The function f(x) = (x - 3)² is invertible on the interval [1,2], as each output is related only with a single input.
The inverse of the function is given as follows:
[tex]f^{-1}(x) = -\sqrt{x} - 3[/tex]
How to obtain the inverse function?The function for this problem is defined as follows:
f(x) = (x - 3)²
The x-coordinate of the vertex of the quadratic function is given as follows:
x = 3.
Hence the function is invertible on the domain [1,2], which is entirely on one side of the vertex, and thus each output is related only with a single input.
The inverse function is obtained exchanging the variables x and y, and then isolating y, thus:
x = (y - 3)²
[tex]y - 3 = -\sqrt{x}[/tex]
[tex]y = -\sqrt{x} - 3[/tex]
[tex]f^{-1}(x) = -\sqrt{x} - 3[/tex]
The negative symbol is because the function is on the left side of the vertex.
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All of the following are true statements. According to your Mini-Lesson, which of them is the more formal definition of a function? A Function is a rule where every input has only one output and every input has an output. A Function describes a particular relationship between two quantities: an input quantity and an output quantity. A Function defines a relationship between two Quantities. A Function is a rule that assigns a single, unique output value to each input value. A Function, when graphed, must pass the vertical line test.
A function is a rule that generates one unique output value for every specified input value. Therefore, the fourth statement is true.
In mathematics, the function is something that relates an input to an output. The output obtained will somehow be related to the input. We write the function as f(x). For example, consider f(x) = x². This means that the function f takes the value of x and squares this value. Like when we input x=4. Then the output is f(x)=4²=16.
So, here see the function assigns a single and a unique output value of 16 for the input value of 4. Therefore, option 4 is correct. That is function gives every input value a unique, separate output value.
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You use beads to make design. Of the beads, 1/3 are red and 1/6 are blue. The rest are white. What fraction of the beads are red red or blue?
Answer:
Step-by-step explanation:
1/3 is 2/6
2/6 + 1/6 = 3/6
3/6 = 1/2
answer 1/2 or half
Answer:
It is 1/2
Step-by-step explanation:
You make 1/3 the same denominator so it becomes 2/6.
2/6+1/6=3/6
Now when you simplify this you get 1/2.
What is 1+0 I have no idea what the answer is
Answer: 1
Step-by-step explanation:
1 + 0 = 1
jared knows that distance his cars travels equals the rate at which he drives multiplied by the time he drives the car on one occasion he drove 15 miles in 0.5 hours what equation can jared use to find the rate at which he drove
Answer:
The equation Jared can use to find the rate at which he drove is:
rate (r) = distance (d) / time (t)
So in this case, Jared can plug in the values he knows into the equation:
r = 15 miles / 0.5 hours
Solving for r, he can find the rate at which he drove:
r = 30 miles per hour.
So Jared drove at a rate of 30 miles per hour.
Step-by-step explanation:
PLEASE GIVE BRAINLIEST
would you expect your answer in (1) to be accurate? explain. hint: show that maximum distance is less than 10%.
The accurate of the function is calculated using the fomula O(n).
Function in math is known as an equation that has only one answer for y for every x and it assigns exactly one output to each input of a specified type.
Here we have given that maximum distance is less than 10%.
As we all know that the basic approach is to check only consecutive pairs of x and y.
Here we have given that for every element x or y, then we have to check the index of the previous occurrence of x or y and if the previous occurring element is not similar to current element update the minimum distance.
Here we have a question arises what if an x is preceded by another x and that is preceded by a y for that situation we have to know how to get the minimum distance between pairs.
Hereby analyzing closely it can be seen that every x followed by a y or vice versa can only be the closest pair so ignore all other pairs.
Complete question:
Write the function to calculate the accuracy? show that maximum distance is less than 10%.
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According to mars, the makers of m&ms their factory produces 25%blue 25%orange, 12.5% for each remaining colors . How do the observed values based on the entire lasso compare to these?25 blue, 25 orange, 12.5 yellow, 12.5 green, 12.5 brown and 12.5 red
Answer: According to Mars, the makers of M&Ms, their factory produces 25% blue, 25% orange, and 12.5% for each of the remaining colors, which are yellow, green, brown, and red.
The observed values based on the entire lasso are 25 blue, 25 orange, 12.5 yellow, 12.5 green, 12.5 brown, and 12.5 red. These values match the expected values according to the manufacturer, which is 25% blue, 25% orange, and 12.5% for each of the remaining colors.
