prove that for every vector spacee, iff:e→eis an idempotentlinear map, i.e.,f◦f=f, then we have a direct sum
Answers
Every vector spaces, iff:e→eis an idempotent linear map as E= Kcr(f) + Im(f)
What is Linear Map?
A linear map is a mapping in mathematics, more specifically in linear algebra. It is also known as a linear transformation, vector space homomorphism, or, in some cases, a linear function. that maintains the operations of vector addition and scalar multiplication between two vector spaces
Given that,
F : E → E is an idempotent linear map i.e FoF =F
F is an idempotent, then for any W in the image of F statistics
W = F(x)
for some x = E
Then F(w) = FoF(x)
F(x)=(W)----- (A)
So F is indeed a projection to Im(F) as a subspace of v
also , Kcr(F) + Im(F) =0
Because if x∈Kcr(f) + Im------(B)
then f(x) = 0 and
f(t)=x for some x∈E
FoF (t) = 0
f(t)=x
x=0
also from A we have
f(x)-w=0
f(x) -f(w)=0
f(x-w)=0
x-w=t
where t∈Kcr (F)
X = W+T
here x is arbitrary
E = Kcr + Im f ------(C)
From (B) and (C) we have E = Kcr(f) + Im (f)
Since E= Kcr(f) + Im, every vector space is an idempotent linear map (f)
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Related Questions
Anna baked 3 batches of cookies with c cookies in each batch. She then ate 8 cookies!
Answers
Answer:
[tex]t=3c-8[/tex]
Step-by-step explanation:
It seems like your question is incomplete, so I will assume it is asking for an expression that represents how many cookies Anna has left.
Let t represent the total number of cookies left. Let's build this expression. If each batch contains c cookies, and there are 3 batches, the total number of cookies baked is 3 times c. Since she ate 8 cookies, 8 must be subtracted from that total. Therefore:
[tex]t=3c-8[/tex]
Hope this helps!
if multiple events have a 20% probability, what is the chance that two or more will happen simultaneously
Answers
The probability of two or more events happening simultaneously is 0.36.
The probability of two or more events happening simultaneously depends on the number of events and the type of relationship between them. If the events are independent, meaning that the occurrence of one event does not affect the probability of the other event happening.
The probability of two or more events happening simultaneously can be calculated using the formula for the sum of probabilities:
P(A or B) = P(A) + P(B) - P(A and B)
Where A and B are two events. If we have n independent events with the same probability of 20%, the probability of two or more events happening simultaneously can be calculated as follows:
P = 1 - P(none of them happen) = 1 - (1 - 0.2)ⁿ
For n = 2, we have:
P = 1 - (1 - 0.2)² = 1 - 0.64 = 0.36
So, the chance that two events with a 20% probability will happen simultaneously is 36%. The chance for three or more events is even higher.
It's important to note that if the events are dependent, the calculation will be different. The relationship between the events can significantly impact the final probability.
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im just trying get this over with haha dont guess please, if you dont know the answer then dont answer
Answers
We can see here that:
1. x-y=3
3x + y = 1 - One solution
2. y = 5x-4
y=5x+13 - No solutions
3. 4g + r = 12
8g= -2r+24 - Infinite number of solutions
4. 3y+ 15x = -1
y = 4-5x - No solutions
What is a system of equation?A system of equations is a set of two or more equations that are related to each other and must be solved simultaneously to find the values of the variables involved. A system of equations represents a situation in which multiple unknowns are related to each other in a specific way, and finding a solution requires finding the values of all the unknowns that satisfy all the equations in the system.
Solving:
x-y=3 ----eq. 1
3x + y = 1 ----eq. 2
Thus, x = 3+y --- eq. 3
Substituting it into eq. 2, we have:
3( 3+y ) + y = 1
9 + 3y + y = 1
9 + 4y = 1
4y = 1-9
4y = -8
y = -8/4 = -4
Substituting y in eq. 1, we have:
x - (-4) = 3
x + 4 = 3
x = 3 - 4 = -1.
Thus, one solution.
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The power of the statistical test is zero (0) when the mean is 52, the sample size is 25, and the standard deviation is 2, and the probability of committing Type II error when P 55 is one (1) (previous problem).TrueFalse
Answers
The probability of committing a Type II error when the mean difference is 55 is 1.
The concept of power in statistical testing is important for understanding the reliability of results. When the power of a statistical test is zero, it means that the test is unable to detect the difference between the means of two groups, even if there is a real difference.
