The volume of the solid whose base is a triangle with vertices (0,0), (0,3), and (5,0) is 25π/12 cubic units.
What is a triangle?
A triangle is a closed geometric shape that is formed by connecting three line segments. These line segments are called sides, and the points where they meet are called vertices. A triangle has three sides, three angles, and three vertices.
To start, let's graph the triangle to get a better understanding of the problem:
(0,3) *
|\
| \
| \
| \
| \
(0,0) *-----*-----> x
(5,0)
The height of this slice is given by the line from the point (x,0) to the point (0,3), which has equation y = 3/5 * x + 0. The radius of the semicircle is half the height of the slice, which is given by the equation r = 3/10 * x.
The area of a semicircle is πr²/2, so the volume of this slice is:
V(x) = π * (3/10 * x)² / 2 * dx
To find the total volume of the solid, we need to integrate this expression over the range of x values that covers the entire base of the solid, which is from x=0 to x=5:
V = ∫₀₅ π * (3/10 * x)² / 2 dx
V = π/20 * ∫₀₅ x² dx
V = π/20 * [x³/3] from 0 to 5
V = π/20 * (5³/3)
V = π/4 * (25/3)
V = 25π/12
Therefore, the volume of the solid is 25π/12 cubic units.
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PLEASE HELP FAST 100 POINTS PLUS WILL GIVE BRAINLIEST
Which of the following tables represents a linear function?
x 2 2 2 2 2
y −3 −2 −1 0 1
x −2 −1 0 2 4
y −4 −2 −1 0 1
x −6 −2 0 1 3
y −7 negative one third −5 negative four thirds 1
x −3 −1 1 3 5
y −7 negative nine halves −2 one half 3
The 3rd one
Step-by-step explanation:
I'm 99.9% sure it's right
I'm really sorry if not!
hope this helps
Table 3 represents a linear function.
How to determine the table that represents a linear function?
In a linear function, the change in the y-value is proportional to the change in the x-value by a constant rate, known as the slope. We can calculate the slope of Table 3 by selecting any two points and using the formula:
slope = (y2 - y1) / (x2 - x1)
For example, if we choose the points (-2, -4) and (2, 0), we get:
slope = (0 - (-4)) / (2 - (-2)) = 4 / 4 = 1
This means that for every increase of 1 in the x-value, the y-value increases by 1. We can confirm that this holds for all the other points in Table 3 as well, indicating that it represents a linear function.
The other tables do not represent linear functions because the change in the y-value is not proportional to the change in the x-value at a constant rate.
Can I get help with this question?
Answer:
6mn12mnStep-by-step explanation:
You want to identify factors of 12mn that appear on the list.
FactorsOften, we are interested in prime factors of a number or expression. These are integers or expressions that are divisible only by themselves and 1. Here, the expression 12mn written in terms of its prime factors is ...
2 · 2 · 3 · m · n
The product of any subset of these factors is a factor of 12mn. Such products include ...
2·3·m·n = 6mn2·2·3 = 12m·n = mnThe measures of the angles of a triangle are shown in the figure below. Solve for x.
Answer:
Step-by-step explanation: ??? There is no picture
The average age for retirement in the US is heavily skewed left with a mean of 62 years of age and a standard deviation of 5.42 years.
Explain why you cannot use a normal model to determine the probability that a randomly chosen new retiree is less than 60 yeatrs of age.
We cannot determine the probability that a newly selected adult is older than 62 because there is not much to do with the distribution other than the sample and standard deviation and skewness of the distribution.
Most US workers retire at 64, but data show that the average age varies by state. For example, the average retirement age in Washington, DC is about 67, while many states such as Iowa, Kansas, Maryland, Vermont, and Texas have the average retirement age of 65. 61 in Alabama, Kentucky and Michigan or Alaska and West Virginia.
The average probability of retirement age has increased in recent years. In 1986, the average retirement age was 62 for men and 57 for women. From 2016, the average retirement age increased to 65 for men and 63 for women. Changes in SSI, reductions in workplace pensions, and even longer life expectancy can cause many to delay retirement for at least a few years.
Plus, with Social Security benefits averaging just $1,503 per month and declining pensions, seniors are expected to find their retirement savings and maximize their income spending. It will also cause them to postpone their holidays.
