Emilio is making soup for a friend. He needs 13 3/4 ounces of chicken broth to make one serving.
If Emilio has 82.5 ounces of chicken broth, how many servings of soup can he make?
Answers
Emilio can make 6 servings of soup with 82.5 ounces of chicken broth.
To determine how many servings of soup Emilio can make with 82.5 ounces of chicken broth, we need to divide the total amount of chicken broth by the amount required per serving.
Emilio needs 13 3/4 ounces of chicken broth for one serving. To convert this mixed number into a proper fraction, we can rewrite 3/4 as 0.75:
13 3/4 ounces = 13 + 3/4 ounces = 13 + 0.75 ounces = 13.75 ounces
Now, we can divide the total amount of chicken broth (82.5 ounces) by the amount required per serving (13.75 ounces):
Number of servings = Total amount of chicken broth / Amount required per serving
Number of servings = 82.5 ounces / 13.75 ounces
Performing the division:
Number of servings = 6
Therefore, Emilio can make 6 servings of soup with 82.5 ounces of chicken broth.
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Related Questions
Which best describes the figure that is represented by the following coordinates?
A(-3, 4), B(3, 4), C(3,-2) and D(-3,-2)
у 4
a trapezoid
10-
8
a square
6
4
a rectangle
10
8
-6
-4
2
8
10%
-20
-2-
a parallelogram
- 10-
Answers
A square with each edge length 6
Can somebody help me please
Answers
The area of figure is 272.52 square units.
The given figure consist:
A parallelogram of,
length = 12
width = 18
Since we know that,
Area of parallelogram = length x width
= 12 x 18
= 216 square units
And it consist of a semicircle of,
radius = 12/2
= 6
Since we know that,
Area of semicircle is = πr²/2
= 3.14 x 6 x 6/2
= 56.52 square units
Thus,
The area of figure is sum of both areas,
⇒ 216 + 56.52
Hence, area is
⇒ 272.52 square units
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Martin was selling tickets for a basketball game in a high school. He sold 1,250 tickets and the total amount collected for the game was $2,750. The student tickets cost $2 each, and adult tickets cost $3 each. How many student and adult tickets were sold?
Answers
Answer:
x = number of students tickets = 1,000
y = number of adults tickets = 250
Step-by-step explanation:
Let
x = number of students tickets
y = number of adults tickets
x + y = 1,250 (1)
2x + 3y = 2,750 (2)
Multiply (1) by 2
x + y = 1,250 (1) * 2
2x + 2y = 2,500 (3)
2x + 3y = 2,750 (2)
Subtract (3) from (2) to eliminate x
3y - 2y = 2,750 - 2,500
y = 250
Substitute y = 250 into (1)
x + y = 1,250
x + 250 = 1,250
x = 1,250 - 250
x = 1,000
x = number of students tickets = 1,000
y = number of adults tickets = 250
All I need is the answer on decimals
Answers
Answer:
3
Step-by-step explanation:
A plumber charges a one-time fee of $75 plus a fixed hourly rate, h, in dollars. The plumber works at a customer's house for 2 hours and charges a total of $205.
Answers
Answer:
hourly fee is $62.5
Step-by-step explanation:
75 is flat fee so it's constant. works 2 hours so flat fee time 2 is 2x.
200=75+2x
200-75=2x
125=2x
62.5=x
find the limit. use l'hospital's rule if appropriate. if there is a more elementary method, consider using it. lim x→[infinity] x tan(2/x)
Answers
The limit is :lim x→[infinity] x tan(2/x)= 2
To determine the limit:
lim x→[infinity] x tan(2/x)
Using derivatives, the L'hospital rule can be used to evaluate limits involving indeterminate forms.
A limit with an ambiguous form is one that does not provide enough information to determine the original limit.
It is a crucial rule in Calculus.
We can use derivatives to find the value of certain types of limits using this rule.
L'Hospital's rule can be applied.
