in a cylinder with a height of 11 yards.The ride goes around the outside of the cylinder, which has a circumference of 513.88 yards. What is the surface area of the cylinder? Estimate to the nearest hundredth, using 3.14. Apply the formula for surface area of a cylinder SA=2B+Ph
Answer:
To calculate the surface area of the cylinder, we need to find the area of the two circular bases and the lateral surface area.
The formula for the circumference of a circle is:
C = 2πr
where C is the circumference and r is the radius. We can rearrange this formula to solve for the radius:
r = C/2π = 513.88/(2 x 3.14) = 81.91 yards
The formula for the area of a circle is:
A = πr^2
where A is the area and r is the radius. We can use this formula to find the area of one of the circular bases:
B = πr^2 = 3.14 x 81.91^2 = 21,237.93 square yards
The formula for the lateral surface area of a cylinder is:
P x h
where P is the perimeter of the base and h is the height. We can use the formula for the circumference to find the perimeter of the base:
P = C = 513.88 yards
Now we can calculate the lateral surface area:
L = P x h = 513.88 x 11 = 5,652.68 square yards
Finally, we can use the formula for the surface area of a cylinder:
SA = 2B + L = 2(21,237.93) + 5,652.68 = 48,128.54 square yards
Therefore, the surface area of the cylinder is approximately 48,128.54 square yards when rounded to the nearest hundredth.
how do you find the base of a rectangle if you only know the height
Step-by-step explanation:
You will have to know more....like the area or the perimeter or the diagonal measure to determine the base if you only have the height.
PLEASE HELP!!
A cone has a radius of 3 inches and a slant height of 12 inches.
What is the exact surface area of a similar cone whose radius is 6 inches?
Enter your answer in the box.
Surface area of similar cone =
___in²
Surface area of similar cone is 44π square inches
What does a cone in maths ?
A cone is referred to as a "right cone" when its vertex is higher than the base's centre (i.e., when the angle formed by the vertex, base center, and any base radius is a right angle); otherwise, the word "oblique" is used. A cone is referred to as an elliptic cone when the base is assumed to be an ellipse rather than a circle.
The surface area of a cone is given by:
surface area = πr(r + l)
where r is the radius and l is the slant height.
surface area of original cone = π(3)(3 + 12) = 45π
slant height of similar cone = (6/3)(12) = 24 inches
surface area of similar cone = π(6)(6 + 18) = 144π
ratio of surface areas = surface area of similar cone / surface area of original cone
= (144π) / (45π)
= 3.2
Therefore, The exact surface area of the similar cone is 6 times the surface area of the original cone,
6 × 45π = 144π square inches
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What is the solution of x=9y and 3y - 3x =24
Answer:
Step-by-step explanation:
The solution to the system of equations `x=9y` and `3y - 3x =24` can be found by substituting the value of `x` from the first equation into the second equation. This gives us `3y - 3(9y) = 24`, which simplifies to `-24y = 24`. Solving for `y`, we get `y = -1`. Substituting this value of `y` into the first equation, we get `x = 9(-1) = -9`. So the solution to the system of equations is `(x,y) = (-9,-1)`.
Is there anything else you would like to know?
At Wynne College, there are 240 students in Year 12.
The interquartile range of the times taken for these students to travel to college is 32 minutes.
How many of these students have travel times within this interquartile range?
About 120 students have travel times within the quoted interquartile range.
InterquartileThe interquartile range (IQR) represents the range of the middle 50% of a set of data, which is the difference between the third quartile (Q3) and the first quartile (Q1).
Assuming that the travel times are normally distributed and that we have information on the quartiles, we can use the formula:
IQR = Q3 - Q1
To find the number of students whose travel times fall within the interquartile range, we need to know the values of Q1 and Q3.
Since the IQR is 32 minutes:
32 = Q3 - Q1
We also know that the total number of students in Year 12 is 240. Assuming that all students have reported their travel times, we can use the interquartile range to estimate the proportion of students whose travel times fall within this range.
If the interquartile range covers the middle 50% of the data, then each quartile should cover 25% of the data. Therefore, we can estimate that the number of students whose travel times fall within the interquartile range is:
Number of students = 240 x 0.50 = 120
Therefore, we estimate that 120 of the 240 Year 12 students have travel times within the interquartile range of 32 minutes.
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Claude has a 10-pound package of hamburger meat and an
11.5-pound package of hamburger meat. Can he make the number of
hamburger patties shown? Choose Yes or No.
A. 87 1/4-pound hamburger patties
B. 45 1/2-pound hamburger patties
C. 30 1/2 -pound and 274-pound hamburger patties
D. 29 3/4 -pound hamburger patties
Answer: We can solve this problem by finding the total weight of the meat and then dividing by the weight of each patty to see if we can make the desired number of patties.
