Answer:
Exact Form:
12 /77
Decimal Form:
0. 155844
Step-by-step explanation:
PLEASE HELP
Given f(x)=4x^2 - 1 and g (x) = 2x- 1 , find each function
1. ( f+g )( x )
2. ( fg ) ( x )
3. ( f^n ) ( x )
The values of the composite functions are
(1) (f + g)(x) = 4x² + 2x - 2
(2) (fg)(x) = 8x³ - 4x² - 2x + 1
(3) (fⁿ)(x) = (4x² - 1)ⁿ
How to evaluate the composite functionsFrom the question, we have the following functions that can be used in our computation:
f(x) = 4x² - 1
g(x) = 2x - 1
Using the above as a guide, we have the following:
(f + g)(x) = f(x) + g(x)
(f + g)(x) = 4x² - 1 + 2x - 1
Evaluate the like terms
(f + g)(x) = 4x² + 2x - 2
(fg)(x) = f(x) * g(x)
(fg)(x) = (4x² - 1) * (2x - 1)
Evaluate the products
(fg)(x) = 8x³ - 4x² - 2x + 1
(fⁿ)(x) = (f(x))ⁿ
So, we have
(fⁿ)(x) = (4x² - 1)ⁿ
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What is the equation for the given graph expressed in the form of y = a|x – h| + k?
The equation of the graph in the required form is y = 5|x| - 5
How to determine the equation for the graphFrom the question, we have the following parameters that can be used in our computation:
The graph (see attachment)
The graph is an absolute value graph
And it can be expressed in the form of y = a|x – h| + k
Where
Vertex = (h, k)
The vertex of the graph is (0, -5)
So, we have
y = a|x| - 5
Using the point on the graph, we have
a|1| - 5 = 0
So, we have
a = 5
This means that the equation of the graph is y = 5|x| - 5
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What is the probability that either event will occur?
Now, find the probability of event A and event B.
A
B
6
6
20
20
P(A and B) = [?]
Enter as a decimal rounded to the nearest hundredth.
Enter
The probability that either event will occur is 0.62
What is the probability that either event will occur?From the question, we have the following parameters that can be used in our computation:
Event A = 6 + 6 = 12
Event B = 20 + 6 = 26
Both A and B = 6
Other Events = 20
Using the above as a guide, we have the following:
Total = A + B + C + Others - Both
So, we have
Total = 12 + 26 - 6 + 20
Evaluate
Total = 52
So, we have
P(A) = 12/52
P(B) = 26/52
Both A and B = 6/52
For either events, we have
P(A or B) = (12 + 26 - 6)/52
Evaluate
P(A or B) = 0.62
Hence, the probability that either event will occur is 0.62
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1. ¬{[Q → (¬S ∧ R)] ∨ ¬P} → (¬P ↔ P) ∴ ¬(Q ∧ P) ∨ (T ∧ R)
2. [(T ∨ Q) → (T ∨ S)] ∨ [(T ∨ Q) → (T ∨ S)]
Given statement solution is :- Regardless of the values of T, Q, and S, the whole Valid Statements & Implications is always true.
Let's analyze the two given statements:
¬{[Q → (¬S ∧ R)] ∨ ¬P} → (¬P ↔ P) ∴ ¬(Q ∧ P) ∨ (T ∧ R)
To prove this statement, we can use logical equivalences and deductions:
¬{[Q → (¬S ∧ R)] ∨ ¬P} → (¬P ↔ P) (Given)
¬{[¬Q ∨ (¬S ∧ R)] ∨ ¬P} → (¬P ↔ P) (Implication equivalence)
¬{[(¬Q ∨ ¬S) ∧ (¬Q ∨ R)] ∨ ¬P} → (¬P ↔ P) (Implication equivalence)
¬{(¬Q ∨ ¬S ∨ ¬P) ∧ (¬Q ∨ R ∨ ¬P)} → (¬P ↔ P) (De Morgan's Law)
(Q ∧ S ∧ P) ∨ (¬Q ∧ P) → (¬P ↔ P) (De Morgan's Law)
At this point, the formula ¬P ↔ P is a contradiction. The left side (¬P) states that P is false, while the right side (P) states that P is true. Therefore, the whole statement is always true, regardless of the values of Q, S, and P.
