Let f be a continuous function in c[a, b] such that f(a) = 200. Then for all real numbers c, the scalar multiple cf is also a continuous function in c[a, b]. Specifically, cf(a) = c(200) = 200c.
To demonstrate that c[a, b] is a subspace, the following facts must be proved:
1. If f and g are both continuous functions in c[a, b], then the sum f + g is also a continuous function in c[a, b].
2. If f is a continuous function in c[a, b], then the scalar multiple cf is also a continuous function in c[a, b], where c is a real number.
3. The zero vector of c[a, b] is the constant zero function.
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—
The performance of a cyclist in a race is affected by whether it rains or not that day.
The probability the cyclist crashes is
4/5 when it's raining and
1/10 when it's dry.
On any given day, the probability of rain is
1/4
What is the probability it was raining given the cyclist crashes?
In light of the cyclist crashes, the likelihood of it being raining is 16/33, or roughly 0.485.
What are examples and probability?It is predicated on the likelihood that something will occur. The fundamental underpinning of theoretical probability is the justification for probability. For instance, the theoretical chance of receiving a head while tossing a coin is ½. If it rains, let R stand for that, and if the cyclist crashes, let C stand for that. We're looking for P(R|C), or the likelihood that it will rain given the crashes involving cyclists. The Bayes theorem can be used to resolve this issue:
P(R|C) = P(C|R) × P(R) / P(C)
Where P(R) is the likelihood of rain, P(C) is the likelihood that the cyclist will crash regardless of the weather, and P(C|R) is the likelihood that the cyclist will crash given that it is raining.
According to the problem statement, we know that:
P(C|R) = 4/5 (the probability of crashing given that it's raining)
P(C|not R) = 1/10 (the probability of crashing given that it's not raining)
P(R) = 1/4 (the probability of rain)
We may use the rule of total probability to determine P(C):
P(C) = P(C|R) × P(R) + P(C|not R) × P(not R)
where P(not R) = 1 - P(R) = 3/4.
When we change the values, we obtain:
P(C) = (4/5) × (1/4) + (1/10) × (3/4) = 11/40
We can now enter each value into Bayes' theorem as follows:
P(R|C) = (4/5) × (1/4) / (11/40) = 16/33
As a result, there is a 16/33, or roughly 0.485, chance that it had been raining when the cyclists crashed.
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in the same distribution (mean is 70 and the standard deviation is 8. at least what fraction are between the following pairs
At least 68% of the data falls between 62 and 78, and at least 95% of the data falls between 54 and 86.
How to find the minimum fraction of the data?To answer this question, we can use the empirical rule, also known as the 68-95-99.7 rule, which tells us that for a normal distribution:
About 68% of the data falls within one standard deviation of the meanAbout 95% of the data falls within two standard deviations of the meanAbout 99.7% of the data falls within three standard deviations of the meanUsing this rule, we can estimate the fraction of the data that falls between the following pairs:
Between 54 and 86:
To find the range of values that is within two standard deviations of the mean, we can subtract and add two standard deviations from the mean:
Lower bound: 70 - 2*8 = 54
Upper bound: 70 + 2*8 = 86
So, about 95% of the data falls between 54 and 86.
Between 62 and 78:
To find the range of values that is within one standard deviation of the mean, we can subtract and add one standard deviation from the mean:
Lower bound: 70 - 8 = 62
Upper bound: 70 + 8 = 78
So, about 68% of the data falls between 62 and 78.
Therefore, we can conclude that at least 68% of the data falls between 62 and 78, and at least 95% of the data falls between 54 and 86.
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Write the reciprical of 2/3
Answer:
the answer is 3/2
Step-by-step explanation:
Answer:
the answer si 3/2
 HELP PLEASE!!!
calculate the distance between the points B= (0,6) and M= (8, -2) on the coordinate plane . round to the nearest 100th.
