The polar coordinates corresponding to the rectangular coordinates (0, 5) are (5, π/2).
To find the distance from the origin to the point (r), we can use the Pythagorean theorem. The distance from the origin to a point (x, y) is given by the formula √(x² + y²).
In this case, since the x-coordinate is 0, we only need to find the distance from the origin to the y-coordinate. Therefore, r = √(0² + 5²) = 5.
To find the angle (θ) that the line from the origin to the point makes with the positive x-axis, we can use trigonometry.
However, this is undefined since we cannot divide by zero. Therefore, we need to use a special case. Since the x-coordinate is 0, the point lies on the y-axis.
Therefore, the angle θ is either π/2 or 3π/2. However, we are given the constraint 0 ≤ θ < 2π. Therefore, the angle θ = π/2 since it satisfies the constraint.
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Complete Question:
Convert the following rectangular coordinates into polar coordinates. Always choose 0 ≤ θ < 2 π . (a) ( 0 , 5 )
Give the coordinates of (,)(△) for (,),(,),eh open , 0 comma 4 , close comma b open , 0 comma 2 , close comma and (−,). C open , negative 3 comma 2 , close
The coordinates of the centroid (,)(△) of triangle ABC are (-1, 8/3).
To find the coordinates of the centroid, we can use the formula:
(x, y) = ((x1 + x2 + x3)/3, (y1 + y2 + y3)/3)
where (x1, y1), (x2, y2), and (x3, y3) are the coordinates of the vertices A, B, and C, respectively.
Plugging in the coordinates of A (0, 4), B (0, 2), and C (-3, 2), we get:
(x, y) = ((0 + 0 - 3)/3, (4 + 2 + 2)/3) = (-1, 8/3)
In a two-dimensional plane, the centroid of a triangle is the point where the three medians of the triangle intersect. A median is a line segment drawn from a vertex of the triangle to the midpoint of the opposite side. The centroid divides each median into two segments, with the ratio of the length of the segment closer to the vertex to the length of the segment closer to the opposite side being 2:1.
In a three-dimensional space, the centroid of a solid can be found by dividing the solid into smaller parts, finding the centroid of each part, and then averaging them. The centroid of a solid is the point where the lines connecting the centroids of each part intersect.
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Complete Question: -
Give the coordinates of the centroid of triangle ABC, which is denoted by (,)(△) and is the point where the medians of the triangle intersect for a (0 , 4) , b (0 ,2) , and C (-3 ,2 ).
A triangle and a parallelogram are constructed on the base such that their areas are equal if the altitude of a parallelogram is 100 m then the altitude of a triangle is
Answer:
Let the base of the triangle and parallelogram be denoted by 'b' and their respective altitudes be denoted by 'h1' and 'h2'.
Given that the area of the triangle is equal to the area of the parallelogram, we have:
Area of triangle = (1/2)bh1 Area of parallelogram = b*h2
Since the areas are equal, we have:
(1/2)bh1 = b*h2
Simplifying the above equation, we get:
h1 = 2*h2
Substituting the given value of h2 as 100 m, we get:
h1 = 2*100 = 200 m
Therefore, the altitude of the triangle is 200 meters
Please help me with these questions. I could solve others but not these ones
1) a) ω = 2πf = 2π(50) = 100π radians per second
b) 96.78m/s.
2) (a) T = 1/f = 1/12 seconds,
(b) ω = 2πf = 2π(12) = 24π radians per second.
3) original length of the pendulum is 9.0m.
4) original length of the spiral spring is 20cm.
5) the initial velocity of the football is 17.32m/s At the highest
What are the radians per second?Angular speed is the speed of the object in rotational motion. Distance traveled is represented as θ and is measured in radians. The time taken is measured in terms of seconds. Therefore, the angular speed is articulated in radians per second or rad/s.
