On solving the provided question, we can say that The pace at which the y-values change relative to the x-values is the average rate of change for a function, average rate = m = 81-9/6 = 72/6 = 12
What is function?Mathematics is the study of quantities and their variations, equations and related structures, shapes and their locations, and the locations where they can be found. The term "function" refers to the relationship between a set of inputs, each with its own output. A function is a connection between inputs and outputs in which each input leads to a single, distinct result. Each function is assigned a domain and a codomain, also known as a scope. f is commonly used to denote functions (x). The input is an x. Based on the following factors, there are four main types of functions available: on functions, one-to-one functions, many-to-one functions, inside functions, and on functions.
f(x) =
x1, x2 = 3,9
average rate = f(x2) - f(x1) / 6
average rate = m = 81-9/6 = 72/6 = 12
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Suppose f(x)= 10x-4
Compute the following:
A.) f(x-2)+f(2x)
A.) f(x-2)+f(2x)
To find this expression, we can substitute the definition of f(x) into the expression and simplify:
f(x-2)+f(2x) = [10(x-2)-4] + [10(2x)-4]
= 10x - 20 - 4 + 20x - 4
= 30x - 28
Therefore, f(x-2)+f(2x) = 30x - 28.
Compute f(2a) - f(a) for any value of a ?Given f(x) = 10x - 4, we need to find f(2a) - f(a).
f(2a) = 10(2a) - 4 = 20a - 4
f(a) = 10a - 4
Therefore,
f(2a) - f(a) = (20a - 4) - (10a - 4)
= 10a
Answer: f(2a) - f(a) = 10a. This means that the difference between the function evaluated at 2a and a is simply 10 times a.
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Find the magnitude of the sum
of these two vectors:
The magnitude of the sum of these two vectors is 2.9486m
What is cosine rule?In trigonometry, the law of cosines (also known as the cosine formula, cosine rule, or al-Kashi's theorem) relates the lengths of the sides of a triangle to the cosine of one of its angles.
The parameters that will help us to find the magnitude are
/OB/ = 3.14m
/OA/ = 2.71 m
Let /AB/= x
Sum of the angles is (-60+30)⁰ = 30⁰
Using cosine rule
x²= 3.14² +2.71² - 2*3.14*2.71 *cos30⁰
x² = 9.8596+7.3441 -8.5094
x²= 17.2037 -8.5094
x²= 8.6943
x = √8.6943
x = 2.9486
Therefore, the magnitude of the sum
of these two vectors is 2.9486m or 2.9m approximately
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ons
Write the equation for this relationship and identify the constant of proportionality (or k value).
X
Y
28 36 40
21
27
30
56
42
Given the letters H A M I L T O N in a game of scrabble:
a) How many ways can you arrange all of the letters if the arrangement must end in a vowel [vowel: a,e,i,o,u]
b) How many arrangements are there if you can only use five letters?
The number of ways to arrange all the letters if the arrangement must end in a vowel is; 25,200.
The number of arrangement using only five letters is; 6720.
What is the number of ways to arrange the letters as described?Since there are 5 options for the last letter, and there a 7 letters left to make an arrangement of 7.
Hence, The number of ways can you arrange all of the letters if the arrangement must end in a vowel is; 5 × 7!
= 5 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 25,200.
b). The number of arrangements if one can only use five letters is; ⁸P₅
= 6720.
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The number of a country's unemployed workers increased from 3.4 million to 5.5 million last year. If the country's population remained constant at 76 million, how did its unemployment rate change last year? A. It increased by about 27%. B. It decreased by about 27%. C. It increased by about 3%. D. It decreased by about 3%.
The rate at which its unemployment changes is, 54.6 %
What is the rate?Rate is a ratio of change in one quantity w.r.t. change in another quantity. The net change in one quantity is written as, Δy and the net change in another quantity is written as, Δx.
So formula for the rate is,
m = change in one quantity/change in another quantity = Δy/Δx
Given that,
The initial number of unemployed workers was 3.4 million and the final number was 5.5 million.