To do so, we take the denominator's and numerator's derivatives with respect to x.
lim x→[infinity] x tan(2/x)
= lim x → ∞ tan(2/x)/1/x
[as x → ∞, 1/x→0]
=>2 lim 1/x → 0 tan(2/x)/2/x
=> 2 lim Ф → 0 sec²2Ф
=> 2× sec²0
=> 2
Therefore,
lim x→[infinity] x tan(2/x)= 2
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Bradley made a house for his dog, Bowser, out of wood with a cube base and a triangular prism top. The dimensions of the dog house are a = 4 feet, b = 1 foot, and c = 2.2 feet.
If Bradley plans to paint the outside of the dog house blue, not including the bottom, how many square feet of paint will he use?
A. 101.6 square feet
B. 85.6 square feet
.
C. 117.6 square feet
Answers
Answer:
The answer to your problem is, 23.6 or 49.8
Step-by-step explanation:
Surface area of cube=4 x a²----> 4 x 2²-----> 16 ft²
Surface area of triangular prism=2*[a*c]+2*[a*b/2]---> 2*[2*1.4]+[2*1]
Surface area of triangular prism=5.6+2----> 7.6 ft²
Surface area of the figure=16 ft²+7.6 ft²----> 23.6 ft²
Thus the answer to your problem is, 23.6 or 49.8
It could be either one did the same quiz but I had different answers every time so choose which one you have :).
A certain hallway in a crowded nightclub has a unique sound property. Every 2 feet closer to the stage you walk, the sound intensity (dB) doubles. If the threshold for pain is 127 dB, and the starting sound level is 2 dB, about how many feet down the hallway can you go until you start to feel pain
Answers
The number of feet down the hallway where you go until you start to feel pain is 14 feet.
What is geometric progression GP?A sort of sequence known as geometric progression (GP) is one in which each following phrase is created by multiplying every preceding term by such a fixed number, or "common ratio."
The general form of GP is
a, ar, ar², ar³ ... arⁿ⁻¹
where, a is the initial term.
r is the common ratio.
According to the question,
The initial term is; a = 2.
The common ratio r = 2.
The threshold value beyond which person start feeling pain is 127.
Thus, last term; arⁿ⁻¹ = 128 (1 feet beyond 127 person start feeling pain)
Last second value is arⁿ⁻².
As, the common ratio is same in GP.
Thus, arⁿ⁻¹ / arⁿ⁻² = ar/a
==> 128/arⁿ⁻² = r
==> 128/arⁿ⁻² = 2
==> 2×2ⁿ⁻² = 64
==> 2ⁿ⁻² = 32
==> n = 7
As, i step is of 2 feet, the total steps taken by the person is 2×7 = 14.
Therefore, the total number of steps beyond which person will feel pain is 14.
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Gwen runs back and forth along straight track: During the time interval 0 < t < 45 seconds, Gwens 250ain velocity; In feet per second, is modeled by the function given by v (t) What is the first time;t1 , that Gwen changes direction? Find Gwens average velocity over the time interval 0 < t
Answers
The average velocity of Gwen over the time interval 0 < t is zero. We need to solve the equation:250sin(πt/45) = 0Solving for t, we get:πt/45 = nπwhere n is an integer.
Given that Gwen runs back and forth along a straight track and her velocity, in feet per second, is modeled by the function v(t) during the time interval 0 < t < 45 seconds; We are to determine the first time at which Gwen changes direction and find her average velocity over the time interval 0 < t.Firstly, we know that velocity is a vector quantity and has both magnitude and direction.
Since she is running back and forth along a straight track, her displacement at any given time t is given by the function s(t), which is the integral of her velocity function v(t).That is, s(t) = ∫v(t)dtWe can find the displacement by taking the definite integral of v(t) from 0 to t. Since Gwen is running back and forth, her displacement will be zero at the times when she changes direction.
Therefore, we need to solve the equation:250sin(πt/45) = 0Solving for t, we get:πt/45 = nπwhere n is an integer. Therefore,t = 45n/πwhere n is an integer. Since we are looking for the first time at which Gwen changes direction, we need to take the smallest positive value of n, which is n = 1.
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what is the product of 2x+ y and 5x-y +3
Answers
Answer:
Step-by-step explanation:
= 10x^2 - 2xy + 6x + 5xy - y^2 + 3y
Now, we can combine like terms: = 10x^2 + 3xy + 6x - y^2 + 3y
So, the product of (2x + y) and (5x - y + 3) is 10x^2 + 3xy + 6x - y^2 + 3y.