1. (a) The angular speed of a body vibrating at 50 cycles per second is:
ω = 2πf = 2π(50) = 100π radians per second
(b) The stone is projected horizontally, so its initial vertical velocity is zero. The time taken for the stone to fall from the top of the tower to the ground is given by:
t = √(2h/g) = √(2×75/10) = √15
The horizontal distance traveled by the stone is:
d = vxt = (25m/s)×(√15) ≈ 96.78m
Therefore, the speed with which the stone strikes the ground is approximately 96.78m/s.
2. (a) The period of the motion is:
T = 1/f = 1/12 seconds
(b) The angular speed of the motion is:
ω = 2πf = 2π(12) = 24π radians per second
(c) The velocity at the middle of oscillation is zero, since this is the point of maximum displacement and therefore maximum potential energy.
(d) The acceleration at the end of oscillation is given by:
a = -ω²x = -(24π)²(0.02) = -11.52 m/s²
where x is the amplitude of the motion.
3. The period of a simple pendulum is given by:
T = 2π√(L/g)
where L is the length of the pendulum and g is the acceleration due to gravity. Therefore, we can write:
17 = 2π√(L/g) (1)
8.5 = 2π√[(L-1.5)/g] (2)
Dividing (2) by (1), we get:
1/2 = √[(L-1.5)/L]
Squaring both sides and simplifying, we get:
L = 9.0m
Therefore, the original length of the pendulum is 9.0m.
4. (a) Let the original length of the spiral spring be x. Then, using Hooke's law, we have:
5N = k(25cm - x) (1)
10N = k(30cm - x) (2)
where k is the spring constant. Solving for k in equation (1), we get:
k = 5N/(25cm - x)
Substituting into equation (2), we get:
10N = [5N/(25cm - x)](30cm - x)
Simplifying and solving for x, we get:
x = 20cm
Therefore, the original length of the spiral spring is 20cm.
(b) The pressure of the trapped air column in the capillary tube is balanced by the pressure of the atmosphere. When the tube is held horizontally, the length of the air column is the same as its length in the vertical position. Therefore, the length of the air column is 15cm.
When the tube is held vertically with the open end underneath, the length of the air column is given by:
L = 76cm - 20cm = 56cm
Therefore, the length of the air column is 56cm.
5. The horizontal and vertical components of the initial velocity of the football are:
v₀x = v₀cos(60°) = (20m/s)cos(60°) = 10m/s
v₀y = v₀sin(60°) = (20m/s)sin(60°) = 17.32m/s
At the highest
Hence, the answer to each question:
1) a) ω = 2πf = 2π(50) = 100π radians per second
b) 96.78m/s.
2) (a) T = 1/f = 1/12 seconds,
(b) ω = 2πf = 2π(12) = 24π radians per second.
3) original length of the pendulum is 9.0m.
4) original length of the spiral spring is 20cm.
5) the initial velocity of the football is 17.32m/s At the highest
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Solve the equation using the one to one property 4^(2x+4) = 16^(3x-6)
The solution to the exponential equation is given as follows:
x = 4.
How to solve the equation?The equation for this problem is defined as follows:
4^(2x+4) = 16^(3x-6).
Looking at the left-side of the equality, we can apply the power of power property, as 16 = 4², hence:
16^(3x-6) = 4^[2(3x - 6)] = 4^(6x - 12)
(the power of power property means that we keep the base then multiply the powers relative to the same base).
Hence the equation can be written as follows, considering the simplification to the right side of the equality.
4^(2x+4) = 4^(6x - 12)
Applying the one to one property, the solution is obtained as follows:
6x - 12 = 2x + 4
4x = 16
x = 16/4
x = 4.
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Write a function that models each variation.
x=0. 2 and y=3 when z=2 , and z varies jointly with x and y.
If x and y both increase or decrease by the same factor, then z will also increase or decrease by the same factor, keeping the ratio between the variables constant, the function model used is z = 33.33xy.
