The work done in moving the particle from the origin to x = 4 under the influence of the force F (x) = 8[tex]x^3[/tex]-5 is 492 units of work.
The work done in moving a particle along a path under the influence of a force, we use the work-energy principle.
This principle states that the work done on a particle by a force is equal to the change in the particle's kinetic energy.
Mathematically this can be expressed as:
W = ΔK
Where
W is the work done,
ΔK is the change in kinetic energy and
Both are scalar quantities.
The work done by a force on a particle along a path is given by the line integral:
W = ∫ C F · ds
Where,
C is the path,
F is the force,
ds is the differential displacement along the path and denotes the dot product.
In the case where the force is a function of position only (i.e., F = F(x,y,z)), we can evaluate the line integral using the parametric equations for the path.
If the path is given by the parameterization r(t) = <x(t), y(t), z(t)>, then we have:
W = ∫ [tex]a^b[/tex] F(r(t)) · r'(t) dt
The work done in moving the particle from the origin to a final position at x = 4. We can evaluate the work done using the definite integral of the force from x = 0 to x = 4, as shown in the solution.
The initial kinetic energy is zero.
The work done by the force in moving the particle from x = 0 to x = 4 is given by the definite integral:
W = ∫ F(x) dx
Substituting the given expression for the force, we have:
W = ∫0 (8x - 5) dx
Integrating with respect to x, we have:
W = [(2x - 5x)]_0
W = (2(4) - 5(4)) - (2(0) - 5(0))
W = 512 - 20
W = 492
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A person was driving their car on an interstate highway
and a rock was kicked up and cracked their windshield
on the passenger side.
The driver wondered if the rock was equally likely to
strike any where on the windshield, what the probability
was that it would have cracked the windshield in his line
of site on the windshield. Determine this probability,
provided that the windshield is a rectangle with the
dimensions 28 inches by 54 inches and his line of site
through the windshield is a rectangle with the
dimensions 30 inches
by 24 inches.
a) 0. 373
b) 0. 139
c) 0. 423
d) 0. 476
There is about a 47.62% or 0.476 chance that the rock hit the windshield in the driver's line of sight. Option D.
To determine the probability that the rock hit the driver's line of sight on the windshield, we need to compare the area of the driver's line of sight rectangle to the total area of the windshield rectangle.
The area of the windshield rectangle is:
A1 = 28 in x 54 in = 1512 sq in
The area of the driver's line of sight rectangle is:
A2 = 30 in x 24 in = 720 sq in
Therefore, the probability that the rock hit the driver's line of sight on the windshield is:
[tex]P= \frac{A2}{A1}= \frac{720 \:sq in}{1512 \:sq in }[/tex] = 0.476 or 47.6%
So, there is about a 47.62% chance that the rock hit the windshield in the driver's line of sight.
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Determine whether the two figures are similar. If so, give the similarity ratio of the smaller figure to the larger figure. The figures are not drawn to scale.
*
Captionless Image
Yes; 3:5
Yes; 2:3
Yes; 2:5
No they are not similar
Determine whether the two figures are similar: D. No, the two figures are not similar.
What are the properties of quadrilaterals?In Geometry, two (2) quadrilaterals are similar when the ratio of their corresponding sides are equal in magnitude and their corresponding angles are congruent.
Additionally, two (2) geometric figures such as quadrilaterals are considered to be congruent only when their corresponding side lengths are congruent (proportional) and the magnitude of their angles are congruent;
Ratio = 12/8 = 10/6 = 10/6
Ratio = 3/2 ≠ 5/3 = 5/3
In conclusion, the two figures are not similar because the ratio of their corresponding sides is not proportional.
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Missing information:
The question is incomplete and the complete question is shown in the attached picture.
On the first swing, the length of the arc through which a pendulum swings is 50 inches. the length of each successive
swing is 80% of the preceding swing. determine whether this sequence is arithmetic or geometric. find the length of the
fourth swing
The length of the fourth swing is 25.6 inches. The sequence is arithmetic or geometric.
The length of the arc through which a pendulum swings is 50 inches. To determine whether the sequence is arithmetic or geometric, and to find the length of the fourth swing, we will analyze the given information.
