The rectangle with the maximum area that can be inscribed within the parabola y=7-x^2 has dimensions length 2√(7/3) and height 14/3.
Explanation:The problem is asking to find the dimensions of a rectangle inscribed within a parabola with maximum area. This is an example of an optimization problem in calculus. Here, the parabola given by y=7-x^2 suggests that the rectangle's corners on the x-axis lie at (-x, 0) and (x, 0). The corners on the parabola would be located at (-x, 7-x^2) and (x, 7-x^2). The dimensions of the rectangle would be 2x for the length (base) and 7-x^2 for the height.
For the maximum area, we need to take the derivative of the area function A = length*width = (2x)*(7-x^2) and set it equal to zero, thus defining the critical points.
The derivative of A is given by A' = 14 - 6x^2. Setting A' equal to zero gives: x = √(14/6) which simplifies to x = √(7/3).
Finally, as we are looking for the dimensions of the rectangle, the length (base) is 2*x = 2√(7/3) and the height is given by y = 7 - x^2 = 7 - (7/3) = 14/3.
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which equation is equivalent to
2x + 4a = 10
A. x= 5/4a
B. x= 5 + 2a
C. x= 10 - 4a
D. x=5 - 2a
An equation is equivalent to 2x + 4a = 10 will be D. x=5-2a.
What are equivalent expressions?Those expressions who might look different but their simplified forms are same expressions are called equivalent expressions.
To derive equivalent expressions of some expression, we can either make it look more complex or simple. Usually, we simplify it.
In all answers we are solving for x. That means x needs to be by itself.
2x + 4a = 10
Subtract 4a;
2x = 10 - 4a
Divide by 2;
x = 5 - 2a
D. x=5-2a
Therefore, an equation is equivalent to 2x + 4a = 10 will be D. x=5-2a.
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Solve. 3t < –15 (1 point)
t < –5
t > –5
t < –45
t > –45
The solution to the inequality 3t < –15 is t < –5, attained by dividing both sides by 3.
Explanation:The student's question involves solving an inequality, specifically 3t < –15. To solve for t, we need to isolate t on one side of the inequality. This can be done by dividing both sides of the inequality by 3. The solution to the inequality is:
Divide both sides by 3: t < –5Therefore, the correct answer to the inequality 3t < –15 is t < –5. The other options such as t > –5, t < –45, and t > –45 are incorrect.
Which data sets have outliers? Check all that apply.
a. 14, 21, 24, 25, 27, 32, 35
b. 15, 30, 35, 41, 44, 50, 78
c. 16, 32, 38, 39, 41, 42, 58
d. 17, 23, 28, 31, 39, 45, 75
e. 18, 30, 34, 38, 43, 45, 68
Answer:
its C and E
Step-by-step explanation:
Data sets b, c, and d have outliers, identified by calculating the Interquartile Range (IQR) and checking for values more than 1.5 times the IQR above the third quartile or below the first quartile. The approach to outliers depends on context, and alternative methods such as the standard deviation may yield different results.
Explanation:To determine which data sets have outliers, we can use an appropriate numerical test involving the Interquartile Range (IQR) to identify outliers. The IQR is the range between the first quartile (Q1) and the third quartile (Q3) of the data set, and an outlier is typically defined as a value that is more than 1.5 times the IQR above the third quartile or below the first quartile.
Let's calculate the IQR for each data set:
After calculating the IQR for each set and checking for outlier definition, we find:
When a data value is identified as an outlier, it’s important to consider the context before deciding what action to take. In some cases, it may be appropriate to investigate further to understand why this value is so different from the others. In other cases, the outlier may be removed or handled statistically if it unduly influences results.
Depending on the data's distribution, using a criterion such as any data values farther than two standard deviations from the mean might yield different results for identifying outliers. This approach is most appropriate for data that is mound-shaped and symmetric.
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Esmeralda likes to listen to music while she works out. She had a playlist in her MP3 player that lasted 40 mintues but she accidentally deleted 25 percent of the music. Does Esmeralda have enough music on her playlist for a 30 mintues workout? Explain your answer.