So, the observed values based on the entire lasso compare to the expected values from the manufacturer. The proportion of colors in the observed sample is the same as the proportion of colors that the factory produces, meaning that the factory is producing the expected ratio of colors.
Step-by-step explanation:
Read the following prompt and type your response in the space provided.
Explain how the associative property can be used to solve this problem.
8/9 / 7/5 x 3/16
The associative property shows that the value of 8/9 / 7/5 x 3/16 is 5/24.
How to calculate the value?The associative property states that the way in which we group the numbers or variables in an arithmetic operation does not affect the result. In other words, we can change the grouping of the numbers or variables by using parentheses without changing the outcome of the calculation.
In this problem, we can use the associative property to group the fractions in a way that makes it easier to simplify.
= 8/9 ÷ 7/5 x 3/16
= (8/9) ÷ (7/5) x 3/16
= 8/9 x 5/7 x 3/16
= 8/9 x 15/112
= 120/1008
= 5 / 24
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A fuel pump at a gasoline station doesn't always dispense the exact amount displayed on the meter. When the meter reads 1.000 L, the amount of fuel a certain pump dispenses is normally distributed with a mean of 1 L and standard deviation of 0.05 L. Let X represent the amount dispensed in a random trial when the meter reads 1.000 L Find P(0.9
The value of P(x < 1) is 0.5.
What is probability?It is the chance of an event to occur from a total number of outcomes.
The formula for probability is given as:
Probability = Number of required events / Total number of outcomes.
We have,
Mean = 1
Standard deviation = 0.05
The P-value of Z when x = 1.
z = (x - mean) / standard deviation
z = (1 - 1)/0.05
z = 0
The P-value when z = 0 is 0.5.
So,
P(x < 1) = 0.5
Thus,
P(x < 1) is 0.5.
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Simplify and write the trigonometric expression as a single squared trig function, with no fractions:
Tanx cscx =
The simplification of the trigonometric function tan x csc x is; sec x
How to simplify trigonometric functions?The six trigonometric functions are defined as;
sine (sin), cosine (cos), tangent (tan), cotangent (cot), cosecant (cosec), and secant (sec).
Trigonometric Ratios are defined as the values of all the trigonometric functions based on the value of the ratio of sides in a right-angled triangle.
Now, we know that;
cot x = cos x/sin x
sec x = 1/cos x = (cos x)⁻¹
csc x = 1/sin x = (sin x)⁻¹
tan x = sin x/cos x
Thus;
tan x csc x = (sin x/cos x) * (1/sin x)
= 1/cos x = (cos x)⁻¹
= sec x
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7/12 simplified
i need this as quick as possible
Answer: 7/12 can't be simplified anymore. Have a nice day ;)
2. Which of these gives the better price for a item priced at $150: a 10/10/10
discount or a 15/15 discount?
Answer:
10/10/10
Step-by-step explanation:
Martin's mom has a large box of toys. The box contains 80 different toys. She wants to divide the large box into smaller boxes. She wants each of the smaller boxes to have 10 toys. After she divided the toys, she put 28 more toys into one of the boxes. How many toys are in that box?
Answer: 38 toys
If each of the smaller boxes has 10 toys, and Martin's mom added 28 more toys into only one of the boxes, that box would have a total of 38 toys.
10 + 28 = 38 toys
A new company president is saidto uave causethe company to do180. Before the new president, the company was losing 10,000 per month. What is the company most likely doing under the new president?
The company, under the new president, is most likely, earning a profit of $ 10, 000 per month.
What does it mean to do a 180 ?When a person is said to have done a 180, it means that they are now doing the opposite of what they used to do. For example, a person who was eating only fast foods doing a 180 would end up eating only healthy food.
This means that the new president doing a 180, means that the company is doing the opposite of what they were doing and instead of making losses of $ 10, 000 monthly, they are making profits of $ 10, 000 every month.
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please help will mark BRAINLIEST
find the volume of the solid generated by revolving the region bounded by the parabola y=-(x^2)/9 and the lines y=-1 about the following lines a. the line y=-1 b. the line y=-2 c. the line y=1
The volume of the solid generated by revolving about the line y=-1 is 324π cubic units and the volume of the solid generated by revolving about the line y=-2 is 216π cubic units.