Since the statement is always true, we can conclude ¬(Q ∧ P) ∨ (T ∧ R) is true as well, making the second part of the question valid.
[(T ∨ Q) → (T ∨ S)] ∨ [(T ∨ Q) → (T ∨ S)]
In this statement, we have two identical sub-statements connected by a logical OR operator.
If we assume that (T ∨ Q) → (T ∨ S) is true, then the whole statement is true, regardless of the values of T, Q, and S.
If we assume that (T ∨ Q) → (T ∨ S) is false, then we need to check the other part of the statement.
Assuming (T ∨ Q) → (T ∨ S) is false means that (T ∨ Q) is true, but (T ∨ S) is false. In this case, the second part of the statement [(T ∨ Q) → (T ∨ S)] is true.
Therefore, regardless of the values of T, Q, and S, the whole Valid Statements & Implications is always true.
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which of the decimals below are less than 0.7? select all that apply.
A) 0.65
B) 0.71
C) 0.54
D) 0.31
E) 0.84
face pard He then decides to push the brick like a loy car 311 Mention the transformation of the brick when it is flipped or turned sideways $72 When the key is pushing the brick, without using or flipping the brick, what * What are the two differences between a rectangle and a paralelogram?
1. All angles of a rectangle are right angles (90 degrees), while the angles of a parallelogram can be any angle other than 90 degrees.
2. The opposite sides of a rectangle are equal in length, whereas in a parallelogram, opposite sides are of equal length as well, but they may not be perpendicular to each other.
The questions are a bit unclear and have some spelling errors, but I will try my best to provide an answer:
- Mention the transformation of the brick when it is flipped or turned sideways:
If the brick is flipped over, the transformation it undergoes is called a reflection. This can also be thought of as a mirror image of the original brick. If the brick is turned sideways, the transformation it undergoes is called a rotation. In a rotation, the object is turned about a point, and the image of the object appears to have been rotated around that point.
- When the key is pushing the brick, without using or flipping the brick, what... (I'm not sure what the question is asking for, as it seems incomplete or there is some text missing.)
- What are the two differences between a rectangle and a parallelogram?
In summary, the transformation of a flipped brick is a reflection, while a turned sideways brick undergoes a rotation. The differences between a rectangle and a parallelogram include the angles (all right angles for a rectangle and any angle other than 90 degrees for a parallelogram) and the opposite sides (equal length and perpendicular for a rectangle, equal length but not necessarily perpendicular for a parallelogram).
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A student’s course grades and their corresponding weights are given in the table.
What is the minimum grade needed on the final exam to earn an overall grade of 85% in the class?
The minimum grade needed to have overall grade of 85% in the class is 90%
What is percentage?A percentage is a number or ratio expressed as a fraction of 100. It is represented by by the symbol %.
Percentage can also be said as per 100
For example if there are total of 100 students in a class and their are 40 girls, the percentage of girls in the class is
40/100 × 100
= 40%
For the student to have a overall grade of 85%, the minimum grade in exam is calculated as
(100+70+80+x)/4 = 85
250+x = 340
x = 340 - 250
x = 90%
Therefore the minimum graded needed is 90%
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Mount Saint Helens, a volcano, erupted on May 18, 1980. Before eruption, Mount St. Helens was 2.95 kilometers high. use the bar diagram to find the difference in height of mount st. helens before and after the eruptions in meters PLEASE I NEED HELP :(((
The difference in height is 400 metres.
What is the difference in height of Mount St. Helens?Height refers to vertical distance from the top to the object's base. Occasionally, it is also labeled as an altitude which measures from down to top of a surface.
To get difference in height, we will subtract the height after the eruption from the height before the eruption:
Given:
Height before eruption = 2.95 kilometers
Height after eruption = 2.55 kilometers
Difference in height = Height before eruption - Height after eruption
Difference in height = 2.95 km - 2.55 km
Difference in height = 0.4 kilometers
0.4 kilometers to metres will be:
= 0.4 * 1000 metres
= 400 metres.
Full question:
Mount St. Helens, located in Washington, erupted on May 18, 1980. Before the eruption, the volcano was 2.95 kilometers high. After the eruption, the volcano was 2.55 kilometers high. Find the difference in height of Mount St. Helens before and after the eruption.
