When skiing down a mountain with a vertical height of 1150 feet and a base-to-top distance of 2300 feet, the angle of elevation from the base to the top of the mountain is approximately 26.56 degrees.
When skiing down a mountain with a vertical height of 1150 feet and a base-to-top distance of 2300 feet, the angle of elevation from the base to the top of the mountain is given by tan⁻¹(1150/2300).
How to find the angle of elevation from the base to the top of the mountain?
The angle of elevation of an object from a point is the angle between the horizontal and the line of sight from the observer to the object above the horizontal. We can use the inverse tangent function to calculate this angle of elevation.
Step 1: Find the tangent of the angle of elevation: Let the height of the mountain be h and the distance from the base to the top of the mountain be d. Then tan(θ) = h/d.
Step 2: Inverse tangent of the height and distance of the mountain: Take the inverse tangent of the quotient to find the angle of elevation from the base to the top of the mountain. θ = tan⁻¹(h/d).Here, h = 1150 feet and d = 2300 feet. Substituting these values in the formula θ = tan⁻¹(1150/2300), we can find the angle of elevation from the base to the top of the mountain.θ = tan⁻¹(1150/2300) = 26.56 degrees.
So, the angle of elevation from the base to the top of the mountain is approximately 26.56 degrees.
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answer the question in the image pls :)....................................
The true statements about the image △A'B'C' include the following:
A. AB is parallel to A'B'.
B. [tex]D_{O, \frac{1}{2} }[/tex] (x, y) = (1/2x, 1/2y)
C. The distance from A' to the origin is half the distance from A to the origin.
What is a dilation?In Mathematics and Geometry, a dilation simply refers to a type of transformation which typically changes the size of a geometric object, but not its shape.
This ultimately implies that, the size of the geometric shape would be increased (stretched or enlarged) or decreased (compressed or reduced) based on the scale factor applied.
Next, we would apply a dilation to the coordinates of the pre-image by using a scale factor of 0.5 centered at the origin as follows:
Ordered pair A (-4, 3) → Ordered pair A' (-4 × 1/2, 3 × 1/2) = Ordered pair B' (-2, 1.5).
Ordered pair B (4, 4) → Ordered pair B' (4 × 1/2, 4 × 1/2) = Ordered pair B' (2, 2).
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eight people are sitting around a circular table, each holding a fair coin. all eight people flip their coins and those who flip heads stand while those who flip tails remain seated. what is the probability that no two adjacent people will stand? (2015amc10a problem 22 or 2015amc12a problem 17) (a) 47 256 (b) 3 16 (c) 49 256 (d) 25 128 (e) 51 256
The probability that no two adjacent people would stand is option C: 49/256, as there are 256 distinct counting rotations.
First, we assume that one person stands up. There are 8 possible ways to select which person does. We can, next, assume that no two adjacent people stand up. There are 4 ways to choose which of the remaining 7 people will stand up (since no two adjacent people can stand up).
Then, we can assume that no two adjacent people stand up again. There are 3 ways to choose which of the remaining 6 people will stand up (since no two adjacent people can stand up).
We continue this process until there is only one person left who can stand up.
The number of arrangements according to the number of people standing.
0 people standing = 1 arrangement.1 people standing = ₈C₁ = 8! / 1!7! = 8 arrangements.2 people standing = ₈C₂ = 28 arrangements, but no two people are next to each other. There are 8 arrangements that two people in this case are standing next to each other. So, it will be 28 - 8 = 20 arrangements. 3 people standing = ₈C₃ = 56 arrangements. In this case, three people standing are next to each other = ₈C₁ = 8 arrangements. Two people standing are next to each other and the third person is not = ₈C₁ × ₄C₁ = 8 × 4 = 32. So, it will be 56 - 8 - 32 = 16 arrangements.4 people standing but no two adjacent people = 2 arrangements. Other than these, 5 people standing and 6 people standing would have one arrangement each.Therefore, the number of arrangements that no two adjacent people will stand.
n(S) = 1 + 8 + 20 + 16 + 2 + 2 = 49
The total number of possible outcomes is:
2⁸ = 256
So, the probability that no two adjacent people will stand would be:
= 49/256.
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Complete question is:
Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand?
(a) 47/256
(b) 3/16
(c) 49/256
(d) 25//128
(e) 51/256
PLEASE HELP!!