To find the volume of the solid generated by revolving the region bounded by the parabola y=-(x^2)/9 and the lines y=-1 about a line, we can use the method of cylindrical shells.
a) The line y=-1 is the axis of revolution, so the volume of the solid generated by revolving the region about this line is:
V = ∫[(-1-(x^2)/9)^2 - (-1)^2] * 2πx dx
where the integral is taken over the interval for which the parabola y=-(x^2)/9 is above the line y=-1. So the limits of the integral are from -3 to 3, the volume is:
V = ∫[(-1-(x^2)/9)^2 - (-1)^2] * 2πx dx from -3 to 3 = 324*π cubic units
b) The line y=-2 is the axis of revolution, so the volume of the solid generated by revolving the region about this line is:
V = ∫[(-2-(x^2)/9)^2 - (-2)^2] * 2πx dx
where the integral is taken over the interval for which the parabola y=-(x^2)/9 is above the line y=-2. So the limits of the integral are from -3 to -sqrt(92) = -3 to -3sqrt(2) and 3*sqrt(2) to 3 , the volume is:
V = ∫[(-2-(x^2)/9)^2 - (-2)^2] * 2πx dx from -3 to -3sqrt(2) + ∫[(-2-(x^2)/9)^2 - (-2)^2] * 2πx dx from 3sqrt(2) to 3 = 216*π cubic units
c) The line y=1 is not on the same side of the parabola y=-(x^2)/9 as the region bounded by the parabola and the lines y=-1 so there is no solid generated by revolving about this line.
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In the 30-60-90 triangle below, side s has a length of________ and side q has a length of_______
A. 16√3, 5
B. 8-5, 16
C. 4, 8√3
D. 4√2, 4√2
E. 4, 4 √3
F. 16√3, 16√3
Answer: Option (E)
4, 4√3
===========================================================
Reason:
The hypotenuse is 8. Half of that is 8/2 = 4, which is the value of "s".
This is the short leg of the right triangle. The short leg is opposite the 30 degree angle. Smallest side opposite the smallest angle.
Then use this 30-60-90 triangle template formula
[tex]\text{long leg} = (\text{short leg})*\sqrt{3}\\\\\text{long leg} = 4\sqrt{3}\\\\[/tex]
determine the values of k such that the given augmented matrix is that of a consistent linear system
a. The value of k from augmented matrix [tex]\left(\begin{array}{ccc}3&- 5&k\\- 9&15&3\end{array}\right)[/tex] is - 1.
b. The value of k from augmented matrix [tex]\left(\begin{array}{ccc}k&1&- 6\\5&- 1&6\end{array}\right)[/tex] is - 5.
The complete question is in the attachment. The augmented matrix is a consistent linear system if
Has at least one solution.The last column in the matrix is not a pivot column.a. The augmented matrix [tex]\left(\begin{array}{ccc}3&- 5&k\\- 9&15&3\end{array}\right)[/tex]
Making the second row become zero R₂ = R₂ + 3R₁
[tex]\left(\begin{array}{ccc}3&- 5&k\\- 9 \:+\: (3 \times 3)&15 \:+\: (3 \times - 5)&3 \:+\: (3 \times k)\end{array}\right)[/tex]
[tex]\left(\begin{array}{ccc}3&- 5&k\\- 9 \:+\: 9&15 \:+\: (- 15)&3 \:+\: 3k\end{array}\right)[/tex]
[tex]\left(\begin{array}{ccc}3&- 5&k\\0&0&3 \:+\: 3k\end{array}\right)[/tex]
From the second row
0x + 0y = 3 + 3k
For the system to be consistent, then
3 + 3k = 0
3k = - 3
k = - 3/3
k = - 1
b. The augmented matrix [tex]\left(\begin{array}{ccc}k&1&- 6\\5&- 1&6\end{array}\right)[/tex]
Making the second row become zero R₂ = R₂ + R₁
[tex]\left(\begin{array}{ccc}k&1&- 6\\5 \:+\: k&- 1 \:+\: 1&6 \:+\: (- 6)\end{array}\right)[/tex]
[tex]\left(\begin{array}{ccc}k&1&- 6\\k \:+\: 5&0&0\end{array}\right)[/tex]
From the second row
(k + 5)x + 0y = 0
For the system to be consistent, then
k + 5 = 0
k = - 5
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When the function f(x) is divided by x-1, the quotient is 1x^2+x-8 and the remainder is -2. Find the function f(x) and write the result in standard form.