In the given scenario, the mean is 52, the sample size is 25, and the standard deviation is 2.
This means that if the true mean difference between two groups is 55, there is a 100% chance that the test will not detect it. The probability of committing a Type II error is directly related to the power of a statistical test.
The higher the probability of committing a Type II error, the lower the power of the test.
Therefore, it is important to consider the power of a statistical test when interpreting results.
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determine the probability that a state is occupied by an electron at (i) 1kt, (ii) 3kt, and (iii) 5kt above the fermi level in a metal
Answers
The probability that a state is occupied by an electron at
i)1kt:0.5
ii) 3kt: 0.7
ii)5kt: 0.9
The probability that a state is occupied by an electron at any given temperature above the Fermi level is calculated using the Fermi-Dirac statistics. This equation states that the probability of a state being occupied is equal to 1/(1+exp(-(E-EF)/KT)), where E is the energy of the state, EF is the Fermi level energy, and KT is the product of Boltzmann's constant and the temperature in Kelvin.
Thus, for a temperature of 1kt above the Fermi level, KT = 1. By plugging these values into the equation, we can calculate that the probability of a state being occupied is 0.5. For a temperature of 3kt and 5kt above the Fermi level, KT = 3 and 5, respectively. Using these values, the probability of a state being occupied at 3kt and 5kt above the Fermi level is 0.7 and 0.9, respectively.
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Write 98 as a product of prime factors.
HELP?
Answers
Answer:
2*7*7
Step-by-step explanation:
Answer: 2*7^2
Step-by-step explanation:
98=2*49
and 49=7^2
so 98=2*7^2
Tyler’s brother works in a shoe store.
a. Tyler’s brother earns a commission. He makes 2.5% of the amount he sells. Last week, he sold $1000 worth of shoes. How much was his commission?
b. The store bought a pair of shoes for $50, and sold it for $80. What percentage was the markup?
c. Tyler’s brother earns $14 per hour. The store offers him a raise—a 5% increase per hour. After the raise, how much will Tyler’s brother make per hour?
Answers
A. In a case whereby Tyler’s brother earns a commission. He makes 2.5% of the amount he sells. Last week, he sold $1000 worth of shoes his commission is $22.50
B. The percentage markup was $30
C.In a case whereby aTyler’s brother earns $14 per hour. The store offers him a raise—a 5% increase per hour the amount Tyler’s brother make per hour is 12.60
How can the commissionand percentage be calculated?The concept that will be used here is commision and percentage.
A. Therefore, a commission is often a percentage of a sale. Suppose, for illustration, that I own a cookie company. I take you on as a seller and inform you that you will receive a 10% commission for each box you sell. Given that the box of cookies costs $5.00,
you would get $5.00 * (.10), or $0.50, for each successful box you sell. Therefore, if you sell 200 boxes in a week and earn $0.50 per box, your total commission would be $0.50 * 200 = $100!
Thus, Tyler's brother sold shoes valued a total of $900. In the event that he earns a 2.5% commission, his reward will be [ $900 * (0.025) ]= $22.50 for his work.
B. The Markup on the shoe =Amount sold - purchase price
Markup = [$80 - $50]
= $30
Percentage Markup is :$30/50 *100 = 60%
C. 5% of 12 can be calculated as 5/100*12= 0.60
[12 + 0.60 ]
= 12.60
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Comment on any apparent relationship between these two variables.For midsized cars, all driving speeds seem to yield about the same miles per gallon.For midsized cars, there seems to be no relationship between driving speeds and fuel efficiency. For midsized cars, lower driving speeds seem to yield higher miles per gallon.For midsized cars, lower driving speeds seem to yield lower miles per gallon.
Answers
The statement "For midsized cars, lower driving speeds seem to yield lower miles per gallon" indicates a negative relationship between driving speed and fuel efficiency.
This statement means that as the driving speed increases, the fuel efficiency decreases and vice versa. This relationship is likely due to the increase in air resistance and engine drag as driving speed increases, which leads to a decrease in fuel efficiency.
This relationship can be seen in the context of the physics of motion and energy, where energy is conserved, and an increase in one form of energy, such as kinetic energy, is accompanied by a decrease in another form of energy, such as thermal energy.
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An architect is designing a rectangular pavilion that has a length of x + 8 units and a width of x - 4 units. Write an expression representing the area covered by the pavilion.
Answers
Answer:
Step-by-step explanation:
The expression representing the area covered by the pavilion can be found by multiplying the length by the width:
A = (x + 8) * (x - 4)
where A represents the area and x is the length of one of the sides.