The change in the number of unemployed workers is:
5.5 million - 3.4 million = 2.1 million
To find the initial number of employed workers, we need to subtract the initial number of unemployed workers from the total population:
76 million - 3.4 million = 72.6 million
Now, we can calculate the initial unemployment rate as:
3.4 million / 72.6 million = 0.0469 or 4.69%
Similarly, the final unemployment rate is:
5.5 million / 76 million = 0.0724 or 7.24%
The change in the unemployment rate is:
(0.0724 - 0.0469) / 0.0469 ≈ 0.546 or 54.6%
Therefore, the unemployment rate increased by about 54.6%, which is closest to answer choice A. It increased by about 27%.
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(Combinatorial analysis question)
A batch of 15 phones contains 3 defective phones. A sample of 4 phones out of 15 in the lot is chosen at random. How many of the possible samples contain at least one defective part?
Book says the answer is 870. Can someone give me a step by step answer please?
Answer:
Step-by-step explanation:
There are several ways to approach this problem, but one common method is to use the complementary counting principle, which states that the number of favorable outcomes can be obtained by subtracting the number of unfavorable outcomes from the total number of outcomes. In this case, the favorable outcome is that the sample contains at least one defective phone, and the unfavorable outcome is that the sample contains no defective phones.
The number of possible samples of 4 phones out of 15 can be calculated as the number of combinations of 4 items from a set of 15, which can be written as:
C(15, 4) = 15! / (4! * (15 - 4)!) = 1365
The number of samples containing no defective phones can be calculated as the number of combinations of 4 phones out of the remaining 12 non-defective phones, which can be written as:
C(12, 4) = 12! / (4! * (12 - 4)!) = 495
So, the number of samples containing at least one defective phone can be calculated as:
1365 - 495 = 870
Therefore, out of the 1365 possible samples of 4 phones, 870 of them contain at least one defective phone.
-3x⁴+5x³+0x²+2x-1÷2x²-x
The quotient is -3x²/2 + 7x/4 + 7/8 and the remainder is 23x/8 - 1. The solution has been obtained by using polynomial long division method.
What is polynomial long division?
A formula for multiplying one polynomial by another of the same degree or lower is known as a long division polynomial. Like with long division of numbers, the components of long division of polynomials include the divisor, quotient, dividend, and remainder.
We are given dividend as -3x⁴+5x³+0x²+2x-1 and divisor as 2x²-x.
Using Polynomial Long Division,
Dividing : -3x⁴+5x³+0x²+2x-1 ("Dividend")
By : 2x²-x ("Divisor")
Dividend -3x⁴ + 5x³ + 0x² + 2x - 1
- divisor * -3x²/2 -3x⁴ + 3x³/2
Remainder + 7x³/2 + 0x² + 2x - 1
- divisor * 7x/4 + 7x³/2 - 7x²/4
Remainder 7x²/4 + 2x - 1
- divisor * 7/8 7x²/4 - 7x/8
Remainder 23x/8 - 1
Quotient : -3x²/2 + 7x/4 + 7/8
Remainder: 23x/8 - 1
Hence, the solution has been obtained.
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The ratio of first- class, second- class and third-class passengers who survived the ship wreak is
101:59:89. How many passengers were saved if the total number of first-class passengers were 202?
Answer: If the ratio of first-class, second-class, and third-class passengers who survived the shipwreck is 101:59:89, then for every 101 first-class passengers who survived, there were 59 second-class and 89 third-class passengers who survived.
If the total number of first-class passengers was 202, then there were 202 * 59/101 = 118 second-class passengers who survived, and 202 * 89/101 = 180 third-class passengers who survived.
Therefore, the total number of passengers who survived the shipwreck is 202 + 118 + 180 = 500 passengers.
Step-by-step explanation:
I need the modulus of: I 9+40i I
I = absolute value
The required modulus is 41 .
Complex Numbers:A real and imaginary number combo with the formula a + bi , where
a and b are real numbers, and
i is the "unit imaginary number" √(−1)
a and b may also have zero values.
We are aware that complex numbers are made up of both actual and fictitious numbers.
The imaginary part, y, is multiplied by I the square root of -1, and the real part, x.