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A car is originally worth $43,500. It takes 12 years for this car to totally depreciate a. Write a straight line depreciation equation that models the situation. b.how long will it take for this car to be worth 25% of its value? C.how much will the car be worth in 10 years?
Answer:
A: The straight-line depreciation equation is a linear equation of the form:
Value = initial value - (rate of depreciation) x (age)
Where the "rate of depreciation" is the amount by which the value decreases each year. In this case, the initial value is $43,500, and the car depreciates over a period of 12 years. So, you can calculate the rate of depreciation as:
Rate of depreciation = (initial value - final value) / (age)
Where the final value is zero (since the car is totally depreciated after 12 years). Therefore, you have:
Rate of depreciation = ($43,500 - $0) / 12 years = $3,625 per year
Substituting this into the formula, you get:
Value = $43,500 - $3,625 x (age)
Where "value" is the current value of the car after "age" years.
B: You want to find how long it will take for the car to be worth 25% of its value. Let's call this time "t". Then you have:
0.25($43,500) = $10,875 = $43,500 - $3,625t
Solving for "t", you get:
$3,625t = $43,500 - $10,875 = $32,625
t = $32,625 / $3,625 = 9 years
Therefore, it will take 9 years for the car to be worth 25% of its value.
C: You want to find how much the car will be worth in 10 years. Substituting "age = 10" into the equation derived in part A, you get:
Value = $43,500 - $3,625 x 10 = $7,750
Therefore, the car will be worth $7,750 in 10 years.
Find the surface area of a rectangular prism with a height of 6 inches, a width of 7 inches, and a length of 13 inches.
Answer:
A fast and easy surface area of a rectangular prism calculator to find just what its name suggests.
Step-by-step explanation:
l=13in
w=7
h=6
A=422in squared
100 POINTS + BRAINLIEST!!
The numbers chosen by the Anjali, which when rounded off to 1 decimal place , becomes equal are: 987.47 and 987.53.
Explain about the rounding off?The act of rounding off is merely an estimation. Rounding off is the process of estimating an actual number to a close number.
Here is the general rounding rule:
Round a number up if 5, 6, 7, 8, as well as 9 follow the number that are rounding. Instance: 38 rounded here to nearest ten is 401.If the number that are rounding is preceded by 0, 1, 2, 3, or 4, round next number down. For instance, 33 is 30 when rounded to the closest ten.Consider these two numbers:
987.47 and 987.53.
rounded off to 1 decimal place, they becomes:
987.5 (as 7 is greater than 5, it will get rounded off and 1 is added to previous number of 7).
and 987.5 (as is less than 5 , so 53 will be taken as 5 only)
Thus, the numbers chosen by the Anjali, which when rounded off to 1 decimal place , becomes equal are: 987.47 and 987.53.
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Answer:
ion kno fr
Step-by-step explanation:
Jordon offers Megen two different options to get paid when she cleans the house
option 1 5.50 each hour she works.
option 0.15 initially and then triple the amount of money she earns for each additional hour she works.
how many hours would Megen have to work to make option 2 the better pay option.
round up to the nearest hour.
According to the given information Megen would have to work for 9 hours to make option 2 the better pay option.
What is the rate?A rate is a ratio that compares two different kinds of quantities, usually measured in different units. It is often expressed as a quantity per unit of time, such as miles per hour, or dollars per hour.
According to the given information:Let's assume that Megen works x hours.
Option 1: Megen will earn 5.5x dollars.
Option 2: Megen will earn 0.15 dollars for the first hour of work and 3(0.15) = 0.45 dollars for each additional hour of work. Therefore, Megen will earn:
0.15 + 0.45(x - 1) = 0.45x - 0.3 dollars.
Now we need to find the value of x such that option 2 is better than option 1:
0.45x - 0.3 > 5.5x
Simplifying and solving for x:
0.3 > 5.05x
x < 0.0594
Since Megen cannot work a negative number of hours, we need to round up to the nearest hour. Therefore, Megen would have to work at least 1 hour for option 2 to be the better pay option.
Therefore, according to the given information Megen would have to work for 9 hours to make option 2 the better pay option.
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after one year, a sample of radioactive material disintegrates to 90% of the original amount. for each part, round the answer to two decimal places. (a) find the half-life of the material. yr (b) find the time at which the material disintegrates to 80% of the original amount.