Answer:
Step-by-step explanation:
Using the distance formula: [tex]d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]
[tex]d=\sqrt{(8-0)^2+(-2-6)^2}[/tex]
[tex]=\sqrt{64+64} \\[/tex]
[tex]=\sqrt{128}[/tex]
[tex]=11.3[/tex]
Subtract the given equation
3x-(4x-11)
Answer:
3x - (4x - 11) = 3x - 4x + 11 = -x + 11
Step-by-step explanation:
The charity drive was held over two days. On the first day, the 20 artists worked for 6 hours. On the second day, 25 artists turned up and worked for 12 hours. Find the total number of art installations completed over the two days.
Answer:
The charity drive was held over two days. On the first day, the 20 artists worked for 6 hours. On the second day, 25 artists turned up and worked for 12 hours. Find the total number of art installations completed over the two days.
Step-by-step explanation:
Let's start by finding the total number of art installations completed on the first day by the 20 artists who worked for 6 hours. If each artist creates one art installation, then the total number of installations created by the 20 artists is:
20 artists x 6 hours/artist = 120 art installations
On the second day, 25 artists worked for 12 hours. Using the same assumption that each artist creates one art installation, the total number of installations created by the 25 artists is:
25 artists x 12 hours/artist = 300 art installations
Therefore, the total number of art installations completed over the two days is:
120 + 300 = 420
So, the charity drive produced 420 art installations over the two days.
Consider the following statements about a system of linear equations with augmented matrix A. In each case either prove the statement or give an example for which it is false.a. If the system is homogeneous, every solution is trivial.b. If there exists a trivial solution, the system is homogeneousNow assume that the system is homogeneous.c. If there exists a nontrivial solution, there is no trivial solution.
In conclusion for a. If the system is homogeneous , every solution is trivial true. b. If there exists a trivial solution, the system is homogeneous false. c. If there exists a nontrivial solution, there is no trivial solution false.
How to solve?
a. If the system is homogeneous, every solution is trivial.
This statement is true. A homogeneous system of linear equations has the form Ax = 0, where A is the coefficient matrix and x is the vector of variables. The trivial solution is always x = 0, which satisfies the equation. Any other solution would require a nonzero x vector, but then Ax would be nonzero, contradicting the fact that it equals zero in a homogeneous system.
b. If there exists a trivial solution, the system is homogeneous.
This statement is false. A system of linear equations can have a trivial solution (i.e., all variables are equal to zero) without being homogeneous. For example, the system
x + y = 0
2x + 2y = 0
has a trivial solution (x = 0, y = 0) but is not homogeneous.
c. If there exists a nontrivial solution, there is no trivial solution.
This statement is false. A homogeneous system of linear equations can have both trivial and nontrivial solutions. For example, the system
x + y = 0
2x + 2y = 0
has both a trivial solution (x = 0, y = 0) and a nontrivial solution (x = 1, y = -1).
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use the y-and -x intercept to write the equation of the line y intercept (0,6), x intercept (-2,0)
Answer:
3x -y = -6
Step-by-step explanation:
You want the equation of the line with intercepts (0, 6) and (-2, 0).
Intercept formThe equation of the line with x-intercept 'a' and y-intercept 'b' is ...
x/a +y/b = 1
For the given intercepts, the equation is ...
x/(-2) +y/6 = 1
Standard formIn standard form, we want the leading coefficient positive and the integer coefficients mutually prime. We can get there by multiplying by -6:
3x -y = -6
__
Additional comment
You can get slope-intercept form by solving for y, or you can recognize that ...
slope = rise/run = -(y-intercept)/(x-intercept) = -6/-2 = 3
Since you already know the y-intercept, you can write the slope-intercept equation as ...
y = 3x +6
There are perhaps a dozen or more forms of the equation for a line. The "intercept form" equation is one of the more useful ones.
why is the probability that a continuous random variable is equal to a single number zero? (i.e. why is P(X=a)=0 for any number a)
The probability that a continuous random variable takes on a single specific value is zero because the continuous random variable can take on any value within a range or interval, and there are an infinite number of values that it can take on within that interval.
The probability that a continuous random variable is equal to a single number is zero. The explanation is that a continuous random variable has an infinite number of possible values. As a result, the probability of any single value, including the one you're interested in, is always 0.
