a) For every integer n, if n^3 is even, then n is even.
b) For every integer n, if n^2−2n+7 is even, then n is odd.
c) For every integer n, if n^2 is not divisible by 4, then n is odd.
d) For every pair of integers x and y, if xy is even, then x is even or y is even.
To prove each statement by contrapositive, we will negate the original statement and prove the negation. If the negation of the statement is true, then the original statement is also true.
a) Original statement: For every integer n, if n^3 is even, then n is even.
Contrapositive statement: For every integer n, if n is not even, then n^3 is not even.
To prove the contrapositive, we need to show that if n is not even, then n^3 is not even.
If n is not even, then it must be odd. Let's assume n = 2k + 1, where k is an integer.
Substituting this value of n into n^3, we get:
n^3 = (2k + 1)^3 = 8k^3 + 12k^2 + 6k + 1
We can see that n^3 is of the form 8k^3 + 12k^2 + 6k + 1, which is an odd number. Therefore, the contrapositive statement is true, and by contrapositive, the original statement is also true.
b) Original statement: For every integer n, if n^2−2n+7 is even, then n is odd.
Contrapositive statement: For every integer n, if n is even, then n^2−2n+7 is not even.
To prove the contrapositive, we need to show that if n is even, then n^2−2n+7 is not even.
If n is even, then it can be written as n = 2k, where k is an integer.
Substituting this value of n into n^2−2n+7, we get:
n^2−2n+7 = (2k)^2−2(2k)+7 = 4k^2−4k+7
We can see that n^2−2n+7 is of the form 4k^2−4k+7, which is an odd number. Therefore, the contrapositive statement is true, and by contrapositive, the original statement is also true.
c) Original statement: For every integer n, if n^2 is not divisible by 4, then n is odd.
Contrapositive statement: For every integer n, if n is even, then n^2 is divisible by 4.
To prove the contrapositive, we need to show that if n is even, then n^2 is divisible by 4.
If n is even, then it can be written as n = 2k, where k is an integer.
Substituting this value of n into n^2, we get:
n^2 = (2k)^2 = 4k^2
We can see that n^2 is of the form 4k^2, which is divisible by 4. Therefore, the contrapositive statement is true, and by contrapositive, the original statement is also true.
d) Original statement: For every pair of integers x and y, if xy is even, then x is even or y is even.
Contrapositive statement: For every pair of integers x and y, if x is odd and y is odd, then xy is not even.
To prove the contrapositive, we need to show that if x is odd and y is odd, then xy is not even.
If x is odd, then it can be written as x = 2k + 1, where k is an integer.
If y is odd, then it can be written as y = 2m + 1, where m is an integer.
Substituting these values of x and y into xy, we get:
xy = (2k + 1)(2m + 1) = 4km + 2k + 2m + 1
We can see that xy is of the form 4km + 2k + 2m + 1, which is an odd number. Therefore, the contrapositive statement is true, and by contrapositive, the original statement is also true.
In summary, we have proven each statement by its contrapositive. The original statements are as follows:
a) For every integer n, if n^3 is even, then n is even.
b) For every integer n, if n^2−2n+7 is even, then n is odd.
c) For every integer n, if n^2 is not divisible by 4, then n is odd.
d) For every pair of integers x and y, if xy is even, then x is even or y is even.
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Find the work done by the force field F in moving an object from A to B. F(x y) = 6y3/2 i + 9x x square root y j A{ 1, 1), B(3, 4)
The integrate is W = 12 ∫[0 to 1] (1 + 3t)^(3/2) dt + 27 ∫[0 to 1] (1 + 2t) √(1 + 3t) dt.
To find the work done by the force field F in moving an object from point A to point B, we can use the line integral of the force field along the path connecting A and B. The line integral is given by the formula:
W = ∫[C] F · dr
where C represents the curve connecting points A and B, F is the force field, dr is the differential displacement vector along the curve, and · denotes the dot product.
First, let's calculate the differential displacement vector dr. Since we are moving from point A(1, 1) to point B(3, 4), the differential displacement vector dr can be expressed as:
dr = dx i + dy j
Now, let's determine the curve C connecting A and B. The curve can be parametrized as follows:
x(t) = 1 + 2t
y(t) = 1 + 3t
where t varies from 0 to 1.
