Describe the graph of the basic rational function, f(x)=1/x. Talk about its domain and range and compare the function to other ones, i.e. linear or quadratic or exponential.
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negative values of x with produce negative values of f(x) and positive values will produce positive function values. so this hyperbola is symmetric with respect to the origin and all negative values are in Q3 and all positive values are in Q1.
as x gets farther and farther from 0 in either direction, you can see that 1/x will be an ever-increasingly smaller distance from the x-axis. indeed, the limit of this function is zero as x approaches either positive or negative infinity.
if x is between -1 and 1, then the value of f(x) increases until x = 0, the limit of the function as x approaches 0 being positive and negative infinity depending on which direction you are coming from.
the domain is all real numbers with the exception of 0
the range also happens to be all real numbers with the exception of zero. all that really means is that zero has no multiplicative inverse.
strictly speaking, this is an exponential function, as the function is f(x) = x^(-1).
Final answer:
The graph of f(x) = 1/x is a hyperbola with vertical and horizontal asymptotes at x = 0 and y = 0, respectively. The domain is all real numbers except 0, and the range is the same. Unlike linear, quadratic, or exponential functions, the hyperbolic shape of 1/x presents its unique characteristics needing careful domain consideration.
Explanation:
The graph of the basic rational function f(x) = 1/x is characterized by two distinct branches, one in the first quadrant (where x and y are both positive) and the other in the third quadrant (where x and y are both negative). This function is a hyperbola, and as the value of x approaches zero from either the positive or negative side, the value of f(x) increases without bound, referred to as a vertical asymptote at x = 0. Additionally, as the value of x becomes very large or very small (approaching positive or negative infinity), the value of f(x) approaches zero, creating a horizontal asymptote at y = 0. Importantly, this function is undefined at x = 0, so the domain is all real numbers except 0, (domain: ℝ \ {0}). The function takes on all real number values for the output, so the range is also all real numbers except 0 (range: ℝ \ {0}).
Comparing to other types of functions, unlike linear functions like f(x) = 7x which produces a straight line or quadratic functions such as f(x) = 2x² which graphs as a parabola, the rational function f(x) = 1/x leads to a non-linear graph with two separate branches. Exponential functions like f(x) = eⁿ¹, however, grow or decline at an increasing rate, which is a different behavior compared to the hyperbolic decline or rise of a rational function. It's vital to note that as with the function |√x|, there are points along the graph of 1/x where the function is not defined, highlighting the importance of considering the domain in function analysis.
The function 1/x² also represents a rational function but with a different characteristic as it exhibits positive blowup on both sides. This means that as x approaches zero, whether from the positive or negative side, the function tends to positive infinity, a behavior different from the reciprocal function 1/x.
Related Questions
Betty correctly determined that the ordered pair (–3, 5) is a solution to the system of linear equations mc004-1.jpg and x + 4y = 17. Based on this information, which statement is correct? (–3, 5) satisfies neither the equation 6x + 5y = 7 nor the equation x + 4y = 17. (–3, 5) satisfies the equation 6x + 5y = 7 but not the equation x + 4y = 17. (–3, 5) satisfies the equation x + 4y = 17 but not the equation 6x + 5y = 7. (–3, 5) satisfies both the equation 6x + 5y = 7 and the equation x + 4y = 17
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Since (-3, 5) is a solution of the system of equations, then (-3, 5) must satisfy both equations.
Answer is choice D.
Find the area of the part of the plane 5x + 4y + z = 20 that lies in the first octant.
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Next, we can parameterize the surface by
[tex]\mathbf s(u,v)=\langle4(1-u)(1-v),5u(1-v),20v\rangle[/tex]
so that the surface element is
[tex]\mathrm dS=\|\mathbf s_u\times\mathbf s_v\|=20\sqrt{42}(1-v)\,\mathrm du\,\mathrm dv[/tex]
Then the area of [tex]\mathcal S[/tex] is given by the surface integral
[tex]\displaystyle\iint_{\mathcal S}\mathrm dS=20\sqrt{42}\int_{u=0}^{u=1}\int_{v=0}^{v=1}(1-v)\,\mathrm dv\,\mathrm du[/tex]
[tex]\displaystyle=20\sqrt{42}\int_{v=0}^{v=1}(1-v)\,\mathrm dv=10\sqrt{42}\approx64.8074[/tex]
The area of the plane 5x + 4y + z = 20 in the first octant is calculated using a double integral over the xy-plane. The limits are defined by the intersection of x and y with the xy-plane when z=0, and the solution is approximately 66.67 square units.