Logical equivalence of two English statements.
Define the following propositions:
j: Sally got the job.
l: Sally was late for her interview
r: Sally updated her resume.
Express each pair of sentences using a logical expression. Then prove whether the two expressions are logically equivalent.
(a)
If Sally did not get the job, then she was late for interview or did not update her resume.
If Sally updated her resume and was not late for her interview, then she got the job.
(b)
If Sally got the job then she was not late for her interview.
If Sally did not get the job, then she was late for her interview.
(c)
If Sally updated her resume or she was not late for her interview, then she got the job.
If Sally got the job, then she updated her resume and was not late for her interview.
Answers
As a result, the two phrases are logically equal. The two formulations are not logically identical, as we can see. The two formulations are not logically identical, as we can see.
What is expression?An expression or mathematical expression is a finite collection of symbols that is well-formed according to context-dependent norms.
Here,
(a) The first statement can be expressed as:
(~j -> (l v ~r))
The second statement can be expressed as:
((r ^ ~l) -> j)
To determine whether the two expressions are logically equivalent, we can use truth tables. A truth table is a method for checking the validity of logical equivalences by comparing the truth values of the two expressions for all possible combinations of inputs.
Truth table for (~j -> (l v ~r)) and ((r ^ ~l) -> j):
j l r (~j -> (l v ~r)) ((r ^ ~l) -> j)
T T T T T
T T F T T
T F T F F
T F F T T
F T T T T
F T F T T
F F T T T
F F F T T
Since both expressions have the same truth values for all possible combinations of inputs, we can conclude that the two expressions are logically equivalent.
(b) The first statement can be expressed as:
(j -> ~l)
The second statement can be expressed as:
(~j -> l)
Truth table for (j -> ~l) and (~j -> l):
j l (j -> ~l) (~j -> l)
T T T F
T F T T
F T T T
F F T T
Since the expressions have different truth values for some combinations of inputs, we can conclude that the two expressions are not logically equivalent.
(c) The first statement can be expressed as:
((r v ~l) -> j)
The second statement can be expressed as:
(j -> (r ^ ~l))
Truth table for ((r v ~l) -> j) and (j -> (r ^ ~l)):
j l r ((r v ~l) -> j) (j -> (r ^ ~l))
T T T T T
T T F T T
T F T T T
T F F T T
F T T T F
F T F T F
F F T T F
F F F T T
Since the expressions have different truth values for some combinations of inputs, we can conclude that the two expressions are not logically equivalent.
We can conclude that the two expressions are logically equivalent. We can conclude that the two expressions are not logically equivalent. We can conclude that the two expressions are not logically equivalent.
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Let you invest the amount of money equal to the last 6 digits of your student id. If the interest earned id 9.95% compounded monthly, what will be the balance in your account after 7 years? What is the effective rate of interest? HINT: If your student id is A00123456, the amount of monev is $123456
Answers
The amount of money equal to the last 6 digits of your student id. If the interest earned id 9.95% compounded monthly the effective rate of interest is approximately 10.47%.
If the amount of money you invest is $123456, and the interest earned is 9.95% compounded monthly, we can calculate the balance in your account after 7 years using the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = Final balance
P = Principal amount (initial investment)
r = Annual interest rate (in decimal form)
n = Number of times the interest is compounded per year
t = Number of years
In this case:
P = $123456
r = 9.95% = 0.0995 (expressed as a decimal)
n = 12 (compounded monthly)
t = 7 years
Plugging in these values, we can calculate the final balance:
A = 123456(1 + 0.0995/12)^(12*7)
A ≈ 123456(1.00829166667)^(84)
A ≈ 123456(1.85972923908)
A ≈ $229,464.03
Therefore, the balance in your account after 7 years would be approximately $229,464.03.
