find the exact length of the curve. x = 8 6t2, y = 7 4t3, 0 ≤ t ≤ 2
Answers
The exact length of the curve x = 8 6t2, y = 7 4t3, 0 ≤ t ≤ 2 is 8π units.
To find the length of the curve defined by the parametric equations x = 8 6t^2, y = 7 4t^3, 0 ≤ t ≤ 2, we can use the formula for arc length:
L = ∫[a,b] √[dx/dt]^2 + [dy/dt]^2 dt
where [a,b] is the interval of t, and dx/dt and dy/dt are the derivatives of x and y with respect to t.
In this case, we have:
dx/dt = 96t
dy/dt = 84t^2
So the length of the curve is:
L = ∫[0,2] √(96t)^2 + (84t^2)^2 dt
Simplifying the integrand, we get:
L = ∫[0,2] √(9216t^2 + 7056t^4) dt
Now, we can use a trigonometric substitution to simplify the integrand further. Let t = (1/2)sinθ, so that dt/dθ = (1/2)cosθ. Then:
L = ∫[0,π] √(9216/4)sin^2θ + (7056/4)sin^4θ) (1/2)cosθ dθ
L = 48 ∫[0,π] √(3 + 3sin^2θ)sin^2θ dθ
Using the substitution u = cosθ, du/dθ = -sinθ, and sin^2θ = 1 - cos^2θ, we can rewrite the integral as:
L = 48 ∫[-1,1] √(3 + 3u^2)(1 - u^2) du
Expanding the integrand and simplifying, we get:
L = 48 ∫[-1,1] √(3/4 - (3/4)u^4) du
Now, we can use the substitution v = (3/4)u^2, dv/du = (3/2)u, and u = ±sqrt(4v/3), to get:
L = 32 ∫[0,3/4] √(1 - v^2) dv
This is a standard integral that evaluates to π/4, so we have:
L = 32(π/4) = 8π
Therefore, the length of the curve is 8π units.
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Related Questions
Minimize z=5x1+4x2 Subject to: 6x1+2x2≤24 x1+2x2≤6 −x1+x2≤1 x2≤2
Answers
By substituting the corner points into the objective function, we can compare the values of z and identify the minimum. The corner point with the lowest z value will give us the optimal solution.
To solve the given linear programming problem completely, we will follow the steps mentioned earlier. Let's go through the process in more detail:
Step 1: Identify the decision variables:
Let x1 and x2 be the decision variables representing the values we want to find that minimize z.
Step 2: Formulate the objective function:
The objective function is given as z = 5x1 + 4x2.
Step 3: Formulate the constraints:
The given constraints are:
6x1 + 2x2 ≤ 24
x1 + 2x2 ≤ 6
-x1 + x2 ≤ 1
x2 ≤ 2
Step 4: Combine all the constraints:
To combine the constraints, we rewrite them in standard form:
6x1 + 2x2 ≤ 24 ---> 6x1 + 2x2 + 0x3 + 0x4 = 24
x1 + 2x2 ≤ 6 ---> x1 + 2x2 + 0x3 + 0x4 = 6
-x1 + x2 ≤ 1 ---> -x1 + x2 + 0x3 + 0x4 = 1
0x1 + x2 ≤ 2 ---> 0x1 + x2 + 0x3 + 0x4 = 2
Step 5: Solve the linear programming problem:
To solve this linear programming problem, we can use various methods such as the Simplex method or graphical method. Here, we'll use the graphical method to find the optimal solution.
First, let's plot the feasible region by graphing the equations of the constraints:
Plotting 6x1 + 2x2 = 24:
x1 | x2
0 | 12
4 | 0
Plotting x1 + 2x2 = 6:
x1 | x2
0 | 3
6 | 0
Plotting -x1 + x2 = 1:
x1 | x2
-1 | 0
0 | 1
Plotting x2 = 2:
x1 | x2
0 | 2
Next, we need to find the feasible region by considering the overlapping shaded area formed by the inequalities. However, I cannot provide a visual representation here. Please refer to a graphing tool or software to plot the equations and find the feasible region.