Differentiating these parametric equations, we obtain:
dx = 2 dt
dy = 3 dt
Substituting these values into the differential displacement vector dr, we have:
dr = 2 dt i + 3 dt j
Next, let's calculate the force field F at each point along the curve C. Given that F(x, y) = 6y^(3/2) i + 9x √y j, we can substitute the parametric equations for x and y into F:
F = 6(1 + 3t)^(3/2) i + 9(1 + 2t) √(1 + 3t) j
Now, let's evaluate the line integral by substituting the values of F and dr into the integral expression:
W = ∫[0 to 1] (6(1 + 3t)^(3/2) i + 9(1 + 2t) √(1 + 3t) j) · (2 dt i + 3 dt j)
Expanding the dot product, we have:
W = ∫[0 to 1] (12(1 + 3t)^(3/2) dt + 27(1 + 2t) √(1 + 3t) dt)
Now, we can simplify and integrate each term separately:
W = 12 ∫[0 to 1] (1 + 3t)^(3/2) dt + 27 ∫[0 to 1] (1 + 2t) √(1 + 3t) dt
To evaluate these integrals, we can use appropriate substitution or integration techniques. After integrating both terms, we will have the value of the work done by the force field F in moving the object from A to B along the curve C.
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help with all of u can, I’d appreciate it. Have a good day
Answer:
a. -12n^3 + 8n^2 - 2n
b. n^4 - n/2
c. 6b^7 + 2b^6 - b^5
Step-by-step explanation:
btw photomath (the app) is really good for this type of stuff
PLEASE HELP
A water park charges an entrance fee of $15 and $3 for each ride. Select the function that models the relationship between the total amount spent (A) and the number of rides purchased (n).
A. A= 15n+ 18
B. A= 15n+3
C. A= 3n+18
D. A=3n+ 15 X
Answer:
D. A = 3n + 15
Step-by-step explanation:
3 should be beside the n variable because n represents each ride that was purchased.
As 15 represents the fee
The A variable represents the total amount spent
Therefore your answer is D.
Your Welcome.
Can someone help me with this. Need answer and explanation/work. Thank you. Will Mark brainliest.
Answer:
The hikers are 5 miles apart.
Step-by-step explanation:
East + West = 3 miles apart (x-axis)
North + South = 4 miles apart (y-axis)
Then use the pythagorean theorem to find the distance between them.
sqrt(distance) = sqrt(3^2+4^2)
distance = sqrt(25)
distance = 5 miles
Tad and his buddies drove to the orange bowl to see their favorite football team win the national championship because of heavy traffic they only avsraged 50 mph however on the return trip they averaged 75 mph if the total round trip took 12 hours how long did it take tad and his buddies to drive to the orange bowl
Answer:
7.2
Step-by-step explanation:
Answer:
Step-by-step explanation:
Distance "D" = rate × time
or d = r × t.
Tad and his buddies drove to the orange bowl(while going) -50mph
Tad and his buddies while returning from a trip -75mph
Total round trip = 12 hours
Distance while going = 50 x t
Distance while returning = 75 (12 - t)
50t = 75(12 -t)
50t = 900 -75t
50t + 75t = 900
125t = 900
t = 900/125
t= 7.2
The total number of hours it takes Tad and his buddies to drive to the orange bowl is 7.2 hours.
Rachel makes $38.25 in 3 hours.
How much money did she make
per hour?
4
HELPPPPP 10 points
I NEED IT NOW
Answer:
1 hour: $12.75
Step-by-step explanation:
Easy divide 38.25 by 3 which is 12.75
HELP NQNSJNDNXJJXnwnfnf
Answer:
C. 14 is the answerStep-by-step explanation:
1. Write the question.
17 - 2x = -11
2. Subtract 17 from both sides
-17 -17
- 2x = -28
3. Negatives cancel
2x = 28
4. Divide both sides by 2
/2 /2
x = 14
5. Check the answer, write the equation.
17 - 2x = -11
6. Substitute x with the value
17 - 2(14) = -11
17 - 28 = -11
-11 = -11
The answer is correct!