Tone Sherburn bought a home with a 10.5% adjustable rate mortgage for 30 years. He paid $9.99 monthly per thousand on his original loan. At the end of 5 years he owes the bank $55,000. Now that interest rates have gone up to 12.5%, the bank will renew the mortgage at this rate or Tone can pay $55,000. Tone decides to renew and will now pay $10.68 monthly per thousand on his loan.
What is the amount of the old monthly payment? What is the percent of increase in his new monthly payment? You can ignore the small amount of principal that has been paid.
old monthly payment = $
new monthly payment = $
% increase
Tone's new monthly payment is 16.16% higher than his old monthly payment To solve the problem, we need to use the formula for the monthly payment of a mortgage loan:
monthly payment = (loan amount x monthly interest rate) / (1 - (1 + monthly interest rate)^(-n))
where:
loan amount = the amount of the loan
monthly interest rate = the annual interest rate divided by 12
n = the total number of payments (in months)
Let's first find the loan amount at the beginning of the mortgage. We know that Tone paid $9.99 per thousand, so we can set up a proportion:
$9.99 / 1000 = monthly payment / loan amount
Solving for the loan amount, we get:
loan amount = monthly payment / ($9.99 / 1000) = $1000 * (monthly payment / $9.99)
Substituting the given values, we get:
loan amount = $1000 * (monthly payment / $9.99) = $55,000
This means that Tone borrowed $55,000 at the beginning of the mortgage.
Now, let's find the old monthly payment. We know that Tone had a 10.5% adjustable rate mortgage for 30 years, which means he made 12 x 30 = 360 monthly payments. We can use the formula above to solve for the monthly payment:
monthly interest rate = 10.5% / 12 = 0.00875
n = 360
monthly payment = ($[tex]55,000 x 0.00875) / (1 - (1 + 0.00875)^(-360)[/tex]) = $512.47
Therefore, Tone's old monthly payment was $512.47.
Next, let's find the new monthly payment. We know that Tone will renew his mortgage at 12.5% interest rate, which means the new monthly interest rate is:
monthly interest rate = 12.5% / 12 = 0.01042
We also know that Tone will pay $10.68 per thousand, so we can set up a new proportion:
$10.68 / 1000 = new monthly payment / loan amount
Solving for the new monthly payment, we get:
new monthly payment = $10.68 / ($1000 / $55,000) = $595.40
Therefore, Tone's new monthly payment is $595.40.
Finally, let's find the percent increase in the new monthly payment compared to the old monthly payment. We can use the percent change formula:
percent increase = ((new value - old value) / old value) x 100%
Substituting the given values, we get:
percent increase = (($595.40 - $512.47) / $512.47) x 100% = 16.16%
Therefore, Tone's new monthly payment is 16.16% higher than his old monthly payment
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what is 59.58333333 rounded off to the nearest ten?
Given:
The number rounded to the nearest tenth is 59.6.
To find:
The approximate value of the given number to the nearest tenth.
Solution:
The required values shown in the below table:
Number Nearest tenth
59.58333333 59.6
I DESPERATE AND I'LL GIVE LOTS OF POINTS PLEASE HELP ME THIS IS URGENT I WILL GIVE BRAINLIEST
Step-by-step explanation:
Top floor is x next floor is 2 x then the bottom is 4 x
= 7x
7x = 245 x = 35
then 35 ft 70 ft 140 ft = 245 ft total
How could you correctly rewrite the equation 4(5 + 3) = 2(22 – 6) using the distributive property?
Answer: 20+12=44-12
Step-by-step explanation:
You can distribute the multiplication on both sides of the equation.
Whenever you see a number before parentheses, it means to multiply that number to all terms in the parentheses.
so the new equation will be: (4 * 5) +(4 *3) = (2 *22) - (2*6)
Next, we simplify: 20 + 12 = 44 - 12
If the above equation is what you are looking for then use that but you can simplify further and add the numbers together so it will be
32=32
a semiconductor manufacturer recently found that 3% of the chips produced at its new plant are defective. assume that the chips are independently defective or non defective. use a normal approximation to determine the probability that a box of 500 chips contains: (a) at least 10 defective chips (b) between 15 and 20 (inclusive) defective chips.
(a) The probability of having at least 10 defective chips is approximately 0.9131. (b) The probability of having between 15 and 20 defective chips is approximately 0.4135.