Finally, we evaluate the objective function z = 5x1 + 4x2 at the corner points of the feasible region to determine the minimum value of z. Each corner point represents a specific combination of x1 and x2 within the feasible region.
By substituting the corner points into the objective function, we can compare the values of z and identify the minimum. The corner point with the lowest z value will give us the optimal solution.
Please note that without the graphical representation, I am unable to provide the exact numerical solution. I recommend using a graphing tool or linear programming software to visualize the feasible region and determine the optimal values of x1 and x2 that minimize z.
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I’ve been trying other ways to try and get it but nothing is helping
Answers
Answer:
x= 104
Step-by-step explanation:
The sum of the angles on a straight line is 180°.
72° +(x +4)°= 180° (adj. ∠s on a str. line)
(x +4)°= 180° -72°
(x +4)°= 108°
x +4= 108
x= 108 -4
x= 104
Answer the following question and will mark Brainliest
Answers
a) f(4) = 3*4 - 4
= 8
b) h(-3) = 5 - ( - 3)
= 5 + 3 = 8
c) h(4) - f(-5) = 1 + ( -19)
= -18
have a nice day <3
root( x-1 ) = 2 - root (x+3)
Answers
Explanation: root x-1 = 2- root x+3
First let’s take out the root by square both side by 2 we get:
x-1= 4-x+3
x-1 = -x+7
x+x = 7+1
2x= 8
x= 8/2=4
Answer:
\(x=1\)
Step-by-step explanation:
\(\sqrt{x-1} =2-\sqrt{x+3}\)
Square both sides.
\(\left(\sqrt{x-1}\right)^2=\left(2-\sqrt{x+3}\right)^2\)
\(x-1=x+7-4\sqrt{x+3}\)
Subtract x on both sides.
\(x-1-x=x+7-4\sqrt{x+3}-x\)
\(-1=-4\sqrt{x+3}+7\)
Subtract 7 on both sides.
\(-1-7=-4\sqrt{x+3}+7-7\)
\(-8=-4\sqrt{x+3}\)
Square both sides.
\(\left(-8\right)^2=\left(-4\sqrt{x+3}\right)^2\)
\(64=16x+48\)
Subtract 48 on both sides.
\(64-48=16x+48-48\)
\(16=16x\)
Divide both sides by 16.
\(\frac{16}{16} =\frac{16x}{16}\)
\(1=x\)
Switch sides.
\(x=1\)
A housewife deposited Rs 10,000 on saving account at 5% p.a. interest compounded
yearly and another sum on fixed deposit account at 8% p.a. interest compounded
half yearly. After one year the interest on fixed deposit account
was Rs 152.80 more
than the interest on the saving account, find the total amount of money in her two
accounts at the end of the year.
Answers
Answer:
Rs. 3923.08
Step-by-step explanation:
First principal invested = Rs. 10000
Interest rate = 5%
The interest is compounded yearly.
Time = 1 year
1 year compound interest is equal to simple interest.
Formula for simple interest:
\(SI = \dfrac{PRT}{100}\)
Interest on first sum = \(\frac{10000\times 5\times 1}{100} = Rs\ 500\)
Another sum is on fixed deposit 8% compounded half yearly.
Let the sum = Rs \(x\)
Formula for compound interest is given as:
\(CI = P(1+\frac{R}{100})^T - P\)
It is compounded half - yearly, therefore T = 2
\(CI = x (1+\frac{8}{100})^2 - x\\\Rightarrow CI = x(1.08)^2-x = 0.1664x\)
As per question statement:
\(0.1664x - 500 = 152.80\\\Rightarrow x = \dfrac{652.80}{0.1664} = Rs\ 3823.08\)
A nutrition company is marketing a low calorie snack brownie. A serving size of the snack is 3 brownies and has a total of 50 calories.