The length of the first swing is 50 inches. Each successive swing is 80% of the preceding swing. This means that to find the length of the next swing, we multiply the length of the current swing by 80% (or 0.8).
Since we are multiplying by a constant factor (0.8) to find the next term in the sequence, this is a geometric sequence, not an arithmetic sequence.
Now, let's find the length of the fourth swing.
1st swing: 50 inches
2nd swing: 50 * 0.8 = 40 inches
3rd swing: 40 * 0.8 = 32 inches
4th swing: 32 * 0.8 = 25.6 inches
The length of the fourth swing is 25.6 inches.
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Why does this limit evaluate to 0 instead of 2?
[tex]\lim_{x\to-\infty}\left(\dfrac{2}{1-\frac{5}{e^x}}\right)[/tex]
You're partially correct, as if x approaches ∞ it would approach 2, as eˣ is exponentially growing if x is positive.
If x is negative, which it is in this case, eˣ would get exponentially smaller. For example, e⁻² = 1/e².
So, in this case [tex]\frac{5}{e^x}[/tex] would get exponentially larger, as it is a number over an increasingly small number, like how [tex]\frac{1}{0.001}[/tex] is larger than [tex]\frac{1}{0.1}[/tex].
Therefore the limit would be equivalent to [tex]\frac{2}{\infty}[/tex], which is equal to 0
[tex] \Large{\boxed{\sf \lim_{x\to-\infty}\left(\dfrac{2}{1-\frac{5}{e^x}}\right) = 0}} [/tex]
[tex] \\ [/tex]
Explanation:
We are trying the find the limit of [tex] \: \sf \dfrac{2}{1 - \dfrac{5}{ {e}^{x} } } \: [/tex] when x tends to -∞.
[tex] \\ [/tex]
Given expression:
[tex] \sf \lim_{x\to-\infty}\left(\dfrac{2}{1-\frac{5}{e^x}}\right) [/tex]
[tex] \\ [/tex]
[tex]\blue{\begin{gathered}\begin{gathered} \\ \boxed { \begin{array}{c c} \\ \blue{ \star \: \sf{\boxed{ \sf Properties\text{:}}}} \\ \\ \sf{ \diamond \: \dfrac{c}{ + \infty} = 0^{ + } \: \: and \: \: \dfrac{c}{ - \infty} = 0^{ - } \: \: , \: where \: c \: is \: a \: positive \: number.} \\ \\ \\ \diamond \: \sf \dfrac{c}{ {0}^{ + } } = + \infty \: \: and \: \: \dfrac{c}{ {0}^{ - } } = - \infty \: \: , \: where \: c \: is \: a \: positive \: number.\\ \\ \\ \diamond \: \sf c - \infty = -\infty \: \: and \: \: c + \infty = \infty \: \: ,\: where \: c \: is \: a \: positive \: number. \\ \\ \\ \sf{ \diamond \: \green{e ^{ - \infty} = 0^{+} \: \: and \: \: e ^{ + \infty} = + \infty} } \\ \end{array}}\\\end{gathered} \end{gathered}}[/tex]
[tex] \\ [/tex]
Substitute -∞ for x[tex] \\ [/tex]
[tex] \sf \lim_{x\to-\infty}\left(\dfrac{2}{1-\frac{5}{e^x}}\right) = \sf \left(\dfrac{2}{1-\frac{5}{e^{ - \infty}}}\right) [/tex]
[tex] \\ [/tex]
Simplify knowing that [tex] \sf e^{-\infty} \\ [/tex] approaches 0 but remains a positive number. This will be written as 0⁺.
[tex] \sf \lim_{x\to-\infty}\left(\dfrac{2}{1-\frac{5}{e^x}}\right) = \left(\dfrac{2}{1-\frac{5}{e^{ - \infty}}}\right) = \left(\dfrac{2}{1-\frac{5 \: \: }{0^{ + } }}\right)[/tex]
[tex] \\ [/tex]
Simplify again knowing that 5/0⁺ = +∞.