First, we need to find the mean (μ) and standard deviation (σ) for the binomial distribution.
μ = np = 500 * 0.03 = 15
σ = sqrt(np(1-p)) = sqrt(15*0.97) ≈ 3.66
(a) To find the probability of at least 10 defective chips, we'll calculate the Z-score and use a Z-table:
Z = (X - μ) / σ = (10 - 15) / 3.66 ≈ -1.36
P(X ≥ 10) = 1 - P(X ≤ 10) = 1 - 0.0869 = 0.9131
So, the probability of having at least 10 defective chips is approximately 0.9131
(b) To find the probability of between 15 and 20 defective chips (inclusive), we'll calculate the Z-scores and use a Z-table:
Z1 = (15 - 15) / 3.66 ≈ 0
Z2 = (20 - 15) / 3.66 ≈ 1.36
P(15 ≤ X ≤ 20) = P(Z1 ≤ Z ≤ Z2) = P(0 ≤ Z ≤ 1.36) = 0.4135
So, the probability of having between 15 and 20 defective chips is approximately 0.4135.
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10. Kyle has a container in a shape of a cone. The container has a radius of 5 inches and a height
of 8 inches.
a) What is the volume of the container? (Leave in terms of pi.)
Answer:
Volume= 66.7π
Step-by-step explanation:
Volume = 1/3 ×π×h×r^2
V = 1/3 ×π ×8× 5 ^2
V = 66.7πinches^ 3
daniel made a rectangle from 2 congruent trapezoids with bases 11 and 8 and height of 6. give the length, width, and area of the rectangle
Since the rectangle is made up of 2 congruent trapezoids, we can find the length and width of the rectangle by combining the lengths and widths of the trapezoids.
First, let's find the length of the rectangle:
The length of each trapezoid is the average of its bases:
length = (11 + 8) / 2 = 9.5
Since the trapezoids are congruent, the length of the rectangle is twice the length of a trapezoid:
length of rectangle = 2 * 9.5 = 19
Next, let's find the width of the rectangle:
The height of the trapezoids is the same as the height of the rectangle:
height = 6
The width of the rectangle is the same as the width of a trapezoid. To find the width of a trapezoid, we need to use the Pythagorean theorem, since the trapezoid has a height of 6 and bases of 11 and 8:
width = sqrt(6^2 + ((11-8)/2)^2) = sqrt(36 + 0.75) = sqrt(36.75)
So, the width of the rectangle is:
width = sqrt(36.75)
Finally, let's find the area of the rectangle:
area = length * width = 19 * sqrt(36.75) ≈ 87.83
Therefore, the length of the rectangle is 19 units, the width is approximately 6.07 units, and the area is approximately 87.83 square units.
Check the picture below.
PLEASE HELP will give brainliest
The value of f(11) for the recursive rule and an explicit rule is 149.
What is arithmetic sequence?An arithmetic sequence is a set of integers where each term is created by multiplying the preceding term by a fixed amount (referred to as the common difference). The nth term is often indicated by an, while the first term of the series is typically marked by a1. The following formula is used to determine the nth term in an arithmetic sequence:
a = a1 + (n-1)d
where d is the common distinction. In mathematics and other disciplines, arithmetic sequences are frequently employed to represent circumstances in which the pace or degree of change remains constant.
The recursive and explicit rule for the arithmetic sequence is given as:
an = a1 + (n-1)d
From the given graph f(1) = 19 and common difference = 32 - 19 = 13
The recursive rule is:
f(n) = f(n-1) + 13
The explicit rule is:
f(n) = 19 + 13(n-1)
Now, f(11) is calculated as:
f(11) = 19 + 13(11-1) = 19 + 130 = 149
Hence, the value of f(11) for the recursive rule and an explicit rule is 149.
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Which expression is the equivalent
The expression that is equivalent to 5[4+3(x−6)] is 15x - 70
How to find the equivalent expression?To simplify the expression 5[4+3(x−6)] , we can first simplify the expression inside the square brackets, which is (x-6) multiplied by 3 and then added to 4. This gives us:4 + 3(x - 6) = 4 + 3x - 18 = 3x -
this expression back into the original equation, we get:5[4+3(x−6)] = 5(4 + 3x - 18) = 5(3x - 14) = 15x - 70
Therefore, the expression 15x - 70 is equivalent to 5[4+3(x−6)] as seen here.
