Step-by-step explanation:
Circumference of a circle = πD, where D is the diameter.
Diameter of circle 1 = 1.125
Circumference of circle 1 = π x 1.125
Diameter of circle 1 = 2.25
Circumference of circle 1 = π x 2.25
[tex]\texttt{Ratio of circumferences = }\frac{\pi \times 1.125}{\pi \times 2.25}\\\\\texttt{Ratio of circumferences = }\frac{1}{2}[/tex]
Circumference of circle 1 : Circumference of circle 2 = 1 : 2
Answer:
1:1
Step-by-step explanation:
A one year membership to metro gym costs $460. There is a fee of $40 when you join, and the rest is paid monthly. Write an equation to represent the situation that can help members find how much they pay per month
Answer: The equation that would help members to find how much they would pay per month is
40 + 12x = 460
Step-by-step explanation:
Let x represent the amount of money that members would pay per month.
Let y represent the number of months for which a member uses the gym.
There is a fee of $40 when you join, and the rest is paid monthly. This means that the cost of using the gym for x months would be
40 + xy
A one year membership to metro gym costs $460. Therefore are 12 months in a year. Therefore,
40 + 12x = 460
12x = 460 - 40 = 420
x = 420/12 = 35
The equation that would help members to find how much they would pay per month is
A prison administration wants to know whether the prisoners think the guards treat them fairly. Explain how each of the following components could be used to produce biased results, versus how each could be used to produce unbiased results.A) Component 2: The researcher who had contact with the parcipants. B) Component 4: The exact nature of the measurements made or question asked.
Answer:
A prison administration wants to know whether the prisoners think the guards treat them fairly. The explanation, of each component could be used to produce biased and unbiased results, is as follow
A) Component 2:
If the prison guards ask the questions from the prisoners directly by themselves then it is highly like to get the biased results. If the person are hired outside from the staff then it will minimize the probability of biased results rather it will allow to get the best results.B) Component 4:
The question asked from prisoner would be biased if it is like "You don't have any complaints about how you are treated, do you?" The same question can be unbiased if asked in such a way "We are interested in your opinion about your treatment by the guards. Do you think you are fairly or unfairly treated by them?"Bias in research on prisoner-guard dynamics can result from the involvement of researchers and the phrasing of questions. To ensure unbiased results, neutral interactions and carefully worded questions are vital.
Biased Research Components in Prisoner-Guard Dynamics
In research, it is crucial to mitigate factors that can lead to bias. Two such components in the study of prisoner-guard dynamics are researcher involvement and the phrasing of measurements and questions. If the researcher has direct contact with participants, they can unknowingly influence responses based on their body language, tone, or preconceived notions.
To produce unbiased results, researchers should maintain a neutral demeanor and minimize unnecessary interaction. The exact nature of measurements and questions can also introduce bias. Leading or emotional language in survey questions can skew results towards a specific answer. To avoid this, questions must be neutrally worded and designed to reflect the respondent's true perspective without influence.
Certain bunny rabbits reproduce at a continuous rate of 200% a year. Assuming that a warren starts with 10 bunny rabbits calculate the number of rabbits after 3 years (assume none of the bunny rabbits meet with misadventure).A. 6072
B. 4034
C. 32
D. 561
E. 23
F. 123
Answer:
The correct option would be 'B. 4034'.
Step-by-step explanation:
If a population grows continuously
Then the final population is
[tex]P=P_{0}e^{rt}[/tex]
Where,
[tex]P_{0}[/tex] = Initial population
t = Number of period
r = rate of increasing per period
We have
[tex]P_{0}[/tex] = 10
r = 200%= 2 [tex](\because 1\%=\frac{1}{100})[/tex]
t = 3 years
Hence, the population of rabbits, after 3 years is,
[tex]P=10e^{2\times 3}[/tex]
[tex]P=10e^6[/tex]
[tex]P\approx 4034[/tex]
Therefore, option B is correct.
The number of rabbits after 3 years is B. 4034.
Explanation:To calculate the number of rabbits after 3 years, we need to apply the continuous growth formula:
New Value = Initial Value x (1 + Growth Rate)^Time
Given that the initial value is 10 and the growth rate is 200%, we can plug in the values to find
Thus, P = 10e6
= 4034
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Surveyors need to measure the distance across a pond. they created similar triangles in this picture.
What is the distance across the pond?
Answer:
Therefore the Distance across the Pond is
[tex]x=391\ ft[/tex]
Step-by-step explanation:
Given:
Triangle are Similar
AB = 90 ft
AC = 170 ft
BD = 207 ft
DE = x
To Find:
x = ?