Modulus of x+iy = [tex]\sqrt{x^2 +y^2}[/tex]
Here, 9 and 40 are substituted for x and y.
i.e. 9+40i
We square the coefficients, add them, and then find the square root to obtain the modulus.
|9+40i| =[tex]\sqrt{9^2 +40^2} =\sqrt{81+1600} =\sqrt{1681}[/tex][tex]=41[/tex]
By long division method we find that
|9+40i| =41
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Kylie has 6 boxes. Each box contains 4 cakes. Kylie writes a calculation to find out how many cakes she has in tota 6×4 (a) Write a different way to calculate the total number of cakes. You must not use multiplication.
Answer:
Step-by-step explanation:
6x4=24
another way of writing it is using addition
add 4, six times
so
4+4+4+4+4+4
6 boxes each has four cakes
Consider function f. f(x)=square x-1 which graph represents f?
The graph that represents the function f(x) = sqrt(x - 1) is given by the following option:
C. Graph Y.
What is a translation?A translation happens when either a figure or a function are moved horizontally or vertically.
The four translation rules for functions are given as follows:
Left a units: f(x + a).Right a units: f(x - a).Up a units: f(x) + a.Down a units: f(x) - a.For this problem, the function is defined as f(x) = sqrt(x - 1), meaning that the parent square root function was moved one unit right, and thus the correct graph is given by graph Y.
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The length of a rectangle is 6 feet less than 4 times the width. If the perimeter is 68 feet, find the length and the width of the rectangle.
Answer:
Length = 26 feet; Width = 8 feet
Step-by-step explanation:
Let:
x = Length of rectangle
y = Width of rectangle
4 times the width = 4y
∴ 6 less than 4 times the width = 4y - 6
Perimeter of a rectangle = 2 × [Length + Width]
68 = 2 × [(4y - 6) + y]
Expand the brackets or parenthesis by applying the Distributive Law and bring the like terms together:
68 = 2 × [4y - 6 + y]
68 = 2 × [5y - 6]
68 = 10y - 12
68 + 12 = 10y
Isolate y by making it the subject of the equation:
80 = 10y
y = [tex]\frac{80}{10}[/tex]
y = Width of rectangle: 8 feet
Substitute y into the equation to calculate x:
x = 4(8) - 6
x = Length of rectangle = [tex](32 - 6)[/tex] feet
x = 26 feet
Multiply
-1/5. (-2/7)
Write answer in simplest form.
Answer:
2/35
Step-by-step explanation:
[tex](-\frac{1}{5})(-\frac{2}{7}) = \frac{2}{35}[/tex]
Find the value of X , Y, and z in the parallelogram below.
Answer:
[tex]x = -64\\\\y = -63\\\\z = -57[/tex]
Step-by-step explanation:
Diagonally opposite angles of a parallelogram are equal
Therefore
-y - 3 = 60
Add 3 to both sides; -3 terms on left side cancel out
- y = 60 + 3
- y = 63
Multiply both sides by -1
-1(-y) = -1(63)
y = - 63
Another property of a parallelogram is that the angles formed on either end by a side must equal 180°.
Rephrased,
The consecutive or adjacent angles are supplementary which means they add up to 180°
We see that -2z + 6 and 60° are adjacent angles
So
- 2z + 6 + 60 = 180
-2z + 66 = 180
-2z = 180 - 66
-2z = 114
z = 114/-2
z = -57
Similarly
-2x - 8 + 60 = 180
-2x + 52 = 180
-2x = 180 - 52
-2x = 128
x = 128/-2
x = -64
12x + 7 < −11
OR
5x-8>40
Answer:
x <-1.5
or
x <9.6
Step-by-step explanation:
12x + 7 < - 11
12x < -11 - 7
12x < - 18
12x/12 < -18/12
x = -1.5
OR.
5x - 8 > 40
5x > 40 + 8
5x > 48
5x/5 > 48/5
x > 9.6.
Find the value of x.