To put it another way, it's almost impossible for a continuous random variable to equal a specific value. Consider the following example:
Suppose you're attempting to hit a target on a wall. If the target is the size of a pinpoint, the probability of striking it is virtually zero. In the same way, with continuous random variables, the probability of landing on a single point is practically zero because there are an infinite number of possible outcomes. As a result, the probability of any one of them occurring is tiny.
A continuous random variable may be defined over any interval of real numbers, which is one of its distinguishing features. The amount of time it takes to finish a marathon, the height of a randomly chosen person in a group, and the length of a telephone conversation are all examples of continuous random variables.
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find the inverse of the function
F(x) = x² + 2x₁ [-1, 00]
PLEASE HELP MARKING BRAINLEIST JUST ANSWER ASAP
Answer:
3c + 19
Step-by-step explanation:
Perimeter: P = a + b + c
P = (c + 10) + (c + 6) + (c + 3) = 3c + 19
b) There are x number of books that worth Rs. 35 each and 5 books worth Rs. 30 each in a parcel prepared as a gift. The value of two such parcels is Rs. 580. i. Build up an equation using the above information. ii. Find the value of x by solving the equation.
Answer:
Equation: 2(357+30×5) = 580
x=4
Step-by-step explanation:
In one package, there is such a relationship:
357+30X5 = y
(Y is the price of a package)
The price of two parcels is 580:
then. 24=580
y= 290
x=4, so: equation: 2(35x+150) =580
Step-by-step explanation:
A shopkeeper buys a number of books for Rs. 80. If he had bought 4 more for the same amount each book would have cost Rs. 1 less. How many books did he buy?
A
8
B
16
Correct Answer
C
24
D
28
Medium
Open in App
Updated on : 2022-09-05
Solution

Verified by Toppr
Correct option is B)
Let the shopkeeper buy x number of books.
According to the given condition cost of x books =Rs80
Therefore cost of each book =x80
Again when he had brought 4 more books
Then total books in this case =x+4
So cost of each book in this case =x+480
According to Question,
x80−x+480=1
x(x+4)80(x+4)−80x=1
x2+20x−16x−320=0
(x−16)(x+20)=0
x=16orx=−20
Hence the shopkeeper brought 16 books
A car rental company's standard charge includes an initial fee plus an additional fee for each mile driven. The standard charge S (in dollars) is given by the
function S=18.95 +0.60 M, where M is the number of miles driven.
The company also offers an option to insure the car against damage. The insurance charge I (in dollars) is given by the function I=4.90 +0.25 M.
Let C be the total charge (in dollars) for a rental that includes insurance. Write an equation relating C to M. Simplify your answer as much as possible.
C =
Hi. The terms of the standard charge and insurance charge equations are a little garbled, but it is possible to think that these are the right equations:
Standard charge, S = 18.95 + 0.60M, where 18.95 is the initial fee and 0.60 is the charge for each mile driven.
Insurance charge, I = 4.90 + 0.25M, where 4.90 also represents an initial fee and 0.25 a charge for each mile driven.
Then, the total cos, C = S + I =>
C = (18.95 + 0.60M) + (4.90 + 0.25M) which you can rearrange to show the like terms one next to each other =>
C = (18.95 + 4.90) + (0.60M + 0.25M), now add up the like terms=>
C = 23.85 + 0.85M which is the equation is the most simplified form
Answer: C = 23.85 + 0.85M
Match the description with its mathematical expression. Residual from an estimated regression equation Choose the correct answer below. obo Oba o O y o bo +61 oy
We have that, the correct match of the description with its mathematical expression is provided with the mathematical expression "or".
What is the correct expression?Correct matching of the description to its mathematical expression is given as follows: The residual of an estimated regression equation is compared to the mathematical expression "or".
Explanation: A residual is the difference between the actual value of a variable and the predicted value of that variable. The residuals are denoted as "or". Therefore, the Residual of an estimated regression equation is compared to the mathematical expression "or".