Solution:
In Δ ABC and Δ DBE
∠A ≅ ∠D …………..{ measure of each angle is 90° given }
∠ABC ≅ ∠DBE ……….....{Vertical Angle Theorem}
Δ ABC ~ Δ DBE ….{Angle-Angle Similarity test}
If two triangles are similar then their sides are in proportion.
[tex]\dfrac{AB}{DB} =\dfrac{AC}{DE} \textrm{corresponding sides of similar triangles are in proportion}\\[/tex]
Substituting the values we get
[tex]\dfrac{90}{207} =\dfrac{170}{x}\\\\x=23\times 17=391\ ft[/tex]
Therefore the Distance across the Pond is
[tex]x=391\ ft[/tex]
The calculated distance across the pond is 391 feet
How to determine the distance across the pond?
From the question, we have the following parameters that can be used in our computation:
The figure
Also, we have that
The figures are similar triangles
Using the proportional equation of similar triangles, we have
x/207 = 170/90
This gives
x = 207 * 170/90
Evaluate
x = 391
Hence, the distance across the pond is 391 feet
William cycles at a sped of 15 miles per hour. He cycles 12 miles from home to school. If he increases his cycling speed ny 5 miles per hour how much faster will he arrive his school
Answer:he would arrive 0.2 miles faster.
Step-by-step explanation:
Distance = speed × time
Time = distance/speed
William cycles at a sped of 15 miles per hour. He cycles 12 miles from home to school. This means that the time it takes William to get to school from home would be
12/15 = 0.8 hours
If he increases his cycling speed by 5 miles per hour, his new speed becomes
15 + 5 = 20 miles per hour
Therefore, the new time it takes William to get to school from home would be
12/20 = 0.6 hours
The difference in both times is
0.8 - 0.6 = 0.2 hours
Therefore, he would arrive 0.2 miles faster.
Tessa is cooking steak for her family. She wants to be sure the steak has an internal temperature of at least 160 degrees Fahrenheit. She uses a thermometer to measure the internal temperature at two randomly chosen places. The minimum reading in the sample is 165 degrees Fahrenheit. Identify the population, the parameter, the sample, and the statistic______________.
Answer:
Population : all the steaks Tessa can cook
Parameter : minimum internal temperature of 160 degrees Fahrenheit
Sample : two random thermometer readings
Statistic : minimum sample reading of 165 degrees Fahrenheit
Step-by-step explanation:
Let's recall the definitions of these statistical concepts and match it with the information that were provided to us:
Populations can be the complete set of all similar items that exist, in our case, all the steaks that Tessa can cook.Parameter is is a value that describes a characteristic of an entire population, such as the minimum temperature of the steaks Tessa is cooking in Fahrenheit degrees.Sample is a subset of the population, in our case, the two random readings of the thermometer Tessa did.Statistic is a characteristic of a sample, for our problem, it's the minimum reading of 165 degrees Fahrenheit.Sean reads on Monday Tuesday and Wednesday. He reads 3 times as many minutes on Tuesday as he does on Monday. He reads 4 times as many minutes on Wednesday as he does on Monday. Sean reads 45 minutes on Tuesday. How many minutes does Sean read on Wednesday?
Sean reads 60 minutes on wednesday
Solution:
Given that,
Let "x" be the number of minutes read on monday
Let "y" be the number of minutes read on tuesday
Let "z" be the number of minutes read on wednesday
He reads 3 times as many minutes on Tuesday as he does on Monday
Number of minutes read on tuesday = 3 times the number of minutes read on monday
y = 3x --------- eqn 1
He reads 4 times as many minutes on Wednesday as he does on Monday
Number of minutes read on wednesday = 4 times the number of minutes read on monday
z = 4x ------- eqn 2
Sean reads 45 minutes on Tuesday
y = 45
Substitute y = 45 in eqn 1
45 = 3x
x = 15
Substitute x = 15 in eqn 2
z = 4(15)
z = 60
Thus he reads 60 minutes on wednesday
Final answer:
Sean reads for 60 minutes on Wednesday, calculated by knowing he reads four times as many minutes on Wednesday as on Monday, and he reads 45 minutes on Tuesday which is three times his Monday reading time.
Explanation:
Since Sean reads 3 times as many minutes on Tuesday as he does on Monday, and he reads 45 minutes on Tuesday, we can determine that he reads 45 minutes ÷ 3 = 15 minutes on Monday. Sean then reads 4 times as many minutes on Wednesday as he does on Monday. So, he must read 15 minutes × 4 = 60 minutes on Wednesday.
Practice simplifying rational expressions with negative exponents.
Answer:
Part 1:- option first is correct
[tex]-\frac{1}{2}ab^{12}[/tex]
Part 1:- option third is correct
[tex]\frac{w^{10}}{3y^{4}}[/tex]
Step-by-step explanation:
Given:
The given ration expressions are.