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Missing parts;
Which expression is the equivalent to 5[4+3(x−6)]
(In a survey of 200 people, the ratio of people who were infected by covid-19 delta variant to omicron variant was 5: 3. The number of people who were infected by both variant was half of those who were infected by only omicron variant. If 60 people were infected by neither of the both variant then find the number of people who were infected by only one variant and show the result in a Venn-diagram.)
Step-by-step explanation:
The given attachments are the steps to getting the answer...
the four vertices of a regular tetrahedron are snipped off, leaving a triangular face in place of each corner and a hexagonal face in place of each original face of the tetrahedron. how many edges will the new polyhedron have?
If 4 vertices of a "regular-tetrahedron" are snipped off which leaves a "triangular-face" in place of each corner, then there will be 18 edges in the new polyhedron.
If we snip off the 4 vertices of a "regular-tetrahedron", leaving a triangular face in place of each corner and a hexagonal-face in place of each original face of the tetrahedron,
We get a truncated tetrahedron.
we know that a truncated-tetrahedron has ⇒ four regular "hexagonal-faces", four equilateral "triangular-faces", 12 vertices and 18 edges.
Therefore, the new polyhedron will have 18 edges.
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what rationale is correct for the nusre to empty a hemovac woudn suction device when it is half full
A nurse should empty a Hemovac wound suction device when it is half full to ensure proper functioning and prevent discomfort or injury to the patient due to the device becoming too heavy.
The rationale for a nurse to empty a Hemovac wound suction device when it is half full is to ensure the device is functioning correctly and to prevent it from becoming too heavy and potentially causing discomfort or injury to the patient.
When a Hemovac wound suction device is half full, it may start to lose suction, which can result in less efficient drainage of fluid from the wound site. By emptying the device, the nurse can ensure that the device is functioning correctly and that the suction pressure is maintained.
Additionally, allowing the device to become too full can make it heavy and uncomfortable for the patient, potentially causing discomfort or injury to the wound site. Therefore, emptying the device when it is half full can help prevent these complications and ensure that the patient remains comfortable throughout the healing process.
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Kathleen bought a used 2017 Honda Civic for
$18,000. The car will depreciate at a rate of 11% per
year.
Write an equation that can be used to predict the
value of Kathleen's Honda Civic over the next few
years by typing values into the blank spaces.
y =
).x
(
Answer: y = 18000(0.89)^x, where x represents the number of years in the future.
Step-by-step explanation:
Answer:
Y= 18,000 X (0.11x)
Step-by-step explanation:
x being years y being car
Michael is paid $450 per week and receives a 4% commission on sales in excess of $1000. What was Michaels sales in a week if he paid $570?
Find the volume of the solid generated when the semicircle below is rotated about its diameter RST. Round your answer to the nearest tenth if necessary.
The volume of the solid generated by the revolution is 33.5 cubic units.
It is given that, the diameter of the semi-circle is 4 and it is rotated to one full rotation around its diameter.
A solid generated when a semicircle is being rotated about its diameter is called a "SPHERE".
Therefore, the volume of the solid generated by the revolution is the volume of the sphere.
The formula for the volume of the sphere is given by,
Volume of sphere = (4/3)πr³
where r is the radius and π has the default value of 3.14
Here, the given diameter is 4.
To find the radius = diameter/2
radius = 4/2 = 2.
Now, to calculate the volume of the sphere substitute r=2 and π=3.14
volume of the sphere = (4/3)×3.14×2³
⇒ (4/3)×3.14×8
⇒ 100.48 / 3
⇒ 33.49 (approximately 33.5)
Therefore, the volume of the solid generated by the revolution is 33.5 cubic units.
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PLEASE HELP!!!!!!!!! ITS OVERDUE!!!!!!
ΔDEF is graphed on the coordinate plane.
1. Complete the algebraic rule for each kind of rotation. (2 points)
Rotation
Algebraic rule
90° clockwise about the origin
(x, y)→
180° about the origin
(x, y) →
270° clockwise about the origin
(x, y) →
2. Draw the image of ΔDEF after a 90° clockwise rotation. (2 points)
3. Draw the image of ΔDEF after a 180° rotation. (2 points)
4. Draw the image of ΔDEF after a 270° clockwise rotation. (2 points)
5. Without drawing a 360° rotation, describe how it would appear. (2 points)
The algebraic rule for each kind of rotation is completed below and a 360° rotation would appear as if the shape didn't rotate at all.