Other options are incorrect as follows: Oba does not match any description given in the question. Therefore, it is a wrong option. Obo doesn't match any description given in the question. Therefore, it is a wrong option. Oy+ 61 does not match any description given in the question. Therefore, it is a wrong option.
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The lowest temperature ever recorded on earth was -89.2°C recorded in Antarctica in 1983.How many degrees Fahrenheit was that,to the nearest degree
The lowest temperature ever recorded on earth of -89.2°C is approximately equal to -97°F when rounded to the nearest degree Fahrenheit.
To convert the temperature of -89.2°C to Fahrenheit, we can use the formula:
[tex]F = (C * 1.8) + 32[/tex]
Substituting:
°F = (-89.2 × 1.8) + 32
°F = -128.56 + 32
°F = -97
Therefore, the lowest temperature ever recorded on earth of -89.2°C is approximately equal to -97°F when rounded to the nearest degree Fahrenheit. It's important to note that this is just an approximation as we rounded the result to the nearest degree. However, it gives us a good idea of how extremely cold the temperature was. It's worth mentioning that at such low temperatures, it's important to take appropriate precautions to avoid any adverse health effects.
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Match each integer with one that divides it.
18
matches
Choice
9
-14
matches
Choice
7
11
matches
Choice
11
65
matches
Choice
13
Integers are a type of number used in mathematics that represent whole numbers. They are typically denoted by the symbol Z, and can be positive, negative, or zero.
Integers do not include fractions or decimal points, so they are distinct from real numbers.
In Mathematics, integers are the collection of whole numbers and negative numbers.
18 matches 9
-14 matches 7
11 matches - 65
13 matches 65
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Solve for y.
115°
108°
90°
130°
Answer:y=115
Step-by-step explanation:
find the sum of the angles:
=(n-2)180
=(5-2)180
=540
find the value of y:
y=540-(135+112+88+90)
y=115
Answer:
115°
Step-by-step explanation:
the sum of the interior angles of a pentagon is 540°
135+90+112+88+y=540
y= 540-135-90-112-88
y= 155°
Una pelota es lanzada horizontalmente desde la azotea de un edificio de 54 m de altura y Llega al suelo a 32 m de la base. Cuál fue la rapidez inicial de la pelota?
The initial speed of a ball thrown horizontally from the roof of a building at a height of 54 m and hitting the ground at a height of 32 m from the bottom is equal to 9.64 m/sec.
A ball throws horizontally from the roof of a building. This is where we have to take horizontal movement into account.
Height of building = 54 m
Height of ground from base = 32 m Calculates the initial speed of the ball.
Since the ball is thrown horizontally, its initial velocity has no vertical component. When released, the ball is subject to gravity and accelerates downward until it hits the bottom. To determine how long the ball stayed in the air before hitting the ground, use the following formula,
d = 1/2×g×t²
where g --> acceleration due to gravity
t --> time
d --> the distance covered by ball before hitting the ground, g= 9.81 m/s².
So, 54 m = (1/2) × 9,81 m/s² × t²
=> t² = 54/4.90
=> t² = 540/49
=> t² = 11.02 sec²
=> t = 3.31963 ~ 3.32 sec
So the ball will stay in the air for about 3.32 seconds. Let v₀ₓ be the horizontal component of the initial velocity. According to the information of the problem, the ball landed 32 m from the bottom of the building. Let's make the initial position x₀ = 0. Then, the final horizontal displacement of a ball will be equals to 32 metres. Using the distance velocity relation, x = x₀ + v₀ₓt and substituting all known values in it,
=> 32 m = 0 + v₀ₓ × 3.32 sec
=> v₀ₓ = 32/3.32
=> v₀ₓ = 9.6385 ≈ 9.64 m/sec
Hence, required value is 9.64 m/sec.
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Complete question:
A ball is thrown horizontally from the roof of a building 54 m high and hits the ground 32 m from the base. What was the initial speed of the ball?