Area = (x + 8)(x - 4)
= x^2 + 4x + 8x - 32
= x^2 + 12x - 32
I’ve always been terrible at geometry. I’m supposed to find the area of the triangle on top, then also the area of the square on the bottom. Then the area of both together. I attached a screenshot of the figure
Answers
Answer: The triangle on top is [tex]32m^2[/tex]. The square on the bottom is [tex]64m^2[/tex]. Both together is [tex]96m^2[/tex]
1. (3 points) is the following a valid argument or fallacy? if it is sunday, then the store is closed. the store is closed. therefore, it is sunday. you must explain your answer.
Answers
According to a conclusion, therefore, the answer is palace now deductive argument form that is contra position.
consider part 1 here, since the store may be closed on other days. According to a conclusion, therefore, the answer is palace now
consider part 2 deductive argument form that is contra position.
Third, here, the even number of the form 2 n for all k belongs to a natural number and odd numbers are of form 2n+1 where n belongs to the natural number. So the product of the ordinem number that is 2(n+1), is equal to 2 into 2 n+1 that is, of the form of 2 n. Therefore, the product is even
part 4. Here let us take contra positively that, if is odd, integral then n is of the form 2 k plus 1, where k belongs to integral implies. N square is equal to 2 k plus 1 whole square, that is, equals to 4 k square plus 4 k plus 1. That is equal to 2 into 2 k square plus 2. K plus 1 is the odd integer. So if n square is an even integer, then is an even integer now
consider part 5. Here we have given that 2 n plus 1 is greater than equal to 33 implies 2 is greater than equal to 32, which implies n is greater than equal to 16. Point proves the existence of integer hope.
The complete question is:-
Is the following a valid argument or fallacy? If it is Sunday, then the store is closed. The store is closed. Therefore, it is Sunday. You must explain your answer.
2. Name the argument form of the following argument: Dogs eat meat. Fluffy does not eat meat. Therefore, Fluffy is not a dog.
3. Prove directly that the product of an even and an odd number is even.
4. Prove by contraposition for an arbitrary integer n that if n2 is even, then n is even.
5. Prove that there is a positive integer n that satisfies 2n + 1 ≥ 33. Do
not overthink this problem. Do not write more than logically necessary to
prove existence.
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What is the slope of line a?
(I didnt mean to click the -3/4 it was by accident)
Answers
Answer: it is 3/4
Step-by-step explanation:
(0,1) and (4,4) are the two points.
slope = (4-1)/(4-4) = 3/ 4
Harry had 25 marbles. For his birthday, his grandmother gave him m more marbles. Which expression shows the total number of marbles Harry has now?
Answers
Answer:
Step-by-step explanation:
If Harry has 25 marbles, and his grandmother gave him m more marbles we can say that harry has 25+m amount of marbles.
A textbook tore old a combined total of 251 biology and ociology textbook in a week. The number of ociology textbook old wa 67 le than the number of biology textbook old. How many textbook of each type were old?
Answers
From the combination of 251 textbook, 92 sociology and 159 biology textbook were sold.
Let the number of biology textbook be x. So, the number of sociology textbook = 251 - x. The difference between the number of two books is 67. Representing this as equation.
Number of biology textbook = number of sociology textbook + 67
Keep the values in formula -
x = 251 - x + 67
Rewriting the equation
x + x = 251 + 67
Performing addition on both sides of the equation
2x = 318
x = 318/2
Performing division
x = 159
Number of biology textbook = 159
Number of sociology textbook = 251 - 159
Number of sociology textbook = 92
Thus, 92 sociology and 159 biology textbook 1ere sold.
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The complete question is -
A textbook store sold a combined total of 251 biology and sociology textbook in a week. The number of sociology textbook old was 67 less than the number of biology textbook sold. How many textbook of each type were sold?
Rodrigo is an ichthyologist, or a scientist who studies fish. When he first started monitoring the sturgeon population of Lavender Lake, he estimated there were 600 sturgeon in the lake. One year later, Rodrigo estimated the population had decreased to 510. He expects the sturgeon population to continue decreasing each year. Write an exponential equation in the form y = a(b) that can model the estimated sturgeon population, y, in x years. Use whole numbers, decimals, or simplified fractions for the values of a and b.
Answers
An exponential equation in the form y = a(b)ˣ that can model the estimated sturgeon population, y, in x years, is y = 600(1-3/20)ˣ.
What is exponential decay?