When three variables are said to vary jointly, it means that they are related in such a way that if any one of them changes, the other two also change proportionally. To model the joint variation of z with x and y when x=0.2 and y=3, we can use the following function:
z = kxy
where k is a constant of proportionality.
To find the value of k, we can substitute the given values of x, y, and z into the equation:
2 = k(0.2)(3)
Solving for k, we get:
k = 33.33
Therefore, the function that models the joint variation of z with x and y is:
z = 33.33xy
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Lin is comparing the graph of two functions g and f. The function g is given by g(x) = f(x−2). Lin thinks the graph of g will be the same as the graph of f, translated to the left by 2. Do you agree with Lin? Explain your reasoning
No, the function g will be the same as the graph of f, translated to the right by 2.
What is transformation?
A point, line, or geometric figure can be changed in four different ways that are all collectively referred to as transformations. The original shape of the object is called the Pre-Image and the final shape and position of the object is the Image under the transformation.
f(x + a)horizontally shift the graph of f(x) left by a units.
The graph of f(x) is horizontally shifted by a units right side by f(x - a).
The graph of f(x) is vertically shifted upward by a unit by f(x)+a.
f(x)- a one-unit vertical shift downward of the f(x) graph.
The given function is
g(x) = f(x-2)
According to rule,
The function g will be the same as the graph of f, translated to the right by 2.
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Sally has $ 190,000 to invest to obtain annual income. She to invest in high-quality, low-risk investments. She invests some of it in Certificates of Deposit yielding a 6% return and the rest in AA bonds yielding 11% returns. How much should she invest in each to earn exactly $18,900 per year?
(a) Define any variables needed as a complete sentence.
(b) Write the equation that gives the total amount invested.
(c) Write the equation that represents the total amount earned from the investment. (Remember: Interest * Investment (or Principle) = Amount Returned)
(d) Solve this system of equations using substitution or elimination
(e) State the solution as a complete sentence in the context of the situation.
a)CD and AA
b)CD + AA = 190,000
c)0.06CD + 0.11AA = 18,900
d)AA = 187,333.17.
(a) Let CD be the amount invested in Certificates of Deposit and AA be the amount invested in AA bonds.
(b) The total amount invested is given by the equation CD + AA = 190,000.
(c) The total amount earned from the investment is given by the equation 0.06CD + 0.11AA = 18,900.
(d) To solve this system of equations using elimination, we multiply the first equation by 0.11 and the second equation by 0.06 and add the resulting equations together, yielding the equation 12.06CD + 10.11AA = 214,084. We then subtract the original equation from this new equation to get 9.06CD = 24,084. Solving for CD, we get CD = 2,666.83. We can then substitute this value into either of the original equations to get AA = 187,333.17.
(e) Sally should invest $2,666.83 in Certificates of Deposit and $187,333.17 in AA bonds to earn exactly $18,900 per year.
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Please help me! Thank youuu
Answer: 3,600
dont quote me on it doing it the way i learned how to lol
Step-by-step explanation: Volume = length x width x height.
Answer:
917.33
Step-by-step explanation:
Top conic volume formula: ⅓ r²h=⅓(16/2)²*(18-12.5)
=⅓8²*5.5
=⅓352
=117.33
Volume formula of a cylinder: V=r²h
=8²*12.5
=800
117.33+800=917.33
given the parent function f(x)=2^(x)and the translated function g(x)=2^((x+3)), determine the effect the transformation has on the maximum value on the interval [-2,2]
The parent function f(x) = 2^(x) is transformed into the function g(x) = 2^((x+3)) by shifting the graph 3 units to the left.
This means that the maximum value on the interval [-2,2] for the parent function will now occur at the point (-2+3) = 1 for the transformed function.
Therefore, the maximum value for the transformed function on the interval [-2,2] will be 2^(1) = 2.
In summary, the transformation shifts the graph of the parent function 3 units to the left, causing the maximum value on the interval [-2,2] to occur at x = 1 and have a value of 2.
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William’s car uses 1 litre of fuel to travel 12km. How much fuel will be needed to travel
420km?