(4x+12) 64°
OA. 19
OB. 3.5
O c. 26
OD. 13
Answer: OD. 13
Step-by-step explanation:
Find an equation of the plane.
the plane that contains the line
x = 2 + t,
y = 4 − t,
z = 3 − 3t
and is parallel to the plane
5x + 2y + z = 3
The equation of the plane containing the line x = 2 + t, y = 4 − t, z = 3 − 3t and is parallel to the plane 5x + 2y + z = 3 is 5x + 2y + z = 21
What is a plane?
Any flat, three-dimensional, infinitely long surface is considered to be a plane. It can alternatively be thought of as the two-dimensional equivalent of a point with zero dimensions, a line with one dimension, and a space with three dimensions.
Given that, a plane is containing the line x = 2 + t, y = 4 − t, z = 3 − 3t and is parallel to the plane 5x + 2y + z = 3
Any plan e parallel to the plane 5x + 2y + z = 3 , should be in the form of 5x + 2y + z = k
Now, this plane should contain the line mentioned above
On substituting the terms,
5(2 + t) + 2(4 - t) + (3 - 3t) = k
10 + 5t + 8 - 2t + 3 -3t = k
21 = k
Therefore, The equation of the required plane is 5x + 2y + z = 21
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The lunch special at Rachel's Restaurant is a sandwich, a drink and a dessert. There are 2 sandwiches, 3 drinks, and 2 desserts to choose from. How many lunch specials are possible?
Answer:
3 lunch specials are possible.
A, B, C and D are four towns.
B is 40 kilometres due East of A.
C is 40 kilometres due North of A.
D is 65 kilometres due South of A.
a) Work out the bearing of B from C.
135
b) Calculate the bearing of D from B.
Answer
+
G
a) The bearing of B from C is; 45° South-East.
b) The bearing of D from B is; 58.39° South Of West
How to calculate the bearing?a) Distance Between B and C is;
BC = √( 40² + 40²)
= 40√2
= 56.57 km
Angle of B from C = Tan⁻¹( 40/40) = 45°
Bearing of B From C = 45° South Of East
Or 90 - 45° = 45°
45° East of South
Thus, that is the bearing of B from C
b) Distance Between B and D is;
BD = √(40² + 65²)
= 76.32 km
Angle of D from B = Tan⁻¹( 65/40)
= 58.39 °
Bearing of D From B = 58.39° South Of West
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Solve 2x − 1 = 5.
A . x = 1
B . x = 2
C . x = 3
D . x = 4
C
The answer to the question is C
Answer:
D is the answer cuz 2x-1=5
2x=5-1
2x=4
X=2
We divide both side by two
If Emily throws the ball at an angle of 30 ∘ below the horizontal with a speed of 12 m/s , how far from the base of the dorm should Allison stand to catch the ball? Assume the vertical distance between where Emily releases the ball and Allison catches it is 8.0 m .
Express your answer with the appropriate units.
The distance at which Allison should stand to catch the ball will be 8.32 m.
What is speed?
Speed of an object is defined as the ratio of distance covered by the object to the time taken by the object to cover that distance. In other words its tell how much times an object takes to cover a particular distance.
Now for the given question:
Emily throw the ball at 30° below the horizontal towards Allison at a speed of 12m/s. So here we will calculate the Horizontal and Vertical components of the speed.
[tex]v_{x}[/tex]=12cos30°=10.4 m/s
[tex]v_{y}[/tex]=12sin30°=6 m/s
Vertical distance between Alice and Emily according the question is 8 m.
s= 8 m.
Now by applying Second equation of motion to calculate time take by ball to move from Emily to Allison.
s=[tex]v_{y}[/tex]×t+[tex]\frac{1}{2}[/tex]×a×[tex]t^{2}[/tex]
4= 6×t+[tex]\frac{1}{2}[/tex]×9.8×[tex]t^{2}[/tex] (a=9.8 m/[tex]s^{2}[/tex] acc. due to gravity a constant)
By solving above quadratic equation we get
t= 0.8 s
To find horizontal distance
d= [tex]v_{x}[/tex]×t
d=10.4×0.8
d= 8.32 m
Hence Allison should stand at a distance of 8.32 m to catch the ball.
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You deposit $9000 in an account earning 5% interest compounded continuously. The amount of money in the account after t years is given by A(t)=9000e^0.05t
How much will you have in the account in 14 years? $ Round your answer to 2 decimal places.