Completing the algebraic rule for each kind of rotationAs a general rule of rotation, the algebraic rule for each kind of rotation are
Rotation Algebraic rule
90° clockwise about the origin (x, y)→ (y, -x)
180° about the origin (x, y) → (-x, -y)
270° clockwise about the origin (x, y) → (-y, x)
The images of the triangles cannot be drawn because they are not given
The description of a 360 degreesA 360° rotation about the origin would result in the original shape returning to its starting position.
It would appear as if the shape didn't rotate at all.
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If we are given an acute ∠A, side a, and side b, and the height of the triangle is h = bsin A, state the criteria needed for the following to happen:
No triangles when...
One triangle when... (2 answers)
Two triangles when...
What is triangle?
A triangle is a three-sided polygon with three vertices. The triangle's internal angle, which is 180 degrees, is constructed.
The given information is an acute angle ∠A, and sides a and b, along with the height of the triangle h = b sin A. The criteria for the number of possible triangles that can be formed are as follows:
No triangles can be formed if the length of side a is less than or equal to the length of the height h. That is, if a ≤ h, then no triangle can be formed. This is because the height is the perpendicular distance from the vertex of the angle to the opposite side, and it is necessary for the opposite side to be longer than the height in order for a triangle to exist.One triangle can be formed if the length of side a is greater than the length of the height h. That is, if a > h, then one triangle can be formed. In this case, the triangle is unique, since the other two sides and the angle are fixed.Two triangles can be formed if the length of side a is greater than the length of the height h, and if sin A is less than or equal to a/b. That is, if a > h and sin A ≤ a/b, then two triangles can be formed. In this case, the angle and the two sides adjacent to the angle are fixed, but the length of the opposite side can vary, which leads to the possibility of two triangles with different lengths for the opposite side.Learn more about triangles on:
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Shown below is a circle inside of a square, ABCD.
The circle touches the 4 sides of the square.
The area of the circle is 105cm²
Find the area of the square, ABCD.
The area of the square ABCD is approximately 420/π cm².
To find the area of the square, follow these steps:
1. Calculate the radius of the circle:
Since the area of the circle is 105 cm², you can use the formula for the area of a circle,
which is A = πr²,
where A is the area and r is the radius.
105 = πr²
Divide both sides by π:
r² = 105/π
Take the square root of both sides:
r = √(105/π)
2. Calculate the side length of the square:
Since the circle touches all 4 sides of the square, the diameter of the circle is equal to the side length of the square. The diameter is twice the radius (d = 2r), so:
d = 2√(105/π)
3. Calculate the area of the square:
Now, use the formula for the area of a square, which is A = s²,
where A is the area and s is the side length. In this case, s = d:
A = (2√(105/π))²
A = 4 * (105/π)
4. Multiply to find the area:
A = 420/π cm².
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What is the surface area of this right triangular prism?
Enter your answer in the box.
in²
Surface area of this right triangular prism=96in²
Define right triangular prismA right triangular prism is a three-dimensional geometric shape that has two parallel triangular bases and three rectangular faces that connect the bases. The triangular bases of a right triangular prism are always perpendicular to its rectangular faces. The prism is called "right" because the rectangular faces meet the bases at right angles, meaning the edges connecting the triangular and rectangular faces form right angles.
In the given right triangular prism
S₁=5in
S₂=5in
S₃=8in
l=4in
b=8in
h=3in
Total surface area of this right triangular prism = (S₁+S₂+S₃)l+(b×h)
=(5+5+8)×4+8×3
=96
Hence, surface area of this right triangular prism=96in².
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An ordinary (fair) die is a cube with the numbers 1 through 6 on the sides (represented by painted spots). Imagine that such a die is rolled twice in succession and that the face values of the two rolls are added together. This sum is recorded as the outcome of a single trial of a random experiment.
Compute the probability of each of the following events.
Event A: The sum is greater than 5.
Event B: The sum is an odd number.
Write your answers as fractions.
The probability of event A, the sum being greater than 5, is 11/36. The probability of event B, the sum being an odd number, is 1/2.
Explanation:To calculate the probability of each event, we need to determine the total number of possible outcomes and the number of favorable outcomes for each event.