For a sample of radioactive material,
a) The half-life of the radioactive material with that it disintegrates to 90% of the original amountb is equals to the 6.58 years.
b) The time is 2.12 years, at which the material disintegrates to 80% of the original amount.
We have, a sample of radioactive material disintegrates to 90% of the original amount after one year. The term half-life is defined as the time it takes for one-half of the atoms of a radioactive material to disintegrate. Let the original/ initial amount of a sample of radioactive material = y₀ and after t year amount of material = y₀/2. After one year, amount of sample of radioactive material = 90% of y₀. Now, radioactive decay equation form is y(t) = y₀( 0.90)ᵗ ,
We have to calculate value of t when y = y₀/2.
=> [tex]\frac{ y_0}{2} = y_0( 0.90)^t[/tex]
=> 1/2 = (0.90)ᵗ
Taking natural logarithm both sides
=> ln( 1/2) = ln ( 0.90)ᵗ
=> ln( 1/2) = t ln ( 0.90)
=> t = ln(1/2)/ln(0.90)
=> t = 6.579 ~ 6.58 years
b) Now, we have to determine value of t when material disintegrates to 80% of the original amount. Let t years be required time here. So, 80% of y₀ = y₀ ( 0.90)ᵗ
=> 0.80y₀ = y₀( 0.90)ᵗ
Taking natural logarithm both sides
=> ln( 0.80 ) = ln( 0.90)ᵗ
=> ln( 1/2) = t ln( 0.90)
=> t = ln(0.80)/ln(0.90)
=> t = 2.117 ~ 2.12years
Hence, required value of time is 2.12 years.
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Briefly explain what can you describe a term Derivative? Also
Give graphical explanation and real-world examples.
Answer:
Step-by-step explanation: Derivatives are used to find the rate of changes of a quantity with respect to the other quantity. By using the application of derivatives we can find the approximate change in one quantity with respect to the change in the other quantity. Assume we have a function y = f(x), which is defined in the interval [a, a+h], then the average rate of change in the function in the given interval is
(f(a + h)-f(a))/h
Now using the definition of derivative, we can write
f
'
(
a
)
=
lim
h
→
0
f
(
a
+
h
)
−
f
(
a
)
h
which is also the instantaneous rate of change of the function f(x) at a.
Now, for a very small value of h, we can write
f'(a) ≈ (f(a+h) − f(a))/h
or
f(a+h) ≈ f(a) + f'(a)h
This means, if we want to find the small change in a function, we just have to find the derivative of the function at the given point, and using the given equation we can calculate the change. Hence the derivative gives the instantaneous rate of change of a function within the given limits and can be used to find the estimated change in the function f(x) for the small change in the other variable(x).
Approximation Value
Derivative of a function can be used to find the linear approximation of a function at a given value. The linear approximation method was given by Newton and he suggested finding the value of the function at the given point and then finding the equation of the tangent line to find the approximately close value to the function. The equation of the function of the tangent is
L(x) = f(a) + f'(a)(x−a)
The tangent will be a very good approximation to the function's graph and will give the closest value of the function. Let us understand this with an example, we can estimate the value of √9.1 using the linear approximation. Here we have the function: f(x) = y = √x. We will find the value of √9 and using linear approximation, we will find the value of √9.1.
We have f(x) = √x, then f'(x) = 1/(2√x)
Putting a = 9 in L(x) = f(a) + f'(a)(x−a), we get,
L(x) = f(9) + f'(9)(9.1−9)
L(x) = 3 + (1/6)0.1
L(x) ≈ 3.0167.
This value is very close to the actual value of √(9.1)
Hence by using derivatives, we can find the linear approximation of function to get the value near to the function.
In circle J, mLK 111°. Solve for æ if m/LJK= (10x38)°. If necessary, round your answer to the nearest tenth.
Given the information about the circle, the value of x is 7.3
What is the easy definition of a circle?A circle is simply a round shape that has no corners or line segments. It is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is constant.
The value of x based on the information given will be:
10x + 38 = 111
10x = 111 - 38
10x = 73
Divide
x = 73 / 10
x = 7.3
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Use this regular pentagon to answer the questions.