Answer:
420km/12km = 35
1×35l = 35lites
rate it 5 stars pls TANXX
Answer:
35 litres
Step-by-step explanation:
We can use a ratio to solve.
1 litre x litres
------------- = ----------------
12 km 420 km
Using cross products
1 * 420 = 12x
Divide each side by 12.
420/12 = x
35 =x
35 litres
I needdd help!! pleeeaaseses
The height of the rectangle is
⇒ Height = 1/2 ft
What is mean by Rectangle?A rectangle is a 2 - dimension figure with 4 sides, 4 corners and 4 right angles. And, Opposite sides of the rectangle are equal and parallel to each other.
We have to given that;
Length of the rectangle = 5/3 ft
Area of the rectangle = 5/6 ft
We know that;
Area of rectangle = Length x Height
Hence, We get;
5/6 ft = 5/3 ft x Height
5/6 x 3/5 = Height
Height = 1/2 ft
Thus, The height of the rectangle is;
⇒ Height = 1/2 ft
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Help
Answer 12 its urgent please
The end behaviour of g(x) = log₁,₃(-x + 4) is as x approaches 4, g(x) approaches ∝ and as x approaches 4, g(-∝) approaches -∝
How to determine the end behaviourFrom the question, we have the following parameters that can be used in our computation:
g(x) = log₁,₃(-x + 4)
The maximum input value of the above function is 4
This is so because
-x + 4 = 0
x = 4
Set the value of x in the function to 4
So, we have
g(4) = log₁,₃(4 + 4)
g(4) => ∝
Set the value of x in the function to -∝
So, we have
g(-∝) = log₁,₃(-∝ + 4)
g(-∝) => -∝
Hence, the end behaviour is as x approaches 4, g(x) approaches ∝
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Write the equation in POINT-SLOPE FORM of the given graph using the given point.
Help me, please
The equation of line in the point-slope form is y - 2 = -1(x + 2)
What is the equation of line?The equation of a straight line is a relationship between x and y coordinates, The point-slope form of the equation of a straight line is,
y-y₁ = m(x-x₁), where m is the slope of the line.
Given that,
A graph having straight line,
and it can be seen in the graph it is passing through (-2, 2) & (-1, 1)
Slope m = (y₂-y₁)/(x₂-x₁)
= (2 - 1)/ (-2 -(-1))
= 1/-1
= -1
So now taking point (-2, 2) and slope is -1
The point slope form of the equation is:
y - 2 = -1 (x - (-2))
y - 2 = -1(x + 2)
Hence, the equation is y - 2 = -1(x + 2)
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O EXPONENTS AND POLYNOMIALS Polynomial long division: Problem Divide. (4x^(3)+16x^(2)+18x+7)-:(2x^(2)+4x) Your answer should give the quotient and
In other words, the quotient is 2x+4 and the remainder is 10x+7.
To solve this problem, we will use polynomial long division. The steps are as follows:
1. Divide the first term of the dividend (4x^(3)) by the first term of the divisor (2x^(2)) to get the first term of the quotient (2x).
2. Multiply the first term of the quotient (2x) by the divisor (2x^(2)+4x) to get (4x^(3)+8x^(2)).
3. Subtract the result from step 2 (4x^(3)+8x^(2)) from the dividend (4x^(3)+16x^(2)+18x+7) to get the remainder (8x^(2)+18x+7).
4. Repeat steps 1-3 with the new dividend (8x^(2)+18x+7) and the same divisor (2x^(2)+4x) until the degree of the remainder is less than the degree of the divisor.
The final quotient and remainder are as follows:
Quotient: 2x+4
Remainder: 10x+7
So, the final answer is:
(4x^(3)+16x^(2)+18x+7)-:(2x^(2)+4x) = 2x+4+(10x+7)-:(2x^(2)+4x)
In other words, the quotient is 2x+4 and the remainder is 10x+7.