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pls solve this question its a nest pyq
After considering the given interval we reach the conclusion that the coefficient of t² is 1/2(3w² - 1), under the condition that the given expression is [tex](1-2 t w+t^{2})^{-1/2}[/tex] with range of (t<<1)
To evaluate the value for the given expression we have to apply the principles of binomial theorem
Then
[tex](1-2 t w+t^{2})^{-1/2} = (1 + (-2 t w + t^{2})/2 + (-2 t w + t^{2})^{2/8} + (-2 t w + t^{2})^{3/16} + ...)[/tex]
= 1 - t w + 3/8 * t² * w² - 5/16 * t³ * w³ +
The coefficient of t is the coefficient of the first term with a power of t.
Therefore, the coefficient of t is 1/2(3w² - 1).
The binomial theorem refers to the statement regarding any positive integer n, the nth power of the sum of two numbers a and b could be expressed as the sum of n + 1 terms of the form. The binomial theorem is applied in algebra and probability theory.
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Line r
goes through points (−5,2)
and (−3,8).
Line s
goes through points (−6,4)
and (2,12).
Which statement is true about lines r
and s?
Responses
Line r
is steeper than line s.
Line r is steeper than line
Line s
has a negative slope.
Line s has a negative slope.
Line s
is steeper than line r.
Line s is steeper than line
Line r
has a negative slope.
Answer: To determine which statement is true about lines r and s, we need to compare their slopes.
The slope of a line can be calculated using the formula:
slope = (change in y-coordinates) / (change in x-coordinates)
For line r, using the points (-5, 2) and (-3, 8):
slope_r = (8 - 2) / (-3 - (-5))
= 6 / 2
= 3
For line s, using the points (-6, 4) and (2, 12):
slope_s = (12 - 4) / (2 - (-6))
= 8 / 8
= 1
Now, let's analyze the statements:
Line r is steeper than line s. (False)
Since the slope of line r (3) is greater than the slope of line s (1), this statement is true.
Line r is steeper than line (Incomplete statement)
This statement is incomplete.
Line s has a negative slope. (False)
The slope of line s (1) is positive, not negative. So, this statement is false.
Line s is steeper than line r. (False)
Since the slope of line s (1) is less than the slope of line r (3), this statement is false.
Line r has a negative slope. (False)
The slope of line r (3) is positive, not negative. So, this statement is false.
Based on the analysis, the true statement about lines r and s is:
Line r is steeper than line s.
Jai spots an airplane on radar that is currently approaching in a straight line, and that will fly directly overhead. The plane maintains a constant altitude of 5375 feet. Jai initially measures an angle of elevation of 15° to the plane at point A. At some later time, he measures an angle of elevation of 30° to the plane at point B. Find the distance the plane traveled from point A to point B. Round your answer to the nearest foot if necessary.
The distance that the plane traveled from point A to point B is given as follows:
10,750 ft.
What are the trigonometric ratios?The three trigonometric ratios are the sine, the cosine and the tangent, and they are defined as follows:
Sine of angle = length of opposite side to the angle divided by the length of the hypotenuse.Cosine of angle = length of adjacent side to the angle divided by the length of the hypotenuse.Tangent of angle = length of opposite side to the angle divided by the length of the adjacent side to the angle.The altitude of 5375 feet is the opposite angle, hence the position A is obtained as follows:
tan(15º) = 5375/A
A = 5375/tangent of 15 degrees
A = 20060 ft.
The position B is obtained as follows:
B = 5375/tangent of 30 degrees
B = 9310 ft.
Hence the distance is of:
20060 - 9310 = 10,750 ft.
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Use the graph to answer the question.
Graph of polygon ABCD with vertices at negative 2 comma negative 1, 0 comma negative 4, 4 comma negative 4, 2 comma negative 1. A second polygon A prime B prime C prime D prime with vertices at 5 comma negative 1, 7 comma negative 4, 11 comma negative 4, 9 comma negative 1.
Determine the translation used to create the image.