Answer: When a polynomial is divided by x-1, the quotient is of the form of ax^2 + bx + c and the remainder is of the form of r. The quotient and the remainder are related by the formula:
f(x) = (x-1)(ax^2 + bx + c) + r
Given that the quotient is 1x^2+x-8 and the remainder is -2, we can substitute these into the above equation:
f(x) = (x-1)(1x^2+x-8) + (-2)
f(x) = x^3 - 9x^2 + (1-8)x +(-2)
f(x) = x^3 - 9x^2 - 7x - 2
Therefore the function f(x) is x^3 - 9x^2 - 7x - 2 in standard form.
Step-by-step explanation:
Each year, the newspaper America at a Glance writes an article about the performance of students on a particular nationwide math test. Included in t
this year was the following frequency distribution. It summarizes the test scores from a study which looked at 49 students who completed preperacion
for the test.
Mathematics test score
550 to 599
600 to 649
650 to 699
700 to 749
750 to 799
Frequency
5
11
14
12
7
Based on the frequency distribution, using the midpoint of each data class, estimate the mean mathematics test score for the students in the study for your
intermediate computations, use four or more decimal places, and round your answer to one decimal place.
Answer: To find the mean test score, we need to calculate the sum of the product of each score and its corresponding frequency, and divide it by the total number of students.
To estimate the mean mathematics test score, we can use the midpoint of each data class as the value for each score and multiply it by the corresponding frequency.
For the first class, 550 to 599, the midpoint is (550+599)/2 = 574.5
For the second class, 600 to 649, the midpoint is (600+649)/2 = 624.5
For the third class, 650 to 699, the midpoint is (650+699)/2 = 674.5
For the fourth class, 700 to 749, the midpoint is (700+749)/2 = 724.5
For the fifth class, 750 to 799, the midpoint is (750+799)/2 = 774.5
The mean test score is:
(5574.5 + 11624.5 + 14674.5 + 12724.5 + 7*774.5) / 49 = 669.3
Therefore, the estimate of the mean mathematics test score for the students in the study is 669.3.
Step-by-step explanation:
The graph of the linear function passes through the points (4, 24) and (6, 30).
What is the equation of the function?
y =x +?
The equation of the graph that passes through the points (4, 24) and (6, 30) is y = 3x + 12
What is the equation of line of the graph?The slope-intercept form is expresses as;
y = mx + b
Where m is slope and b is y-intercept.
The point-slope form is expressed as;
y - y₁ = m( x - x₁ )
Given that the the graph passes through the points (4, 24) and (6, 30).
First, determine the slope.
Slope m = ( y₂ - y₁ )/( x₂ - x₁ )
Slope m = ( 30 - 24 )/( 6 - 4 )
Slope m = ( 6 )/( 2 )
Slope m = 3
Next, plug the slope m = 3 and point 1 ( 4,24) into the point-slope form and simplify.
y - y₁ = m( x - x₁ )
y - 24 = 3( x - 4 )
y - 24 = 3x - 12
y = 3x - 12 + 24
y = 3x + 12
Therefore, the equation of line is y = 3x + 12,
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In the article Sweets for the Sweets On Study Island
The information about the acids that are created from the sugar in sweets and bacteria from teeth comes from
A. Smile Magazine .
B. Science and Service Magazine.
C. Jaws.
D. The journal of the American Medical Association.
Correct option is C, The information about the acids that are created from the sugar in sweets and bacteria from teeth comes from Jaws.
What Candy Is Not So Bad for Your Teeth?gum, lollipops, and hard sweets without additional sugar.
Gum with a high Xylitol content, like Icebreaker Ice Cubes, makes you salivate more, which helps kill dangerous bacteria. Hard sweets and sugar-free lollipops can also stimulate the salivary glands.
Black chocolate offers more antioxidants than milk chocolate and less sugar.
As long as it is eaten in moderation and leaves your teeth bushed, the occasional candy piece is acceptable.
The greatest strategy to stop sugar's harmful effects on teeth is to always pay close attention. We advise you to limit your sugar intake, regularly brush away the bacteria-filled plaque on your teeth, and consume nutritional meals to maintain strong teeth.
Combining routine dental exams with fluoride treatments from our dental team will give you and your family the best chance of preventing tooth decay.