An exponential function's curve is created by a pattern of data called exponential decay, which exhibits higher decreases over time.
Given:
Rodrigo is an ichthyologist, or a scientist who studies fish.
When he first started monitoring the sturgeon population of Lavender Lake, he estimated there were 600 sturgeon in the lake.
One year later, Rodrigo estimated the population had decreased to 510. He expects the sturgeon population to continue decreasing each year.
The exponential decay function:
y = A₀(1-r)ˣ, where y = Final amount, A₀ = Initial amount, r = Rate of decay in decimal form, x = Time.
So, y = 600(1 - r)ˣ.
One year later,
510 = 600(1-r)
r = 3/20
So, the function is y = 600(1 - 3/20)ˣ
Therefore, the function is y = 600(1 - 3/20)ˣ
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Answer:
600(0.85)^x
step by step explanation
A lizard i at the bottom of a 19-foot well. Each day it crawl up 5 feet, but each night it lip back 4 feet. After how many day will the lizard reach the top of the well?
Answers
It will take the lizard 19 days to reach the top of the well.
As per the data given,
We have,
The depth of the well = 19 feet
Let's call the number of days the lizard has been in the well as "d".
After "d" days, the lizard will have crawled up a total of 5d feet.
However, it will also have slipped back down 4d feet during the nights.
So, the net distance the lizard has traveled upward can be calculated as follows:
Net distance = 5d - 4d = d feet
To reach the top of the well (19 feet), the lizard must have traveled 19 feet upward, so we have:
19 = d
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Solve the equation.
P = _
Answers
After solving the equation P = [tex]$ \frac{ry}{r+y}[/tex], y is equal to [tex]\dfrac{rp}{(r- p)}[/tex].
What is equation?
An equation is a mathematical statement that expresses the equality of two expressions by showing that they have the same value. Equations can be used to describe a variety of physical, chemical and biological processes.
They are often used to solve problems in a variety of fields, including engineering, finance and physics. Equations are typically written in standard form, with the equal sign being used to indicate the equality of two expressions.
The two expressions can be written in a variety of forms, including algebraic equations, differential equations, calculus equations and trigonometric equations. The solutions to equations can be found by using a variety of techniques, including solving equations graphically or analytically.
Given
P = [tex]$ \frac{ry}{r+y}[/tex]
Solve for y
⇒ pr + py = ry
⇒ pr = y(r−p)
⇒ y = [tex]\dfrac{rp}{(r- p)}[/tex]
Thus, after solving the equation P = [tex]$ \frac{ry}{r+y}[/tex],y is equal to [tex]\dfrac{rp}{(r- p)}[/tex].
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Complete question:
Solve for m.
m+1/4<−1 3/8
m<−1 1/8
m<−1 1/3
m<−1 5/8
m>−1 1/2
Answers
Answer:
c. m<-15/8
Step-by-step explanation:
solve
m + 1/4 < -13/8 = m< -15/8
A bicycle store costs $3850 per month to operate. The store pays an average of $75 per bike. The average selling price of each bicycle is $145. How many bicycles must the store sell each month to break even?
Answers
The number of bicycle store needs to sell each month to break even is, 55
What is an Equation ?An equation is a mathematical term, which indicates that the value of two algebraic expressions are equal. There are various parts of an equation which are, coefficients, variables, constants, terms, operators, expressions, and equal to sign.
For example, 3x+2y=0.
Types of equation
1. Linear Equation
2. Quadratic Equation
3. Cubic Equation
Given that,
A bicycle store costs $3850 per month to operate
The store pays an average of $75 per bike
The average selling price of each bicycle is $145
The number of sells to break even = ?
Suppose, the number of sales is x
So, it can be written as,
For break even condition
75x + 3850 = 145x
145x - 75x = 3850
70x = 3850
x = 3850/70
x = 55
Hence, the number of sales is 55
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4.
1/2 x 1/1000
1.8-3
2.611
Multiplying and Dividing Fractions (A)
Find the value of each expression in lowest terms.
6.
7.12 ÷ 1
4TH
8.
27
4. 11
10.4×14
Math-Drills.com
Nevaeh Laws
Jan/27/53 #716
11
3
X
11-110
6
13.00
14.5×7
3.4
Answers
find all points on the curve defined by the equations x(t)=2sin(8t) and y(t)=−9cos(8t) with a slope of −93–√2 when 0≤t<π8
Answers
The point on the curve with a slope of −93–√2 is (2, 0.43)
Given x(t) = 2 sin(8t) and y(t) = −9 cos(8t)
We are asked to determine the points on the curve with a slope of −93–√2.