Explanation:To solve for the area of the part of the plane that lies in the first octant, we first need to isolate z in our equation. z = 20 - 5x - 4y. The limits for x and y in the first octant are from 0 to positive infinity, but in this case, they will be limited by the plane defined by z = 0 (the xy-plane). x and y will range from 0 to the point where they meet the plane. Therefore, we set z = 0 and solve for both x and y, giving us x = 4 and y = 5.
Now in order to calculate the area, we must interpret this integral on the xyz-plane as a double integral on the xy-plane. Now, we integrate over the region in the xy-plane that x and y range over:
Area = ∫ from 0 to 4 ∫ from 0 to (5 - 1.25x) (20 - 5x - 4y) dy dx
And that is the integral you need to calculate to find the area. Attempting to calculate this integral results in the area ≈ 66.67 square units.
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What happens to the average kinetic energy of water molecules as water freezes? A. It decreases as the water releases energy to its surroundings.
B. It increases as the water releases energy to its surroundings.
C. It increases as the water absorbs energy from its surroundings.
D. It decreases as the water absorbs energy from its surroundings.
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Answer: Option (A) is the correct answer.
Explanation:
Kinetic energy is defined as the energy possessed by an object because of it's motion.
Whereas average kinetic energy is the sum of kinetic energy of all the particles of a substance.
Therefore, when water freezes then there will decrease in kinetic energy of particles and thus, particles will gain potential energy.
Hence, we can conclude that when water freezes average kinetic energy of water molecules decreases as the water releases energy to its surroundings.
The average "kinetic-energy" of water molecules decreases as water freezes, because water releases energy to its surroundings, option (a) is correct.
When water freezes, it undergoes a phase transition from a liquid to a solid state. During this process, water molecules lose energy and slow down, causing decrease in their average kinetic energy.
As temperature decreases, water molecules arrange themselves into a more structured pattern due to the formation of hydrogen bonds.
In order for water to freeze, it needs to release energy to its surroundings, in form of heat. This release of energy is necessary to facilitate transformation from a higher-energy liquid state to lower-energy solid state. As a result, average kinetic energy of water molecules decreases as water freezes.
Therefore, the correct option is (a).
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The distance from Talamunda to Velcratia is 700 miles. The train takes 5 hours to travel the distance. At what unit rate is the train traveling?
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Answer: 140 miles per hour
Step-by-step explanation:
For which real numbers x and y is it true that x + y = x+y?
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x + y = y + x
For sure
Find the limit. use l'hospital's rule if appropriate. if there is a more elementary method, consider using it. lim x→0 2x − sin(2x) 2x − tan(2x)
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Using the L'Hospital's Rule, we differentiate the numerator and denominator of the given function separately, substitute these derivatives back into the function, and then attempt to evaluate the limit as x approaches 0.
Explanation:To find the limit of the given function as x approaches 0, we will use the L'Hospital's Rule since the function gets an indeterminate form 0/0 as x approaches 0. L'Hospital's Rule states that the limit of a quotient of two functions as x approaches a certain value is equal to the limit of the quotients of their derivatives.
First, we differentiate the numerator and the denominator separately. For the numerator, the derivative of 2x is 2, and the derivative of sin(2x) is 2cos(2x). So the derivative of the numerator is 2 - 2cos(2x).
For the denominator, the derivative of 2x is 2, and the derivative of tan(2x) is 2sec²(2x), since the derivative of tan(x) is sec²(x) and because of the chain rule, we multiply by 2. So, the derivative of the denominator is 2 - 2sec²(2x).
Now, we substitute these derivatives back into the original function and take the limit as x approaches 0: lim x→0 (2 - 2cos(2x)) / (2 - 2sec²(2x)).
After further simplifying the above expression, you can evaluate the limit.
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The limit is 0.
We need to find the limit:
lim x→0 (2x − sin(2x)) / (2x − tan(2x))
Both the numerator and the denominator approach 0 as x approaches 0, which means it is in the indeterminate form 0/0. This suggests that we can use L'Hôpital's Rule.
According to L'Hôpital's Rule, we can differentiate the numerator and the denominator and then take the limit of the resulting fraction.
First, let's compute the derivative of the numerator:
Numerator: 2x - sin(2x)Derivative: 2 - 2cos(2x)Next, let's compute the derivative of the denominator:
Denominator: 2x - tan(2x)Derivative: 2 - 2sec²(2x)Now, we apply L'Hôpital's Rule:
lim x→0 (2 - 2cos(2x)) / (2 - 2sec²(2x))
As x approaches 0, cos(2x) approaches 1 and sec²(2x) also approaches 1, so:
lim x→0 (2 - 2(1)) / (2 - 2(1)) = 0 / 0
This fraction again gives an indeterminate form. Applying L'Hôpital's Rule a second time will be helpful. We need to differentiate the numerator and the denominator again:
Second derivative of the numerator: d/dx [2 - 2cos(2x)] = 4sin(2x)
Second derivative of the denominator: d/dx [2 - 2sec²(2x)] = -8sec²(2x)tan(2x)
Applying L'Hôpital's Rule once more, we get:
lim x→0 (4sin(2x)) / (-8sec²(2x)tan(2x))
Substitute x = 0:
lim x→0 (4sin(2x)) / (-8sec²(2x)tan(2x)) = 0
Therefore, the limit is 0.
A student has three mangos, two papayas, and two kiwi fruits. if the student eats one piece of fruit each day, and only the type of fruit matters, in how many different ways can these fruits be consumed
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7! / ( 3! * 2! * 2! ) = 210
There are 210 different ways for the fruits can be consumed.
What is Multiplication?
To multiply means to add a number to itself a particular number of times. Multiplication can be viewed as a process of repeated addition.
Given that;
A student have 3 mangos, 2 papayas, and 2 kiwi fruits.
And, The student eats one piece of fruit each day, and only the type of fruit matters.
Now,
Total number of fruits = 3 + 2 + 2
= 7
And, There are 3 mangos, 2 papayas, and 2 kiwi fruits.
So, Number of ways for the fruits can be consumed, is calculated as;
= 7! / 3! 2! 2!
= 7 x 6 x 5 x 4 x 3! / 3! 2! 2
= 7 x 6 x 5 x 4 / 2 x 1 x 2 x 1
= 7 x 6 x 5
= 210
Thus, There are 210 different ways for the fruits can be consumed.
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How long will it take for the object to fall all the way down to the ground? The function for objects dropped from a height where t is the time in seconds, h is the height in feet at time t, and h is the intial height is h(t)=-16t^2+h. The building is 162 feet tall.
Thanks so much!
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-16t^2+162, or 16t^2 = 162. Taking the + square root of both sides,
we get t = 162/16 seconds, or 10 1/8 seconds.
Adina is constructing a line perpendicular to YJ . She has already constructed two arcs as shown. She moves her compass point to Y to construct an arc above the line. What must be true about the width of Adina’s compass opening before she draws the arc?
A. It must be narrower than it was when constructing the first two arcs. B. It must be the same as it was when constructing the first two arcs.
C. It must be wider than it was when constructing the first two arcs.
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The arc is every two-point smooth curve. The length of an arc is called the length of its arc. In a chart, a chart arc is an adjacent pair of vertices. A semicircle is called an arc, whose endpoints lie on a circle diameter.
It must be broader than the first two arcs or you'll only get two circles alongside. The length of a point between the c and j is measured. The arches are a good way to cross.The compass shall be set to a length broader than the length of YC when drawing arcs below and above the YJ line.The arc production passes, not giving a line perpendicular to YJ.Therefore, the final answer is "Option C".
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solve this system of linear equations. separate the x- and y- values with a coma. 6x+20y=-62
3x-9y=-12
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To solve the system of linear equations, use the method of substitution by solving for one variable and substituting it into the other equation.
Explanation:To solve the system of linear equations:
6x + 20y = -62
3x - 9y = -12
We can use the method of substitution:
From the first equation, solve for x: x = (-62 - 20y) / 6Substitute the value of x into the second equation: 3((-62 - 20y) / 6) - 9y = -12Simplify and solve for y:After finding the value of y, substitute it back into the first equation to solve for x. The solution is: x = -2, y = 4.
find the equation in general form of the circle with center at the origin and radius equal 6.
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What is 2 2/5 ÷ (- 1/4) And how?
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first you turn both fractions into decimals, then you just divide the two.
On a quiz worth 5 points, three students earned a 5, five students earned a 4, four students earned a 3,six students earned a 2 , nine earned a 1 and eight students earned a zero . What is the class average on this quiz
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The ice cream Palace received 3 gallons of strawberry ice cream, 5 pints of mocha ice cream, and 1 quart of vanilla ice cream today. How many pints of ice cream did they receive?
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The fifth term of a geometric sequence is 781.25. Each previous term is 1/5 of the value of the current term. Which recursive formula represents the situation?
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Answer:
[tex]a_n=5a_{n-1}[/tex]
Step-by-step explanation:
Given fifth term of a geometric sequence is 781.25.
[tex]a_5= 781.25[/tex].
Also, each previous term is 1/5 of the value of the current term.
Therefore, common ratio would be 5.
But we just need to find the recursive formula .
Recursive formula of a geometric sequence is given by
[tex]a_n=ra_{n-1}[/tex]
Plugging value of r in above formula, we get
[tex]a_n=5a_{n-1}[/tex]
Therefore, recursive formula would be [tex]a_n=5a_{n-1}[/tex] represents the situation.Answer: c
Step-by-step explanation:
buddy beneath me is blabbin to much
The speed of the car was 45 mph. A driver noticed that while moving with this speed it took him 40 seconds to cross a bridge. On the way back crossing the same bridge, it took him 30 seconds. What was the speed of the car on the way back?
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Each time the car travels the same distance across the bridge.
By setting the distances equal you get the following equation:
[tex]30x = 40*45 \\ \\ x = \frac{40*45}{30} = 60[/tex]
Answer: Car was going 60 mph
Answer:
Speed of car on the way back = 30 mph
Step-by-step explanation:
Speed of car on the way = 45 mph = 45 x 1.6 = 72 kmph = 20 m/s
Time taken to cross bridge on the way = 40 seconds
Length of bridge = Speed of car on the way x Time taken on the way = 20 x 40 = 800 m
Time taken to cross bridge on the way back = 30 seconds
Length of bridge = Speed of car on the way back x Time taken on the way back
800 = Speed of car on the way back x 30
Speed of car on the way back = 26.67 m/s =96 kmph = 30 mph
Speed of car on the way back = 30 mph
What are the values of the mean and standard deviation after converting?
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Find three positive numbers whose sum is 100 and whose product is a maximum. (enter your answers as a comma-separated list.)
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Let the two positive integers be x and y.
Then x + y = 100, and xy = the product.
Let's eliminate x. Solve x + y = 100 for x: x = 100 - y. Now subst. this last result into P = xy: P = product = (100 - y)(y) = 100y - y^2
Differentiating, dP/dy = 100 - 2y. Set this = to 0 and solve for y: -2y = -100, and y = 50. Since x + y = 100, x is thus also = to 50.
Solution set: (50,50).
Now suppose that three positive integers add up to 100, and that we want to maximize their product.
Then x + y + z = 100. Let's maximize f(x,y,z) = xyz (the product of x, y and z).
Since x + y + z = 100, we can eliminate z by solving x + y + z = 100 for z and subst. the result back into f(x,y,z) = xyz:
We get f(x,y) =xy(100-x-y), a function of two variables instead of three.
I won't go through the entire procedure of maximizing a function in three variables, but will get you started:
Find the 'partial of f with respect to x' and then the 'partial of f with respect to y'. Set each of these partial derivatives = to 0:
f = 0 = (partial of xy(100-x-y) with respect to x
x
= xy(partial of 100-x-y with respect to x) + (100-x-y)(partial of xy with respect to x)
= xy(-1) + (100-x-y)(y)
We must set this partial = to 0: -xy+100y-xy-y^2 = 0
-2xy + 100y - y^2 = 0
or y(-2x + 100 - y) = 0
of which y=0 is one solution and in which -2x + 100 - y = 0
You must now go through the same procedure with respect to the partials with respect to y.
If you'd like to continue this discussion, please respond with questions and comments.
A local little league has a total of 85 players, of whom 80% are left-handed. How many left-handed players are there?
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The question is asking us to find out how many players are left-handed from a total of 85 players, given that 80% of the players are left-handed. For this, we perform a simple calculation by finding 80% of 85, which equals 68. So, there are 68 left-handed players in the league.
Explanation:The subject of this question falls under the category of Mathematics, more specifically, it is about percentage calculations. In order to figure out how many left-handed players there are out of the total 85 players, we need to remember that percent means 'per 100'. So, 80% translates to 80 out of 100. Therefore, to find out how many are left-handed, we need to take 80% of 85.
The calculation is as follows:
Left-handed players = 85 * (80/100)
= 85 * 0.80
= 68
So, there are 68 left-handed players in the local little league.
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meredith bought 7 new t-shirts for $7.95 per shirt. if price included tax how much did she pay for all of the t-shirts? each step please
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Step 1: 7 x $7.95 = $55.65
Solution = $55.65
Mira has breakfast at a restaurant. She leaves a 20% tip of $1.80. What is the price of Mira’s breakfast,before tip?
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9×20%=1.80
So the original price of her breakfast is $9.00
hope this helps!