1. [tex]\frac{-2a^{2}b^{4}}{4ab^{-8}}[/tex]
2. [tex]\frac{-5w^{4}y^{-2}}{-15w^{-6}y^{2}}[/tex]
We need to simplify the given expressions.
Solution:
Part 1:-
Given expression is
[tex]=\frac{-2a^{2}b^{4}}{4ab^{-8}}[/tex]
Using law of exponents [tex]\frac{x^{m} }{x^{n} } = x^{(m-n)}[/tex]
[tex]=-\frac{a^{2-1}b^{4-(-8)}}{2}[/tex]
[tex]=-\frac{a^{1}b^{4+8}}{2}[/tex]
[tex]=-\frac{ab^{12}}{2}[/tex]
[tex]=-\frac{1}{2}ab^{12}[/tex]
Therefore, option first [tex]-\frac{1}{2}ab^{12}[/tex] is correct.
Part 2:-
Given expression is.
[tex]=\frac{-5w^{4}y^{-2}}{-15w^{-6}y^{2}}[/tex]
Using law of exponents [tex]\frac{x^{m} }{x^{n} } = x^{(m-n)}[/tex]
[tex]=\frac{w^{4-(-6)}y^{-2-2}}{3}[/tex]
[tex]=\frac{w^{4+6}y^{-4}}{3}[/tex]
[tex]=\frac{w^{10}y^{-4}}{3}[/tex]
[tex]=\frac{w^{10}}{3y^{4}}[/tex]
Therefore, third option [tex]\frac{w^{10}}{3y^{4}}[/tex] is correct.
Pam is taking a train from the town of Rome to the town of Florence. Rome is located 40 miles due west of the town of Paris. Florence is 35 miles east, and 55 miles north of Rome. On her trip, how close does Pam get to Paris?
Answer:
Pan is closest to Paris when she gets to Florence when she is 35 miles away
Step-by-step explanation:
With Paris at (0,0), Rome is 40miles west which is on the negative x axis.
Florence is 35 miles closer (east to Paris) so we have -45 + 35 = -10
and 55 mikes north of Rome which is on the positive y-axis. So Florence is at point (-10,55)
The distance between the two points Florence and Paris is √(x2 - x1)^2 + (y2 - y1)^2
x1 = 0, y1 = 0
x2 = -10, y2 = 55
So we have
√(-10-0)^2 + (55-0)^2
= √(-10)^2 + (55)^2
= √ 100 + 3025
= √3125
= 55.9 mikes from Paris
Pam is closest to Paris when she gets to Florence when she is 35 miles away
Pam gets closest to Paris when she is 5 miles away, which occurs when she travels directly east from Rome before heading north to Florence.
Explanation:Pam is taking a train from Rome to Florence, and we need to find out how close she gets to Paris along the way. Rome is 40 miles due west of Paris, and Florence is 35 miles east and 55 miles north of Rome. While traveling from Rome to Florence, Pam would initially get closer to Paris as she travels east, but as she moves north, the closest point to Paris would be directly east from Rome before moving north.
Step-by-Step ExplanationDetermine the initial distance of Rome to Paris, which is 40 miles to the east.Since Florence is 35 miles east of Rome, Pam would at least come within 5 miles of Paris (40 miles - 35 miles).However, when traveling north to reach Florence, Pam's distance to Paris increases. Thus, the closest Pam gets to Paris on her trip is 5 miles, when she is directly east of Rome, before heading north.Sample annual salaries (in thousands of dollars) for public elementary school teachers are listed. Find the sample standard deviation. Round to two decimal places if necessary
23.9
15.3
40.2
30.3
21.0
22.8
ID: ES6L 2.4.1-9+
Answer:
The standard deviation is 9.34
Step-by-step explanation:
First we need to find mean of the sample:
[tex]m=(23.9+15.3+40.2+30.3+12+22.8)/6=24.08[/tex]
Standard deviation will be like below:
[tex]d=\sqrt{(x-m^2)/6}=9.34[/tex]
The endpoints of a diameter of a circle are (3, -2) and (-5, 8). What is the equation of the circle in standard form?
Answer:
(x + 1)² + (y − 3)² = 41
Step-by-step explanation:
The center of the circle is the midpoint of the diameter.
(h, k) = ((x₂ + x₁)/2, (y₂ + y₁)/2)
(h, k) = ((3 + -5)/2, (-2 + 8)/2)
(h, k) = (-1, 3)
The radius is half the length of the diameter.
r = ½ √((x₂ − x₁)² + (y₂ − y₁)²)
r = ½ √((3 − -5)² + (-2 − 8)²)
r = ½ √(8² + 10²)
r = √41
Therefore, the equation of the circle is:
(x − h)² + (y − k)² = r²
(x + 1)² + (y − 3)² = 41
The standard form of equation of the circle is (x + 1) + (y - 3) = (6.40)².