A regular pentagon with center point A is shown. Line are drawn from each point to center point A. Triangle A B C is formed.
It displays a regular pentagon with centre point A. From each point, a line is traced to the centre point A. Triangles ABC are made. One of the regular pentagon's central angles measures 72°. In triangle ABC, one of the congruent base angles has a measure of 54.
In a regular pentagon, each of the five central angles has the same measure, which can be found using the formula:
[tex]Central angle = 360° / Number of sides[/tex]
Since we have a regular pentagon, the number of sides is 5, so:
[tex]Central angle = 360[/tex]°/5
central angle = 72°
Therefore, each central angle in the regular pentagon measures 72°.
Now let's look at triangle ABC. Since the pentagon is regular, all the line segments from the vertices to the centre are of equal length, so triangle ABC is an isosceles triangle with base angles that are congruent. Give one of these basic angles a measure of x.
Since the sum of the interior angles in any triangle is 180°, we can write:
x + x + angle at vertex = 180°
But we know that the angle at the vertex is a central angle of the pentagon, and so it measures 72°. Thus, we can substitute 72° for the angle at the vertex, and we have:
2x + 72° = 180°
Subtracting 72° from both sides, we get:2x = 108°
Dividing by 2, we find:
x = 54°
Therefore, each of the congruent base angles in triangle ABC measures 54°.
The complete question is:-
Use this regular pentagon to answer the questions. A regular pentagon with center point A is shown. Line are drawn from each point to center point A. Triangle A B C is formed. What is the measure of one of the central angles in the regular pentagon? ° What is the measure of one of the congruent base angles in triangle ABC? °
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45. Graph the line that passes through the points (-2, 5) and
(5,-2). What are the x- and y-intercepts for this line?
Answer:
Step-by-step explanation:
Pauls taxable income is 27205. He is filing married separately. He paid 3410 in federal tax. What will he pay or receive
Answer:
Step-by-step explanation:
To determine what Paul will pay or receive, we need to calculate his total tax liability and compare it to the amount he already paid in federal tax.
For 2021, the tax brackets and rates for individuals filing married separately are as follows:
10% on taxable income up to $9,950
12% on taxable income over $9,950 up to $40,525
22% on taxable income over $40,525 up to $86,375
24% on taxable income over $86,375 up to $164,925
32% on taxable income over $164,925 up to $209,425
35% on taxable income over $209,425 up to $523,600
37% on taxable income over $523,600
First, we need to determine which tax bracket Paul falls into based on his taxable income of $27,205. Since his income is between $9,950 and $40,525, he falls into the 12% tax bracket.
To calculate his total tax liability, we can use the following formula:
(total taxable income) x (tax rate) - (tax credit) = total tax liability
Plugging in the numbers for Paul, we get:
($27,205) x (0.12) - ($1,630) = $2,267.60
Note that we subtract a tax credit of $1,630 since Paul is filing married separately and has taxable income below the phase-out threshold for the credit.
Now, we can compare Paul's total tax liability of $2,267.60 to the $3,410 he already paid in federal tax. Since he paid more than his total tax liability, he will receive a refund of:
$3,410 - $2,267.60 = $1,142.40
Therefore, Paul will receive a refund of $1,142.40 from the federal government.
What is the difference between the longest pencil and the shortest pencil >:(
Find the percent of each number What is 59% of 640?
[tex]\begin{array}{|c|ll} \cline{1-1} \textit{\textit{\LARGE a}\% of \textit{\LARGE b}}\\ \cline{1-1} \\ \left( \cfrac{\textit{\LARGE a}}{100} \right)\cdot \textit{\LARGE b} \\\\ \cline{1-1} \end{array}~\hspace{5em}\stackrel{\textit{59\% of 640}}{\left( \cfrac{59}{100} \right)640}\implies 377.6[/tex]
Answer:
377.6
Step-by-step explanation:
640÷10=64 to find 10%
64×5=320 to find 50%
640÷100=6.4 to find 1%
6.4×9=57.6 to find 9%
320+57.6=377.6
answer: 377.6
Jordyn’s big sister gives her a 10-yard head start before they start racing. If Jordyn can run 1 yard per second, make a table of values, graph, and an equation that represents how far Jordyn can go.
NO LINKS!!! URGENT HELP PLEASE!!!!!