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Can anyone help with the fractions ?
The tabs can be completed in the following way:
1. Fraction for 25% = 25/100 = 5/20 = 1/4
2. 90 percent: Decimal 0.90: Fraction: 90/100 =9/10
3. 60 percent: Decimal 0.60: Fraction: 60/100 = 6/10 = 3/5
4. 35 percent: Decimal 0.35: Fraction: 35/100 = 7/20
5. 33 percent: Decimal 0.33: Fraction: 33/100
6. 65 percent: Decimal 0.65: Fraction: 65/100 = 13/20
What is a percentage?A percentage is a value that is obtained as a fraction of the number 100. In the above question, we are given a list of values in percentages and told to convert them to decimal and fraction forms.
To do this, we need to divide the number 90 by 100 for the second expression and express this as a decimal. The resultant figure is 0.9. When converted to a fraction, the value now has a denominator and numerator.
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How many liters of water must be evaporated from 10L of a 40% salt solution to produce a 50% solution?
2L of water must be evaporated from the 10L of 40% salt solution to produce a 50% solution.
To produce a 50% salt solution from a 40% salt solution, we must evaporate some water to increase the concentration of the salt. We can use the formula:
C1V1 = C2V2
Where C1 is the initial concentration, V1 is the initial volume, C2 is the final concentration, and V2 is the final volume.
Plugging in the given values:
0.40(10L) = 0.50(V2)
Simplifying:
4L = 0.50V2
Dividing both sides by 0.50:
V2 = 8L
So the final volume of the solution must be 8L in order to have a 50% concentration of salt. To find the amount of water that must be evaporated, we can subtract the final volume from the initial volume:
10L - 8L = 2L
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Find the intersection and union of the given sets. A = {h, o, m, e} , B = {h, o, u, s, e}
The intersection of sets A and B
The intersection of two sets is the set of elements that are common to both sets. To find the intersection of sets A and B, we simply look for the elements that are in both sets. In this case, the intersection of A and B is {h, o, e}.
The union of two sets is the set of elements that are in either one of the sets or both. To find the union of sets A and B, we simply combine the elements from both sets, making sure to not include any duplicates. In this case, the union of A and B is {h, o, m, e, u, s}.
So, the intersection of sets A and B is {h, o, e} and the union of sets A and B is {h, o, m, e, u, s}.
Here is the solution in HTML:
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please help! i’ll give brainly!!
[tex]-\dfrac{3}{4}x-\dfrac{1}{3}+\dfrac{7}{8}x-\dfrac{1}{2} = -\dfrac{3}{4}x+\dfrac{7}{8}x -\dfrac{1}{3}-\dfrac{1}{2} = \\[/tex]
[tex]= -\dfrac{3(2)}{4(2)}x+\dfrac{7}{8}x -\dfrac{1(2)}{3(2)}-\dfrac{1(3)}{2(3)} = -\dfrac{6}{8}x+\dfrac{7}{8}x -\dfrac{2}{6}-\dfrac{3}{6} \\[/tex]
[tex]= \dfrac{-6+7}{8}x -\dfrac{2+3}{6} = \dfrac{1}{8}x -\dfrac{5}{6} \\[/tex]
her body fat is 14.9% she weighs 172 pounds how much of her weight is made up of fat?
25.628 pounds of her weight is made up of fat.
We can use the formula Weight of Fat = Body Fat % x Body Weight. To find out how much of her weight is made up of fat, we need to multiply her body fat percentage by her weight. We can do this by using the following formula:
Body fat weight = Body fat percentage x Weight
In this case, the body fat percentage is 14.9% and the weight is 172 pounds. Plugging these values into the formula, we get:
Body fat weight = 14.9% x 172 pounds
Body fat weight = 0.149 x 172 pounds
Body fat weight = 25.628 pounds
In this case, 14.9% of 172 pounds is 25.628 pounds.
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What association is shown in the given scatter plot?