What is distance between two points?The length of the line segment bridging two points on a plane is known as the distance between the points.
The formula to find the distance between the two points is usually given by d=√{(x₂-x₁)² + (y₂-y₁)²}
Given is that the endpoints of a diameter of a circle are (3, -2) and (-5, 8).
The center: (3-5/2, -2+8/2) = (-1, 3)
The diameter d = √{(x₂-x₁)² + (y₂-y₁)²}
d = √{(-5-3)² + (8 + 2)²}
d = √{64 + 100}
d = √164
d = 12.81
The radius r = 6.40
The standard form of equation of the circle is,
(x + 1) + (y - 3) = (6.40)²
Therefore, the standard form of equation of the circle is,
(x + 1) + (y - 3) = (6.40)²
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The driver of a 810.0 kg car decides to double the speed from 23.6 m/s to 47.2 m/s. What effect would this have on the amount of work required to stop the car, that is, on the kinetic energy of the car? KEi= × 105 J KEf= × 105 J times as much work must be done to stop the car.
Answer:
KEi = 2.256×10^5 JKEf = 9.023×10^5 J4 times as much workStep-by-step explanation:
The kinetic energy for a given mass and velocity is ...
KE = (1/2)mv^2 . . . . . m is mass
At its initial speed, the kinetic energy of the car is ...
KEi = (1/2)(810 kg)(23.6 m/s)^2 ≈ 2.256×10^5 J . . . . . m is meters
At its final speed, the kinetic energy of the car is ...
KEf = (1/2)(810 kg)(47.2 m/s)^2 ≈ 9.023×10^5 J
The ratio of final to initial kinetic energy is ...
(9.023×10^5)/(2.256×10^5) = 4
4 times as much work must be done to stop the car.
_____
You know this without computing the kinetic energy. KE is proportional to the square of speed, so when the speed doubles, the KE is multiplied by 2^2 = 4.
Graph g(x)=3x2−12x−3 .
Answer:
See the image.Step-by-step explanation:
The function is given by [tex]g(x) = 3x^{2} - 12x - 3[/tex].
Differentiating the function, we get [tex]\frac{d g(x)}{dx} = 6x - 12[/tex].
Now, at x = 2, 6x - 12 will be 0.
Hence, at x = 2, either the function will have maximum or minimum value.
g(2) = 12 - 24 -3 = -15.
g(1) = 3 -12 -3 = -12.
g(0) = -3.
Hence, the given function passes through (2, -15), (1, -12) and (0, -3).
A sample of 40 employees from the local Honda plant was obtained and the length of time (in months) worked was recorded for each employee. A stemplot of these data follows. In the stemplot, 5|2 represents 52 months.What would be a better way to represent this data set?
A. Display the data in a time plot.
B. Split the stems.
C. Use a pie chart.
D. Use a histogram with class width equal to 10.
Answer:
Option D = Use a histogram with class width equal to 10
Step-by-step explanation:
A histogram will be better to represent the data. A histogram is a graphical representation of a frequency distribution, where you have rectangular bars placed side by side. the vertical axis represent the frequency while the horizontal axis represent the variable being represented which is the length of time worked by the employees.
One of the advantages of the histogram is that it has no gaps between the bars and it is mostly used for grouped data. This explains why the best representation ad the best option is to use the HISTOGRAM.
Final answer:
Option D, a histogram with class width equal to 10, would likely be the best representation for the data set of the length of time worked by employees, as it would facilitate easy viewing of data distribution and patterns.
Explanation:
The question asks for a better way to represent a data set of the length of time worked by employees at a local Honda plant. The provided options include a time plot, split stems, a pie chart, and a histogram with a class width of 10. Looking at the options:
A time plot is more suitable when the data involves changes over time and when you're looking for trends.Splitting the stems in a stemplot can provide a finer breakdown of the data, which might be helpful if the data is quite detailed or clustered. However, it may not necessarily be a better representation.A pie chart is not ideal for this type of numerical, continuous data as pie charts are best for categorical data and showing proportions/parts of a whole.A histogram with class width equal to 10 would be beneficial because it groups the data into intervals, making it easier to see the distribution and identify patterns such as the range of months employees have worked, any concentrations of data points, and outliers.In conclusion, option D, which suggests using a histogram with a class width equal to 10, would likely be the best graphical representation for this data set.
What is the Translation Postulate?
Answer: to suggest or accept that a theory or idea is true as a starting point for reasoning or discussion.