To calculate the effective rate of interest, we can use the formula:
Effective Rate = (1 + r/n)^n - 1
Plugging in the values:
r = 9.95% = 0.0995 (expressed as a decimal)
n = 12 (compounded monthly)
Effective Rate = (1 + 0.0995/12)^12 - 1
Effective Rate ≈ (1.00829166667)^12 - 1
Effective Rate ≈ 1.10471160007 - 1
Effective Rate ≈ 0.10471160007
Therefore,
If the amount of money you invest is $123456, and the interest earned is 9.95% compounded monthly, we can calculate the balance in your account after 7 years using the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = Final balance
P = Principal amount (initial investment)
r = Annual interest rate (in decimal form)
n = Number of times the interest is compounded per year
t = Number of years
In this case:
P = $123456
r = 9.95% = 0.0995 (expressed as a decimal)
n = 12 (compounded monthly)
t = 7 years
Plugging in these values, we can calculate the final balance:
A = 123456(1 + 0.0995/12)^(12*7)
A ≈ 123456(1.00829166667)^(84)
A ≈ 123456(1.85972923908)
A ≈ $229,464.03
Therefore, the balance in your account after 7 years would be approximately $229,464.03.
To calculate the effective rate of interest, we can use the formula:
Effective Rate = (1 + r/n)^n - 1
Plugging in the values:
r = 9.95% = 0.0995 (expressed as a decimal)
n = 12 (compounded monthly)
Effective Rate = (1 + 0.0995/12)^12 - 1
Effective Rate ≈ (1.00829166667)^12 - 1
Effective Rate ≈ 1.10471160007 - 1
Effective Rate ≈ 0.10471160007
Therefore, the effective rate of interest is approximately 10.47%.
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Write a program that will solve a quadratic equation: ax2 + bx + c = 0. The program must ask the user to enter the values for a, b and c. It should then determine the discriminant of the parabola and based on the discriminant it should display the real roots: Discriminant = b2 - 4ac > 0 means there will be two distinct real roots. Discriminant = b2 - 4ac = 0 means there will be only one real root. Discriminant = b2 - 4ac < 0 means there will be no real roots. Use the if - then else structure in your program to decide the outcome of the program. User must then use the quadratic equation to solve for x: = − ± √ −
Answers
Answer:
x= -b ±√b²- 4ac/2a
Step-by-step explanation:
ax2 + bx + c = 0
Dividing the quadratic equation by a gives
x2 + b/ax + c/a = 0
x2 + b/ax = - c/a
Applying the square formula
(x)^2 + 2(x) (b/2a) + (b/2a)^2= + (b/2a)^2- c/a
(x+b/2a)^2= b²/4a² - c/a
(x+b/2a)^2= b²/4a² - 4ac/4a²
(x+b/2a)^2= b²- 4ac/4a²
Taking square root of both sides gives
(x+b/2a)= ±√b²- 4ac/2a
x= -b/2a±√b²- 4ac/2a
x= -b ±√b²- 4ac/2a
This is called the quadratic formula and it can solve the value for x for the given value of a, b and c for the quadratic equation.
From this it is concluded that
Discriminant = b2 - 4ac > 0 means there will be two distinct real roots. Discriminant = b2 - 4ac = 0 means there will be only one real root. Discriminant = b2 - 4ac < 0 means there will be no real roots.
This is because a square root of a value greater than 0 is possible and gives two values for roots of x.
The square root of a value less than 0 gives a negative number and therefore the roots are an imaginary number.
The square root of a value greater equal to 0 gives a zero leaving behind only one real root.
what equations are equivalent to 60% of 25
Answers
Step-by-step explanation:
Since,
\(60\%25 = 15\)
Another equation that equals 15 is
\(25 \times 0.6 = 15\)
please answer, click on file, thank you
Answers
Answer:
3x²
im lazy to solve right now
hope it helps
Answer:
Step-by-step explanation:
half of the circle is 180 degrees
so x + 3x = 180
4x = 180
x = 45
What is x rounded to the nearest hundredth?
Answers
Answer:
Step-by-step explanation:
7/6x=140
x=140*6/7
x=120
What are the coordinates of Q' if PQR is translated two units right and three units down ?
Answers
EXPLANATION:
Q is currently at 0,5.
If we go 2 units right, then it is 0 + 2.
Now we have 2,5.
If we go 3 units down, then its 5 - 3.
Now we have 2,2.