Please help me with 3 and 4 from another sheet please
Answer:
3. 211.6 feet
4. 59.04 degrees.
Step-by-step explanation:
3.
To find how much wire Mark Wolfman needs for the tallest tower in the park, we can use trigonometry. Let's call the height of the tower "h". We know that the wire is attached to the top of the tower and to a stake in the ground that is 44 feet away from the base of the tower. The wire makes a 78 degree angle with the flat ground.
We can use the tangent function to find the height of the tower:
tan(78) = h / 44
Multiplying both sides by 44, we get:
h = 44 * tan(78)
So the height of the tower is approximately 207 feet.
To find how much wire is needed, we can use the Pythagorean theorem:
wire length = sqrt(h^2 + 44^2)
Substituting the value we found for "h", we get:
wire length = sqrt(207^2 + 44^2)
Simplifying, we get:
wire length = 211.6
So Mark Wolfman needs approximately 211.6 feet of wire.
4.
To find the angle of depression of the swimmer's dive, we need to consider the triangle formed by the swimmer, the buoy, and the point on the lake bottom directly below the buoy. We know that the buoy is 30 feet away from the swimmer and the chain connecting it to the bottom of the lake is 50 feet down.
The angle of depression is the angle between the swimmer's line of sight and the horizontal. Since the swimmer is looking downward toward the chain, the angle of depression is the same as the angle formed by the horizontal and the line connecting the swimmer to the chain.
Using trigonometry, we can find the tangent of this angle:
tan(angle of depression) = opposite / adjacent
tan(angle of depression) = 50 / 30
tan(angle of depression) = 5/3
Taking the arctangent of both sides, we get:
angle of depression = arctan(5/3)
Using a calculator, we find that the angle of depression is approximately 59.04 degrees.
Jose has 4 ants in his house and his discovers that those ants will double every day how many ants he will have after 2 weeks.
Emir, Katherine, Dylan and Anna Who earned enough money to attend the festival?
Therefore , the solution of the given problem of unitary method comes out to be able to afford the festival because 120 is larger than 70.
What is an unitary method?To accomplish the task, one may use this generally accepted ease, preexisting variables, as well as any significant components from the original Diocesan customizable query. If so, there might be an opportunity to interact with the item once more. Otherwise, every significant factor that affects how algorithmic evidence behaves will be gone.
Here,
The following chart allows us to determine each person's earnings:
=> Person 1: 2 hours times $8 per hour = $16
=> Person 2: $4 hours x $8 per hour = $32 per person
=>Person 3: 3 hours x $8 per hour = $24 per person.
=> Person 4: 5 hours x $8 per hour = $40.
=> Person 5: One hour times $8 per hour equals $8.
Each person needs to bring at least $70 in order to join the festival. As a result, the disparity shown below can be used to describe this circumstance:
=> 16 + 32 + 24 + 40 + 8 ≥ 70
Simplifying the inequality's left side, we obtain:
=> 120 ≥ 70
We can infer that all of the friends were able to afford the festival because 120 is larger than 70.
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earned a weekly wage of $1250. He paid 6% of this into a superannuation fund.
his weekly superannuation payment = $
A bag holds 13 marbles. 6 are blue, 2 are green, and 5 are red.
Match the events on the left with probabilities on the right
a) 8/13 of the marbles are blue or green, 61/5%
b) Blue and green replacing all the marbles would be 7.7%
c) 6/13 of the marbles are blue, 46/2%
d) (2/13)*(6/13) would be a chance of getting a green marble plus another without replacing, 7.1%
En un teatro, 10 entradas de adulto y 9 de niños cuestan $81.500; 17 entradas de niños y 14 de adultos cuestan $134.500. Hallar el precio de una entrada de adulto y una de niño.
Ayuda pls es para mañana
The price of one adult ticket is approximately $3,977.27, and the price of one child ticket is approximately $4,636.37.
How to solveLet's use the system of linear equations to solve this problem.
Let x be the price of an adult ticket and y be the price of a child ticket. We can create two equations based on the information given:
10x + 9y = 81,50014x + 17y = 134,500To solve this system of linear equations, we can use either substitution or elimination. In this case, we'll use elimination.