So, 14 is the answerHope this helped,
Kavitha
Find the volume of compostie figure
Step-by-step explanation:
The first composite shape is a combination of a rectangular prism and a pyramid. To find the volume of the entire shape you find the volume of each individual shape and add them together. The second figure consists of a cylinder and a hemisphere.
Hope that helps :)
what is the equation of a line that is parallel to the line 2x 5y = 10 and passes through the point (–5, 1)? check all that apply.
A. y = −x − 1
B. 2x 5y = −5
C. y = −x − 3
D. 2x 5y = −15 y
E. − 1= −(x 5)
The equation of a line that is parallel to the line 2x - 5y = 10 and passes through the point (-5, 1) is B. 2x - 5y = -5.
To find the equation of a line that is parallel to the line 2x - 5y = 10, we need to determine the slope of the given line first. The equation is in the form of Ax + By = C, where A = 2, B = -5, and C = 10.
To find the slope, we can rearrange the equation into slope-intercept form (y = mx + b), where m represents the slope.
2x - 5y = 10
-5y = -2x + 10
y = (2/5)x - 2
From this equation, we can see that the slope of the given line is 2/5.
A line that is parallel to this line will have the same slope. Therefore, the equation of the parallel line can be determined using the point-slope form (y - y1 = m(x - x1)), where (x1, y1) represents the coordinates of the given point (-5, 1).
Using the slope of 2/5 and the point (-5, 1), we can now check the options to see which ones satisfy the conditions:
A. y = -x - 1: This equation has a slope of -1, not 2/5. It is not parallel to the given line.
B. 2x - 5y = -5: This equation has the same slope of 2/5 and passes through the point (-5, 1). It satisfies the conditions and is parallel to the given line.
C. y = -x - 3: This equation has a slope of -1, not 2/5. It is not parallel to the given line.
D. 2x - 5y = -15y: This equation has a slope of 2/20, which simplifies to 1/10. It is not parallel to the given line.
E. -1 = -(x - 5): This equation does not represent a line. It is not a valid option.
Therefore, the equation of a line that is parallel to the line 2x - 5y = 10 and passes through the point (-5, 1) is B. 2x - 5y = -5.
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If the probability of surviving the zombie apocalypse for 1 year is 20% a. what are the odds of surviving (explain)? (1 pt) b. What are the odds of not surviving (explain)? (1 pt) c. If I bet $25 on you to survive, and you do, how much would you pay based on the odds? (1 pt)
a) The odds of surviving is 1/4
b) The odds of not surviving is 4.
c) You would receive a payout of $6.25 if you bet $25 on surviving and you successfully survive the zombie apocalypse.
a) The odds of surviving the zombie apocalypse can be calculated by taking the probability of survival (20%) and dividing it by the probability of not surviving
100% - 20% = 80%
So the odds of surviving would be,
20%/80% = 1/4
This means that for every one chance of surviving, there are four chances of not surviving.
b) The odds of not surviving the zombie apocalypse can be calculated by taking the probability of not surviving (80%) and dividing it by the probability of survival (20%).
So the odds of not surviving would be
80%/20% = 4/1
This means that for every four chances of not surviving, there is one chance of surviving.
c) If you bet $25 on surviving and you do survive, based on the odds of 1/4, you would win the bet. The payout can be calculated by multiplying the bet amount by the odds of surviving.
So the payout would be
$25 * 1/4 = $6.25
Therefore, based on the odds, you would receive a payout of $6.25 if you bet $25 on surviving and you successfully survive the zombie apocalypse.
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use base conversion method, convert the decimal expansion (25679),, to an Octal expansion.
The decimal number (25679)₁₀ is equivalent to the octal number (62117)₈. The base conversion method, also known as the radix conversion method, is a systematic approach to converting numbers from one base to another.
To convert the decimal expansion (25679)₁₀ to an octal expansion, we can use the base conversion method.
Step 1: Divide the decimal number by 8 and keep track of the remainders.
25679 ÷ 8 = 3209 remainder 7
3209 ÷ 8 = 401 remainder 1
401 ÷ 8 = 50 remainder 1
50 ÷ 8 = 6 remainder 2
6 ÷ 8 = 0 remainder 6
Step 2: The remainders from bottom to top form the octal expansion.