[tex] \\ [/tex]
[tex] \sf \lim_{x\to-\infty}\left(\dfrac{2}{1-\frac{5}{e^x}}\right) = \left(\dfrac{2}{1-\frac{5 \: \: }{0^{ + } }}\right) = \sf \left(\dfrac{2}{1 - \infty}\right) = \dfrac{2}{ - \infty} [/tex]
[tex] \\ [/tex]
Conclusion[tex] \\ [/tex]
[tex] \sf \lim_{x\to-\infty}\left(\dfrac{2}{1-\frac{5}{e^x}}\right) = \dfrac{2}{ - \infty} = 0^{-} \\ \\ \\ \implies \boxed{ \boxed{ \sf \lim_{x\to-\infty}\left(\dfrac{2}{1-\frac{5}{e^x}}\right) =0}}[/tex]
[tex] \\ \\ \\ [/tex]
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4. A person wants to buy a car from Toyota Company. If the price of car
including VAT is Birr 5,000,000 then,
a) What is the price of car before VAT?
b) What is the value of VAT?
Answer:
Step-by-step explanation:a) To find the price of the car before VAT, we need to first calculate the percentage of VAT included in the price:
VAT% = (VAT / Total Price) x 100
where VAT% is the percentage of VAT, VAT is the value of VAT, and Total Price is the price of the car including VAT.
From the given information, we have:
Total Price = Birr 5,000,000
VAT% = 15% (assuming a VAT rate of 15% in Ethiopia)
Therefore, we can solve for the value of the car before VAT as follows:
Total Price = Car Price + VAT
Birr 5,000,000 = Car Price + 0.15Car Price
Birr 5,000,000 = 1.15Car Price
Car Price = Birr 4,347,826.09
So the price of the car before VAT is Birr 4,347,826.09.
b) To find the value of VAT, we can use the same formula as above and solve for VAT:
Total Price = Car Price + VAT
Birr 5,000,000 = Birr 4,347,826.09 + VAT
VAT = Birr 652,173.91
Therefore, the value of VAT is Birr 652,173.91.
The area of a square with side length s is s2. Meg crocheted a baby blanket for her new cousin. The blanket is a square with 30-inch sides. What is the area of the baby blanket? Write your answer as a whole number or decimal
The area of the baby blanket with side length of 30 inches is equal to 900 square inches.
Let 'A' represents the area of the square.
And s represents the side length of the square.
The area of a square is given by the formula
A = s^2.
For Meg's baby blanket,
The side length of the baby blanket is equal to 30 inches,
Substitute the values in the area formula we get,
A = s^2
⇒ A = 30^2
⇒ A = 900 square inches
Therefore, the area of Meg's baby blanket is equal to 900 square inches.
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A lab technician is filling vitamin C capsules. He has 2.87 ounces of vitamin C and is putting 0.014 ounces of vitamin C into each capsule. How many capsules will the lab technician be able to fill with vitamin C? A. 3 B. 25 C. 402 D. 205 
Answer:
D) 205
Step-by-step explanation:
If the technician has a total of 2.87 oz, and can have a max of 0.014 oz in each capsule, we have to divide the total amount by the max amount per bottle.
2.87/0.014
=205
This means that the technician can fill 205 capsules with 0.014 oz of vitamin C.
Hope this helps!
Which of the following is equivalent to [tex]\sqrt{x} 12qr^{2}[/tex]
The calculated value of the expression that is equivalent to √(x¹²qr²) is x⁶r√q
Calculating the expression that is equivalent to √(x¹²qr²)From the question, we have the following parameters that can be used in our computation:
√x12qr²
Express properly
So, we have
√(x¹²qr²)
Evaluating the expression in the brackets using the law of indices
So, we have
√(x¹²qr²) = x⁶r√(q)
Next, we open the brackets
This gives
√(x¹²qr²) = x⁶r√q
Hence, the expression that is equivalent to √(x¹²qr²) is x⁶r√q
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Select the correct answer.
consider this equation.
cos(ф) = [tex]\frac{8}{9}[/tex]
if ф is an angle in quadrant iv, what is the value of tan(ф)?