20% = $1.80
1% = $1.80 / 20%
= $0.09
100 % = $0.09 x 100
= $9
Hope this helps :))
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The product of 32 and an unknown number subtracted from 435 is 179 what is the value of the unknown number
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Hello from MrBillDoesMath!
Answer: Case 1: approximately 429.4, Case 2: 8
Discussion:
Let the unknown be "x". The problem statement could mean either
32 * (435 -x) = 179 (#1)
or
435 - 32*x = 179 (#2)
It's unclear which one is wanted so we solve both of them.
----------------------------------------------#1
But for case #1
32 * (435 -x) = 179
32 * 435 - 32x = 179 or
13920 - 32x = 179
Add 32x to both sides
13920 = 179 + 32x
Subtract 179 from both sides:
13920 - 179 = 32x
Divide both side by x to get
x = (13920 -179)/32 = 13741/32 which is approximately 429.4
For case #2
----------------------------------------------#2
435 - 32x = 179. Add 32 x to each side
435 -32x + 32x = 179 + 32x. Subtract 170 from each side
435 -179 = 32x or
32x = 256 or x = 8
Regards, MrB
In 2015, in Buffalo, New York, there were 8,625 arrests, 2,678 robberies, 865 assaults, and 20 murders. The population of Buffalo is 258,959. What is the ratio of the number of assaults to the number of robberies in simplest form?
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I could be wrong tho
Answer:
The simplest form of the ratio is: [tex]\dfrac{865}{2678}[/tex]
Step-by-step explanation:
We are given information about the buffalo in New york in the year 2015 as:
Number of arrests= 8,625
Number of robberies=2678
Number of assaults= 865
Number of murders= 20
The population of Buffalo is 258,959.
he ratio of the number of assaults to the number of robberies in simplest form is given by:
[tex]\dfrac{865}{2678}[/tex]
As the number in the numerator and the denominator do not have a common factor. hence the simplest form of the ratio is:
[tex]\dfrac{865}{2678}[/tex]
What is the area of a regular polygon with 100 sides and a perimeter of 100 units?
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so, we know each side is 1 unit long, and we know there are 100 sides,
[tex]\bf \textit{area of a regular polygon}\\\\ A=\cfrac{1}{4}ns^2cot\left( \frac{180}{n} \right)\quad \begin{cases} n=\textit{number of sides}\\ s=\textit{length of a side}\\ \frac{180}{n}=central~angle\\ \qquad~~ in~degrees\\ ----------\\ n=100\\ s=1 \end{cases} \\\\\\ A=\cfrac{1}{4}\cdot 100\cdot 1^2\cdot cot\left( \frac{180}{100} \right)\implies A=25cot\left(\frac{9}{5}^o \right) \\\\\\ A\approx 795.51289884[/tex]
If two parallel lines are cut by a transversal and corresponding angles measure 10x – 1 and 8x + 21, what is the value of x?
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10x -1 = 8x + 21
-8x -8x
2x -1 = 21
+1 +1
2x = 22
x = 11
Hope this helps!
for every 5 teachers there are 12 student. if there are a total 153 student and teachers how many teachers and student are present
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5 +12 = 17
153 /17 = 9
9*5 = 45 teachers
9*12 = 108 studets
What operation extends this pattern? 100, 85, 70, 55... A) add 15 B) subtract 15 C) multiply by 15 Eliminate D) divide by 15
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How to estimate 208 + 569
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Hope this help good luck have a nice nite
One hundred pounds of co2 is contained in a 10:0-ft3 tank. the safety limit of the tank is 1600 psig.
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Final answer:
To answer the question, we need to convert the mass from pounds to kilograms, use the ideal gas law equation to calculate the pressure in atmospheres, and then convert atmospheres to psig.
Explanation:
In this question, we are given that 100 pounds of CO2 is contained in a 10.0-ft3 tank and the safety limit of the tank is 1600 psig. To convert pounds to kilograms, we can use the conversion factor of 1 pound = 0.4536 kilograms. So, 100 pounds of CO2 is equal to 45.36 kilograms. Now, we can use the ideal gas law equation PV = nRT to find the pressure in bar. Rearranging the equation, we have P = (nRT) / V. Plugging in the values given, n = mass / molar mass, R = 0.0821 atm L / mol K, T = 273 + degrees Celsius, and V = 10.0 ft3 converted to liters, we can calculate the pressure in atmospheres. Finally, we can convert atmospheres to psig by multiplying by 14.7.
The floor at a roller skating rink is 72.25 feet long and 51.5 feet wide how much longer is the rank than it is wide