A. Clustering
B. Linear
C. Negative
D. None of the above
The polygons are regular.
X = ?
help how do I start this and get the answer to this
The figure for the given net diagram is square pyramid.
What is a net diagram?Net diagram is a 2-dimensional plane figure which can be folded to form a 3-dimensional figure. Or we can say net diagrams are the figures which obtained by unfolding some 3D figures.
The given net diagram has a square and 4 triangles.
If we fold all the faces and make a solid figure, we get square pyramid.
Therefore, the figure is square pyramid.
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The scatter plot shows the number of hits and home runs for 20 baseball players who had at least 10 hits last season. The table shows the values for 15 of those players.
The model, represented by y=0.15x−1.5, is graphed with a scatter plot.
Use the graph and the table to answer the questions.
Player A had 154 hits in 2015. How many home runs did he have? How many was he predicted to have?
Player B was the player who most outperformed the prediction. How many hits did Player B have last season?
What would you expect to see in the graph for a player who hit many fewer home runs than the model predicted?
Tο depict the relatiοnship between twο numerical variables using Data visualizatiοn that is knοwn as a scatterplοt.
What is a Scatterplοt?Using this scatterplοt, we graphed a line using linear regressiοn. This line can use as οur reference pοint and ignοre the rest οf the pοints οn the graph.
We will examine this graph and try tο find apprοximately what value οf y is given when we prοvide an x value.
Since the number οf hits is the x value, tο find hοw high the graph gοes at that x value we can lοοk at the graph tοο.
Right between 100 and 150 is 125, sο we can nοw start there and mοve up.
When we mοve up, we get a line that's between 10 and 20. We can alsο view it's in the tοp half οf that bοx. That means the value οf y fοr this will be mοre than 15.
Since it's nοt lying directly οn 20, we have the οnly οptiοn left is 18.
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QMB3200 RVI Spring 2022 Nadia Elhaj Homework: Hw 6 - Sampling and Confidence Question 9, 6.5.30 > HW Score: 46.88%, 7.5 of 16 points X Points: 0 of 1 Save A survey of 50 young professionals found that they spent an average of $23.76 when dining out, with a standard deviation of $13.54. Can you conclude statistically that the population mean is greater than $21? Use a 95% confidence interval. The 95% confidence interval is ___, ____ . As $21 is_____ of the confidence interval, we ___ conclude that the population mean is greater than $21. (Use ascending order. Round to four decimal places )
The population mean is greater than $21.
The 95% confidence interval is $13.3896, $34.1304. As $21 is outside of the confidence interval, we can conclude that the population mean is greater than $21.
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find q.
write your answer in simplest radical form
The required measure of q in the given triangle is 3.
What are trigonometric ratios?Trigonometric ratios are mathematical functions used to relate the angles of a right-angled triangle to the lengths of its sides. There are three basic trigonometric ratios: sine (sin), cosine (cos), and tangent (tan), which can be defined as follows:
Sine (sin) = opposite/hypotenuse
Cosine (cos) = adjacent/hypotenuse
Tangent (tan) = opposite/adjacent
Here,
Applying the Sine rule,
sin45 = q/3√2
q = 3
Thus, the required measure of q in the given triangle.
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O a
Ob
● C
Od
For the following boxplot, what are the upper and lower
fences?
2
3
4 5
Lower Fence: 0 Upper Fence: 8
Lower Fence: -1 Upper Fence: 12
Lower Fence: 3 Upper Fence: 5
Lower Fence: 1
Upper Fence: 8
9
Answer:
Therefore, the lower fence is 1.5 and the upper fence is 5.5.