The value of tan(ф) when cos(ф) = 8/9 in quadrant IV is - √17/8
To solve for the value of tan(ф), we need to use the trigonometric identity: tan(ф) = sin(ф)/cos(ф).
Since ф is in quadrant IV, we know that the cosine value is positive (due to cosine being positive in the adjacent side of quadrant IV) and the sine value is negative (due to sine being negative in the opposite side of quadrant IV).
We are given the value of the cosine, which is cos(ф) = 8/9. To find the sine, we can use the Pythagorean identity: sin²(ф) + cos²(ф) = 1.
Plugging in the given value of the cosine, we get: sin²(ф) + (8/9)² = 1. Solving for sin(ф), we get sin(ф) = - √(1 - (64/81)) = - √(17/81) = - √17/9.
Now that we have the values of sin(ф) and cos(ф), we can substitute them into the tan(ф) equation: tan(ф) = sin(ф)/cos(ф) = (- √17/9)/(8/9) = - √17/8.
Therefore, the value of tan(ф) when cos(ф) = 8/9 in quadrant IV is - √17/8.
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using mad when do you see a moderate amount of overlap in these two graphs when the mad is $20
Lower Bound: 20 - 29.67 ≈ -9.67
Upper Bound: 20 + 29.67 ≈ 49.67.
How to solveThe conversion of MAD to the standard deviation for normal distributions can be obtained using the given relationship:
Standard Deviation (σ) = MAD / 0.6745
For both distributions, the MAD holds a value of $20, thus arriving at σ ≈ 29.67.
There are two normal distributions now defined by their parameters as follows:
Mean (µ1) = $100 and Standard Deviation (σ1) = 29.67
Mean (µ2) = $120 and Standard Deviation (σ2) = 29.67.
Since both distributions share an equivalent standard deviation, we can perform a comparison of means to determine the overlap between them.
4
Typically there is observed moderate overlapping within one standard deviation from the difference in means.
The calculation of the difference in means indicates µ2 - µ1 = 120 - 100 = 20. Taking one standard deviation (which equates to 29.67) into consideration with respect to the difference of the means leads us to this range:
Lower Bound: 20 - 29.67 ≈ -9.67
Upper Bound: 20 + 29.67 ≈ 49.67.
It's noteworthy that negative values would not make sense within this context leading us to assume that the approximate overlap range is situated between $0 and $50 resulting in these normal distributions manifesting a sensible amount of overlap therein.
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The Complete Question
When comparing two normal distribution graphs with a mean of $100 and $120 respectively, and both having a MAD of $20, at what range do you see a moderate amount of overlap between the two distributions?
John earns $8. 50 per hour proofreading advertisements at a local newspaper. Write a function in function notation. Use d as your variable to represent days
The function notation is E(h) = 8.5h where h represents the number of hours worked so the domain is {0, 1, 2, 3, 4, 5} and the range is {0, 8.5, 17, 25.5, 34, 42.5}.
Let E(t) be John's earnings in dollars after working t hours, where t is in the domain 0 ≤ t ≤ 5. Then E(t) = 8.50t, since John earns $8.50 per hour proofreading ads.
The domain of the function is 0 ≤ t ≤ 5, since John works no more than 5 hours per day.
The range of the function is 0 ≤ E(t) ≤ 42.50 since John earns $8.50 per hour and works no more than 5 hours per day.
Therefore, the maximum earnings he can make in one day is 5 hours multiplied by $8.50 per hour, which equals $42.50.
The minimum earnings are $0, which would occur if John does not work at all.
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The question is -
John can earn $8.50 per hour proofreading adverse at a local newspaper. He works no more than 5 hours a day. Write a function in function notation and find a reasonable domain and range of his earnings.
x+2y=6
-7x+3y=-8 (using substitution)
Answer:
point form - (2,2)
x=2 y=2
Find the measure of the question marked arc (view photo )
The arc angle indicated with ? is derived as 230° using the angle between intersecting tangents.