Step-by-step explanation:
Here's an example: Astronomers postulate that the comet will reappear in 4000 years.
Deep Blue, a deep sea fishing company, bought a boat for $250,000. After 9 years, Deep Blue plans to sell it for a scrap value of $95,000. Assume linear depreciation.
Answer:
Therefore, we use the linear depreciation and we get is 17222.22 .
Step-by-step explanation:
From Exercise we have that is boat $250,000.
The straight line depreciation for a boat would be calculated as follows:
Cost boat is $250,000.
For $95,000 Deep Blue plans to sell it after 9 years.
We use the formula and we calculate :
(250000-95000)/9=155000/9=17222.22
Therefore, we use the linear depreciation and we get is 17222.22 .
To calculate the annual depreciation expense for Deep Blue's boat using the straight-line method, subtract the salvage value from the purchase price and divide by the number of years of useful life, resulting in an annual depreciation expense of $17,222.22.
To determine the annual depreciation, subtract the salvage value from the purchase price and divide by the useful life of the asset in years:
Subtract the salvage value from the purchase price: $250,000 - $95,000 = $155,000.
Divide the result by the number of years of useful life: $155,000 / 9 years = $17,222.22.
Therefore, Deep Blue would record an annual depreciation expense of $17,222.22.
A baseball is thrown into the air and follows a parabolic path given by the equation s = -16t 2 + v0t, where s is feet above ground, t is the time in seconds and v0 is the initial velocity. If the ball is thrown with an initial velocity of 64 feet per second, how high will it travel?
16 ft.
64 ft.
128 ft.
Answer: 64 ft
Step-by-step explanation:
We are given the following equation that models the baseball's parabolic path:
[tex]s=16t^{2}+V_{o}t[/tex] (1)
Where:
[tex]s[/tex] is the ball maximum height
[tex]t[/tex] is the time
[tex]V_{o}=64 ft/s[/tex] is the initial height
With this information the equation is rewritten as:
[tex]s=16t^{2}+(64)t[/tex] (2)
Now, we have to find [tex]t[/tex], and this will be posible with the following formula:
[tex]V=V_{o}-gt[/tex] (3)
Where:
[tex]V=0 ft/s[/tex] is the final velocity of the ball at the point where its height is the maximum
[tex]g=32 ft/s^{2}[/tex] is the acceleration due gravity
Isolating [tex]t[/tex]:
[tex]t=\frac{V_{o}}{g}[/tex] (4)
[tex]t=\frac{64 ft/s}{32 ft/s^{2}}[/tex] (5)
[tex]t=2 s[/tex] (6)
Substituting (6) in (2):
[tex]s=16 ft/s^{2}(2 s)^{2}+(64 ft/s)(2 s)[/tex] (7)
Finally:
[tex]s=64 ft[/tex]
The Derby Dragons have a mean height of 72.0 inches and a standard deviation of 1.2. The Aviston Aces have a mean height of 70.8 inches and a standard deviation of 0.7. The Ballwin Bears have a mean height of 73 inches and a standard deviation of 1.0. On average, which team is a taller? Which team has players whose heights are more consistent?
Answer: Ballwin Bears
Step-by-step explanation:
Answer:
The Ballwin Bears are taller on average, and the Aviation Aces have players whose are more consistent.
Step-by-step explanation:
(This is the correct answer on Knewton-alta)
Sporting goods company sells tents in two styles, shown below. The sides and floor of each tent are made of nylon. Which tent requires less nylon to manufacture? Use 3.14 for pi and round to the hundredths place.
Missing portion of the question:
Question has missing the images for both type of tents hence the lengths were also missing which is attached in the picture.
Answer:
Tunnel Tent requires less nylon as its area (214.80ft²) is less than the area of Pup tent (247.40ft²)
Step-by-step explanation:
Pup Tent:
Area for Pup Tent: (Perimeter)(Height) + 2 (Area of front triangular side)
Perimeter=sum of all sides = 8+8+8=24
Height = 8
Area of Triangle = (1/2) length * height
length=8
height can be calculated by Pythagorean Theorem:
hyp²=base²+perp²
height = perp=4√3
So,
Area of Triangle = (1/2) length * height
=27.7
Hence,
Area for Pup Tent: (Perimeter)(Height) + 2 (Area of front triangular side)
=(24)(8)+(2)(27.7)
=247.40m²
Tunnel Tent:
Area of tunnel tent = πrh + πr² + 2rh
π=constant=3.14
r=radius=4
h=height=8
Area of tunnel tent = πrh + πr² + 2rh
=(3.14)(4)(8)+(3.14)(4)²+2(4)(8)
=214.80m²
Please help ASAP!!