7 units to the right
3 units to the right
7 units to the left
3 units to the left
The translation used to create the image A'B'C'D', from the pre image ABCD is T(7, 0), which corresponds with the option;
7 units to the right
What is a translation translation transformation?The coordinates of the vertices of the polygon ABCD are;
A(-2, -1), B(0, -4), C(4, -4), D(2, -1)
The coordinates of the vertices of the polygon A'B'C'D' are;
A'(5, -1), B'(7, -4), C'(11, -4), D'(9, -1)
The coordinates of the vertices of the image indicates;
The difference between the coordinates are;
A'(5, -1) - A(-2, -1) = (5 - (-2), -1 - (-1)) = (7, 0)
B'(7, -4) - B(0, -4) = (7 - 0, -4 - (-4)) = (7, 0)
C'(11, -4) - C(4, -4) = (11 - 4, -4 - (-4)) = (7, 0)
D'(9, -1) - D(2, -1) = (9 - 2, -1 - (-1)) = (7, 0)
The translation that is used to create the image is therefore;
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Read the description of g g below, and then use the drop-down menus to complete an explanation of why g g is or is not a function. g g relates a student to each course the student takes in a school year.
g is a function because each student is uniquely mapped to each course the student takes in a school year.
What is a function?In Mathematics and Geometry, a function refers to any mathematical equation which is typically used to define and represent a relationship that exists between two or more variables such as an ordered pair, points on a graph or table.
Based on the description of g, we can reasonably infer and logically deduce that the relation g represent a function because the input values are uniquely mapped to the output values.
In this context, we can conclude that the description of g represents a function because a student represent the input values that is being to each course (output value) the student takes in a school year.
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I’m am so lost please help thank you all
Answer:
Step-by-step explanation:
148 people
area of a rectangular piece of cloth 3 1/4m by 25 cm
Answer:
8125 cm² (or 0.8125 m²)
Step-by-step explanation:
area of a rectangular piece of cloth 3 1/4m by 25 cm
the area of the rectangle is found by making the base by the height, we convert the meters into centimeters
3 1/4 m = 3.25m = 325 cm
we find the area
325 × 25 = 8125 cm² (or 0.8125 m²)
Please help me important question in image
The following property must be given to prove that ΔCJG ~ ΔCEA: B. GJ║AE.
What are the properties of similar triangles?In Mathematics and Geometry, two triangles are said to be similar when the ratio of their corresponding side lengths are equal and their corresponding angles are congruent.
Additionally, the lengths of three (3) pairs of corresponding sides or corresponding side lengths are proportional to the lengths of corresponding altitudes when two (2) triangles are similar.
Based on the side, side, side (SSS) similarity theorem, we can logically deduce the following congruent and similar triangles:
ΔCJG ≅ ΔCEA (line segment GJ is parallel to line segment AE)
ΔBIJ ≅ ΔBDF (line segment IJ is parallel to line segment DF).
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If D = 2w? - w - 7 and C = 3w - 6, find an expression that equals D + 3C in
standard form.
The expression that equals D + 3C in standard form is 2w² + 8w - 25.
To find an expression that equals D + 3C in standard form, we first need to simplify D and C.
Starting with D = 2w² - w - 7, we can rearrange the terms to put it in standard form:
D = 2w² - w - 7
D = 2w² - 2w + w - 7
D = 2w(w - 1) + (w - 7)
Next, simplifying C = 3w - 6:
C = 3w - 6
Now, we can substitute these expressions into D + 3C:
D + 3C = (2w² - w - 7) + 3(3w - 6)
Expanding and simplifying:
D + 3C = 2w² - w - 7 + 9w - 18
D + 3C = 2w² + 8w - 25
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the area of a trapizoid is 150 square millimeters 7mm 16mm 13mm what is the trapezoids height
The height of the trapezoid is approximately 13.043 mm.
Understanding How to Solve TrapezoidTo find the height of a trapezoid, you can use the formula:
Area = 1/2 * (base1 + base2) * height
Given:
Area = 150 square millimeters
base1 = 7 mm , w
base2 = 16 mm
150 = 1/2 * (7 + 16) * height
150 =1/2 * 23 * height
Make height the subject of the formula
300 = 23 * height
height = 300 / 23
height = 13.043 mm (rounded to three decimal places)
Therefore, the height of the trapezoid is approximately 13.043 mm.
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What is the unit rate if you drove a go-kart 9 miles in 36 minutes?
Answer:
Step-by-step explanation:
To find the unit rate, we need to divide the distance traveled by the time taken.
In this case, the distance traveled is 9 miles, and the time taken is 36 minutes.