If c represents the number of calories and b represents the number of brownies write a proportional relationship involving c and b and solve it for c.
Answers
The relationship between the variables c and b is c = \(\frac{50}{3}b\)
What is Variation?A variation is a relation between a set of values of one variable and a set of values of other variables. Direct variation. In the equation y = mx + b, if m is a nonzero constant and b = 0, then you have the function y = mx (often written y = kx), which is called a direct variation.
if c ∝ b
Then,
c = kb where k is a constant
When c = 50, b = 3
substituting into the equation above,
50 = 3k
k = 50/3
substitute k into the equation
c = \(\frac{50}{3}b\)
In conclusion c is related to b by the equation c = \(\frac{50}{3}b\)
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Find the 96th term of the arithmetic sequence 1, -12, -25, ...1,−12,−25
Answers
96th term of the arithmetic sequence 1, -12, -25, ...1,−12,−25 is -1234
arithmetic sequence 1, -12, -25, .. .1,−12,−25
an arithmetic sequence can be written as
a , a + d , a + 2d , a + 3d , . . . . . a + (n-1)d
nth tern of an arithmetic sequence is
aₙ= a+ (n-1)d
a= first term of an arithmetic sequence
d = common difference of an arithmetic sequence
n = number of terms in an arithmetic sequence
So in the above arithmetic sequence
a= 1
d= -13
n= 96
a₉₆= 1+ (96-1)(-13)
= 1- (95)13
= 1- 1235
= -1234
96th term of the arithmetic sequence 1, -12, -25, ...1,−12,−25 is -1234
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good amount of points!
pls help
what is -3+7?
(negative three plus seven)
Answers
Answer: 4 :)
Step-by-step explanation:
-3+7 its is positive 4
I NEED HELP, WILL GIVE BRAINLIEST
Answers
Answer:
Figure ABCD was formed by first translating figure ABCD \(\boxed{5}\) units left and \(\boxed{4}\) units down, and then reflecting across the \(\boxed{\textrm{x}}\) -axis
Step-by-step explanation:
To determine the translated units take any point in the pre-image ABCD and see the relative position of the corresponding point in the translated image A'B'C'D'
All other points and hence the entire pre-image will be translated by the same amount
Let's take point C in ABCD and point C' in A'BC'D C coordinates are (4, 1) and C' coordinates are (-1, -3)So horizontal translation is (-1 -4 ) = - 5 which is a translation of 5 units leftVertical translation is (-3 - 1) = -4 which is a translation of 4 units downThis translation puts the segment B'D' on the x-axis and since it is a symmetrical figure across the x-axis. So reflecting it across the x-axis will not change its shape or its size or position.So the answer is 5 units left, 4 units down and reflected across the x-axis
Let an be the number of ways to climb n stairs if a person climbing the stairs can take one stair or two stairs at a time. Identify the initial condition for the recurrence relation in the previous question. (You must provide an answer before moving to the next part.)
Answers
The total number of ways 'n' stairs can be climbed in are given by:\(a_{n} = a_{n-1} + a_{n-2}, n\geq 2\)
What is Addition Rule of Fundamental Counting Principle?The principle states that if we have P number of ways of doing something and Q number of ways of doing another thing and we can not do both at the same time, then there are P+Q ways to do things by one of the methods.
Given: \(a_{n}\) = Number of ways to climb 'n' stairs, if a person is taking one or two stairs at a time.