Therefore, (25679)₁₀ = (62117)₈.
Hence, the decimal number (25679)₁₀ is equivalent to the octal number (62117)₈.
The question should be:
Use base conversion method, convert the decimal expansion (25679)₁₀ to an Octal expansion.
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What information is needed in order to apply the hypotenuse-leg (HL) theorem
Answer:
This theorem states that 'if the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle, then the triangles are congruent. ' This is kind of like the SAS, or side-angle-side postulate. But SAS requires you to know two sides and the included angle.
Step-by-step explanation:
i need help asap.!!!!!
what is the probability rule for deciding whether events a and b are independent
Events A and B are considered independent if and only if the probability of their intersection (P(A ∩ B)) is equal to the product of their individual probabilities (P(A) * P(B)).
Independence: Two events A and B are independent if the occurrence or non-occurrence of one event does not affect the probability of the other event.
Joint Probability: The joint probability P(A ∩ B) represents the probability of both events A and B occurring together.
Multiplication Rule: According to the multiplication rule for independent events, if events A and B are independent, the probability of their intersection is equal to the product of their individual probabilities.
P(A ∩ B) = P(A) * P(B)
Interpretation: If the equation holds true, events A and B are considered independent since the probability of their intersection can be determined solely by multiplying their individual probabilities.
Dependence: If the equation does not hold true, it implies that the occurrence of one event affects the probability of the other event, indicating dependence between the two events.
In summary, the multiplication rule for independent events states that events A and B are independent if and only if P(A ∩ B) = P(A) * P(B).
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What is the area of this
Answer:
area of trapezoid = 1/2×(a+b)×height
area of trapezoid =1/2×(15m+7m)×7m
area of trapezoid =1/2×22m×7
area of trapezoid =77m²
area of trapezoid =1/2×(a+b)×height
area of trapezoid =1/2×(9ft+12ft)×4ft
area of trapezoid =1/2×21ft×4ft
area of trapezoid =42ft²
Raymond bought a car for $40,000. He took a 20,000 loan from a bank at an interest rate of 15% per year for a 5 - year period. What is the total amount ( interest and loan) that he would have to pay rhe bank at the end of 5 years
Answer:
$35,000
Step-by-step explanation:
The amount that would be repaid = amount borrowed + interest earned on loan
interest earned on deposit can be determined by determining the simple interest
Simple interest = principal x time x interest rate
principal = the amount deposited = 20,000
Time = the duration of the deposit =5
interest rate = the percentage on deposit that would be earned = 15
20,000 x 5 x 0.15 = $15,000
total = 20,000 + 15,000 = $35,000
Which is NOT an equation?
a
x + 14 = 32
b
3 + x
c
10 – 2x = 10 + 5x
d
12 = 0.5x – 3
Answer:
B because it does not contain an = sign
P and Q are points on the line 4x+2y=8
b) Draw the line 4x+2y=8 for values of x from -2 to 3.
WINNER WILL GET BRAINLIEST
Please help me.
There is no picture but it would be great if you could help me.
Answer:
[tex]4 \sqrt{2 - 3 = 1[/tex]
Answer:
(-2,8), (-1,6), (0,4), (1,2), (2,0), (3,-2)
Step-by-step explanation:
Insert these points in graph and draw a line. You will get the answer. ^_^
sorry had to edit because of a blunder
Given a smooth function / such that f(-0.2) = -0.91736, f(0) = -1 and f(0.2) = -1.04277. Using the 2-point forward difference formula to calculate an approximated value of f'(0) with h = 0.2, we obtain: f'(0) = -0.21385 f'(0) = -1.802 f'(o) -2.87073 f'(0) = -0.9802
The approximated value of f'(0) using the 2-point forward difference formula with h = 0.2 is -0.21385. So, first option is the correct answer.