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a room 27 feet by 32 feet is to be carpeted all to wall. the width of the carpet is 27 inches. the length, in yards, of the carpet needed for this room is
The length in yards, of the carpet needed for this room is 128 yards.
length of the room = 32 ft
breadth of the room = 27 ft
so we find area now,
area = length X breadth
area = 32 X 27
area = 864ft^2
The width of the carpet in feet is = 2.25ft
so , to find the length of the carpet needed, you must apply the formula for calculate the area of a rectangle and solve for the length, as following,
area = length X width
864 = length X 2.25
length = 864/ 2.25
length = 384ft
length = 128yards
so the length, in yards, of the carpet needed for this room is 128 yards.
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Question 15 of 25
Which relation is not a function?
OA. X 126
18 24 36
y 6 12 6 30 24
OB. ((7,3), (3, 7), (1, 5), (5, 4), (0, 6)}
OC. {(6,5), (3, 2), (2, 4), (6, 3), (1, 0))
O D.
·0€
7
34
Answer:
Step-by-step explanation:oc (6,5), (3,2), (2,4) (6, 3), (1, 0)
given directed line segment ab determine point x on the graph such that x partitions the segment from a to b in the ratio 1:4 and give it's coordinates
The coordinates of X are (-6.4, 5.75)
How to determine the coordinatesFrom the question, we have the following parameters that can be used in our computation:
A = (-10,7)
B = (8,-5).
m : n = 1 : 4
The coordinate is then calculated as
X = 1/(m + n) * (mx2 + nx1, my2 + ny1)
So, we have
X = 1/5 * (1 * 8 + 4 * -10, 1 * -5 + 4 * 7)
Evaluate
X = (-6.4, 5.75)
Hence, the coordinate is (-6.4, 5.75)
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Complete question
given directed line segment ab determine point x on the graph such that x partitions the segment from a(-10,7) to b(8,-5) in the ratio 1:4 and give it's coordinates
Use dimensional analysis to convert the quantity to the indicated units. If necessary, round the answer to two
decimal places.
100 in. to ft
O 1200 ft
8.33 ft
O 16.67 ft
33.33 ft
The dimensional analysis conversion of 100 inch to feet is 8.33 feet.
What is dimensional analysis?Dimensional analysis is the study of the relationships between different physical quantities in engineering and science by identifying their base quantities and units of measurement and tracking these dimensions as calculations or comparisons are performed. when there is insufficient information to set up precise equations, a method of analysis in which physical quantities are expressed in terms of their fundamental dimensions. Dimensional Analysis (also known as the Factor-Label Method or the Unit Factor Method) is a problem-solving strategy that takes advantage of the fact that any number or expression can be multiplied by one without changing its value.
Here,
100 inch to feet,
=100/12
=8.33 feet
The conversion of 100 inch to feet using dimensional analysis is 8.33 feet.
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in 2012 the population of california was 38,041,430, and the population living in the united states but outside california was 275,872,610. during the year, it is estimated that 748,252 persons moved from california to elsewhere in the united states, while 493,641 persons moved to california from elsewhere in the united states.
The migration matrix that has been found is
[0.98033 0.00179
0.01967 0.99821 ] by matrix method.
The 2012 populations of California and the US minus California are 38,041,430 and 275,872,610, respectively, according to the exercise statement. Let's represent these quantities by the variables c and u. What we want are two vectors m1 and m2 such that
cm 1+um 2 = [ 2013 population of California
2013 population of US minus California ]
m1 = [ 0.98033 0.01967 ] and m2 = [ 0.00179 0.99821]
Now that we have the two vectors for equation ,we use the fact that a vector equation of the form
au + bv = w
is equivalent to a matrix equation of the form
[ u v ] [a b] = w
to get our migration matrix for this system.
[0.98033 0.00179
0.01967 0.99821 ]
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The complete question is :
In 2012 the population of California was 38,041,430, and the population living in the United States but outside California was 275,872,610. During the year, it is estimated that 748,252 persons moved from California to elsewhere in the United States, while 493,641 persons moved to California from elsewhere in the United States. Set up the migration matrix for this situation, using five decimal places for the migration rates into and out of California. Let your work show how you produced the migration matrix.
a particle follows the path where x and y are in feet. if the particle starts from rest when , and accelerates along the path at 0.5 ft/s2, determine the radius of curvature, , the velocity,, the normal acceleration, , and the magnitude of the acceleration, , after it travels 30 feet along the path to the point (2.806, 142). (1410 ft, 5.48 ft/s, 0.0213 ft/s2, 0.500 ft/s2)
The radius of curvature is 1410 ft, the velocity is 5.48 ft/s, the normal acceleration is 0.0213 ft/s2, and the magnitude of the acceleration is 0.500 ft/s2 after the particle travels 30 ft along the path to (2.806, 142).