Then, this can be done in two ways:
When the person climbs one stair at a time, the remaining (n-1) stairs can be climbed in \(a_{n-1}\) ways.Similarly, when the person climbs two stairs at a time, the remaining (n-2) stairs can be climbed in \(a_{n-2}\) ways.Thus, the total number of ways 'n' stairs can be climbed in are given by:
\(a_{n} = a_{n-1} + a_{n-2}, n\geq 2\)
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Helpppppppppppppppppppppppp
Answers
the actual amount is 78000 dollar
Step-by-step explanation:
one pack cigarettes cost 25 dollar
3 pack cigarettes costs 75 dollar
20 years × 52 =1040
1040 weeks
1040 × 75 dollar
78000 dollar
Consider the polygon GEOM with coordinate G(0 -2) E(-12), O(-5, 1), M(-5,-6). If we rotate GEOM 90° clockwise about the origin, then reflect over the y-axis, what are the coordinate of G"E"O"M ?
Answers
Cup o' Coffee buys its coffee for $1.25 a cup. The coffee shop then sells each cup for $3.75. What is the percent markup for a cup of coffee?
Answers
Answer:
47%
Step-by-step explanation:
.375 × .125 = .046875
rounding up, 5 keeps the 7 as is, 7 is greater than 5, therefore the 8, becomes a 9, 9 is greater than 5, so the 6 becomes a 7, .047.
move the decimal three digits to the left and you get, 47%.
Percentage markup is the percent of the difference of the selling price to the buying price of a product. The percent markup for a cup of coffee is 200%.
The percentage markup has to be find out for the cup of coffee.
What is percentage markup?Percentage markup is the percent of the difference of the selling price to the buying price of a product. It can be given as,
\(M=\dfrac{\rm SP-C}{C} \times 100\)
Here, \(SP\) is the selling price and \(C\) is the cost of the product.
Given information-
The Cup o Coffee buys its coffee for $1.25 a cup.
The Cup o Coffee sells its coffee for $1.25 a cup.
To find out the percentage markup for the cup of coffee, put the values in the above formula. Thus,
\(M=\dfrac{3.75-1.25}{1.25} \times 100\\M=\dfrac{2.5}{1.25} \times 100\\M=200\)
Hence the percent markup for a cup of coffee is 200%.
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Can someone help me with this Question.
Answers
The formula we need to use is given above. In this formula, we will substitute the desired values. Let's start.
\(P=3W+D\)A) First, we can start by analyzing the first premise. The team has \(8\) wins and \(5\) losses. It earned \(8 \times 3 = 24\) points in total from the matches it won and \(1\times5=5\) points in total from the matches it drew. Therefore, it earned \(24+5=29\) points.
B) After \(39\) matches, the team managed to earn \(54\) points in total. \(12\) of these matches have ended in draws. Therefore, this team has won and lost a total of \(39-12=27\) matches. This number includes all matches won and lost. In total, the team earned \(12\times1=12\) points from the \(12\) matches that ended in a draw.
\(54-12=42\) points is the points earned after \(27\) matches. By dividing \(42\) by \(3\) ( because \(3\) points is the score obtained as a result of the matches won), we find how many matches team won. \(42\div3=14\) matches won.
That leaves \(27-14=13\) matches. These represent the matches team lost.
Finally, the answers are below.
\(A)29\)
\(B)13\)
Answer:
a) 29 points
b) 13 losses
Step-by-step explanation:
You want to know points and losses for different teams using the formula P = 3W +D, where W is wins and D is draws.
A 8 wins, 5 drawsThe number of points the team has is ...
P = 3W +D
P = 3(8) +(5) = 29
The team has 29 points.
B 54 pointsYou want the number of losses the team has if it has 54 points and 12 draws after 39 games.
The number of wins is given by ...
P = 3W +D
54 = 3W +12
42 = 3W
14 = W
Then the number of losses is ...
W +D +L = 39
14 +12 +L = 39 . . . substitute the known values
L = 13 . . . . . . . . . . subtract 26 from both sides
The team lost 13 games.
__
Additional comment
In part B, we can solve for the number of losses directly, using 39-12-x as the number of wins when there are x losses. Simplifying 3W +D -P = 0 can make it easy to solve for x. (In the attached, we let the calculator do the simplification.)