Esmeralda deleted 10 minutes of music from her 40-minute playlist. With the remaining 30 minutes of music, she has enough for a 30-minute workout.
Explanation:To determine if Esmeralda has enough music on her playlist for a 30 minute workout, we need to calculate how much music she has remaining after deleting 25%.
Let's start by finding 25% of the original music duration. To do this, we multiply the original duration (40 minutes) by 0.25:
25% of 40 minutes = 40 minutes * 0.25 = 10 minutes
Therefore, Esmeralda deleted 10 minutes of music.
To find out how much music she has left, we subtract the deleted music from the original duration:
Remaining music = Original duration - Deleted music = 40 minutes - 10 minutes = 30 minutes
Since the remaining music duration is exactly 30 minutes, Esmeralda has enough music on her playlist to last for a 30-minute workout.
Inez has 699 pennies and 198 nickels estimate how many more pennies than nickels she has.
what is point O on MN called if MO=ON
What is half of 9 1/2 inches?
As per linear equation, the half of 9(1/2) inches is 4(3/4) inches or 4.75 inches.
What is a linear equation?A linear equation is an equation that has one or multiple variables with the highest power of the variable is 1.
The given measurement is 9(1/2) inches.
Therefore, the half of the given measurement is
= [9(1/2) ÷ 2] inches
= [19/2 ÷ 2] inches
= 19/4 inches
= 4(3/4) inches
= 4.75 inches
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a country's population in 1992 was 72 million in 1998 it was 76 million estimate the population in 2012
1. The figure attached is a regular octagon with radii and an apothem drawn. What is the measure of angle 1?
A. 22.5 degrees
B. 45 degrees
C. 60 degrees
D. 67.5 degrees
lottie needs a driver. Drive A is offering his services for an initial $200 in addition to $80 per hour. Driver B is offering this service for an intal $230 and an addition of $70 per hour When will the two drivers charge the same amount
Driver A and Driver B will charge the same amount after 3 hours of service.
Explanation:The subject of this question is a problem involving linear equations, a topic in mathematics. Specifically, it pertains to the intersection point of two linear functions, representing the costs charged by the two drivers over time. Let's denote the driving time as 't' (in hours) and the total cost as 'C' to solve it.
Driver A's cost function could be formulated as C = 200 + 80t, while Driver B's as C = 230 + 70t.
To find out when both drivers will charge the same amount, set these two equations equal to each other:
200 + 80t = 230 + 70t
This simplifies to 10t = 30 after subtracting 200 from both sides and 70t from both sides. Finally, dividing by 10 gives us t = 3.
Therefore, both drivers would charge the same amount after 3 hours of service.
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What are the terms in the expression 7x+4y+17x+4y+1 ?
Evaluate the function as indicated determine its domain and range.
2x+1 , X<0
FX=
2x+2 ,X>or equal to 0
a. F(-1) b.f(0) c.F(2) d.F(t^2+1) ...?
Which statement can be used to prove that a given parallelogram is a rectangle?
A.The opposite sides of the parallelogram are congruent.
B.The opposite angles of the parallelogram are congruent.
C.The diagonals of the parallelogram bisect the angles.
D.he diagonals of the parallelogram are congruent.
A cab charges $1.45 for the flat fee and $0.55 for each mile. Write and solve an inequality to determine how many miles Ariel can travel if she has $35 to spend. please help!
BC ___ AC
Choose the relationship symbol that makes the statement true.
Answer:
BC≅AC
Step-by-step explanation:
From the given figure, triangle ABC is such that ∠CAB=∠CBA=70°, therefore the sides of the triangle that are AC and BC will be congruent to each other as if two angles of the same triangle are equal then the sides opposite to those angles are also equal to each other, therefore
BC is congruent to AC that is BC≅AC.
on a certain clock the minute hand is 4 in long and the hour hand is 3 in long. How fast is the distance between the tips of the hands changing at 4 PM? ...?