What is an angle between intersecting tangentsThe angle between two tangent lines which intersect at a point is 180 degrees minus the measure of the arc between the two points of tangency.
angle G = 180° - arc angle HF
arc angle HF = 180° - 50°
arc angle HF = 130°
so the arc angle indicated with ? is;
? = 360° - 130°
? = 230°
Therefore, using the angle between the intersecting tangents, the arc angle indicated with ? is 230°.
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Write a function to model the volume of a rectangular prism if the length is 26cm and the sum of the width and height is 32cm. what is the maximum possible volume of the prism?
To model the volume of a rectangular prism with length 26cm and width w and height h such that the sum of the width and height is 32cm, we can use the following function:
V(w, h) = 26wh
subject to the constraint:
w + h = 32
We can solve for one of the variables in the constraint equation and substitute it into the volume equation, giving us:
w + h = 32 => h = 32 - w
V(w) = 26w(32 - w) = 832w - 26w^2
To find the maximum possible volume, we can take the derivative of this function with respect to w and set it equal to zero
dv/dw= 832 - 52w = 0
Solving for w, we get:
w = 16
Substituting this value back into the constraint equation, we get:
h = 32 - w = 16
Therefore, the maximum possible volume of the prism is:
V(16, 16) = 26(16)(16) = 6656 cubic cm
So the function to model the volume of the rectangular prism is V(w) = 832w - 26w^2, and the maximum possible volume is 6656 cubic cm when the width and height are both 16cm.
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Line x is parallel to line y. Line z intersect lines x and y. Determine whether each statement is Always True.
Line x is perpendicular to line y. Line z crosses lines x and y. Only statements 3 and 4 are true.
∠6 = ∠8 is not true because they both lie on the same plane and makes an angle of 180° and can never be true. ∠6 = ∠1 is also not true because ∠1 is clearly obtuse angle and ∠6 is clearly acute angle so they cannot be equal. Hence, statement a and b are false.
∠7 = ∠3 is always true because they are corresponding angles and corresponding angles are always equal. m∠2 + m∠4 = 180° is also true because they lie on same plane and have common vertex and hence, they are supplementary angles and make a sum of 180°. Hence, statement 3 and 4 is always true.
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A national math competition advances to the second round only the top 5% of all participants based on scores from a first round exam. Their scores are normally distributed with a mean of 76. 2 and a standard deviation of 17. 1. What score, to the nearest whole number, would be necessary to make it to the second round? To start, determine the z-value that corresponds to the top 5%
To make it to the second round, a participant needs to score approximately 92 (nearest whole number).
To determine the z-value that corresponds to the top 5%, we use the standard normal distribution table. Since we want to find the top 5%, we subtract 5% from 100%, which gives us 95%. The area under the standard normal distribution curve for z-values corresponding to 95% is 1.645 (from the table).
We can use the formula z = (x - μ) / σ to find the score (x) that corresponds to a z-value of 1.645. Plugging in the given values, we get:
1.645 = (x - 76.2) / 17.1
Solving for x, we get x ≈ 91.8. Since we need the score to the nearest whole number, we round up to 92. Therefore, a participant needs to score approximately 92 to make it to the second round.
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Answer the following questions for the function f(x) = x Sqrt (x^2 + 4) defined on the interval - 5 ≤ x ≤ 5. f(x) is concave down on the interval x = to x =
f(x) is concave up on the interval x = to x = The inflection point for this function is at x = The minimum for this function occurs at x = The maximum for this function occurs at x =
The function f(x) = x Sqrt (x^2 + 4) is concave down on the entire interval. The inflection point is at x is equal to 0. There is no minimum or maximum in the interval.
To find the concavity, we need to find the second derivative
f(x) = x√(x^2+4)
f'(x) = √(x^2+4) + x^2/√(x^2+4)
f''(x) = -4x^3/(x^2+4)^(3/2)
The second derivative is negative for all x, which means the function is concave down on the entire interval.
To find the inflection point, we need to solve
f''(x) = 0
-4x^3/(x^2+4)^(3/2) = 0
This is true only when x = 0. Therefore, the inflection point is at x = 0.