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The marketing manager for a nationally franchised lawn service company would like to study the characteristics that differentiate home owners who do and do not have a lawn service. A random sample of 30 home owners located in a suburban area near a large city was selected; 11 did not have a lawn service (code 0) and 19 had a lawn service (code 1). Additional information available concerning these 30 home owners includes family income (Income, in thousands of dollars) and lawn size (Lawn Size, in thousands of square feet). The PHStat output is given below: Binary Logistic Regression Z p Predictor Intercept Income Lawn Size Coefficients -7.8562 0.0304 1.2804 SE Coef 3.8224 0.0133 0.6971 -2.0553 2.2897 1.8368 -Value 0.0398 0.0220 0.0662 Deviance 25.3089 Which of the following is the correct expression for the estimated model? In (estimated odds ratio) = -7.8562 +0.0304 Income + 1.2404 Lawnsize In (odds ratio) = -7.8562 +0.0304 Income + 1.2804 Lawnsize Y - -7.8562 +0.0304 Income + 1.2804 Lawnsize Y = -7.8562 +0.0304 Income + 1.2804 Lawnsize
Answers
The correct expression for the estimated model is: In (odds ratio) = -7.8562 +0.0304 Income + 1.2804 Lawnsize. This model was created using binary logistic regression analysis to study the characteristics that differentiate home owners who have a lawn service (code 1) and those who do not (code 0).
The additional information available for these 30 home owners includes family income (Income, in thousands of dollars) and lawn size (Lawn Size, in thousands of square feet). The estimated model shows that for every one unit increase in income,
the odds of having a lawn service increase by 0.0304, and for every one unit increase in lawn size, the odds of having a lawn service increase by 1.2804. This information can be useful for the marketing manager to target potential customers based on their family income and lawn size.
The correct expression for the estimated model in this case is:
ln(odds ratio) = -7.8562 + 0.0304 Income + 1.2804 Lawn Size
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find a formula for the nth term, an, of the sequence assuming that the indicated pattern continues.
1/5, -4/11, 9/17, -16/23, ...
a_n =
Answers
The formula for the nth term, an, of the given sequence is an = (-1)ⁿ⁺¹ * n² / (6n + 5), where the numerator alternates between positive and negative perfect squares, and the denominator increases by a constant difference of 6.
To find the formula for the nth term, we need to analyze the pattern in the given sequence.
The numerators alternate between positive and negative perfect squares: 1, -4, 9, -16, ...
The denominators increase by a constant difference of 6: 5, 11, 17, 23, ...
Based on this pattern, we can observe that the numerator is given by (-1)ⁿ⁺¹ * n². The exponent (n+1) ensures that the sign alternates between positive and negative.
The denominator is given by 6n + 5.
Putting it all together, the formula for the nth term, an, is:
an = (-1)ⁿ⁺¹ * n² / (6n + 5).
This formula will give you the value of each term in the sequence based on the position of n.
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Bus A and Bus B leave the the bus depot at 2 pm.
Bus A takes 15 minutes to complete its route once and bus B takes 25 minutes to complete its route once.
If both buses continue to repeat their route, at what time will they be back at the bus depot together?
Assume the buses have have no breaks in between routes.
Give your answer as a 12-hour clock time.
Answers
Answer:
3:15 pm
Step-by-step explanation:
You can imagine that they go in a circle (we can do whatever we want in theoretical math and physics). By doing so we know we can obtain a cosinusoidal wave by projecting the motion on a plane. Each one of the buses has a period (minimum time after what they're in the same point again). For the first one it's 15 minutes, for the second one it's 20 minutes. We have to take the least common multiple of the periods, that is 75 minutes. So They'll meet every 75 minutes. Add 75 minutes (1h and 15 minutes) to 2pm and there you go.