The angle between the minute and hour hands of a clock at 4 PM decreases at a speed of 330 degrees per hour. However, the rate at which the distance between the tips of the hands changes depends on the specific time and must be calculated using trigonometric differentiation.
Explanation:To solve this problem, we first need to understand the rate at which an hour hand and a minute hand of a clock cover the distance. The hour hand takes 12 hours to complete a 360 degree rotation, i.e., it moves at a rate of 30 degrees per hour. The minute hand covers a 360 degree rotation in 60 minutes (or 1 hour), i.e., it moves at a rate of 360 degrees per hour.
At 4 PM, the hour hand will be at 120 degrees (4 hours * 30 degrees/hour) and the minute hand will be at the top of the clock, or at 0 degrees. Therefore, the minute and hour hands are 120 degrees apart. However, this angle decreases over time since the minute hand moves faster than the hour hand. To determine how fast this angle is decreasing, we calculate the relative angular velocity of the minute hand with respect to the hour hand, which is the difference of their velocities, 360 degrees - 30 degrees = 330 degrees per hour. This is the speed at which the angle between the two hands is changing.
Trigonometry is used to calculate the distance between the tips of the hands, however, the variation of this distance with respect to time is complex and depends on the specific time. So at any given time, the rate of change in the distance between the tips of the hands should be calculated using principles of trigonometric differentiation.
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The rate at which the distance between the tips of the hands of a clock is changing at 4 PM can be calculated using the rates at which the minute and hour hands are moving. At 4 PM, the rate is approximately -30 degrees per hour. Using the formula for the distance between two points on a circle, the rate of change of the distance can be calculated to be approximately 8*sqrt(3) inches per hour.
Explanation:To find the rate at which the distance between the tips of the hands is changing, we need to consider the speeds at which each hand is moving. The minute hand moves at a constant speed of 1 revolution per hour, or 360 degrees per hour. The hour hand moves at a speed of 1/12 revolution per hour, or 30 degrees per hour.
At 4 PM, the minute hand is pointing at the 12 and the hour hand is pointing at the 4. The angle between the minute hand and the hour hand is 120 degrees. To find the rate at which this angle is changing, we take the derivative of the angle with respect to time, which gives us -30 degrees per hour.
Since the lengths of the hands are given in inches, the rate at which the distance between the tips of the hands is changing will be in inches per hour. Using the formula for the distance between two points on a circle, which is given by 2r*sin(theta/2), where r is the radius of the circle (in this case, the length of the minute hand) and theta is the angle between the two points (in this case, the angle between the minute hand and the hour hand), we can calculate the rate of change of the distance. Plugging in the values, we have 2*4*sin(120/2) = 2*4*sin(60) = 2*4*sqrt(3)/2 = 8*sqrt(3) inches per hour.
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Pleasee help x.x 1. Solve the system by elimination-
-2x+2y+3z=0
-2x-y+z=-3
2x+3y+3z=5
2.Solve using substitution-
x-y-z=-8
-4x+4y+5z=7
2x+2z=4
Thanks for any help :o
A horizontal pull A pulls two wagons over a horizontal frictionless floor, the first wagon is 500N, the second is 2000 N. The tension in the light horizontal rope connecting the wagons is.
the options are:
a) equal to A, by Newton’s third law.
b) equal to 2000 N.
c) greater than A.
d) less than A.
Scientists are studying the temperature on a distant planet. Let
y
represent the temperature (in degrees Celsius). Let
x
represent the height above the surface (in kilometers). Suppose that
x
and
y
are related by the equation
y=41-3x
.
Answer the questions below.
Note that a change can be an increase or a decrease.
For an increase, use a positive number. For a decrease, use a negative number.
what is the temperature on the surface of the planet?
what is the temperature change for each kilometer we go up from the surface?
Answer:
The temperature on the surface of the planet is 41 °C.The tempearture decreases 3 °C when the height increases 1 km.Step-by-step explanation:
The given equation is
[tex]y=41-3x[/tex]
Where [tex]x[/tex] is the height above the surface and [tex]y[/tex] is the temperature in degrees Celsius.