How long will you have in the account in 14 years? ______years. Round your answer to 2 decimals
How long does it take for the money in the account to double? ____ years. Round your answer to 2 decimal places
The amount of money which would be in the account after 14 years is; $18,123.77.
The number of years it'd take for the money to double is; 13.86 years.
What is the doubling time of the compound interest arrangement?Since the given model of the account is;
A (t) = 9000e^0.05t
After 14 years, it follows that; t = 14.
Therefore;
A (14) = 9000e^0.05(14)
A (14) = 9000 × 2.013752707
A (14) = $18,123.77.
Also, the doubling time implies that; A (t) = 18,000
Therefore, 18000 = 9000e^0.05t
2 = e^0.05t
0.05t = ln (2)
t = ln (2) / 0.05
t = 13.86 years.
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Which of the following expressions is equivalent to the logarithmic expression below? log4 7/x^3
Answer:
D
Step-by-step explanation:
using the rules of logarithms
• log a - log b = log ([tex]\frac{a}{b}[/tex] )
• log [tex]x^{n}[/tex] = nlog x
then
[tex]log_{4}[/tex] [tex]\frac{7}{x^3}[/tex]
= [tex]log_{4}[/tex] 7 - [tex]log_{4}[/tex] x³
= [tex]log_{4}[/tex] 7 - 3[tex]log_{4}[/tex] x
Can anyone help me solve this? Thank you!
Your friend sings every Saturday night at your favorite bar in San Diego. During his show, people show up at the rate of 1 person every minute.
a) What is the probability that on a random minute, at least 3 people show up?
b) What is the probability that he will have to wait at least 3 minutes for the next person to show up?
c) If he has already waited for 1 minute for the next person to show up, what is the probability that he will have to wait at least 3 more minutes for the next person to show up?
a) The probability that on a random minute, at least 3 people show up is 0.080.
b) The probability that he will have to wait at least 3 minutes for the next person to show up is 1 - (5/2e) .
c) The probability that he will have to wait at least 3 more minutes for the next person to show up is 2/e.
What is Poisson Probability?A probability distribution called the Poisson distribution is used to indicate how frequently an event is likely to happen over a given timeframe.
Given:
a) Let X be the number of people who show up during a random minute.
Then X follows a Poisson distribution with parameter λ = 1.
The probability of at least 3 people showing up in a random minute is:
P(X ≥ 3) = 1 - P(X < 3)
= 1 - (P(X = 0) + P(X = 1) + P(X = 2))
We can use the Poisson probability mass function to find P(X = k) for k = 0, 1, 2:
P(X = k) = ([tex]e^{-\lambda[/tex]) x ([tex]\lambda^k[/tex]) / k!
Plugging in λ = 1, we get:
P(X = 0) = [tex]e^{-1[/tex] / 0! = 1/e
P(X = 1) = [tex]e^{-1[/tex] x 1 / 1! = 1/e
P(X = 2) = [tex]e^{-1[/tex] x 1 / 2! = 1/2e
So, the probability of at least 3 people showing up in a random minute is:
= 1 - (P(X = 0) + P(X = 1) + P(X = 2))
= 1 - (1/e + 1/e + 1/2e)
= 1 - (2/e + 1/2e)
= 1 - (5/2e)
= 0.080
b) The probability of having to wait at least 3 minutes for the next person to show up
= 1 - P(X <= 2)
= 1 - (5/2e)
c) Now,
P(X ≤ 1)
= P(X = 0) + P(X = 1)
= (1/e) + (1/e)
= 2/e
So, the probability of having to wait at least 3 more minutes for the next person to show up
= 1 - (2/e)
= 1 - P(X <= 1).
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The default rate on government-guaranteed student loans at a certain private 4-year institution is 7 percent. The college extends 10 such loans. (a) What is the probability that none of them will default? (b) That at least three will default? (c) What is the expected number of defaults?
The probability that none of them will default 0.0066
What is probability?Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true.
(a) If 1,000 student loans are made, what is the probability of fewer than 50 defaults?