A)40%
B)45%
C)5%
D)10%
Answers
Answer:
A) 40%
Step-by-step explanation:
25% of yards have 6-8 trees; 10% of yards have 8-10 trees; 5% of yards have more than 10 trees. So, a total of 25% +10% +5% = 40% of yards have 6 trees or more. If a yard is randomly chosen, the probability it will have 6 or more trees is 40%.
what is the answer for 7(-5+m)
Answers
7−35
hope this helps:)
"What set of reflections would carry hexagon ABCDEF onto itself?. . Hexagon ABCDEF on the coordinate plane with pointA at negative 1, 1, pointB at negative 3, 1, pointC at negative 4, 2, pointD at negative 3, 3, pointE at negative 1, 3, and pointF at 0, 2. . .x-axis, y=x, x-axis, y=x .. y=x, x-axis, y=x, y-axis .. y-axis, x-axis, y-axis .. x-axis, y-axis, y-axis ."
Answers
A set of reflections that would carry hexagon ABCDEF onto itself would be "x-axis, y=x, x-axis" or "y=x, x-axis, y-axis".
What is a combination of reflections?
Combination of Two Reflections. A point or object once reflected can further be reflected to form a new image. The axes of these reflections may be parallel to each other or they intersect each other at a point.
Given the coordinates of hexagon ABCDEF, it can be determined that a set of reflections that would carry the hexagon onto itself would be a combination of reflections over the x-axis and y-axis.
One possibility would be to reflect over the x-axis, then reflect over the y=x line, and finally reflect over the x-axis again.
This would take the hexagon from its original position to itself.
Another possibility would be to reflect over the y = x line, then reflect over the x-axis, and finally reflect over the y-axis.
This would also take the hexagon from its original position to itself.
Hence, a set of reflections that would carry hexagon ABCDEF onto itself would be "x-axis, y=x, x-axis" or "y=x, x-axis, y-axis".
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Parallel lines have so much in common, It’s a shame they’ll never meet.
Answers
Answer:
This is actually very true
Step-by-step explanation:
lol
The National Junior Honor Society made $238at their cake sale. They sold circular Shaped cakes for $7 and heart shaped cakes for $5. If they Sold twice as many heart cakes as circular ones. how many heart cakes did they sell?
Answers
The number of heart cakes that were sold at the cake sale was 28
According to the question,
Cost of circular-shaped cake = $7
Cost of heart shaped cake = $5
Total sale = $238
Let the number of circular cakes shaped be x
Since the number of heart cakes was sold twice as many as the circular cake, the number of heart cakes will be 2x
Cost of x circular-shaped cake = $7 * x
Cost of 2x heart shaped cake = $5 * 2x
Total sale = Cost of x circular cake + Cost of 2x heart cake
Thus, we get the following equation,
238 = 7x + 10x
238 = 17x
x = 14
Thus, the number of heart cakes sold was 2x that is 28.
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What is the range of this relation?
Answers
Answer:
{- 1, 0, 4 }
Step-by-step explanation:
the range is the y- coordinates of the plotted points
the coordinates of the points are
(- 4, - 1 ) , (1, 0 ) , (1, 4 ) with y- coordinates - 1, 0, 4
then range is { - 1, 0, 4 }
solve the equation -4(1.75 + x) = 18
Answers
Answer:
x= -25/4
Step-by-step explanation:
Simplify both sides of the equation:
-4(1.75+x)=18
(-4)(1.75)+ (-4)(x)=18 | Distribute
-7+-4x=18
-4x-7=18 | Then solve.
Add 7 to both sides:
-4x-7+7=18+7
-4x=25
Then divide both sides by -4
Which statement explains why 49 is a perfect square?
49 can be multiplied by 49.
49 is equal to 7 plus 7
49 is between 36 and 64
49 is equal to 7 times 7
I need this one asap to..
Answers
Answer:
49 is equal to 7 times 7
Step-by-step explanation:
49 can be multiplied by 49 does not explain why 49 is a perfect square.
49 is not equal to 7 plus 7.
49 is between 36 and 64 does not help in explaining why 49 is a perfect square.