Hope this helps! Gl!
Step-by-step explanation:
Please help!! 1/6+1/3+1/6
Answers
Answer:
2/3
Step-by-step explanation:
1/6+1/6= 2/6, both divided by 2 gives you 1/3
1/3+1/3= 2/3
Good luck luv
Hope this helps
Answer:
2/3
Step-by-step explanation:
To add fractions with different denominators, change the denominator to the least common multiple of the denominators.
The denominators here are 6, 3, and 6. The least common multiple of these three numbers is 6. Now, convert each of the fractions into fractions with a denominator of 6, specifically only 1/3.
1/6+2/6+1/6
Now add up the numerators,
1/6+2/6+1/6 = 4/6
Simplify.
4/6 = 2/3
Find all solutions of the equation in the interval [0, 21). tan²0-2 sec 0 = −1 Write your answer in radians in terms of . If there is more than one solution, separate them with commas. 0 = 0 П 0,0
Answers
The solution to the equation tan²θ - 2secθ = -1 in the interval [0, 21) is θ = 0, π.
Interval's equation solutions within [0, 21)?To solve the equation tan²θ - 2secθ = -1 in the interval [0, 21), we'll apply trigonometric identities and algebraic manipulation. First, we'll rewrite secθ as 1/cosθ and substitute it into the equation:
tan²θ - 2/cosθ = -1
Next, we'll convert tan²θ to its equivalent in terms of sin and cos:
(sinθ/cosθ)² - 2/cosθ = -1
Simplifying the equation further, we obtain:
(sin²θ - 2cosθ)/cos²θ = -1
Multiplying through by cos²θ, we have:
sin²θ - 2cosθ = -cos²θ
Rearranging the terms, we get:
sin²θ + cos²θ - 2cosθ = 0
Using the Pythagorean identity sin²θ + cos²θ = 1, we can rewrite the equation as:
1 - 2cosθ = 0
Solving for cosθ, we find:
cosθ = 1/2
Since we're interested in solutions within the interval [0, 21), we need to find the values of θ for which cosθ = 1/2 within this range. The cosine of π/3 and 5π/3 is indeed 1/2. However, only π/3 lies within the interval [0, 21), so it is the solution to the equation.
Hence, the solution to the equation tan²θ - 2secθ = -1 in the interval [0, 21) is θ = π/3.
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what is X in the equation. -38+6x = -12x+52
Answers
Answer:
x=5
Step-by-step explanation:
add 38 to to the pther side
combine 38+52=90
add 12 to the other side
12+6=18
divide 90 by 18= 5
1. What is the domain of the function below? (1 point)
{(0, 2), (3, 1), (5,2), (8,4)}?
{1, 2, 4)
{0,3,5,8)
O{0, 1, 2, 3, 4, 5, 87
Of(0, 2), (3, 1), (5,2), (8,4)}
者
Answers
Answer:
0,3,5,8
Step-by-step explanation:
Domain numbers are always on the left
Range is always on the right
Hope this helps!:)
hey please!! Thank you <33
Answers
Answer:
h÷2
Step-by-step explanation:
\(2\:gallons =1\:hour\\2 \: gallons = h \:hours\\2h = 2\\\frac{h}{2} = \frac{2}{2}\)
what method of probability assessment would most likely be used to assess the probability that a major earthquake will occur in california in the next three years?
Answers
Seismic hazard analysis is a method of probability assessment that would most likely be used to assess the probability that a major earthquake will occur in California in the next three years.
Seismic hazard analysis is a method used to assess the probability of earthquakes occurring in a specific location, and it involves analyzing the historical records of earthquakes, geological data, and the tectonic plate movements in the area. This method is used to predict the likelihood of future earthquakes and their potential magnitude and intensity.
In the case of California, a seismic hazard analysis would likely be used to assess the probability of a major earthquake occurring in the next three years. The data collected from this analysis would help the government and the public prepare for potential earthquakes and minimize the damage caused by such events.
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The regular price of a sweater is $32. For a sale, the price of the sweater is marked down 30%. What is the sale price of the sweater?
Answers
Answer: $22.40
Step-by-step explanation:
30% of 32, or 0.3(32) is equal to 9.6. Then you have to subtract that amount, which is 32 - 9.6, which then gets us to 22.40.
A bag contains 8 red marbles, 9 yellow marbles, and
7 green marbles. How many additional red marbles
must be added to the 24 marbles already in the bag so
that the probability of randomly drawing a red marble
is 3/5?
A. 11
B. 16
C. 20
D. 24
E. 32
Answers
Answer:
x = 16
Step-by-step explanation:
\(\frac{8+x}{24+x}=\frac{3}{5}\\ \\ 5(8+x)=3(24+x)\\\\40+5x=72+3x\\\\5x=32+3x\\\\2x=32\\\\x=16\)
ANSWER ALL THESE QUESTIONNN WITH THE WORK AND YOU WILL GET 25 POINTS AND BRAINILEST
Answers
Answer:
1: 16 2: 15 3: 25 4: 12
I don't know any more than that but I hope this helps a little
Step-by-step explanation:
BRAINLY AND 20 POINTS IF ANSWERED!!!!!! roberto is walking. The distance, D, in meters, he walks can be found using the equation D=1. 4t, where t is time in seconds
[ ] meters per second
Answers
1.4m/s is the rate that Roberto is walking. We know the formula for calculating the time i.e. t= d/r.
The term "distance" refers to how far we move. The rate is a measurement of our trip speed. Time is measured by how far we travel. The distance an object will travel over time and at a specific average rate is the subject of rate problems.
Given,
Distance = D
D= 1.4t
Rate= ?
Substituting the given values in the formula t= d/r
where,
t= time in seconds
d= distance
r= rate
We get,
t= 1.4t/r
t/1.4t= 1/r
t gets cancelled
so we have,
1/1.4= 1/r
r= 1.4m/s
Therefore, 1.4m/s is the rate at which Roberto is walking.
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The complete question is as follows:
Roberto is walking. The distance, D, in meters, he walks can be found using the equation D=1. 4t, where t is time in seconds.
What is the rate that Roberto's walking in meters per second?
a consumer electronics company is comparing the brightness of two different types of picture tubes for use in its television sets. tube type a has mean brightness of 100 and standard deviation of 16, and tube type b has unknown mean brightness, but the standard deviation is assumed to be identical to that for type a. a random sample of tubes of each type is selected, and is computed. if equals or exceeds , the manufacturer would like to adopt type b for use. the observed difference is .
Answers
The probability that , Xb exceeds Xa , by 3.0 or more if ub and ua, are equal is 0.2537.
The probability of an occurrence is a figure that represents how likely it is that the event will take place. In terms of percentage notation, it is stated as a number between 0 and 1, or between 0% and 100%. The higher the likelihood, the more probable it is that the event will take place.
Probability is a way to gauge how likely something is to happen. Several things are difficult to forecast with absolute confidence. With it, we can only make predictions about the likelihood of an event happening, or how likely it is.
n1 = n2 = 25,
hypothesis,
standard error for difference,
\(\sqrt{\frac{16^2}{25} +\frac{16^2}{25} }\)
=4.525
z =(3-0)/4.525
z=0.663
P(z ≥ 0.663) = 0.2537.
No, there is not strong evidence that \(\mu _B\) is greater than \(\mu _A\).
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Complete question;
A consumer electronics company is comparing the brightness of two different types of picture tubes for use in its television sets. Tube type A has mean brightness of 100 and standard deviation of 16, and tube type B has unknown mean brightness, but the standard deviation is assumed to be identical to that for type A. A random sample of n = 25 tubes of each type is selected, and X -X, is computed. If u, equals or exceeds u,, the manufacturer would like to adopt type B for use. The observed difference is X,X, - 3.0. a. What is the probability that , exceeds X, by 3.0 or more if ug and u, are equal? b. Is there strong evidence that ug is greater than u,?