Find the variance and standard deviation of the given set of data to the nearest tenth. {530, 150, 320, 500, 200, 690, 770}
A. variance = 48,326.5, standard deviation = 219.8
B. variance = 56,381, standard deviation = 237.4
C. variance = 219.8, standard deviation = 48,326.5
D. variance = 48,326.5, standard deviation = 24,163.3
Answer:
Step-by-step explanation:
variance = 48,326.5, standard deviation = 219.8
Answer: variance = 48,326.5, standard deviation = 219.8
Step-by-step explanation: I completed the quiz, and option A. variance = 48,326.5, standard deviation = 219.8 was the correct answer!
On a canoe trip, Rita paddled upstream (against the current) at an average speed of 2 miles per hour relative to the riverbank. On the return trip downstream (with the current) . Her average speed was 3 miles per hour. Find Rita's paddling speed in still water and the speed of the river's current.
Answer: Rita's paddling speed in still water is 2.5 miles pet hour and the speed of the river's current is 0.5 miles per hour.
Step-by-step explanation:
Let x represent Rita's paddling speed in still water.
Let y represent the speed of the river's current.
On a canoe trip, Rita paddled upstream (against the current) at an average speed of 2 miles per hour relative to the riverbank. This means that
x - y = 2 - - - - - - - - - - - - -1
On the return trip downstream (with the current) . Her average speed was 3 miles per hour. This means that
x + y = 3 - - - - - - - - - - - - -2
Adding equation 1 and equation 2, it becomes
2x = 5
x = 5/2 = 2.5 miles per hour.
Substituting x = 2.5 into equation 1, it becomes
2.5 - y = 2
y = 2.5 - 2 = 0.5 miles per hour
Answer:
per hour relative to the riverbank. On the return trip downstream (with the current) . Her average speed was 3 miles per hour. Find Rita's paddling
Step-by-step explanation:
The expression $x^2 15x 54$ can be written as $(x a)(x b),$ and the expression $x^2 - 17x 72$ written as $(x - b)(x - c)$, where $a$, $b$, and $c$ are integers. What is the value of $a b c$?
Answer: a=6 b=9 c=8
Step-by-step explanation:
The problem consists of finding the roots of the quadratic equations:
[tex]x^2+15x+54=0\\[/tex]
[tex]x^2-17x+72=0\\[/tex]
The roots can be found with the following equation for solving quadratic equations:
[tex]x_{12}=\frac{-B\pm\sqrt{B^2-4AC}}{2A}[/tex] for equation: [tex]Ax^2+Bx+C=0[/tex]
After solving the equations you can write the result as:
[tex]x^2+15x+54=(x+6)(x+9)\\[/tex]
[tex]x^2-17x+72=(x-9)(x-8)[/tex]
Answer:
b = 9
a = 6
c= 8
Therefore, a,b,c = 6,9,8
Question:
The expression $x^2 - 15x - 54$ can be written as $(x-a)(x-b),$ and the expression $x^2 - 17x + 72$ written as $(x - b)(x - c)$, where $a$, $b$, and $c$ are integers. What is the value of $a b c$?
Step-by-step explanation:
To determine the values of a,b and c.
We need factorize the two equations.
i) x^2 -15x +54
x^2 -6x -9x +54
x(x-6) -9(x-6)
=(x-6)(x-9)
ii) x^2 -17x +72
x^2 -9x -8x +72
x(x-9) -8(x-9)
=(x-9)(x-8)
From the question:
Comparing
(x-a)(x-b) to (x-6)(x-9)
And
(x-b)(x-c) to (x-9)(x-8)
We can see that b is common in the two cases and also 9 is common in the two factorised equations.
So,
b = 9
a = 6
c= 8
Therefore, a,b,c = 6,9,8
How confident are you with using quadratic equations to model and interpret real world problems do you see quadratic equations as relevant to real world situations reflection
Answer:The following states how confident are you? So Quadratic Functions are simple functions listen to this example Multiplying (x+4) and (x−1) together (called Expanding) gets x2 + 3x − 4 :
expand vs factor quadratic
So (x+4) and (x−1) are factors of x2 + 3x − 4
Just to be sure, let us check:
Step-by-step explanation:
Yes, (x+4) and (x−1) are definitely factors of x2 + 3x − 4
A ball is thrown straight up with a speed of 12 meters per second near the surface of Earth. What is the maximum height reached by the ball?