Step-by-step explanation:
To determine the upper and lower fences for a boxplot, we first need to calculate the interquartile range (IQR), which is the difference between the third quartile (Q3) and the first quartile (Q1). Then, we can use the following formulas:
Lower Fence = Q1 - 1.5 * IQR
Upper Fence = Q3 + 1.5 * IQR
In the given boxplot, the box extends from 2 to 5, with the median (Q2) at 3.5. The first quartile (Q1) is 3 and the third quartile (Q3) is 4. Therefore, the IQR is:
IQR = Q3 - Q1 = 4 - 3 = 1
Using the formulas above, we can calculate the lower and upper fences:
Lower Fence = Q1 - 1.5 * IQR = 3 - 1.5 * 1 = 1.5
Upper Fence = Q3 + 1.5 * IQR = 4 + 1.5 * 1 = 5.5
Therefore, the lower fence is 1.5 and the upper fence is 5.5.
A cylinder has a radius of 16 yards. Its volume is 6,430.72 cubic yards. What is the height of the cylinder?
Answer:
We can use the formula for the volume of a cylinder:
V = πr^2h
where V is the volume, r is the radius, and h is the height.
Plugging in the given values, we get:
6,430.72 = π(16)^2h
Simplifying:
6,430.72 = 256πh
Dividing both sides by 256π:
h = 6,430.72 / (256π)
h ≈ 8.05
Therefore, the height of the cylinder is approximately 8.05 yards.
For each of the following, find the formula for an exponential
function that passes through the two points given.
a. (0, 6) and (2, 294)
f(x)=
b. (0,1280) and (2, 20)
g(x) =
The formulas for the exponential functions that pass through the two points given are:
f(x) = 6 * 7^x
g(x) = 1280 * 0.125^x
To find the formula for an exponential function that passes through two points, we can use the general form of an exponential function, which is f(x) = a * b^x, where a and b are constants. We can plug in the x and y values from the two points and solve for a and b.
For part a:
f(x) = a * b^x
6 = a * b^0 (from point (0,6))
294 = a * b^2 (from point (2,294))
From the first equation, we can see that a = 6. We can substitute this value into the second equation:
294 = 6 * b^2
49 = b^2
b = 7
So the formula for the exponential function that passes through the two points is f(x) = 6 * 7^x.
For part b:
g(x) = a * b^x
1280 = a * b^0 (from point (0,1280))
20 = a * b^2 (from point (2,20))
From the first equation, we can see that a = 1280. We can substitute this value into the second equation:
20 = 1280 * b^2
0.015625 = b^2
b = 0.125
So the formula for the exponential function that passes through the two points is g(x) = 1280 * 0.125^x.
In conclusion, the formulas for the exponential functions that pass through the two points given are:
f(x) = 6 * 7^x
g(x) = 1280 * 0.125^x
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Lines ℓ and m are intersected by transversal t. ℓ ∥ m.
There are two parallel horizontal lines l and m intersected by another line t making angles 1 and 3 with l and 5, 7 with m. 1 ,4 and 2 , 3 are opposite angles at the point of intersection of l and t. 5, 8 and 6, 7 are opposite angles at the point of intersection of m and t.
If m∠3 = 78°, what is m∠6?
Answer:
m<6 = 78°
Step-by-step explanation:
I hope your drawing of this problem looks somewhat like this:
^ t
/
/
1 / 2
<-----------------------------------------------------------------------------> ℓ
3 / 4
/
5 / 6
<------------------------------------------------------------------------------> m
7 / 8
/
/
V
Sorry about the terrible drawing, but I hope I got the angle numbers correctly written.
Angles 3 and 6 are are called alternate interior angles.
They are "interior angles" because they are on the inside of lines l and m. They are "alternate angles" because they are on different sides of the transversal, t.
There is a Geometry theorem about this situation.
Theorem:
If parallel lines are cut by a transversal, then alternate interior angles are congruent.
In this case, since the pair of angles 3 and 6 is a pair of alternate interior angles, then by the theorem above, they are congruent.
m<6 = m<3
Since m<3 = 78°, then m<6 = 78°.
Carlos constructed 3 parallel lines as part of an art project. He also drew a line passing through each of the parallel lines. Some of the angles formed by the intersection of line t and lines l, m, and n are numbered in the diagram below.