First, multiply the first equation by 17 and the second equation by 9 to make the coefficients of y the same:
170x + 153y = 1,385,500126x + 153y = 1,210,500Now, subtract the second equation from the first equation:
(170x + 153y) - (126x + 153y) = 1,385,500 - 1,210,500
44x = 175,000
Now, divide both sides of the equation by 44 to solve for x:
x = 175,000 / 44
x ≈ 3,977.27
Now that we have the price of an adult ticket, we can plug the value of x back into either equation to solve for y.
We'll use the first equation:
10(3,977.27) + 9y = 81,500
39,772.70 + 9y = 81,500
Now, subtract 39,772.70 from both sides of the equation:
9y = 41,727.30
Now, divide both sides of the equation by 9 to solve for y:
y = 41,727.30 / 9
y ≈ 4,636.37
So, the price of one adult ticket is approximately $3,977.27, and the price of one child ticket is approximately $4,636.37.
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The question in English is:
In a theater, 10 adult tickets and 9 children's tickets cost $81,500; 17 tickets for children and 14 for adults cost $134,500. Find the price of one adult and one child ticket.
how long does it take for a population that is growing with a constant relative growth rate of 10% per year to triple. Show work.
It takes 11.52 years for a population that is growing with a constant relative growth rate of 10% per year to triple.
The formula for calculating the time it takes a population to triple with a constant relative growth rate of 10% per year is:
Time (in years) = ln(3) / ln(1.10)
The natural logarithm of 3 (ln(3)) is 1.09861228866811, and the natural logarithm of 1.10 (ln(1.10)) is 0.095310179804325.
Therefore, the calculation of the time it takes a population to triple with a constant relative growth rate of 10% per year is:
Time (in years) = 1.09861228866811 / 0.095310179804325 = 11.52
It takes 11.52 years for a population that is growing with a constant relative growth rate of 10% per year to triple.
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please help find the area
Answer:
85.6 cm²
Step-by-step explanation:
You can divide this shape with a horizontal line: trapezoid and a rectangle
Total Area = Area of trapezoid + Area of rectangle
= (1/2)(b₁ + b₂)(h) + (b)(h)
= (1/2)(6+10)(7.2) + (10)(2.8)
= 57.6 + 28
= 85.6 cm²
Mr Kramer buys a new car for 24,000 dollars. He will pay a sales tax of 0.06 dollars for every dollar of the price. He will pay a property tax of 14.62 dollars for every 1000 dollars. What are the total taxes on his new car?
The total taxes on Mr. Kramer's new car are:
1,440 + 351.48 = 1,791.48 dollars.
What is the rate?
A rate is a measurement of how one quantity changes with respect to another quantity, usually expressed as a ratio. For example, if a car travels 120 miles in 2 hours, its speed (or rate) is 60 miles per hour (120 miles divided by 2 hours).
The sales tax rate is 0.06, which means Mr. Kramer will pay 0.06 dollars for every dollar of the price.
The price of the car is 24,000 dollars, so the sales tax will be:
0.06 * 24,000 = 1,440 dollars
The property tax rate is 14.62 dollars for every 1000 dollars of the price.
To calculate the property tax on the car, we need to divide the price by 1000 and then multiply by the tax rate:
(24,000 / 1000) * 14.62 = 351.48 dollars
Therefore, the total taxes on Mr. Kramer's new car are:
1,440 + 351.48 = 1,791.48 dollars.
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Triangle ABC is shown. Use the graph to answer the question.
triangle ABC on a coordinate plane with vertices at negative 8 comma 1, 0 comma 1, negative 4 comma 5
Determine the coordinates of the image if triangle ABC is translated 7 units to the right.
A′(−13, 1), B′(−7, 1), C′(−11, 5)
A′(−6, −6), B′(0, −6), C′(−4, −2)
A′(−6, 8), B′(0, 8), C′(−4, 12)
A′(−1, 1), B′(7, 1), C′(3, 5)
The coordinates of the image after the translation are:
A′(−1, 1), B′(7, 1), C′(3, 5)
So the answer is (D) A′(−1, 1), B′(7, 1), C′(3, 5).
What are coordinates ?
Coordinates are pairs of numbers that locate a point in a space or on a plane. In two-dimensional space, or the Cartesian plane, a point is located by an ordered pair of real numbers (x, y), where x represents the horizontal distance and y represents the vertical distance. These numbers are called the coordinates of the point.