Answer:
maximum height =7.347m
Step-by-step explanation:
maximum height = (U²sin²θ)/2g
where θ = 90° as the ball is thrown straight up.
sin90°=1 , so our formula reduces to;
H= U²/2g
U=12m/s , g=9.8m/s²
H= 12²/(2*9.8)
H=7.347m
Answer:H=7.339m
Step-by-step explanation:
The formula to find maximum height is :
H= v^2/(2× g)
V means speed=12 m/s
g means acceleration due gravity (constant)= 9.81m/s^2
Apply the formula
H=(12^2)/(2×9.81)
H=144/19.62
H=7.339m
Use the distributive property to remove the parentheses.
(8x-10)1/2=
Answer:
The answer to your question is 4x - 5
Step-by-step explanation:
[tex](8x -10)\frac{1}{2}[/tex]
Distributive property, this property allows us to multiply two terms separately. That means that in this problem [tex]\frac{1}{2}[/tex] will multiply the first term, and after that the second term. One half multiply 8x and also - 10.
[tex]\frac{8x}{2} - \frac{10}{2}[/tex]
Simplify and result
[tex]4x - 5[/tex]
Using the distributive property on [tex](8x - 10) \times \frac{1}{2}[/tex], we get [tex]4x-5[/tex]
Given: [tex](8x - 10) \times \frac{1}{2}[/tex]
To find: simplify using distributive property
Distributive property can be depicted by [tex]x(a+b) = xa + xb[/tex]
We can use this property and solve as follows:
[tex](8x - 10) \times \frac{1}{2} = [\frac{1}{2} \times 8x] - [10 \times \frac{1}{2}] = 4x -5[/tex]
We get the expression without parentheses [tex]4x-5[/tex]
2.) Fill in the blank.
csc^2 0 = cot ^2 0 + ___
A.) sec ^2 0
B.) -cos ^2 0
C.) -tan ^2 0
D.) 0
E.) 1
Answer:
E.) 1
Step-by-step explanation:
Firstly we will solve for L.H.S.
L.H.S. =[tex]Csc^2\theta[/tex]
Since we know that [tex]Csc^2\theta[/tex] is the inverse of [tex]Sin^2\theta[/tex].
So we can say that;
[tex]csc^2\theta=\frac{1}{sin^2\theta}[/tex]
Now For R.H.S.
[tex]Cot^2\theta+1[/tex]
Since we can rewrite [tex]cot^2\theta[/tex] as [tex]\frac{cos^2\theta}{sin^2\theta}[/tex].
Now we can say that the R.H.S. is;
[tex]\frac{cos^2\theta}{sin^2\theta}+1[/tex]
Now we add the fraction and get;
[tex]\frac{cos^2\theta+sin^2\theta}{sin^2\theta}[/tex]
Now according to trigonometric identity;
[tex]cos^2\theta+sin^2\theta=1[/tex]
So, [tex]\frac{cos^2\theta+sin^2\theta}{sin^2\theta}=\frac{1}{sin^2\theta}[/tex]
Here,
[tex]csc^2\theta=\frac{1}{sin^2\theta}[/tex] and [tex]Cot^2\theta+1[/tex] = [tex]\frac{1}{sin^2\theta}[/tex]
L.H.S. = R.H.S.
Hence [tex]csc^2\theta=cot^2\theta+1[/tex]
From a group of 8 women and 6 men, a committee consisting of 3 men and 3 women is to be formed. How many different committees are possible if a. 2 of the men refuse to serve together? b. 2 of the women refuse to serve together? c. 1 man and 1 woman refuse to serve together?
To find the number of different committees, we can use combinations. For part a, subtract the committees with the 2 men from the total number. For part b, use the same method with the women. For part c, subtract the committees with both the man and woman from the total number.
Explanation:To find the number of different committees that are possible, we can use combinations. In this case, we want to choose 3 men from a group of 6 and 3 women from a group of 8. The number of combinations can be calculated using the formula:
C(n, r) = n! / (r! * (n - r)!)
a. If 2 of the men refuse to serve together, we need to subtract the number of committees with these 2 men from the total number of committees. The number of committees with the 2 men can be calculated by choosing 1 man from the remaining 4, and then choosing 2 men from the remaining 3. So the number of committees without the 2 men is C(6, 3) * C(8, 3), and the number of committees with the 2 men is C(4, 1) * C(3, 2). The total number of committees is the difference between these two values.
b. If 2 of the women refuse to serve together, we use the same method as in part a, but with the women instead of the men.
c. If 1 man and 1 woman refuse to serve together, we can subtract the number of committees with both of them from the total number of committees. The number of committees with both of them can be calculated by choosing 2 women from the remaining 6 and choosing 2 men from the remaining 5. The total number of committees is the difference between C(6, 3) * C(8, 3) and C(6, 2) * C(5, 2).
(100pts.) A triangle has all integer side lengths and two of those sides have lengths 9 and 16. Consider the altitudes to the three sides. What is the largest possible value of the ratio of any two of those altitudes?