To find the minimum and maximum, we need to find the critical points. The critical points are found by setting the first derivative equal to zero
f'(x) = √(x^2+4) + x^2/√(x^2+4) = 0
Multiplying both sides by √(x^2+4), we get
x^2+4 + x^2 = 0
2x^2 = -4
x^2 = -2
This equation has no real solutions, which means there are no critical points in the interval. Therefore, there is no minimum or maximum in the interval.
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Out of 300 people sampled, 33 received flu vaccinations this year. Based on this, construct a 95% confidence interval for the true population proportion of people who received flu vaccinations this year. Give your answers as decimals, to three places < p <
A 95% confidence interval for the true population proportion of people who received flu vaccinations this year is 0.067 < p < 0.133.
To construct a 95% confidence interval for the true population proportion of people who received flu vaccinations, we can use the formula:
CI = p ± z√((p(1-p))/n)
where:
CI is the confidence interval
p is the sample proportion (33/300 = 0.11)
z is the z-score associated with a 95% confidence level, which is approximately 1.96
n is the sample size (300)
Substituting the values, we get:
CI = 0.11 ± 1.96√((0.11(1-0.11))/300)
CI = 0.11 ± 0.043
CI = (0.067, 0.133)
Therefore, the 95% confidence interval for the true population proportion of people who received flu vaccinations is 0.067 < p < 0.133.
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If P (A) = 0. 5, P(B) = 0. 1, and A and B are mutually exclusive, find P(A or B).
If P (A) = 0. 5, P(B) = 0. 1, and A and B are mutually exclusive, then P(A or B) is 0.6 or 60%.
To find the probability of A or B occurring, we use the formula P(A or B) = P(A) + P(B) - P(A and B). However, since A and B are mutually exclusive events, they cannot occur together. This means that the probability of A and B occurring together is zero. Therefore, we can simplify the formula to P(A or B) = P(A) + P(B).
Using the given values, we have P(A) = 0.5 and P(B) = 0.1. Plugging these values into the formula, we get:
P(A or B) = 0.5 + 0.1
P(A or B) = 0.6
Therefore, the probability of A or B occurring is 0.6 or 60%. This means that there is a 60% chance of either A or B happening, but not both at the same time since they are mutually exclusive.
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Please help
jkl will be rotated 90° clockwise around the origin to form j'k'l
what are the coordinates of point j' after the rotation?
The coordinates of point j' after the rotation are (-y, x), where x and y are the original coordinates of point j
The coordinates of point j' after rotating point jkl 90° clockwise around the origin can be found by applying the following transformation:
j' = (cos(90°) * x - sin(90°) * y, sin(90°) * x + cos(90°) * y)
Since we are rotating 90° clockwise around the origin, we have cos(90°) = 0 and sin(90°) = 1, so the transformation simplifies to:
j' = (0 * x - 1 * y, 1 * x + 0 * y)
j' = (-y, x)
Therefore, the coordinates of point j' after the rotation are (-y, x), where x and y are the original coordinates of point j.
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HELP!!!
Find the Area of a Rectengle with the base of 3x+1 in and a height of 2x-3 in.
A.5x^2-2 in^2
B.6x^2+7x-2 in^2
C.10x-4 in
D.6x^2-7x-3 in^2
The area of a rectangle with base of 3x+1 in and a height of 2x-3 in is given as follows:
D. A = 6x² - 7x - 3 in².
How to obtain the area of a rectangle?The area of a rectangle of length l and width w is given by the multiplication of dimensions, as follows:
A = lw.
The dimensions for this problem are given as follows:
w = 3x + 1.l = 2x - 3.Hence the expression for the area of the rectangle is given as follows:
A = (3x + 1)(2x - 3)
A = 6x² - 9x + 2x - 3
A = 6x² - 7x - 3 in².
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4) Your phone needs to be charged every other day and your tablet needs to be charged every third day. If you charge both today, how many days will it be until you need to charge both on the same day?
Answer:
If they were both charged today, it would be on the sixth day that they were actually charging at the same time
The difference between the record high and low temperatures in Chicago, Illinois is 109°F. The record low temperature was -5°F. Use an equation to find the record high temperature.