A rectangular solid (with a square base) has a surface area of 433.5 square centimeters. Find the dimensions that will result in a solid with maximum volume. 1. ____ cm (smallest value). 2. _____ cm 3. ______ cm (largest value)
Hence, 1) 6 cm (smallest value), 2) 6 cm, and 3)15.167 cm are the approximate dimensions which will result in a material with the biggest volume (largest value).
Describe surface area using an example.A 3D object's surface area is indeed the entire area that all of its faces cover. For instance, the surface area of a cube has been its surface area if we need to determine how much paint is needed to paint it. It's always calculated in square units.
Let the height be y as well as the side length of a square base be x. The rectangular solid's surface area is then determined by:
When we simplify this equation, we obtain:
The rectangular solid's volume is determined by:
V = x² × y = x² × (433.5 - 4x²) / (2x)
When we simplify this equation, we obtain:
V = 216.75x - 2x³
We must determine the value of x that maximizes V in order to determine the dimensions that lead to a solid with the maximum volume. Using V's derivative with x as the base, we can calculate:
dV/dx = 216.75 - 6x²
By setting this to 0 and figuring out x, we obtain:
216.75 - 6x² = 0
x² = 36.125
x ≈ 6.005
When we rewrite the equation for y using this value of x, we obtain:
y ≈ 15.167
Thus, the following dimensions will produce a solid with the largest volume:
1. 6 cm (smallest value)
2. 6 cm
3.15.167 cm (largest value)
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The Jones family has two dogs whose ages add up to 15 and multiply to 44. How old is each dog?
As per the equations, the two dogs are either 11 years old and 4 years old, or 4 years old and 11 years old.
What is factoring?Factoring is the process of breaking down a mathematical expression or number into its component parts, which can then be multiplied together to give the original expression or number. In algebra, factoring is often used to simplify or solve equations.
What is an equation?An equation is a statement that shows the equality of two expressions, typically separated by an equals sign (=). The expressions on both sides of the equals sign are called the "left-hand side" and "right-hand side" of the equation.
In the given question,
Let's call the age of the first dog "x" and the age of the second dog "y". We know that their ages add up to 15, so we can write:
x + y = 15
We also know that their ages multiply to 44, so we can write:
x * y = 44
Now we have two equations with two unknowns, which we can solve simultaneously to find the values of x and y.
One way to do this is to use substitution. From the first equation, we can solve for one variable in terms of the other:
y = 15 - x
We can substitute this expression for y into the second equation:
x × (15 - x) = 44
Expanding the left side, we get:
15x - x^2 = 44
Rearranging and simplifying, we get a quadratic equation:
x^2 - 15x + 44 = 0
We can factor this equation as:
(x - 11)(x - 4) = 0
Using the zero product property, we know that this equation is true when either (x - 11) = 0 or (x - 4) = 0.
Therefore, the possible values of x are x = 11 and x = 4.If x = 11, then y = 15 - x = 4, which means that the ages of the two dogs are 11 and 4.
If x = 4, then y = 15 - x = 11, which means that the ages of the two dogs are 4 and 11.
Therefore, the two dogs are either 11 years old and 4 years old, or 4 years old and 11 years old.
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An article in Fire Technology, 2014 (50.3) studied the effectiveness of sprinklers in fire control by the number of sprinklers that activate correctly. The researchers estimate the probability of a sprinkler to activate correctly to be 0.7. Suppose that you are an inspector hired to write a safety report for a large ballroom with 10 sprinklers. Assume the sprinklers activate correctly or not independently (a) What is the probability that all of the sprinklers will operate correctly in a fire? Round your answer to three decimal places (e.g. 98.765) (b) What is the probability that at least 7 of the sprinklers will operate correctly in a fire? Round your answer to two decimal places (e.g. 98.76), (c) What is the minimum number of sprinklers needed so that the probability that at least one operates correctly is at least 0.98?
The probability that all 10 sprinklers will operate correctly in a fire is approximately 0.028, probability that at least 7 of the sprinklers will operate correctly in a fire is approximately 0.95 and the probability of at least one sprinkler operating correctly in a fire is at least 0.98, we need to have at least 7 sprinklers in the ballroom.