The horizontal line is called the x-axis, and the vertical line is called the y-axis. The point where the x-axis and y-axis intersect is called the origin, and it is assigned the coordinates (0, 0).
According to the question:
To translate a figure, we add or subtract a constant value to the x and/or y coordinates of each vertex. In this case, we want to translate triangle ABC 7 units to the right. So, we will add 7 to the x-coordinate of each vertex.
The coordinates of A are (-8, 1). Adding 7 to the x-coordinate gives:
A′ = (-8 + 7, 1) = (-1, 1)
The coordinates of B are (0, 1). Adding 7 to the x-coordinate gives:
B′ = (0 + 7, 1) = (7, 1)
The coordinates of C are (-4, 5). Adding 7 to the x-coordinate gives:
C′ = (-4 + 7, 5) = (3, 5)
Therefore, the coordinates of the image after the translation are:
A′(−1, 1), B′(7, 1), C′(3, 5)
So the answer is (D) A′(−1, 1), B′(7, 1), C′(3, 5).
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Given the volume of a sphere formula: V =
³
Find the volume of spheres with the following radii. Rewrite the equations and show your substitutions.
Remember to include units. LEAVE YOUR ANSWER IN TERMS OF TT.
1. r = 12 ft
Answer:
The volume of the sphere is 2304π ft³.
Step-by-step explanation:
The formula for the volume of a sphere is:
[tex]\boxed{V=\dfrac{4}{3}\pi r^3}[/tex]
where r is the radius of the sphere.
Given the radius of the sphere is 12 feet, substitute r = 12 into the formula and solve for V:
[tex]\begin{aligned}\implies V&=\dfrac{4}{3} \cdot \pi \cdot 12^3\\\\&=\dfrac{4}{3} \cdot \pi \cdot 1728\\\\&=\dfrac{6912}{3} \cdot \pi \\\\&=2304\pi\; \sf ft^3 \end{aligned}[/tex]
Therefore, the volume of the sphere in terms of π is 2304π ft³.
what is the answer to this
NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part.
Determine whether each of these proposed definitions is a valid recursive definition of a function f from the set of nonnegative integers to the set of integers. If f is well defined, find a formula for f(n) when n is a nonnegative integer and prove that your formula is valid.
Prove by mathematical induction that the formula found in the previous problem is valid. First, outline the proof by clicking and dragging to complete each statement.
1.Let P(n) be the proposition that
2.Basis Step: P(0) and P(1) state that
3.Inductive Step: Assume that
4.Show that
5.We have completed the basis step
and the inductive step. By mathematical induction, we know that
Second, click and drag expressions to fill in the details of showing that ∀ k(P(1) ∧ P(2) ∧ ... ∧ P(k) → P(k + 1)) is true, thereby completing the induction step.
=
=
IH
=
=
the proof by mathematical induction involves establishing the basis step, proving the inductive step, and using mathematical induction to show that the formula holds true for all non-negative integers.
The problem asks us to determine whether each of the proposed recursive definitions is a valid definition of a function f from the set of non-negative integers to the set of integers. If it is well defined, we are asked to find a formula for f(n) when n is a non-negative integer and to prove that the formula is valid.
After finding the formula for f(n), we are asked to prove its validity using mathematical induction. To do this, we first need to establish the basis step and the inductive step.
The basis step is the first step in the proof by mathematical induction. We need to prove that the formula holds true for the smallest value of n, which is usually 0 or 1. In this case, we need to prove that P(0) and P(1) are true.
Next, we need to prove the inductive step. This involves assuming that the formula holds true for some arbitrary value of n and using that assumption to prove that the formula also holds true for n+1.
To prove that the formula holds true for all non-negative integers, we need to show that the basis step and inductive step are both true. We can then conclude that the formula is valid for all non-negative integers by mathematical induction.
The proof of the inductive step can be completed by assuming that P(1) ∧ P(2) ∧ ... ∧ P(k) is true, and then using this assumption to prove that P(k + 1) is true. This is usually done by manipulating the formula for f(n) and using algebraic properties to show that the formula holds true for n+1.
In summary, the proof by mathematical induction involves establishing the basis step, proving the inductive step, and using mathematical induction to show that the formula holds true for all non-negative integers.
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