Let's add the weights of the two packages of meat:
10 + 11.5 = 21.5
For option A, we need 87 1/4-pound patties.
Each pound of meat will yield 4 patties of this size, so 21.5 pounds of meat will yield:
21.5 x 4 = 86 patties
Since we need 87 patties, Claude does not have enough meat for option A.
For option B, we need 45 1/2-pound patties.
Each pound of meat will yield 2 patties of this size, so 21.5 pounds of meat will yield:
21.5 x 2 = 43 patties
Since we need 45 patties, Claude does not have enough meat for option B.
For option C, we need 30 1/2-pound patties and 2 74-pound patties.
Each pound of meat will yield 2 patties of the smaller size and 1 patty of the larger size, so 21.5 pounds of meat will yield:
(21.5 x 2) + (2 x 1) = 43 + 2 = 45 patties
Since Claude only has 21.5 pounds of meat, he does not have enough for option C.
For option D, we need 29 3/4-pound patties.
Each pound of meat will yield 1.5 patties of this size, so 21.5 pounds of meat will yield:
21.5 x 1.5 = 32.25 patties
Since we only need 29.75 patties, Claude has enough meat for option D.
Therefore, the answer is Yes, Claude can make 29 3/4-pound hamburger patties.
You spin the spinner once. 234 What is P(not greater than 2)? Write your answer as a fraction or whole number.
When the sun is at a certain angle in the sky, a 100 foot building will cast a 25 foot shadow , how tall is a person if he casts a 1.5 foot shadow at the same time
The height of the person that cast a shadow of 1.5 foot is 6 foot.
How to solve proportional relationship?When the sun is at a certain angle in the sky, a 100 foot building will cast a 25 foot shadow. A person cast a shadow of 1.5 foot at the same time.
Therefore, the height of the person can be found by setting up a proportional relationship between the length of shadow and the actual height of the building and person.
Hence,
let
x = height of the person
Therefore,
100 / 25 = x / 1.5
cross multiply
25x = 100 × 1.5
25x = 150
divide both sides of the equation by 25
x = 150 / 25
x = 6
Therefore,
height of the person = 6 foot.
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What rate of interest, to the nearest tenth of a percent, compounded quarterly is needed for an investment of $1600 to grow to $2400 in 11 years
Compound interest is the interest earned on the principal and the interest previously accumulated. It is given by
[tex]A=P(1+r/n)^{nt}[/tex]
where P = Principal, r = annual rate of interest, n = the number of times interest is compounded per year & t = time in years.
The given principal is $1600 for 11 years & amount is $2400 compounded quarterly.
To find the rate of interest compounded quarterly for 11 years substitute the given values in the above formula i.e
[tex]2400=1600(1+r/4)^{11*4}[/tex]
r=3.70%.
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A state's labor department conducted a study regarding the mean yearly earnings per household in the state. The department randomly selected 533 households in the state to be the sample. The population mean is $49,829.00. The department also found the standard deviation to be $933.00. What is the approximate standard error of the mean, assuming a 99.7% confidence level? 1.75 40.41 17.45 23.09
The approximate standard error of the mean, assuming a 99.7% confidence level is 40.41.
Given population mean (µ) = $49,829
population standard deviation (σ) = $933
the randomly selected households (n) = 533 households
the confidence level = 99.7%
To find the standard error of the mean at a 99.7% of confidence level, we will substitute the above values in the below equation,
Standard error of the mean = σ/√n = 933/√533 ≅ 40.41
From the above analysis, we can conclude that the standard error of the mean at a 99.7% confidence level is 40.41.
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Answer:
40.41
Step-by-step explanation:
Plato/Edmentum
Pls help i rlly need help
Answer:
C. Luisa is incorrect, it's not 0. It's only zero when you get "x=0" as an answer but the variable x got canceled here. And as 3=3, it is a true statement so it includes all real numbers. If you had gotten 3=8 for example, it would be a fake statement, so then no solution.
And for the second part,
1. No solution
5x+24=5x+25
24=25
It's false
2. One solution
12p-7=-3p+8
15p=15
P=1
3. One solution.
3x+20=5x
20 = 2x
X = 10
Today's clients are ages 15, 72, 8, 89, 14, 16, 24, and 38. What is the range of the clients' ages?
Answer:
81
Step-by-step explanation:
The difference between the maximum and minimum values in a collection of data is known as the range. We must first determine the lowest and maximum values before we can determine the age range of the clients.
the minimum age is 8, and the maximum age is 89.
Therefore, the range of the clients' ages is:
Max age - Min age = 89 - 8 = 81
So the range of the clients' ages is 81.