Which conjecture can Carlos make about the angles formed by line t and lines l, m, and n?
a) Angles 1, 2, and 3 are congruent.
b) Angles 1, 3, and 5 are congruent.
c) Angles 2 and 4 are supplementary.
d) Angles 1 and 5 are supplementary.
A conjecture which Carlos can make about the angles formed by line t and lines l, m, and n include the following: B. angles 1, 3, and 5 are congruent.
What is corresponding angles postulate?In Mathematics, corresponding angles postulate simply refers to a theorem which states that corresponding angles are always congruent (equal) if the transversal intersects two parallel lines.
This ultimately implies that, the corresponding angles would always be congruent (equal) if a transversal intersects two (2) parallel lines.
By applying corresponding angles postulate to both lines l, m, and n, we can reasonably infer and logically deduce that the following angles are congruent:
∠1 ≅ ∠3
∠3 ≅ ∠5
∠1 ≅ ∠5
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Help me Please I beg of you
Answer:
1st page:
To determine if Jason and Arianna made a mistake in their solution, we can examine the slopes and y-intercepts of the two equations.
The first equation, 5x - 3y = -1, can be written in slope-intercept form as y = (5/3)x + 1/3, where the slope is 5/3 and the y-intercept is 1/3.
The second equation, 3x + 2y = 7, can be written in slope-intercept form as y = (-3/2)x + 7/2, where the slope is -3/2 and the y-intercept is 7/2.
To solve the system of equations, we need to find the point of intersection of the two lines. We can see from the slopes that the lines are not parallel, so they must intersect at some point. However, the slopes are not perpendicular either, so they do not intersect at a right angle.
By graphing the two equations on the same coordinate plane, we can see that the point of intersection is (2, 2), not (4.04, 7.31) as Jason and Arianna calculated. Therefore, they must have made a mistake in their solution.
In summary, we can tell that Jason and Arianna made a mistake in their solution by examining the slopes and y-intercepts of the two equations and graphing them to find the actual point of intersection.
2nd page:
To determine if Jason and Arianna made a mistake in their solution, we can graph the two linear equations on the same coordinate plane and look for the point of intersection.
We can begin by rearranging the equations into slope-intercept form:
5x - 3y = -1 → y = (5/3)x + 1/3
3x + 2y = 7 → y = (-3/2)x + 7/2
Now we can graph the two lines. We can plot two points for each line and connect them with a straight line to obtain the graphs.
For the first equation, when x = 0, we get y = 1/3. When x = 3, we get y = 6/3 = 2. Plotting these two points and connecting them, we get:
Graph of the first equation
For the second equation, when x = 0, we get y = 7/2. When x = 2, we get y = 1/2. Plotting these two points and connecting them, we get:
Graph of the second equation
We can see from the graphs that the lines intersect at the point (2, 2), not at (4.04, 7.31) as Jason and Arianna found. Therefore, they must have made a mistake in their solution.
In summary, we can tell from the graphs of the equations that Jason and Arianna must have made a mistake because the lines do not intersect at the point they found.
3rd page:
We can determine if there is a unique solution to the system of linear equations by examining the slopes of the two equations.
The slope of the first equation, 5x - 3y = -1, can be found by rearranging the equation into slope-intercept form y = (5/3)x + 1/3. We can see that the slope of the line is positive and not equal to the slope of the second equation, which is -3/2.
Similarly, the slope of the second equation, 3x + 2y = 7, can be found by rearranging the equation into slope-intercept form y = (-3/2)x + 7/2. Again, we can see that the slope of the line is negative and not equal to the slope of the first equation, which is 5/3.
Since the slopes of the two lines are not equal, they will intersect at a unique point. In other words, there is only one solution to the system of equations.
Therefore, we can conclude that there is a unique solution to the system of linear equations given by 5x - 3y = -1 and 3x + 2y = 7, based on the slopes of the graphs of the equations in the system.
Step-by-step explanation: