Answer:
[tex]x^{2}+\frac{1}{2} x + \frac{33}{16}= 2 + (x + \frac{1}{4})^2[/tex]
Step-by-step explanation:
[tex]x^{2}+\frac{1}{2} x=x^{2}+2 \times \frac{1}{4} \times x +(\frac{1}{4})^2 - (\frac{1}{4})^2[/tex]
[tex]=[x^{2}+2 \times \frac{1}{4} \times x +(\frac{1}{4})^2] - (\frac{1}{16})[/tex]
[tex]=[x + \frac{1}{4}]^2 - (\frac{1}{16})[/tex]
____________________
then
[tex]x^{2}+\frac{1}{2} x = [x + \frac{1}{4}]^2 - (\frac{1}{16}) -2 + 2[/tex]
then
[tex]x^{2}+\frac{1}{2} x = [(x + \frac{1}{4})^2 - (\frac{1}{16}) -2] + 2[/tex]
then
[tex]x^{2}+\frac{1}{2} x = [(x + \frac{1}{4})^2 - (\frac{33}{16}) ] + 2[/tex]
then
[tex]x^{2}+\frac{1}{2} x = (x + \frac{1}{4})^2 - \frac{33}{16} + 2[/tex]
then
[tex]x^{2}+\frac{1}{2} x + \frac{33}{16}= (x + \frac{1}{4})^2 + 2[/tex]
then
[tex]x^{2}+\frac{1}{2} x + \frac{33}{16}= 2 + (x + \frac{1}{4})^2[/tex]
In 2016, the city of Rio de Janeiro had a population density of 5377 people/km2
What was the population density of Rio de Janeiro in people per square meter
Newton's Law of Cooling states that the rate of change of the temperature of an object, T, is proportional to the difference of T and the temperature of the region, TR or dT over dt equals k times the quantity T minus T sub R end quantity. An object with core temperature of 1200°F is removed from a fire and placed in a region with a constant temperature of 80°F. After 1 hour, its core temperature is 830°F. What is the object's core temperature 4 hours after it is taken off the fire? (1 point)
Answer:
305 °F
Step-by-step explanation:
The core temperature of the object after 4 hours can be found using an exponential decay formula to model the decay of the difference between core temperature and ambient.
Cooling ModelThe solution to the differential equation described by Newton's law of cooling is the exponential equation ...
y = ab^t +c
where 'a' is the initial core temperature difference from ambient, 'b' is the decay factor of that difference in 1 unit of time period t. 'c' is the ambient temperature.
For this problem, the ambient temperature is c=80, and the differences of interest are ...
a = 1200 -80 = 1120
b = (830 -80)/1120 = 75/112
Using these values in the model gives ...
y = 1120(75/112)^t +80 . . . . . . where y(t) is the core temperature at time t
Note that units of time are hours.
ApplicationWe want y when t=4.
y = 1120(75/112)^4 +80 ≈ 1120(0.20108) +80 ≈ 305.212
The core temperature after 4 hours is about 305 °F.
__
Additional comment
The differential equation will have a solution of the form ...
[tex]T-T_R=(T_0-T_R)e^{kt}[/tex]
where k = ln(75/112) ≈ -0.40101
In the above, we defined b = e^k = 75/112. Accuracy with this fraction can be better than using a truncated value of k.
train travelling at a uniform speed covers a distance of 255 km in Find the speed of the train and 3 4/5 1 1/2 the distance covered in hours. hours.
The speed of the train is 150 km/hour.
The distance covered by the train in 3 4/5 hours is 570 km.
The speed of a body is calculated using the formula, speed = distance/time.
The distance covered is calculated using the formula, distance = speed*time.
The time taken by a body is calculated using the formula, time = distance/speed.
In the question, we are asked for the speed of a train, when it covers a distance of 225 km in 1 1/2 hours.
Distance = 225 km.
Time = 1 1/2 hours = 1.5 hours.
Speed = Distance/Time = 225/1.5 km/hour = 150 km/hour.
Now, we are asked to calculate the distance covered by the train in 3 4/5 hours.
Speed = 150 km/hour.
Time = 3 4/5 hours = 3.8 hours.
Distance = Speed*Time = 150*3.8 km = 570 km.
Thus, the speed of the train is 150 km/hour.
The distance covered by the train in 3 4/5 hours is 570 km.
The provided question is incorrect. The correct question is:
"A train at a uniform speed covers a distance of 225 km in 1 1/2 hours. Find the speed of the train and the distance covered in 3 4/5 hours.
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A Big Year
By Bob Kowalski
Would you go to the ends of the earth to see a bird? What if it were a really special bird: one with beautiful feathers, an entrancing call, or a silly dance? What if seeing that one special bird would allow you to win a contest?
If that contest doesn't get you on television or win you any cash prizes, would you still do it? For those who participate in the "Big Year," the honor of beating the previous record is the only reward they get or even want.
A "Big Year" is a year in which a person attempts to see as many different species of birds as possible within a particular region. For most in North America who participate in a "Big Year," this region is the lower 48 American states, plus Alaska, Canada, and a couple of French islands off the Canadian coast.
You may be thinking that looking at birds is silly, but just think about the numbers of the recent record holders and the commitment it takes to get these numbers. One recent "Big Year" winner managed to see 744 birds in one year, missing the record by just one bird. Big Year birders travel by train, plane, boat, car, bicycle, and of course, by foot. They can cover over 150 thousand miles to get numbers of sightings this high. They can also spend a small fortune.
Just to clarify, the birds these contestants are counting are the number that they see in a particular year. You see, the contest is based on an honor system. No pictures or other evidence is required as proof of a sighting. Most birders take great pride in their reputation and their abilities to see or hear and then identify a bird. Usually, important sightings of the rare birds needed to get counts in the 700s are visited by hundreds of birders. It is pretty hard to cheat your way to a record-breaking year, but in general, few are interested in cheating.
This honesty comes from the fact that most people who want to break such a record know the greatest rewards are not necessarily in winning. Such rewards are in being able to commit a year of your life to doing something you love. Rewards are found in seeing amazing, inspiring creatures like the California Condor or the Magnificent Frigate bird. Rewards also come in spending time with people who, like you, want to spend their time looking to the skies and trees for glimpses of emerald, crimson, or cerulean blue feathered jewels.
You don't have to be able to travel a continent to have a big birding experience though. Have a big month. Or a big weekend. Set a personal record, learn to identify the species that live in your part of the world, or try to learn the calls of just two species of birds. You will soon find looking at birds isn't such a strange way to spend your time.
Extra! Extra! Backyard Birding
Many schools, families, and young birders across the country participate in the "Great Backyard Bird Count." While not as long as a "Big Year," the "Great Backyard Bird Count" happens every year. It depends on birders and families across the country to watch feeders and other areas in their yards and count the number of birds they see. Unlike the "Big Year," the goal is not to see who can count the most birds. Instead, participants in this event work together to help bird experts get a good idea of how birds are doing. Participants are given checklists and enter their sightings on a website. Called a "citizen-science" project, this event is open to anyone, requires no travel, and happens every year over one weekend in February.
Read this sentence from the article:
You may be thinking that looking at birds is silly, but just think about the numbers of the recent record holders and the commitment it takes to get these numbers.
Why would the author suggest that looking at birds is silly?
To create a sense of doubt in the reader
To distance himself from what seem to be rather strange people
To show readers that some people do not think birding is interesting
To suggest he has never birded himself but might someday
Ill give brainliest
The inference is that the author suggest that looking at birds is silly A. to create a sense of doubt in the reader.
What is an inference?It should be noted that that an inference simply means the conclusion that can be deduced based on the information given in a story.
In this case, A Big Year" is a year in which a person attempts to see as many different species of birds as possible within a particular region. For most in North America who participate in a "Big Year," this region is the lower 48 American states, plus Alaska, Canada, and a couple of French islands off the Canadian coast.
The inference is that the author suggest that looking at birds is silly is to create a sense of doubt in the reader. Therefore, the correct option is A.
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z=2cispi/3 in rectangular form
The conversion of "z = 2(cos(π/3))" in polar form to rectangular form is equal to 1.
What is a polar coordinate?A polar coordinate can be defined as a two-dimensional coordinate system, wherein each point on a plane is typically determined by a distance (r) from the pole (origin) and an angle (θ) from a reference direction (polar axis).
How to transform polar coordinates to rectangular coordinates?In geometry, the relationship between a polar coordinate (r, θ) and a rectangular coordinate (x, y) based on the conversion rules is given by the following polar functions:
a = rcos(θ) ....equation 1.
b = rsin(θ) ....equation 2.
Where:
θ is the angle.r is the radius of a circle.Note: The exact value of cos(π/3) is equal to ½.
Substituting the given parameters into the formula, we have;
z = 2(½)
z = 2/2
z = 1.
In conclusion, we can logically deduce that the conversion of "z = 2(cos(π/3))" in polar form to rectangular form is equal to 1.
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Complete Question:
Convert z = 2(cos(π/3)) in rectangular form
Anyone know the answer to this?
Answer: y = 3x+5, 3, (-1,2)
Step-by-step explanation:
y = mx + c where m is the slope, so we must find the slope
Slope = rise/run = (5-2)/(0-(-1)) = 3
So y = 3x + c
To find c, we have to substitute one of the coordinates on the graph into the equation, (-1,2), the point that is marked
2 = 3(-1) + c
c = 5
So the equation of the graph is y = 3x + 5
What is the area of the actual square window
shown in the scale drawing?
Answer:
2.25m²
Step-by-step explanation:
if 1in = 2m then 0.5in = 1m
so 0.75in = 1.5m
area = 1.5 x 1.5
a = 2.25m²
(2 Fruit juice A is composed of 2 parts apple juice and 1 part grape juice.
Fruit juice B is composed of 1 part apple juice and 2 parts grape juice.
When A and B are mixed, the new mixture has 10 kg of apple juice and
8 kg of grape juice. How many kilograms does each type of original
fruit juice weigh
The amount of kilograms of each type of original fruit juice weighs; 12 kg of Apple and 6 kg of grape
How to Solve Algebra Word Problems?
The concentrations of apple and grape juice in the two different juice types are given as;
Juice A:
(2/3) apple
(1/3) grape
Juice B:
(1/3) apple
(2/3) grape
Let a be how much juice A is in the mixture;
Let b be how much juice B is in the mix.
Thus;
²/₃a + ¹/₃b = 10 ------(eq 1)
¹/₃a + ²/₃b = 8 ------(eq 2)
Solving both equations simultaneously gives us;
a = 12 and b = 6
Thus, the amount of kilograms of each type of original fruit juice weighs; 12 kg of Apple and 6 kg of grape
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Find the gradient of the line passing through the points (– 2,– 4) and (3,5).
Answer:
Gradient of the line is choice D.9/5
Step-by-step explanation:
Hello!Slope between two points:slope=(y₂-y₁)/(x₂-x₁)
(x₁.y₁)=(-2,-4)(x₂.y₂)=(3,5)slope(m)=
[tex] \frac{5 - ( - 4)}{3 - ( - 2)} \\ refine \: (m )= \frac{9}{5} [/tex]
Problem 9: Determine is the following function is one-to-one (explain your answer); F = {(-2,1), (-5,-1), (3,-5), (1,-2), (0,5), (-1,6), (-6,7), (7,-6)}
Answer:
YES! it's one-to-one function.
Step-by-step explanation:
Hello!One to one function or one to one mapping states that each element of one set, say Set (A) is mapped with a unique element of another set, say, Set (B), where A and B are two different sets. It is also written as 1-1. In terms of function, it is stated as if f(x) = f(y) implies x = y, then f is one to one.
Thus, when we look at this given function
X Y
[tex] - 2 \: \: \: \: \: \: 1 \\ - 5 \: \: \: \: \:- 1 \\ 3 \: \: \: \: \: \: - 5 \\ 1 \: \: \: \: \: \: - 2 \\ 0 \: \: \: \: \: \: 5 \\ - 1 \: \: \: \: \: \: 6 \\ - 6 \: \: \: \: \: \: \: 7 \\ \: \: 7 \: \: \: \: \: \: - 6[/tex]
So, when you look at each value of x is attaches with different value of y and there is no repetition of elements means that one element of x is directly attached to one element of y.
Hope it helps!
A storage shed is to be built in the shape of a box with a square base. It is to have a volume of 729 cubic feet. The concrete for the base cost $8 per square foot, The material for the roof costs $3 Per square foot, and the material for the sides costs $5.50 Per square foot. Find the dimensions of the most economical shed.
The dimensions of the most economical shed are 9 ft in length, 9 ft in width, and 9 ft in height if the storage shed is to be built in the shape of a box with a square base.
What is volume?It is defined as a three-dimensional space enclosed by an object or thing.
It is given that:
A storage shed is to be built in the shape of a box with a square base.
It is to have a volume of 729 cubic feet.
Let x be the length of the base
Let y be the height of the box.
V = 729 cubic feet
The base is square:
l = w = x
V = x²y
y = 729/x²
Cost of the base = $8x²
Cost of the roof = 3x²
Cost of the sides = 4(5.50)xy
= 22xy
Total cost
C= 8x² + 3x² + 22xy
C = 11x² + 22x(729/x²)
C = 11x² + 16038/x
Find the first derivative:
dC/dx = 22x - 16038/x²
Equate; dC/dx = 0
22x - 16038/x² = 0
22x = 16038/x²
22x³ =16038
x³ = 729
x = 9
y = 729/(9)² = 9
Length of base = 9 ft
Width of base = 9 ft
Height of box = 9 ft
Thus, the dimensions of the most economical shed are 9 ft in length, 9 ft in width, and 9 ft in height.
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Find a polynomial $f(x)$ of degree $5$ such that both of these properties hold:
$\bullet$ $f(x)$ is divisible by $x^3$.
$\bullet$ $f(x)+2$ is divisible by $(x+1)^3$.
Write your answer in expanded form (that is, do not factor $f(x)$).
Answer:
Step-by-step explanation:
f(x)=ax³
ax³+2=b(x+1)³
ax³+2=b(x³+3x²+3x+1)
(a-b)x³-3bx²-3bx-b=0
Paolo helped in the community garden for 2 3/4 hours this week. This was 1 5/6 equal length shift, because Paolo stopped early one day when it started to rain. How long is a single shift?
If 2+3/4 hours is equal to 1+5/5 equal length shift then the single shift is 1.5 hours long.
Given that Paolo helped in the community garden for 2 +3/4 hours this week which is 1+5/6 times length shift.
We are required to find the length of a single shift.
First we have to find the proper fraction of the given mixed fractions.
Fraction is combination of numerator and denominator.The number that is present above the division sign is known as numerator and the number that is present below the division sign is known as denominator.
2+3/4=(8+3)/4=11/4
1+5/6=(6+5)/6=11/6
let the length of a shift is x hours.So,
11/4=11*x/6
11/4=11/6 x
66/44=x
x=3/2
x=1.5 hours.
Hence if 2+3/4 hours is equal to 1+5/5 equal length shift then the single shift is 1.5 hours long.
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please help jajajajaja
a tree 20 feet tall with a circumference of 3 ft has a vine wound around 7 times. How long is the vine?
The length of the vine is 21 ft
what is the circumference of a circle?
the circumference is the perimeter of a circle
the perimeter of the circle = 2πr = 3 ft
the vine is 7 times the perimeter
length of vine = 7×3= 21 ft
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If the tree is 20 feet tall with circumference 3 feet then the length of vince is around 21 feet.
Given the height of tree is 20 feet and the circumference of tree is 3 feet.
We have to find the length of vine.
Circumference is the perimeter of a circular object. Because the trunk of a tree is in shape of circle so the perimeter of the trunk is 2πr.
We have been given the circumference of tree be 3 feet.
Circumference=2πr
Because vine is 7 times the circumference so the length of vine being:
Length of vine=7*3
=21 feet.
Hence if the tree is 20 feet tall with circumference 3 feet then the vince is around 21 feet long.
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Newton's Law of Cooling states that the rate of change of the temperature of an object, T, is proportional to the difference of T and the temperature of its surrounding environment. A pot of chili with temperature 23°C is placed into a −18°C freezer. After 2 hours, the temperature of the chili is 7°C.
Part A: Assuming the temperature T of the chili follows Newton's Law of Cooling, write a differential equation for T. (10 points)
Part B: What is the temperature of the chili after 4 hours? (20 points)
Part C: At what time, t, will the chili's temperature be −10°C? (10 points)
For this, let's go through each problem carefully and step-by-step.
According to the question, the rate of change of the temperature of any object that is defined by T, is directly proportional to the difference of T and the temperature of the environment around it, which we'll denote as X.
[tex]\frac{dT}{dt}[/tex][tex]= k (T-X)[/tex]
K is a constant of proportionality here. And the temperature of the surrounding environment is said to be (-18°C). Thus,
[tex]\frac{dT}{dt} = k(T+18)[/tex].
For part A, in order to find the differential equation for T, we need to solve for k. So we separate the variables and then integrate to solve the equation.
[tex]\int\limits{\frac{dT}{T+18} } = \int\limits {k} \, dt[/tex]
[tex]ln(T+18) = kt+c[/tex]
Now thw inital temperature of a pot of chili is 23°C, so at [tex]t = 0, T_0 = 23*C[/tex].
Substituting 23 for T and 0 for t, we have the following:
[tex]ln(23+18) = k(0)+c[/tex]
[tex]ln(41) = c[/tex]
We know the temperature of chili after 2 hours is 7°C, so we know that when [tex]t = 2, T_1 = 7[/tex]
Substituting t for 2, and T for 7, we get:
[tex]ln(7+18) = 2k+ln(41)[/tex]
[tex]ln(25) = 2k + ln(41)[/tex]
Solving for 2k
[tex]2k = ln(25) -ln(41)[/tex]
[tex]2k = ln(\frac{25}{41})[/tex]
[tex]k = \frac{1}{2}ln(\frac{25}{41})[/tex].
Substituting the value of [tex]\frac{dT}{dt} = k (T+18)[/tex], the differential equation obtained is [tex]\frac{dT}{dt} = \frac{1}{2}ln(\frac{25}{41})(T+18)[/tex].
For part B, to find the temperature of the chili after four hours, we first need to solve the above differential equation.
The solution of the differential equation is given by the equation [tex]ln(T+18) = kt+c[/tex]. Substituting the values of k and c, we have:
[tex]ln(T+18) = \frac{1}{2}ln(\frac{25}{41})t+ln(41)[/tex].
Using the above relation, at any time (t), the temperature (T) can be found out in the following.
At [tex]t = 4, T_2 = \phi[/tex]
[tex]ln(T_2+18)=\frac{1}{2}ln(\frac{25}{41})*4+ln(41)[/tex]
[tex]ln(T_2+18)=2ln(\frac{25}{41})+ln(41)[/tex]
[tex]ln(T_2+18)=-0.989 + 3.714[/tex]
[tex]ln(T_2+18)[/tex] ≅ [tex]2.725[/tex]
Solving the natural logarithm,
[tex]T_2+18 = e^{2.725} = 15.256[/tex]
[tex]T_2 =15.256 - 18[/tex]
[tex]T_2 = -2.744[/tex].
So the temperature of the chili after four hours would be -2.744°C approximately.