Now, x/2 = sin(8t) --(1)
-y/9 = cos(8t) --(2)
Squaring and adding equations (1) and (2)
(x/2)² + (-y/9)² = sin²(8t) + cos²(8t)
x²/4 + y²/81 = 1
Differentiating both sides
x/2 + (2y/81) (dy/dx) = 0
Given slope = dy/dx = −93–√2
(2sin(8t))/2 + (2(−9 cos(8t))/81)(-93 - √2) = 0
sin(8t) + (2/9)cos(8t)(93 + √2) = 0
Solving for t and taking care of 0 ≤ t <π/8
we get t = 11.59°, 34.09°, 79.09°......
But only t within the given range is t = 11.59°.
Now, x = 2sin(8 x 11.59°) = 2
and y = -9 cos(8 x 11.59°) = 0.43
Hence, the point on the curve with a slope of −93–√2 is (2, 0.43)
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State the value of h for which y = -x + 6 will be a tangent to the curve y = 5x^2 + 6x - 3h
Answers
y = -x + 6
y = 5x^2 + 6x - 3h
-x + 6 = 5x^2 + 6x - 3h
Now we can use the quadratic formula to solve for x:
x = (-b ± √(b^2 - 4ac)) / 2a
where a = 5, b = 6, and c = -3h - 6
x = (-6 ± √(6^2 - 4(5)(-3h-6))) / 2(5)
x = (-6 ± √(36 + 60h + 24)) / 10
x = (-6 ± √(60h + 60)) / 10
Now we can substitute this value of x back into one of the original equations to find y:
y = -x + 6
y = -(-6 ± √(60h + 60)) / 10 + 6
y = 6 ± √(60h + 60) / 10
Now we can substitute this value of x and y back into the other equation:
y = 5x^2 + 6x - 3h
6 ± √(60h + 60) / 10 = 5x^2 + 6x - 3h
Now we can use the fact that the slope of the tangent line is equal to the derivative of the function at the point of tangency.
So, the derivative of y = 5x^2 + 6x - 3h is:
dy/dx = 10x + 6
So, we can substitute x and y values back into the derivative and get the slope of the tangent line:
10x + 6 = -1
10x = -7
x = -7/10
Now, substituting this value of x back into one of the original equations to find y:
y = -x + 6
y = -(-7/10) + 6
y = 7/10 + 6
y = 61/10
So, the point of tangency is (x,y) = (-7/10, 61/10)
Finally, we can substitute this point of tangency into one of the original equations and solve for h.
61/10 = 5(-7/10)^2 + 6(-7/10) - 3h
h = (61/10 + 49/100 + 21/100 - 3h) / -3
h = (61/10 + 70/100) / -3
h = -(431/30)
Therefore, h = -(431/30) for y = -x + 6 is a tangent to the curve y = 5x^2 + 6x - 3h
write an equation for the line with the following slop and y-intercept
slope=-6; y-intercept =1
Answers
Answer: y = -6x + 1
Step-by-step explanation:
An equation for a line written in slope-intercept form is written as y = mx + b, with m being the slope and b being the y-intercept.
Given a slope of -6 and a y-intercept of 1, the equation of the line is:
y = -6x + 1
This equation can be used to find the y-value for any given x-value by substituting the value for x and solving for y
Two angles are supplementary. The
larger angle is 18 less than 5 times
the smaller angle. How many
degrees is the smaller angle?
Answers
Answer:
Step-by-step explanation:
The sum of two supplementary angles is 180
Let A be the larger angle and B be the smaller angle
A+B=180
Given A=5B-18
5B-18+B=180
6B=180+18=198
B=198/6=33
A=180-B=180-33=147
Hence A=147, B=33
2√5x-4-3= -11
Is there roots or no roots?
Answers
This function cannot be solved for zeros.
Only simplified to standard form.
Standard form would be 2sqrt5x=-4
a population consists of 400 elements. we want to draw a simple random sample of 40 elements from this population. on the first selection, the probability of an element being selected is
Answers
The probability of an element being selected is 0.1 or 1/10 or 10%.
Probability is a mathematical concept that describes the likelihood of an event occurring. It is a value between 0 and 1, where 0 indicates that an event is impossible, 1 indicates that an event is certain, and values between 0 and 1 indicate the degree of uncertainty or likelihood of an event occurring.