The radius of curvature, r, can be calculated using the formula r = v2/a, where v is the velocity and a is the normal acceleration. The velocity is calculated by integrating the acceleration with respect to time. The normal acceleration is calculated by taking the derivative of the velocity with respect to time. The magnitude of the acceleration is the same as the acceleration given in the problem.
1. Calculate the velocity, v, by integrating the acceleration (0.5 ft/s2) with respect to time:
v = 0.5 * t = 0.5 * 30 s = 15 ft/s
2. Calculate the normal acceleration, a, by taking the derivative of the velocity with respect to time:
a = dv/dt = 0.5 ft/s2
3. Calculate the radius of curvature, r, using the formula r = v2/a:
r = 15 ft/s2/0.5 ft/s2 = 30 ft = 1410 ft
4. The magnitude of the acceleration is the same as the acceleration given in the problem, 0.5 ft/s2.
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on drawing line 1/4-inch long represent a lenght of 1 foot on the same drawing what length represent 4 feet
The line represents 1 foot, so 4 feet would require a line that is 4 times as long as the 1/4 inch line, which is 1 inch.
The formula for finding the length of a line to represent a certain number of feet is L = (1/4)*F, where L is the length of the line, and F is the number of feet. In this case, to find the length of a line to represent 4 feet, the calculation would be L = (1/4)*4 = 1. Therefore, the line would need to be 1 inch long to represent 4 feet. This is because 1/4 inch of the line represents 1 foot, so 4 feet would require a line that is 4 times as long as the 1/4 inch line, which is 1 inch.Therefore, to represent a length of 4 feet on the same drawing, a line 1 inch long needs to be drawn. The calculation is as follows: 1/4 inch line = 1 foot, therefore 4/4 inch line = 4 feet. Therefore, 1 inch line = 4 feet.
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Since the cylinder and the prism share the same height, what is the volume of the prism?
Diameter of cylinder base - 19.8
base area of cylinder - 307.8
volume of cylinder - 9.234
base area of the prism - 196
Answer:
The volume of the prism is 6.186
Step-by-step explanation:
1. The volume of a cylinder can be calculated using the formula V = πr^2h where r is the radius of the cylinder base, h is the height and π is a mathematical constant.
2. The diameter of the cylinder base is given as 19.8 cm and the base area of the cylinder is given as 307.8 cm^2.
3. The volume of the cylinder is given as 9.234 cm^3.
4. If the cylinder and the prism share the same height, the volume of the prism can be calculated by using the base area of the prism.
5. The base area of the prism is given as 196 cm^2.
6. The volume of the prism can be calculated by multiplying the base area with the height of the prism.
7. The volume of the prism is 196cm^2 * h = 6.186 cm^3
8. So the volume of the prism is 6.186 cm^3
let v denote the set of all differentiable real-valued functions defined on the real line. prove that v is a vector sp ace with the operations of addition and scalar multiplication defined in example 3.
The set of all differentiable real-valued functions defined on the real line. Proved that V is a vector space with the operations of addition and scalar multiplication defined as:
(f + g)(s) = f(s) + g(s)
(c f)(s) = c[f(s)].
Let V denote the set of all differentiable real-valued functions defined on the real line. Prove that V is a vector space with the operations of addition and scalar multiplication defined as:
(f + g)(s) = f(s) + g(s)
(c f)(s) = c[f(s)].
To prove vector space, we need to hold eight properties of vector space.
1) Closed
(f + g)(s) = f(s) + g(s) (given)
so, closed holds with respect to given defined operation.
2) Associative law
let defined three function, f,g,h in V,
(f+g)(s)+h(s)=f(s)+g(s)+h(s)
= f(s)+(g+h)(s)
hence associative law hold.
3) Existence of identity,
(f+o)(s)=f(s)
f(s)+o(s)=f(s)
Hence, O(s) is identity for this function,
4) Existences of Inverse,
(-f)(s) is inverse for f(s)
5) Commutative law,
(f+g)(s)=(g+h)(s)
6)( cf)(s)=c[f(s)]g
7) c(f+g)(s)=cf(s)+g(s)
8) c(fg)(s)=(cf(s))g(s)
Hence, the given function is defined Operations.
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