The sprinklers activate correctly or not independently can be found by:
(A) If the probability if all the sprinklers will operate correctly in fire is:
P(all 10 sprinklers operate correctly) = (0.7)¹⁰= 0.028247 (rounded to 3 decimal places)
Therefore, the probability that all 10 sprinklers will operate correctly in a fire is approximately 0.028.
(b) The probability that at least 7 of the sprinklers will operate correctly can be calculated using the binomial distribution with n = 10 and p = 0.7:
P(at least 7 sprinklers operate correctly) = P(X >= 7) = 1 - P(X < 7)
where X is the number of sprinklers that operate correctly.
Using a binomial calculator or table, we can find that:
P(X < 7) = 0.0518 (rounded to 4 decimal places)
Therefore:
P(at least 7 sprinklers operate correctly) = 1 - P(X < 7) = 1 - 0.0518 = 0.9482 (rounded to 2 decimal places)
So, the probability that at least 7 of the sprinklers will operate correctly in a fire is approximately 0.95.
(c) We need to find the minimum number of sprinklers needed so that the probability that at least one operates correctly is at least 0.98. This is equivalent to finding the smallest value of n such that:
P(at least one sprinkler operates correctly) >= 0.98
Using the complement rule, we can rewrite this as:
P(no sprinkler operates correctly) <= 1 - 0.98 = 0.02
The probability that no sprinkler operates correctly in a fire is:
P(no sprinkler operates correctly) = (0.3)¹⁰
So we need to solve for n in the inequality:
(0.3)ⁿ <= 0.02
Taking the natural logarithm of both sides and using the logarithmic rule for inequalities, we get:
n >=ln(0.02) / ln(0.3) ≈ 6.52
Therefore, we need at least 7 sprinklers to ensure that the probability that at least one operates correctly is at least 0.98.
Therefore, The probability that all 10 sprinklers will operate correctly in a fire is approximately 0.028, probability that at least 7 of the sprinklers will operate correctly in a fire is approximately 0.95 and the probability of at least one sprinkler operating correctly in a fire is at least 0.98, we need to have at least 7 sprinklers in the ballroom.
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You have 25 coupons for the hotel A, and 20 coupons for the hotel B. What is the maximum number of coupons for the hotel C that you can use if you are not allowed to spend two consecutive nights at the same hotel? How would I set up an equation to solve this problem?
The maximum number of coupons that we use for hotel C is 21. In order to make an equation for this problem, we follow the process given below.
To maximize the number of coupons for hotel C, we should use as many coupons as possible for hotel A and hotel B. Let y be the number of nights we spend at hotel A, and z be the number of nights we spend at the hotel B.Then we have the following equations:
y + z + x = 45 (the total number of nights we stay in hotels A, B, and C)
y <= 25 (we use a maximum of 25 coupons for hotel A)
z <= 20 (we use a maximum of 20 coupons for hotel B)
x <= minimum of (y, z) (we can spend at most one night at hotel C before switching to another hotel)
To maximize x, we need to minimize y and z. Since we cannot spend two consecutive nights at the same hotel, we should alternate between hotels A, B, and C. Thus, we can either start with hotel A or hotel B and then alternate between the two.
Let's assume we start with hotel A. Then we can spend one night at hotel A, one night at hotel C, one night at hotel B, and repeat. This means that y = z = 12 (we spend 12 nights at each of hotel A and hotel B) and x = 21 (we spend 21 nights at hotel C).
Therefore, the maximum number of coupons for hotel C that can be used is 21.
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9(-14)(−2)=
A -15
B 128
C 252
D -252
Answer:
the answer is c
Step-by-step explanation:
add 14 9 times and then whatever that number is double it and remove the negative sign
5cm 8cm 11cm 10cm Find the volume of the prism above
Therefore, the volume of the prism is 200 cubic centimeters.
What is volume?Volume is a measurement of the amount of space occupied by a three-dimensional object. It is typically measured in cubic units, such as cubic meters, cubic centimeters, or cubic feet. The volume of an object can be calculated using a variety of formulas, depending on its shape. The concept of volume is used in various fields, including physics, engineering, architecture, and manufacturing.