Unit rate = distance/time
Unit rate = 9 miles / 36 minutes
Simplifying the above expression by dividing both numerator and denominator by 9, we get:
Unit rate = 1 mile / 4 minutes
Therefore, the unit rate at which the go-kart is traveling is 1 mile per 4 minutes.
select all answers that are similar to polygon K
Answer:
B, C, D
Step-by-step explanation:
You want to identify the shapes that are similar to polygon K.
Right trianglePolygon K is a right triangle that has leg lengths in the ratio 4:2 = 2:1.
Polygons B, C, and D are right triangles similar to polygon K:
B is a dilation by a factor of 1/2C is a reflection of BD is a rotation of KOthersPolygons A and E are also right triangles, but have different ratios of leg lengths. In A they are 1/1, and in E they are 4/3. These triangles are not similar to triangle K.
__
Additional comment
Polygons similar to the same polygon are similar to each other. Rigid transformations such as reflection and rotation result in figures congruent to the original. Congruent figures are also similar.
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Can someone answer this question
Answer:
The correct answer is (d): (x - 5) is a factor of f(x).
Step-by-step explanation:
100 Points! Algebra question. Photo attached. Graph the function. Thank you!
The graph of the function is attached and the amplitude and the period are 5 and 2π
How to determine the amplitude and period of the functionFrom the question, we have the following parameters that can be used in our computation:
f(x) = 5cos(θ)
A sinusoidal function is represented as
f(x) = Acos(B(x + C)) + D
Where
Amplitude = APeriod = 2π/BSo, we have
A = 5
Period = 2π/1
Evaluate
A = 5
Period = 2π
Hence, the amplitude is 5 and the period is 2π
The graph of the function is attached
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A rectangle has 2 sides that are each 6 centimeters lo[ng. The perimeter is 22 centimeters. How long are the other sides?
Answer:
The other sides = 5 centimeters.
Step-by-step explanation:
Let's assume the length of the other two sides of the rectangle is "x" centimeters.
A rectangle has two pairs of equal sides. Therefore, we can set up the equation for the perimeter of the rectangle:
Perimeter = 2(length + width)
Given:
Length = 6 centimeters
Perimeter = 22 centimeters
Using the equation, we can substitute the given values:
22 = 2(6 + x)
Simplifying further:
22 = 12 + 2x
Subtracting 12 from both sides:
10 = 2x
Dividing both sides by 2:
5 = x
Therefore, the length of the other two sides of the rectangle is 5 centimeters.
What is the sum of the interior angles of the polygon shown below?
Answer: 360 degrees.
Step-by-step explanation: The sum of the interior angles in a quadrilateral is always 360.
(you can check using the formula 180(n-2), where n is the number of sides.)
Work out the integer values
Please help, I attached a photo.
Step by step explanation if possible :)))
The integers that satisfy the inequality are -1, 0, 1, and 2.
To find the integer values that satisfy the inequality x < 6/(x - 1), we can start by analyzing the domain of the expression.
First, note that the denominator (x - 1) cannot be equal to zero since division by zero is undefined. Therefore, we must exclude x = 1 from the domain.
Next, consider the sign of the expression 6/(x - 1). When the denominator (x - 1) is positive, the expression will be positive. Similarly, when the denominator is negative, the expression will be negative.
Considering the inequality x < 6/(x - 1), we can divide the problem into two cases based on the sign of (x - 1):
Case 1: (x - 1) > 0
In this case, the expression 6/(x - 1) is positive. To satisfy the inequality x < 6/(x - 1), x must be less than the positive value of 6/(x - 1). Since the denominator is positive, we can drop the absolute value sign. Therefore, the inequality becomes:
x < 6/(x - 1)
Case 2: (x - 1) < 0
In this case, the expression 6/(x - 1) is negative. To satisfy the inequality x < 6/(x - 1), x must be greater than the negative value of 6/(x - 1). Since the denominator is negative, we need to flip the inequality sign and change the direction. Therefore, the inequality becomes:
x > 6/(x - 1)
Now, let's solve each case separately:
Case 1: (x - 1) > 0
Since the denominator (x - 1) is positive, we can multiply both sides of the inequality by (x - 1) without changing the direction of the inequality. This gives:
x(x - 1) < 6
Expanding the left side:
x² - x < 6
Rearranging the inequality:
x² - x - 6 < 0
Now we can factorize the quadratic:
(x - 3)(x + 2) < 0
To determine the solution, we need to consider the sign of the expression for different intervals:
When x < -2, both factors are negative, so the expression is positive.