To calculate an approximate value of f'(0) using the 2-point forward difference formula with h = 0.2, we can use the given function values:
f(-0.2) = -0.91736
f(0) = -1
f(0.2) = -1.04277
Using the 2-point forward difference formula, we have:
f'(0) ≈ (f(h) - f(0)) / h
Substituting the values:
f'(0) ≈ (f(0.2) - f(0)) / 0.2
f'(0) ≈ (-1.04277 - (-1)) / 0.2
f'(0) ≈ (-0.04277) / 0.2
f'(0) ≈ -0.21385
Therefore, the approximated value of f'(0) using the 2-point forward difference formula with h = 0.2 is -0.21385. Therefore, the correct answer is first option: f'(0) ≈ -0.21385.
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How many different ways can you write a fraction that has a numerator of 2 as a sum of fractions? Explain.
Answer:
Step-by-step explanation:
1/5 + 1/5 = 2/5
1/7 + 1/7 = 2/7
1/3 + 1/3 = 2/3
There are an infinite number of these fractions. They must be 1 and 1 in the numerator, and the denominator must be relatively prime to 2. The examples I have picked are prime in the denominator, but the rule is not without many exceptions. For example
1/9 + 1/9 = 2/9
I don't think you can pick an even denominator because it will reduce when put with two. Oh wait 2/18 + 2/18 = 4/18 = 2/9 But these could be reduced before adding. Still, it might count. It depends on who is marking the question.
What about an odd and even denominator?
1/9 + 1/18 = 3/18 = 1/6 There must be something that works, but I can't come up with an example.
SOMEONE PLEASE HELP!!!!!!!!!!!
Answer:
40
Step-by-step explanation:
This rhombus is divided into 4 triangles. UL= upper left LL= lower left UR= upper right LR= Lower Right
UL: ½bh = ½(5)(4)=10
LL: ½bh = ½(5)(4)=10
UR: ½bh = ½(5)(4)=10
LR: ½bh = ½(5)(4)=10
Add up all the areas of each individual triangle (10+10+10+10 =40) and that's your final answer.
2x + 7 = 15
What does x equal?
The growth of an apple tree was measured. The tree grew 15 inches per year.
On average, how many millimeters did the tree grow per day?
Answer:
1.04 mill
Step-by-step explanation:
15 inches to mill = 381
[tex]365\sqrt{381}[/tex] = 1.04 ( rounded )
Can someone help me in proving theorems in square pleasee ASAP
Step-by-step explanation:
[tex]90 = 4x - 6 \\ 96 = 4x \\ x = 24[/tex]
[tex]2x - 8 = \\ = 48 - 8 = \\ = 40[/tex]
Which expression is equivalent to 3t - 5 + 8t + 3 ?
Answer:
11t -2
Step-by-step explanation:
3t - 5 + 8t + 3
Combine like terms
3t +8t- 5 + 3
11t -2
Start by changing the problem to 3t + -5 + 8t + 3.
Now combine like terms.
Combine like terms → your numbers
and your "t" terms.
So 3t + 8t is 11t and -5 + 3 is -2.
So the answer is 11t - 2.
Please help! I need this answer right ! And I’m unsure can you LMK ASAP???
Answer:B
Step-by-step explanation:
The longer piece of wood has dimensions of 1 inch by 1 inch by 8 inches. The
Short piece of wood has dimensions of 1 inch by 1 inch by 4 inches. How many
square inches of paint would be needed to cover all sides of this object?
Easy Points! Explain Well!
Answer:
I am guessing it is 67 as for maybe the whole thing is 180 degrees. So 180 subtracted by 67 and 46 equals 67!!!
67+46+?=180
113+?=180
?=180-113
?=67
The sheet of metal has dimensions 56 cm by 33 cm. It sis melted down
and recast into discs of the same thickness and radius 7 cm. How many
disces will be cast ?
Answer:
12
Step-by-step explanation:
To determine the number of discs, divide the area of the metal sheet by the area of the discs
The metal sheets are in the shape of a rectangle, thus the area of a rectangle would be used to determine the area of the sheet
Area of a rectangle = length x breadth
56 x 33 = 1848 cm^2
The disc is in the shape of a circle.
area of a circle = πr²
π = 3.14
r = radius
3.14 x 7² = 153.86 cm²
Number of discs = 1848 /153.86 = 12
1,3x + 3,9x = 0
Pls someone explain the steps for this. Like how do I find the x1 and x2.
Answer:
x=0
I used phone math!!!!!!!!