49 is equal to 7 times 7 explains why 49 is a perfect square. A perfect square is an integer that is a square of an integer.
7² = 49
Help pls!!
Which expression best represents the product of two consecutive integers?
Answers
Answer:
C
Step-by-step explanation:
Because it is the only one that makes sense.
A remote controlled car travels 8 feet in 2 seconds Label the other tick marks on the double number line with equivalent rates for the constant speed of the car
Answers
The car travels 4 feet in 1 second, 16 feet in 4 seconds, 24 feet in 6 seconds, 32 feet in 8 seconds.
Describe Distance?Distance is a measure of the amount of space between two objects or points. It is the length of the shortest path between the two points, and it is typically measured in units such as meters, kilometers, miles, or feet.
In mathematics, distance is often calculated using the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides.
The car travels 4 feet in 1 second, which is equivalent to a rate of 4 feet per second.
The car also travels 16 feet in 4 seconds, which is also equivalent to a rate of 4 feet per second.
Similarly, the car travels 24 feet in 6 seconds, 32 feet in 8 seconds, and so on, all at a constant rate of 4 feet per second.
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What is the average rate of change of the line?
Answers
Answer:
- 12
Step-by-step explanation:
The slope of the line is a measure of the average rate of change
The equation of a line in point- slope form is
y - b = m(x - a)
where m is the slope and (a, b) a point on the line
y - 3 = - 12(x + 4) ← is in point- slope form
with slope m = - 12 ← average rate of change
5/6 x 2 1/5 will be. Less greater or equal to 2 1/5
Answers
Answer:
Less greater
Step-by-step explanation:
Firstly, let's solve:
\(\frac{5}{6}\times2\frac{1}{5}\)
Convert to improper fraction
\(=\frac{5}{6}\cdot \frac{11}{5}\)
Cancel common factor: 5
\(\frac{11}{6}\)
Mixed fraction
\(=1\frac{5}{6}\)
2 1/5 > 1 5/6
Therefore, 1 5/6 is less greater than 2 1/5
~Lenvy~
problem 5 (30 points, each 10 points). in a chemical plant, 24 holding tanks are used for final product storage. four tanks are selected at random and without replacement. suppose that four of the tanks contain material in which the viscosity exceeds the customer requirements. 1. what is the probability that exactly one tank in the sample contains high-viscosity material? 2. what is the probability that at least one tank in the sample contains high-viscosity material? 3. in addition to the four tanks with high-viscosity levels, four different tanks contain material with high impurities. what is the probability that exactly one tank in the sample contains high-viscosity material and exactly one tank in the sample contains material with high impurities?
Answers
1. The probability of selecting exactly one tank with high-viscosity material is 0.
2. The probability of selecting at least one tank with high-viscosity material is 1.
3. The probability of selecting exactly one tank with high-viscosity material and exactly one tank with high impurities is 0.25.
1. The probability of selecting exactly one tank with high-viscosity material is calculated by the binomial distribution formula, P(X = n) = (nCx)p^x(1-p)^n-x, where n is the number of trials, x is the number of successes, and p is the probability of success. In this case, n = 4, x = 1, and p = 24/24 = 1. Therefore, P(X = 1) = (4C1)1^1(1-1)^4-1 = 0.
2. The probability of selecting at least one tank with high-viscosity material is calculated by the complement rule, P(X > 0) = 1 - P(X = 0). In this case, P(X > 0) = 1 - (4C0)1^0(1-1)^4-0 = 1.
3. The probability of selecting exactly one tank with high-viscosity material and exactly one tank with high impurities is calculated by the binomial distribution formula, P(X = n) = (nCx)p^x(1-p)^n-x, where n is the number of trials, x is the number of successes, and p is the probability of success. In this case, n = 8, x = 2, and p = 24/24 = 1. Therefore, P(X = 2) = (8C2)1^2(1-1)^8-2 = 0.25.
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