Now, the temperature on the surface can be find when the height above the surface is zero, that is [tex]x=0[/tex]. So, we replace this value and solve for [tex]y[/tex]
[tex]y=41-3x=41-3(0)\\y=41[/tex]
Therefore, the temperature on the surface of the planet is 41 °C.
The second question is about the constant rate of change of the given relation, which is also the slope of the linear expression. That constant rate of change is always the coefficient of variable [tex]x[/tex]. Therefore, the ratio of change is [tex]r=-3[/tex]. Which means the tempearture decreases 3 °C when the height increases 1 km.
The decreasing behaviour is deduct from the ratio of change, when it's negative, that means the function decreases, like in this case.
find the limit as x approaches 0 of x/sin3x
8 is 15% of what number?
15% of 18.95 is what number?
what percent of 64 is 326?
10% of what number is 40?
Electronic baseball games manufactured by Tempco Electronics are shipped in lots of 36. Before shipping, a quality-control inspector randomly selects a sample of 8 from each lot for testing. If the sample contains any defective games, the entire lot is rejected. What is the probability that a lot containing exactly 2 defective games will still be shipped? (Round your answer to three decimal places.)
The probability that a lot containing exactly two defective electronic baseball games manufactured by Tempco will still be shipped is approximately 0.821 or 82.1%. This problem is solved using the concept of hypergeometric distribution in probability theory.
Explanation:This problem is a case of hypergeometric distribution in probability theory. In the given situation, the total number of games in the lot is 36, of which 34 are okay and two are defective. The inspector selects eight games out of these 36. We must calculate the probability that neither of the two defective games is in the selected 8.
This can be calculated as follows: The probability of selecting a good game first is 34/36. After we choose a good one, we now have 33 good games left in a set of 35 games, so the probability of choosing another good one is 33/35. We continue this way until we have selected all eight games. Thus, the probability that all eight games selected will be good, and therefore the lot will be shipped, is (34/36) * (33/35) * (32/34) * (31/33) * (30/32) * (29/31) * (28/30) * (27/29) = 0.821. So, the probability that a lot containing exactly two defective electronic baseball games manufactured by Tempco will still be shipped is approximately 0.821, or 82.1%, when expressed as a percentage.
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During a 52 week period, a company paid overtime wages for 19 weeks and hired temp help for 11 weeks. During 5 weeks, the company paid overtime AND hired temp help. If an auditor randomly examined the payroll for only one week, whats the probability that the payroll for that week contained overtime wages or temp help wages? ...?
The probability that the payroll for one week contained overtime wages or temporary help is [tex]\boxed{\frac{10}{13}}[/tex].
Further explanation:
It is given that during a [tex]52[/tex] week period, a company paid overtime wages for [tex]19[/tex] weeks and hired temporary help for [tex]11[/tex] weeks.
During [tex]5[/tex] weeks, the company paid overtime and hired temporary help.
Calculation:
A company paid overtime wages for [tex]19[/tex] weeks and [tex]5[/tex] weeks during the [tex]52[/tex] week period.
Therefore, the probability of overtime wages [tex]P(O)[/tex] is calculated as follows:
[tex]\begin{aligned}P(O)&=\dfrac{19}{52}+\dfrac{5}{52}\\&=\dfrac{19+5}{52}\\&=\dfrac{24}{52}\end{aligned}[/tex]
The probability of overtime wages is [tex]\frac{24}{52}[/tex].
The same company hired temporary help for [tex]11[/tex] weeks and [tex]5[/tex] weeks during the [tex]52[/tex] week period.
Therefore, the probability of temporary help [tex]P(H)[/tex] is calculated as follows:
[tex]\begin{aligned}P(H)&=\dfrac{11}{52}+\dfrac{5}{52}\\&=\dfrac{11+5}{52}\\&=\dfrac{16}{52}\end{aligned}[/tex]
The probability of temporary help is [tex]\frac{16}{52}[/tex].