It's a binomial problem but the large n requires using a normal approximation.
mean = np = 1000*0.07 = 70
std = sqrt(npq) = sqrt(70*0.93) = 8.0685
z(50) = (50-70)/[8.0685] = -2.4788
P(# of defaults < 50) = P(z<-2.4788) = 0.0066
(b) More than 100?
z(100) = (100-70)/8.0685 = 3.7182
P(defaults > 100) = P(z>3.7182) = 0.00010035
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Correct question
The default rate on government-guaranteed student loans at a certaing public 4-year institution is 7% (a) If 1,000 student loans are made, what is the probability of fewer than 50 defaults? (b) More than 100? Show your work carefully.
Devaughn is 8 years older than Sydney. The sum of their ages is 94. What is Sydney's age?
Answer:
Step-by-step explanation:
S+8=D
S+D equals 94.
94/2= 45+2=47
47-8=39
Sydney is 39
Answer:
43
Step-by-step explanation:
94-8= 86
86/2= 43
43+8= 51
a:43
prove it :
51+43= 94
Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.
After 1 day, each woman would have made 8 scarves.
How many scarves would each woman have made?The expression that represents the number of scarves that Greta can make each day is:
Number of scarves she can make = number of scarves already made + (number of scarves she can make per day x number of days)
= 7 + d
The expression that represents the number of scarves that Evelyn can make each day is:
Number of days x number of scarves she can make per day
d x 8 = 8d
When both of them make the same number of scarves, the two expressions above would be equal.
7 + d = 8d
7 = 8d - d
7 = 7d
d = 7/7
d = 1 day
Number of scarves that would be made in 1 day = 8 x 1 = 8 scarves
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The lengths of the three sides of a triangle are x + 15 , 2x + 15 and x + 12 (where x cannot equal zero). What is the perimeter of the triangle?.
The perimeter of the triangle is 3x + 42.This can bean represented mathematically as (xd + 15) + (2x + 15) + (x + 12) = 3x + 42.
The perimeter of a triangle is the sum of the lengths of its three sides. In this problem, the lengths of the three sides of the triangle are given as x + 15, 2x + 15, and x + 12, where x cannot equal zero. To calculate the perimeter of the triangle, we need to add these three lengths together. This can bean represented mathematically as (xd + 15) + (2x + 15) + (x + 12) = 3x + 42. Therefore, the perimeter of the triangle is 3x + 42.
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Plot and connect the points in the order listed below. When you are done, find the area of the resulting figure.
A(-5,4)A(−5,4), B(4,4)B(4,4), C(4,1)C(4,1), D(1,1)D(1,1), E(4,-3)E(4,−3), F(4,-6)F(4,−6), G(-5, -6)G(−5,−6)
The total area of the figure with points A(-5,4), B(4,4), C(4,1), D(1,1), E(4,-3) F(4,-6), G(-5, -6) is 51 square units.
What is Graph?Graph is a mathematical representation of a network and it describes the relationship between lines and points.
Divide the figure is into two rectangles and a triangle.
The points B, C, E, and F form a rectangle with base BC and height 7 units.
The points A, D, and G form a rectangle with base AG and height 5 units. The points C, D, and E form a triangle with base CE and height 4 units.
The length of BC is 4 - 4 = 0, so the area of the rectangle with base BC is 0.
The area of the rectangle with base AG is (4 - (-5)) x 5 = 45 square units. The area of the triangle with base CE is (4 - 1) x 4 / 2 = 6 square units.
Therefore, the total area of the figure with points A(-5,4), B(4,4), C(4,1), D(1,1), E(4,-3) F(4,-6), G(-5, -6) is 45 + 6 = 51 square units.
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BIL
K-7 ft-X-8 ft-
K
-20 ft-
✈
The simplified expression of K-7 ft-X-8 ft-K-20 ft is -35ft-X
What is Expression?An expression is combination of variables, numbers and operators.
The given expression is K-7 ft-X-8 ft-K-20 ft
We need to simplify the given expression.
Add the like terms
-7 ft-X-8 ft-20 ft
-27ft-X-8 ft
-35ft-X
Hence, -35ft-X is the simplified expression of K-7 ft-X-8 ft-K-20 ft
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