To find part C in what time the chili would be 10°C, we need to substitute again.
[tex]t = \phi[/tex][tex], T = -10[/tex]
[tex]ln(-10 + 18) = \frac{1}{2}ln(\frac{25}{41})t + ln(41)[/tex]
[tex]ln(8) = \frac{1}{2}ln(\frac{25}{41})t + ln(41)[/tex]
Solving for [tex]\frac{1}{2}ln(\frac{25}{41})t[/tex],
[tex]\frac{1}{2}ln(\frac{25}{41})t = ln(8) - ln(41)[/tex]
[tex]\frac{1}{2}ln(\frac{25}{41})t = ln(\frac{8}{41})[/tex]
[tex]\frac{1}{2}ln(\frac{25}{41})t = -1.634[/tex]
[tex]ln(\frac{25}{41})t[/tex][tex]= -1.634 * 2[/tex]
[tex](-0.494)t=-3.268[/tex]
[tex]t = \frac{-3.268}{-0.494}[/tex]
[tex]t=6.615[/tex] hours, approximately.
Thus, the chili would reach -10°C at around 6.615 hours.
Hope this helped. This took me a long time.
Which function best fits the following points?
A.=-12.84032+0.0225x
O B. y=65.0778-772.9605*
O C. y=-197.0571x2+ 245.6243x + 6.0321
O D. None of the above
The function that best fits the graph points is; B: y = 65.0778 * 772.9605ˣ
How to Interpret Function Graphs?
From the given graph, we can see that is parabolic form and as such we can say it is an exponential function.
Looking at the options, only option B is in exponential form and as such, we will take one point on the graph to check if this is the right function.
Let us use the coordinate (0.9, 26000)
y = 65.0778 * 772.9605ˣ
y = 65.0778 * 772.9605^(0.9)
y = 25837.76
This is very close and as such is the correct option.
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Name the set(s) to which each number belongs.*
-20√/10
*More than one answer may be correct mark all correct answers.
A. reals
B. whole
C. integer
D. rationals
E. irrationals
F. natural
[tex] \qquad \qquad \bf \huge\star \: \: \large{ \underline{Answer} } \huge \: \: \star[/tex]
The given number is :
[tex]\qquad❖ \: \sf \: \cfrac{ - 20}{ \sqrt{10} } [/tex]
since 10 isn't a perfect square, root 10 is an irrational number, and therefore the whole number is irrational as well, and we already know that irrational numbers are also part of real numbers.
[tex] \qquad \large \sf {Conclusion} : [/tex]
So the Correct choices are :
A. RealE. IrrationalWhat’s the slope please help
By secant line formula, the slope of the line represented in the table is equal to - 2.
How to determine the slope of a linear function based on a table
In this problem we must determine the slope of a function based on the secant line formula. After a quick glance at the table, we find that the slope formula can be used as Δx and Δy are constant at every pair consecutive points. Then, the slope of the line is:
m = (- 3 - 9)/(9 - 3)
m = - 12/6
m = - 2
By secant line formula, the slope of the line represented in the table is equal to - 2.
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The mean, median, and mode of the five numbers 5,7,8,A,B are equal. Find all possible values of A+B
If the mean, median, and mode of the five numbers 5, 7, 8, A, and B are equal, the possible values of A+B are 20, where A is 8, and B is 12.
What are the measures of central tendency?The measures of central tendency include the mode, the median, and the mean.
Statistically, a central tendency is a typical value for a probability distribution.
The mode's value appears most often in a set of data values. The median is the middle value of a dataset. The mean represents the average.
To determine the possible values of A and B, we can start by finding the median value, which is the middle value in ascending order.
Data and Calculations:Median (Middle value) = 8
Mode = 8
Total value = 40 (8 x 5)
A = 8
B = 12.
Mean = 8 (40/5)
Thus, if the mean, median, and mode of the five numbers 5, 7, 8, A, and B are equal, the possible values of A+B are 20, where A is 8, and B is 12.
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