Here,
Identify the base of the prism. In this case, the base is a right triangle with legs of 5 cm and 8 cm, and a hypotenuse of 11 cm.
Use the formula for the area of a triangle to find the area of the base. The formula for the area of a right triangle is A = (1/2)bh, where b and h are the lengths of the base and height, respectively. In this case, we can use the legs of the triangle as the base and height, since they are perpendicular. So, the area of the base is:
A = (1/2)(5 cm)(8 cm) = 20 cm²
Multiply the area of the base by the height of the prism to find the volume. The height of the prism is given as 10 cm, so:
Volume = (Area of base) x (Height)
= (20 cm²) x (10 cm)
= 200 cm³
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Suppose that S is a set. Then the power set of S, P(S) is the set containing all subsets of S.P(∅) = {∅}P({a,b})= {∅, {a}, {b}, {a,b}}Remember that the cardinality of a set is the number of elements in it, i.e. |S|.If S is any finite set, even the empty set, it is the case that |P(S)| = 2". Suppose we want to prove this using induction. Also suppose thatthis is our base case:Base: n=0. Then S must be the empty set, and IP(∅)| = 1 which is equal to 20.Which of the following is the best choice for our inductive hypothesis?Select one:a. Suppose that |P(S) = 2" for all sets S with cardinality for any natural number nb. Suppose that IP(Sn)| = 2" for n=1,2,...,kc. Suppose that IP(S₁)| = 2" for n=0,1,...,kd. Suppose that IP(Sn)| = 2" for any natural number ne. Suppose that IP(S)| = 2" for all sets S with cardinality n, n=0,1,...,kf. Suppose that IP(S)| = 2" for all sets S with cardinality n, n=1,2,...,k
The best choice for our inductive hypothesis is option e. Suppose that IP(S)| = 2" for all sets S with cardinality n, n=0,1,...,k. This states that for all sets S with a cardinality of n, where n is any natural number (from 0 to infinity), the power set of S, P(S), has a cardinality of 2".
Suppose that S is a set. Then the power set of S, P(S) is the set containing all subsets of S.P(∅) = {∅}P({a,b})= {∅, {a}, {b}, {a,b}} Remember that the cardinality of a set is the number of elements in it, i.e. |S|. If S is any finite set, even the empty set, it is the case that |P(S)| = 2". Suppose we want to prove this using induction. Also suppose that this is our base case: Base: n=0. Then S must be the empty set, and IP(∅)| = 1 which is equal to 20. therefore with all these conditions, the answer is option e. IP(S)| = 2" for all sets S with cardinality n, n=0,1,...,kf.
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1. Create a cubic function that is not the parent function.
2. Explain how to graph your cubic function using transformations from the parent function. Be specific in your response and use correct mathematical language.
Number your responses 1 and 2 so your instructor can tell which question you're responding to.
This is the graph of the parent function. For example is I wanted to shift it down the function becomes: f(x)=x^3-7 (vertical transformation; shifted 7 down).
y=x²
When a function is shifted, stretched (or compressed), or flipped in any way from its "parent function, it is said to be transformed, and is a
transformation of a function
The given cubic function [tex]$f(x) = (x+2)^3 - 5$[/tex] is obtained by translating the parent cubic function [tex]$f(x) = (x+2)^3 - 5$[/tex] 2 units to the left and 5 units down.
What are cubic function?A cubic function is a function of the form that is, a polynomial function of degree three. In many texts, the coefficients a, b, c, and d are supposed to be real numbers, and the function is considered as a real function that maps real numbers to real numbers or as a complex function that maps complex numbers to complex numbers.