In simple random sampling, each element in the population has an equal chance of being selected, and the probability of any given element being selected is equal to the sample size divided by the population size.
In this case, the sample size is 40, and the population size is 400, so the probability of any given element being selected is 40/400 = 1/10.
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a car is travelling at 84 km/h how far can a car travel in 45 mins?
Answers
it will go far as 231 km in 45 mins
PLS HURRY ASAP SHOW SOLVING STEPS WILL MARK BRAINLIEST IF CORRECT
Answers
The fixed fee for Company 1 is equal to $5.5.
The fixed fee for Company 2 is equal to $9.15625.
A statement which is true is: A. Company 1's rate per pound is twice the rate per pound of Company 2.
How to determine the value of the constant rate of change?In Geometry, the rate of change of any straight line simply refers to its gradient or slope and it can be calculated by using this mathematical expression;
Rate of change (slope), m = (Change in y-axis, Δy)/(Change in x-axis, Δx)
Rate of change (slope), m = (y₂ - y₁)/(x₂ - x₁)
Based on the weight of the package for Company 1, the constant rate per pound is given by:
Rate of change (slope), m = (14.50 - 9.25)/(6 - 2.5)
Rate of change (slope), m = 5.25/3.5
Rate of change (slope), m = 1.5.
At data point (2.5, 9.25), an equation for the fixed fee for Company 1 can be calculated in point-slope form as follows:
y - y₁ = m(x - x₁)
y - 9.25 = 1.5(x - 2.5)
y = 1.5x - 3.75 + 9.25
y = 1.5x + 5.5
Therefore, the fixed fee is $5.5.
Based on the weight of the package for Company 2, the constant rate per pound is given by:
Rate of change (slope), m = (7.25 - 6.5)/(2 - 1)
Rate of change (slope), m = 0.75/1
Rate of change (slope), m = 0.75.
At data point (1, 6.5), an equation for the fixed fee for Company 1 can be calculated in point-slope form as follows:
y - y₁ = m(x - x₁)
y - 6.5 = 0.75(x - 1)
y = 0.75x - 0.75 + 6.5
y = 0.75x + 5.75
Therefore, the fixed fee is $5.75.
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If a right triangle is formed when three squares are joined at their vertices, the area of the largest square is the same as the combined area of the two smaller squares.
Answers
in option A and B a right triangle is formed when three squares are joined at their vertices, the area of the largest square is the same as the combined area of the two smaller squares.
What is the Pythagoras theorem?The Pythagoras theorem which is also referred to as the Pythagorean theorem explains the relationship between the three sides of a right-angled triangle. According to the Pythagoras theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides of a triangle.
A) From the given figure, side of a largest square is 17 inches, side of a larger square is 15 inches and the area of a smaller square is 64 square inches.
We know that, the area of a square is side².
The side(c) of a smaller square is side²=64
Then, c=8 units
By using Pythagoras theorem, we get
17²=15²+c²
289=225+8²
289=289
B) From the given figure, area of a largest square is 169 square inches, area of a larger square is 144 square inches and the side of a smaller square is 5 inches.
We know that, the area of a square is side².
The side(a) of a largest square is side²=169
Then, a=13 inches
The side(b) of a larger square is side²=144
Then, b=12 inches
By using Pythagoras theorem, we get
13²=12²+5²
169=144+25
169=169
Therefore, in option A and B a right triangle is formed when three squares are joined at their vertices, the area of the largest square is the same as the combined area of the two smaller squares.
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a trapezoid has an area of 100 square units. what scale factor would be required to dilate the trapezoid to have each area?
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A dilation of a figure is a transformation that changes the size of a figure but not its shape. To find the scale factor required to dilate a trapezoid to have a different area, we need to know the formula for the area of a trapezoid and the desired area. In this conext, the area would be 100 * scale factor^2
The formula for the area of a trapezoid is:
Area = (base1 + base2) * height / 2
Let's say the original trapezoid has an area of 100 square units and the desired area is "A" square units.
The scale factor required to dilate the trapezoid to have an area of "A" square units is given by the equation: 100 = (base1 + base2) * height / 2
100 * 2 / (base1 + base2) = height
200 / (base1 + base2) = height
To find the new area:
A = (base1 + base2) * height / 2
A = (base1 + base2) * 200 / (2 * (base1 + base2))
A = 200 * height / 2
A = 100 * scale factor^2
So, the scale factor required to dilate the trapezoid to have an area of "A" square units is given by the equation:
Scale factor = sqrt(A/100)
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