10^3
10^14=10negative^11
true or false
heeellllp asap
Answers
Answer:
False
Step-by-step explanation:
Using the rule of exponents
\(a^{m}\) × \(a^{n}\) ⇔ \(a^{(m+n)}\) , then
\(10^{3}\) × \(10^{14}\) = \(10^{(3+14)}\) = \(10^{27}\)
a. Find 5
∑
n=0
(9(2) n)−7(−3) n)
b. Given the following premises are p→q,¬p→r, and r→s. Prove that ¬q→s
c. Show that ¬(p∨¬q) and q∧¬p are equivalent by:
Answers
By using the same logic and identity, we can also say that ¬(p∨¬q) is equivalent to q∧¬p.
a. To find the given series i.e., 5∑n=0(9(2)n)−7(−3)nTo find 5∑n=0(9(2)n)−7(−3)n,
we need to find the first five terms of the series. The given series is,
5∑n=0(9(2)n)−7(−3)n5[(9(2)0)−7(−3)0] + [(9(2)1)−7(−3)1] + [(9(2)2)−7(−3)2] + [(9(2)3)−7(−3)3] + [(9(2)4)−7(−3)4]
After evaluating, we get:
5[(9*1) - 7*1] + [(9*2) - 7*(-3)] + [(9*4) - 7*9] + [(9*8) - 7*(-27)] + [(9*16) - 7*81]15 + 57 + 263 + 1089 + 4131= 5555b.
Given premises: p → q, ¬p → r, r → s.
We are to prove that ¬q → s. i.e.,
Premises: (p → q), (¬p → r), (r → s)
Conclusion: ¬q → s
To prove ¬q → s,
we need to assume ¬q and show that s follows.
Then we use the premises to derive s.
Proof:
1. ¬q Assumption
2. ¬(¬q) Double negation
3. p Modus tollens 2,1 & p → q
4. ¬¬p Double negation
5. ¬p Modus ponens 4,3 (Conditional elimination)
6. r Modus ponens 5,2 (Conditional elimination)
7. s Modus ponens 6,3 (Conditional elimination)
8. ¬q → s Conditional introduction (Implication)
Thus, ¬q → s is proven.
c. To show that ¬(p∨¬q) and q∧¬p are equivalent, we need to show that their negation is equivalent. i.e.,
we show that (p ∨ ¬q) ↔ ¬(q ∧ ¬p)Negation of (p ∨ ¬q) = ¬p ∧ q Negation of (q ∧ ¬p) = ¬q ∨ p
Thus, we are to show that (p ∨ ¬q) ↔ ¬(q ∧ ¬p) is equivalent to ¬p ∧ q ↔ ¬q ∨ p
Proof:
¬(q ∧ ¬p) ≡ ¬q ∨ p Negation of (q ∧ ¬p)(p ∨ ¬q) ≡ ¬(q ∧ ¬p)
De Morgan's laws ∴ (p ∨ ¬q) ≡ ¬q ∨ p
By using the same logic and identity, we can also say that ¬(p∨¬q) is equivalent to q∧¬p.
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a. The formula given is, ∑n=0(9(2)n)−7(−3)n. Let’s find out the first five terms of the given formula as follows:
First term at n = \(0:9(2)^0-7(-3)^0= 9 + 7= 16\)
Second term at n = \(1:9(2)^1-7(-3)^1= 18 + 21= 39\)
Third term at n = \(2:9(2)^2-7(-3)^2= 36 + 63= 99\)
Fourth term at n = \(3:9(2)^3-7(-3)^3= 72 + 189= 261\)
Fifth term at n = \(4:9(2)^4-7(-3)^4= 144 + 567= 711\)
Therefore, the first five terms of the given formula are: 16, 39, 99, 261, 711.
b. To prove that ¬q→s from p→q, ¬p→r, and r→s,
we need to use the law of contrapositive for p→q as follows:
¬q→¬p (Contrapositive of p→q)¬p→r (Given)
∴ ¬q→r (Using transitivity of implication) r→s (Given)
∴ ¬q→s (Using transitivity of implication)
Therefore, ¬q→s is proved.
c. To show that ¬(p∨¬q) and q∧¬p are equivalent,
we need to use the De Morgan’s laws as follows:
¬(p∨¬q) ≡ ¬p∧q (Using De Morgan’s law)
≡ q∧¬p (Commutative property of ∧)
Therefore, ¬(p∨¬q) and q∧¬p are equivalent.
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