The value of the record high temperature is 114°F
Using an equation to find the record high temperature.From the question, we have the following parameters that can be used in our computation:
The difference between the record high and low temperatures in Chicago, Illinois is 109°F. The record low temperature was -5°F.This means that
High - Low = 109
Substitute the known values in the above equation, so, we have the following representation
High - 5 = 109
Add 5 to both sides
High = 114
Hence, the record high temperature is 114°F
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Mollie drew mol and ted drew ted. they measured a few parts of their triangles
and found that ml = td, ol = ed, and l = d. what postulate can mollie and ted
use to justify why their triangles must be congruenta
Mollie and Ted can use the Side-Side-Side (SSS) postulate to justify why their triangles must be congruent.
According to the given information, the two triangles share three corresponding sides of equal length: ML = TD, OL = ED, and L = D.
The SSS postulate states that if three corresponding sides of two triangles are congruent, then the triangles are congruent. Therefore, because Mollie's triangle and Ted's triangle share three corresponding sides of equal length, they are congruent by the SSS postulate.
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-8
Find the distance, d, of AB.
A = (-7, -7) B = (-3,-1)
-6 -4
A
-2
B -2
-4
-6
-8
d = √x2-x1² + y2 - Y₁|²
d = [?]
Round to the nearest tenth.
Distance
Step-by-step explanation:
Using the distance formula:
d = √[(x2 - x1)² + (y2 - y1)²]
where A = (x1, y1) and B = (x2, y2), we can find the distance between points A and B as follows:
d = √[(-3 - (-7))² + (-1 - (-7))²]
d = √[4² + 6²]
d = √52
d ≈ 7.2
Therefore, the distance between points A and B is approximately 7.2 units, rounded to the nearest tenth.
Evaluate the definite integral
∫ (t^5 - 2t^2)/t^4 dt
To evaluate the definite integral of the given function, ∫ (t^5 - 2t^2)/t^4 dt, follow these steps:
1. Simplify the integrand: Divide each term by t^4.
(t^5/t^4) - (2t^2/t^4) = t - 2t^(-2)
2. Integrate each term with respect to t.
∫(t dt) - ∫(2t^(-2) dt) = (1/2)t^2 + 2∫(t^(-2) dt)
3. Apply the power rule to the remaining integral.
(1/2)t^2 + 2(∫t^(-2+1) dt) = (1/2)t^2 + 2(∫t^(-1) dt)
4. Integrate t^(-1) with respect to t.
(1/2)t^2 + 2(ln|t|)
Now, since we need to evaluate the definite integral, we should have the limits of integration. Let's assume the limits of integration are a and b. Then, apply the Fundamental Theorem of Calculus:
[(1/2)b^2 + 2(ln|b|)] - [(1/2)a^2 + 2(ln|a|)]
This expression gives the value of the definite integral for the given function within the limits a and b.
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A consumer group is investigating two brands of popcorn, R and S. The population proportion of kernels that will pop for Brand R is 0. 90. The population proportion of kernels that will pop for Brand S is 0. 85. Two independent random samples were taken from the population. The following table shows the sample statistics. Number of Kernels in Samples Proportion from Sample that Popped Brand R 100 0. 92 Brand S 200 0. 89 The consumer group claims that for all samples of size 100 kernels from Brand R and 200 kernels from Brand S, the mean of all possible differences in sample proportions (Brand R minus Brand S) is 0. 3. Is the consumer group’s claim correct? Yes. The mean is 0. 92−0. 89=0. 3. Yes. The mean is 0. 92 minus 0. 89 equals 0. 3. A No. The mean is 0. 92+0. 892=0. 905. No. The mean is the fraction 0. 92 plus 0. 89 over 2 equals 0. 905. B No. The mean is 0. 92−0. 892=0. 15. No. The mean is the fraction 0. 92 minus 0. 89 over 2 equals 0. 15. C No. The mean is 0. 90+0. 852=0. 875. No. The mean is the fraction 0. 90 plus 0. 85 over 2 equals 0. 875. D No. The mean is 0. 90−0. 85=0. 5
The consumer group's claim that the mean of all possible differences in sample proportions (Brand R minus Brand S) is 0.3 is correct.