Event A: The sum is greater than 5. There are 11 favorable outcomes (6,5), (6,4), (6,3), (6,2), (6,1), (5,6), (4,6), (3,6), (2,6), (1,6), and (5,5). The total number of possible outcomes is 36. So, the probability of event A is 11/36.
Event B: The sum is an odd number. There are 18 favorable outcomes (1,3), (1,5), (1,5), (2,1), (2,3), (2,5), (2,3), (3,1), (3,3), (3,5), (4,1), (4,3), (4,5), (5,1), (5,3), (5,5), (6,1), (6,3), and (6,5). The probability of event B is 18/36 or 1/2.
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find the value of g(7). g(x)=7/8x- 1/2
Answer:
Sure, I can help you with that! To find the value of g(7), we simply need to substitute x=7 into the expression for g(x): g(7) = (7/8)*7 - 1/2 g(7) = 49/8 - 1/2 g(7) = 95/8 Therefore, g(7) = 95/8.
0 / 350
Answer:
9. The weights, in pounds, of five pumpkins from a pumpkin patch are 17, 18, 17, 15, and 13.
What is the MAD of the weights?
Step-by-step explanation:
9. The weights, in pounds, of five pumpkins from a pumpkin patch are 17, 18, 17, 15, and 13.
What is the MAD of the weights?
what is the mean of 41,52,56,65,65,76,89
the mean of 41, 52, 56, 65, 65, 76, and 89 is 63.43.
Step-by-step explanation:
The mean of the value is 63.43ITS THE LAST DAY OF THE SEMESTER AND IF I DONT GET THIS RIGHT IM GONNA FAIL PLEASE HELP ASAPPP!!!!
According to the puzzle the specified values are;
1. TU = 156.15m
2. Area of rectangle = 12335.85 m²
3. GH = 182.33m
4. Area of triangle = 8934.17 m²
What is a triangle?Three line segments meet at three non-collinear points to form a triangle, a geometric shape. The three points where the line segments meet are referred to as the vertices of the triangle, while the three line segments themselves are referred to as the sides of the triangle.
1. Given triangle TUV apply Pythagoras theorem,
175² = 79² + TU² after simplification,
TU = 156.15m (option A is correct)
2. Area of rectangle = length × width
Area of rectangle = TU × UV = 156.15 × 79
Area of rectangle = 12335.85 m² (option G is correct)
3. Given triangle GHI apply Pythagoras theorem,
207² = 98² + GH² after simplification,
GH = 182.33m (option I is correct)
4. Area of triangle = [tex]\frac{1}{2}[/tex] × base × height
Area of triangle = [tex]\frac{1}{2}[/tex] × HI × GH
Area of triangle = [tex]\frac{1}{2}[/tex] × 98m × 182.33m
Area of triangle = 8934.17 m²
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A golfer measured the speed, in miles per hours (mph) of several drives with the same golf club. The frequency table tells how often each speed occurred. What is the median speed of the drives in miles per hour?
Answer: 90
Step-by-step explanation:
To find the median speed of the drives, list the speeds in order from lowest to highest and find the middle value. If there's an even number of speeds, find the average of the two in the middle.
Explanation:To find the median speed of the drives in miles per hour, you would first arrange the speeds in order from smallest to largest and then identify the speed that sits exactly in the middle of this list. If there is an even number of observations, the median is the average of the two middle speeds.
For instance, if the speed of the drives listed in the table are 50 mph, 60 mph, 70 mph, 80 mph, and 90 mph, the median would be 70 mph. If the speeds were 50 mph, 70 mph, 80 mph, and 90 mph, the median would be the average of 70 and 80 mph, which is 75 mph.
The frequency table is useful in identifying how often each speed occurred, but it won't directly affect your calculation of the median. It's important to understand this concept as part of a broader understanding of statistics and data analysis.
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i need help with this question?
Answer:
D. [tex]\sqrt{3}[/tex] ; 2
Hope this helps!
Step-by-step explanation:
At the beginning of 2005 there were 670 deer living in a nature reserve. The population is declining by x% each year and after 4 years has reduced to 557. Find the value of x. Give your answer correct to 2 decimal places.
The annual rate of decline is 5.3%.
What is exponential ?
Exponential refers to a mathematical function where the variable is in the exponent. Exponential functions have the general form:
[tex]f(x) = a^{x}[/tex]
here "a" is a constant called the base, and "x" is the variable. When the base "a" is a positive number greater than 1, the function grows exponentially as "x" increases. When the base "a" is a number between 0 and 1, the function decays exponentially as "x" increases.