Improper answers will be reported
Answer:
8/3
Step-by-step explanation:
The range of possible lengths for the third side is 16±9, or 7 to 25. For lengths of 7 and 25, the area of the triangle will be zero, so the ratio of altitudes will be infinite (actually, undefined, as division by 0 is involved).
For positive triangle area and integer side lengths, the range of side lengths can be from 8 to 24. For any given triangle, the ratio of maximum altitude to minimum altitude will be the same as the ratio of the maximum side length to the minimum side length.
For the triangles under consideration, the shortest side length we can have is 8. For that 8-9-16 triangle, the ratio of the maximum to minimum side lengths is 16/8 = 2.
The longest side length we can have is 24. For that 9-16-24 triangle, the ratio of maximum to minimum side lengths is 24/9 = 8/3 = 2 2/3. This is more than 2, so 8/3 is the largest possible ratio of any two altitudes.
_____
More explanation
The area of a triangle is given by the formula ...
A = (1/2)bh
Then the altitude for a given base (b) is ...
h = 2A/b
That is, the altitude is inversely proportional to the base length for a triangle of a given area. Once you choose the three sides of the triangle, the area is fixed, so the ratio of altitudes is the inverse of the ratio of base lengths. The ratio of maximum altitude to minimum altitude is the ratio of the inverse of the minimum base length to the inverse of the maximum base length, which is to say it is the same as the ratio of maximum to minimum base lengths.
Answer:
8/3
Step-by-step explanation:
Imogene's car travels 294 mi averaging a certain speed. If the car had gone 7 mph faster, the trip would have taken 1 hour less. Find the average speed
Answer:
42 mph
Step-by-step explanation:
If you start with the assumption that the answer is an integer, you can solve a problem like this by looking at the factors of 294.
294 = 2·3·7² = 6·49 = 7·42
At 42 mph, the 294-mile trip took ...
time = distance/speed = 294 mi/(42 mi/h) = 7 h
At a speed 7 mph faster, the 294-mile trip took ...
(294 mi)/(49 mi/h) = 6 h . . . . . 1 hour less
The average speed of Imogene's car for the 294-mile trip was 42 miles per hour.
_____
Alternative solution
If you let s represent Imogene's speed, you can use the above time and distance relationship to write an equation relating the trip times:
294/s = 294/(s+7) +1
Multiplying by s(s+7), we get ...
294(s+7) = 294s +s(s+7)
2058 = s^2 +7s . . . . . . . . . . . subtract 294s
We can complete the square by adding (7/2)^2 = 12.25 to both sides
2070.25 = s^2 +7s +12.25 = (s +3.5)^2
±45.5 = s +3.5 . . . . . take the square root; next subtract 3.5
-3.5 +45.5 = s = 42 . . . . use the positive solution
Imogene's average speed was 42 mph.
Imogene's average speed was 294 mph. We solve this by using the relationship between distance, speed, and time, and applying it to the specific constraints of the problem. By setting s = speed, we use two equations representing each scenario and solve for s.
Explanation:Let's assign 's' to Imogene's average speed for the trip. The time taken at this speed is the distance traveled (294 mi) divided by 's' (speed = distance/time), making the time = 294/s. If the car was faster by 7 mph, the speed would be s+7 mph, and the time taken would then be 294/(s+7). The problem states that the second scenario would take 1 hour less, so 294/s = 294/(s+7) + 1.
Cross-multiplication and simplification of this equation result in (294s + 2058) = 294(s +7) or 294s + 2058 = 294s + 2058, which simplifies to 2058 = 7s, or s = 294 mph. Therefore, Imogene's average speed was 294 mph.
Learn more about Average Speed here:https://brainly.com/question/17661499
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Find the height of a right square pyramid that has a rectangular base area of 70 square units and a volume of 140 cubic units. A. 2 units B. 6 units C. 1 units D. 18 units
Answer:
B) 6 units
Step-by-step explanation:
The formula of the volume of the Pyramid is 1/3×area of the base×height
1/3×70×height=140
Dividing both sides by 70
1/3×height=2
Multiplying both sides by 3
Height = 6
Final answer:
The height of a right square pyramid with a rectangular base area of 70 square units and a volume of 140 cubic units is 6 units, by using the volume formula of the pyramid.
Explanation:
The question asks us to find the height of a right square pyramid with a given base area and volume. To find the height, we need to use the formula for the volume of a pyramid, which is given by the formula: Volume = (Base Area * Height) / 3. We are given the base area as 70 square units and the volume as 140 cubic units.
Using the formula to solve for the height we get:
Height = (Volume * 3) / Base Area = (140 * 3) / 70 = 6 units. Therefore, the height of the pyramid is 6 units.