When -2 < x < 3, the first factor (x - 3) is negative, and the second factor (x + 2) is positive, so the expression is negative.
When x > 3, both factors are positive, so the expression is positive.
Therefore, the solution to Case 1 is:
-2 < x < 3
Case 2: (x - 1) < 0
Since the denominator (x - 1) is negative, we need to flip the inequality sign. Multiplying both sides of the inequality by (x - 1) changes the direction of the inequality. This gives:
x(x - 1) > 6
Expanding the left side:
x² - x > 6
Rearranging the inequality:
x² - x - 6 > 0
Factoring the quadratic:
(x - 3)(x + 2) > 0
Considering the sign of the expression for different intervals:
When x < -2, both factors are negative, so the expression is positive.
When -2 < x < 3, the first factor (x - 3) is negative, and the second factor (x + 2) is positive, so the expression is negative.
When x > 3, both factors are positive, so the expression is positive.
Therefore, there are no solutions in Case 2.
Combining the solutions from both cases, we find that the integer values satisfying the inequality x < 6/(x - 1) are:
-2 < x < 3
In other words, the integers that satisfy the inequality are -1, 0, 1, and 2.
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Now, let’s rewrite our point-slope equation in slope-intercept form.
y−6=−2(x+2)
y−6=
x−4
y=−2x+2
Is this the same equation we got when we wrote it in slope-intercept form earlier? YES!
Any equation written in point-slope form can be rewritten in slope-intercept form and most of the time, we will rewrite them. Slope-intercept form is the most popular and used form of linear functions/equations. In Slope-intercept form, all like terms have been combined, therefore it is in simplified form. Remember that we always want the most simplified answer possible, unless otherwise directed!
The equation in slope-intercept form is,
y = - 2x + 2
We have to given that;
The point-slope equation is,
y - 6 = - 2 (x + 2)
Now, We can rewrite the equation in slope-intercept form as;
y - 6 = - 2 (x + 2)
y - 6 = - 2x - 4
y = - 2x - 4 + 6
y = - 2x + 2
Thus, the equation in slope-intercept form is,
y = - 2x + 2
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how do you find out the area of a tent, floor included
To find the area of a tent, including the floor, we need to measure or determine the dimensions of both the tent's floor and any additional areas such as vestibules or extensions.
Floor Area: Measure the length and width of the tent's floor in the same unit of measurement (e.g., feet or meters). Multiply the length by the width to calculate the floor area. For example, if the length is 8 feet and the width is 6 feet, the floor area would be 8 feet * 6 feet = 48 square feet.Additional Areas: If the tent has vestibules, extensions, or any other separate areas, measure each area's dimensions and calculate their individual areas separately.Total Area: Once you have calculated the individual areas of the floor and any additional areas, simply add them together to find the total area of the tent, including the floor. For example, if the floor area is 48 square feet and there is an additional vestibule area of 10 square feet, the total area would be 48 square feet + 10 square feet = 58 square feet.Learn more about the area of the tent here:
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48 as a tree diagram prime factor
Answer:
48 prime factors are: 2 x 2 x 2 x 2 x 3
Find the measure of AB.
20,
E
61%
D
B
A
21. B
65°
D
C
22. A
B
E
D
91
To find the measure of angle AB, we can use the fact that the sum of the angles in a triangle is 180 degrees.
Given:
Angle A = 65 degrees
Angle B = 91 degrees
We can subtract the sum of these two angles from 180 degrees to find angle AB:
Angle AB = 180 - (Angle A + Angle B)
Angle AB = 180 - (65 + 91)
Angle AB = 180 - 156
Angle AB = 24 degrees
Therefore, the measure of angle AB is 24 degrees.
List all numbers from the given set that are a. natural numbers, b. whole numbers, c. integers, d. rational numbers, e. irrational numbers, and f. real numbers.
Answer:
See attached
Step-by-step explanation:
You want the given numbers classified as to Natural, Whole, Integer, Rational, Irrational, and/or Real.
RealAll the given numbers are real.
IrrationalThe number √2 is irrational. All the others are rational.
IntegerThe values √16 = 4, 0, and -5 are integer values. Whole numbers are non-negative integers, so exclude -5. Natural numbers are positive integers, so exclude 0 and -5.
The Xs indicate the categories each number belongs to.