Now, the combined probability [tex]P[/tex] that the payroll for one week contained overtime wages or temporary help is calculated by adding the probabilities of overtime wages and temporary help as follows:
[tex]\begin{aligned}P&=\dfrac{24}{52}+\dfrac{16}{52}\\&=\dfrac{24+16}{52}\\&=\dfrac{40}{52}\\&=\dfrac{10}{13}\end{aligned}[/tex]
The probability that the payroll for one week contained overtime wages or temporary help is [tex]\boxed{\dfrac{10}{13}}[/tex].
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Answer details:
Grade: Middle school
Subject: Mathematics
Chapter: Probability
Keywords: Probability, events, outcome, sample, experiment, equally likely, random, complementary events, joint probability, independent events, mutually exclusive events, conditional probability.
Find parametric equations for the tangent line at the point (cos(3pi/6),sin(3pi/6), (3pi/6))
on the curve x=cost, y= sint, z=t
x(t)=?
y(t)=?
z(t)=?
Malia works at a movie theater and is paid hourly. One month, she earned $388.80. The total amount she earns can be found using the equation y = 8.1x, where x is the number of hours she works and y is the total amount she earns in dollars. Bill works at a golf course and is also paid hourly. The table shows the total amount he is paid for different numbers of hours worked. The number of hours:18, 28, 35. Total number earned:144.90, 225.40, 281.75. How do the units compare?
Answer:
Malia's unit rate is greater than Bill's by $0.05.
Step-by-step explanation:
Consider the provided information.
The total amount she earns can be found using the equation y = 8.1x, where x is the number of hours she works and y is the total amount she earns in dollars.
The unit rate, when written as a fraction, has a denominator equal to one.
The unit price of Malia is $8.1
For Bill works:
Number of hours Total amount earned
18 $144.90
28 $225.40
35 $281.75
Unit price would be:
[tex]\dfrac{144.90}{18}=\dfrac{y}{x}\\\\8.05=\dfrac{y}{x}\\\\y=8.05x[/tex]
Hence, unit price of Bill is $8.05
Malia's unit rate is 8.1 whereas Bill's unit rate is 8.05,
Thus, Malia's unit rate is greater than Bill's by 8.10-8.05=0.05.
The perimeter of a rectangle is 96 ft. The ratio of its length to its width is 5 : 3. What are the dimensions of the rectangle?
write a polynomial function with rational coefficients so that p(x)=0 has the given roots: -5 and 3i
The polynomial with rational coefficients and roots -5 and 3i will be [tex]p(x)=x^3+5x^2+9x+45=0[/tex].
Given information:Polynomial p(x)=0 has the roots: -5 and 3i
It is required to write a polynomial with rational coefficient and the given roots.
The polynomial has a complex root 0+3i. So, there will be one more root of the polynomial which is 0-3i.
Now, the roots of the polynomial are
[tex]\alpha =-5\\\beta=0+3i\\\gamma=0-3i[/tex]
So, the required polynomial can be written as,
[tex](x-\alpha)(x-\beta)(x-\gamma)=(x-(-5))(x-(0+3i))(x-(0-3i))\\=(x+5)(x-3i)(x+3i)\\=(x+5)(x^2-(3i)^2)\\=(x+5)(x^2+9)\\=x(x^2+9)+5(x^2+9)\\=x^3+5x^2+9x+45[/tex]
Therefore, the polynomial with rational coefficients and roots -5 and 3i will be [tex]p(x)=x^3+5x^2+9x+45=0[/tex].
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Marcus states that angle ORP and angle LRP are a linear pair. Which best describes his statement?
He is correct. The angles share a common vertex so they are a linear pair.
He is correct. The angles share a common ray so they are a linear pair.
He is incorrect. Angle ORP does not have a linear pair shown on the diagram.
He is incorrect. Ray RO and ray RL are not opposite rays.
Answer: D is the answer
example of why division is not associative.
Division is not associative, as if you look at one of the operands to a nest of divisions, the result will vary either in proportion to it or inversely to it depending on whether it's to the right of an odd or even number of divisions. And the basic associative law changes the number of operations the rightmost operand is to the right of by 1.
if AB is congruent to ED and BC is congruent to DC, how is AC congruent to EC