We can start by using the formula for exponential decay:
[tex]N(t) = N0 * (1 - r)^{t}[/tex]
where N(t) is the population after t years, N0 is the initial population, r is the annual rate of decay (as a decimal), and t is the time in years.
We are given that the initial population is 670, so N0 = 670. After 4 years, the population has reduced to 557, so N(4) = 557. We can plug these values into the formula and solve for r:
[tex]557 = 670 * (1 - r)^{4}[/tex]
[tex](557/670)^{1/4} = 1 - r[/tex]
[tex]0.947 = 1 - r[/tex]
[tex]r = 0.053[/tex]
So the annual rate of decline is 5.3%.
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Given a function f(x) = x+y, 0≤x+2y≤2 , otherwise = 0 (a) Show that f is a PDF. (b) Find the marginal of X and Y . (c) Find the Cov(X, Y ).
a) f(x) is a PDF. b) the marginal of X and Y is (y/2 + 1) / 2 c) the covariance of X and Y is: -1/18
What is meant by PDF?
In probability theory, a probability density function (PDF) is a function that describes the relative likelihood for a continuous random variable to take on a given value.
What is covariance?
Covariance is a statistical measure that quantifies the degree to which two random variables are linearly associated.
According to given information:(a) To show that f(x) is a probability density function (PDF), we need to show that it satisfies the following two conditions:
Non-negativity: f(x) is non-negative for all x in its domain.
Normalization: The integral of f(x) over its domain is equal to 1.
The domain of f(x) is given by the inequality 0 ≤ x + 2y ≤ 2. To find the integral of f(x) over its domain, we need to integrate it with respect to y from (0-x/2) to (2-x/2), and then integrate the result with respect to x from 0 to 2:
∫(0 to 2) ∫(0-x/2 to 2-x/2) (x+y) dy dx
Solving the inner integral with respect to y, we get:
∫(0 to 2) [xy + [tex]y^2[/tex]/2] |_0-x/[tex]2^{(2-x/2)[/tex] dx
= ∫(0 to 2) ([tex]x^2[/tex]/4 - [tex]x^3[/tex]/12 + 1) dx
= [[tex]x^3[/tex]/12 - [tex]x^4[/tex]/48 + x] |_[tex]0^2[/tex]
= 2 - 2/3 + 2 = 8/3
Since the integral is finite and positive, the first condition of non-negativity is satisfied. To satisfy the normalization condition, we divide the function by the integral:
f(x) = (x+y) / (8/3)
Therefore, f(x) is a PDF.
(b) To find the marginal of X, we integrate f(x,y) over the range of y:
f(x) = ∫(0-x/2 to 2-x/2) (x+y) / (8/3) dy
= (x/2 + 1) / 2
Similarly, to find the marginal of Y, we integrate f(x,y) over the range of x:
f(y) = ∫(0 to 2) (x+y) / (8/3) dx
= (y/2 + 1) / 2
(c) To find the covariance of X and Y, we use the formula:
Cov(X, Y) = E[XY] - E[X]E[Y]
To find E[XY], we integrate xy*f(x,y) over the range of x and y:
E[XY] = ∫(0 to 2) ∫(0-x/2 to 2-x/2) xy*(x+y)/(8/3) dy dx
= ∫(0 to 2) [[tex]x^3[/tex]/6 - [tex]x^4[/tex]/24 + [tex]x^2[/tex]/4] dx
= 2/3
To find E[X] and E[Y], we integrate xf(x) and yf(y) over their respective ranges:
E[X] = ∫(0 to 2) x*(x/2+1)/2 dx
= 7/3
E[Y] = ∫(0 to 2) y*(y/2+1)/2 dy
= 7/6
Therefore, the covariance of X and Y is:
Cov(X, Y) = E[XY] - E[X]E[Y] = 2/3 - (7/3)*(7/6) = -1/18
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Last night the temperature on a thermometer was −4°F. The temperature this morning was colder than last night. Which could represent the temperature on the thermometer this morning?
Select all the correct answers.
Answer:
-5, -12
Step-by-step explanation:
We just have to find somthing colder than -4f so it has to be less than -4 so it would be -5 and -12.
The correct answers to your question would be -5 and - 15.