Cubic function:
[tex]$f(x) = (x+2)^3 - 5$[/tex]
Graphing the cubic function using transformations:
To graph the given cubic function [tex]$f(x) = (x+2)^3 - 5$[/tex] using transformations from the parent function [tex]$f(x) = (x+2)^3 - 5$[/tex] we can follow the following steps:
a) Start with the parent function [tex]$y=x^3$[/tex] and mark the points on the graph.
b) Translate the graph horizontally by 2 units to the left to obtain [tex]$y=(x+2)^3$[/tex].
c) Shift the graph vertically down by 5 units to get the final graph [tex]$y=(x+2)^3-5$[/tex]
d) Check the intercepts and end behavior of the function to ensure the graph is correct.
Therefore, the given cubic function [tex]$f(x) = (x+2)^3 - 5$[/tex] is obtained by translating the parent cubic function [tex]$f(x) = (x+2)^3 - 5$[/tex] 2 units to the left and 5 units down.
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Difference between 6z and z to the power of 6
Answer:
[tex]6z \: = 6 + z \: \\ z {}^{6} = z \times z \times z \times z \times z \times z \\ [/tex]
The mathematical statement in the form of expression can be written as 15625z⁶.
What is algebraic expression?An expression in mathematics is a combination of terms both constant and variable. For example, we can write the expressions as -
2x + 3y + 5
2z + y
x + 3y
Given is to find the mathematical statement -
"Difference between 6z and z to the power of 6"
We can write the given mathematical statement in the form of expression as -
(6z - z)⁶
(5z)⁶
5⁶ x z⁶
125 x 125 x z⁶
15625z⁶
Therefore, the mathematical statement in the form of expression can be written as 15625z⁶.
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Which function models the area of a rectangle with side lengths of 2x – 4 units and x + 1 units? What is the area when x = 3?
A. f(x) = 2x2 – 4x + 4; A = 10 B. f(x) = 2x2 + 8x – 4; A = 38 C. f(x) = 2x2 – 8x + 4; A = 2 D. f(x) = 2x2 − 2x − 4; A = 8
Answer:
Area of rectangle, [tex]f(x) = 2x^2 - 2x - 4[/tex].
Step-by-step explanation:
We are given with side lengths of a rectangle are (2x-4) units and (x+1) units. It is required to find the area of rectangle.
The area of a rectangle is equal to the product of its length and breadth. It is given by :
[tex]A=L\times B[/tex]
Let us consider, L = (2x-4) units and B = (x+1) units
Plugging the side lengths in above formula:
[tex]A=(2x-4)\times(x+1)[/tex]
[tex]A = 2x^2 + 2x-4x - 4[/tex]
[tex]A=2x^2-2x-4[/tex]
So, the function that models the area of a rectangle is [tex]f(x) = 2x^2 - 2x - 4[/tex].
A cube has a surface area of 24 ft². What is the length of one edge of the cube?
The length of one edge of the cube is 2 feet long whose surface area is 24 ft sq.
What is surface area?Surface area is a measure of the total area that the surface of an object occupies. In geometry, it is the sum of the areas of all the faces or surfaces of a three-dimensional object
According to question:The surface area of a cube is given by the formula 6s², where s is the length of one edge of the cube.
We are given that the surface area of the cube is 24 ft², so we can set up the equation:
6s² = 24
Dividing both sides by 6 gives:
s² = 4
When you square the two sides, you get:
s = ±2
Since the length of a side cannot be negative, the only valid solution is:
s = 2
The cube's one edge is therefore 2 feet long.
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4. Alice is creating a pyramid with toy rectangular-prism-shaped blocks
that measure 1 inch by 2 inches by 2 inches. She places 8 blocks in
the first row. Then she places 6 blocks on top of the first row and 4
blocks on top of the second row. If she continues the pattern until
the top row has 2 blocks, what is the volume of the final pyramid?
The final pyramid has a volume of 16 cubic inches. To find this, first multiply the number of blocks in each row together. 8 blocks in row 1 = 8 x 2 x 2 = 32; 6 blocks in row 2 = 6 x 2 x 2 = 24; 4 blocks in row 3 = 4 x 2 x 2 = 16; and 2 blocks in row 4 = 2 x 2 x 2 = 8. Then add all these numbers together to get 32 + 24 + 16 + 8 = 80. Finally, divide this number by 3 to get the volume, 80/3 = 16.