This can be calculated by subtracting the sample proportion of Brand S from the sample proportion of Brand R, resulting in a difference of 0.03 or 3%. This matches the consumer group's claim that the mean of all possible differences in sample proportions is 0.3. It is important to note that this result only applies to the specific samples taken and cannot be generalized to all samples from these brands.
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Use the summation formulas to rewrite the expression without the summation notation.
∑ 8i+7/n^2
The expression without the summation notation for ∑ 8i+7/n²2 using the summation formulas is (4n + 3)/2n.
To rewrite the expression without the summation notation, we need to use the summation formulas. We can start by expanding the given summation:
∑ 8i+7/n²2 = 8(1)/n²2 + 8(2)/n²2 + 8(3)/n²2 + ... + 8(n)/n²2 + 7/n²2
Next, we can simplify each term by factoring out 8/n²2:
= (8/n²2)(1 + 2 + 3 + ... + n) + 7/n²2
Using the formula for the sum of the first n positive integers, we have:
= (8/n²2)(n(n+1)/2) + 7/n²2
= (4n² + 4n)/2n² + 7/n²2
= (4n + 3)/2n
Therefore, the expression is (4n + 3)/2n.
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There are four spaces each so you can put either parentasis or brakets
In the given function the domain is [-1, ∞]
Range is [-3, ∞]
The interval when function is positive [0, ∞]
The domain of a function is the set of values that we are allowed to plug into our function.
This set is the x values in a function such as f(x).
The range of a function is the set of values that the function assumes
In the given function the domain is [-1, ∞]
Range is [-3, ∞]
The interval when function is positive [0, ∞]
The interval when function is negative [-∞, -1]
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Find the moment of inertia about the y-axis of the
first-quadrant area bounded by the curve y=9−x^2
and the coordinate axes find ly (answer as a fraction)
To find the moment of inertia about the y-axis of the first-quadrant area bounded by the curve y=9−x^2 and the coordinate axes, we can use the formula:
I = ∫y² dA
where I is the moment of inertia, y is the distance from the y-axis to the infinitesimal element of area dA, and the integral is taken over the first-quadrant area.
To set up the integral, we need to express y in terms of x for the curve y=9−x². Solving for y, we get:
y = 9 - x²
The area element dA is given by:
dA = y dx
Substituting y in terms of x, we get:
dA = (9 - x²) dx
Now we can express the moment of inertia as an integral:
I = ∫y² dA
= ∫(9 - x²)² dx (limits of integration: x = 0 to x = 3)
To evaluate the integral, we can expand the integrand using the binomial theorem:
I = ∫(81 - 36x² + x⁴) dx
= 81x - 12x³ + (1/5)x⁵ (limits of integration: x = 0 to x = 3)
Finally, we can substitute the limits of integration and simplify:
I = (81(3) - 12(3)³ + (1/5)(3)⁵) - 0
= 243 - 108 + 27
= 162
Therefore, the moment of inertia about the y-axis is 162 units^4.
To find the moment of inertia (Iy) about the y-axis for the first-quadrant area bounded by the curve y = 9 - x^2 and the coordinate axes, we need to integrate the expression for the moment of inertia using the limits of the region.
The curve intersects the x-axis when y = 0, so:
0 = 9 - x²
x² = 9
x = ±3
Since we're in the first quadrant, we're interested in x = 3.
The moment of inertia about the y-axis is given by the expression Iy = ∫x²dA, where dA is the area element. In this case, we'll use a vertical strip with thickness dx and height y = 9 - x². Therefore, dA = y dx.
Now, let's integrate Iy:
Iy = ∫x²(9 - x²) dx from 0 to 3
To solve this integral, you may need to use polynomial expansion and integration techniques:
Iy = ∫(9x² - x⁴) dx from 0 to 3
Iy = [3x³/3 - x⁵/5] from 0 to 3
Iy = (3(3)³/3 - (3)⁵/5) - (0)
Iy = (81 - 243/5)
Iy = (405 - 243)/5
Iy = 162/5
So the moment of inertia about the y-axis for the first-quadrant area bounded by the curve y = 9 - x^2 and the coordinate axes is Iy = 162/5.
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