To solve for the volume of the rotated region we will use the integration and its formula is V=π∫ba(r21−r22)h where r1 and r2 are the radii of the washer and h is the height. Note that the limits a and b is determined by finding the volume of the solid .
Using the washer method, the volume of the solid is
V=π∫ba[(x14)2−(x4)2]dx
Find the limits, equate the two equations in terms of x
4√x=x4
Raising both sides to fourth power,
x=x8x8−x=0
Factor out x
x(x7−1)=0
x=0,1
Thus, the limits is from x =
V=π∫10[(x^1/4)^2−(x4)^2]dx
Integrate this equation with in limits 0 to 1
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Please help ASAP or a huge cockroach will crawl into in your ear
In this figure, GH=12, GP=16, and FP=2.
What is EF?
Enter your answer in the box.
EF
Answer:
This is nonsense, if GP = 16 and GH = 12 then HP = 4 (16 - 12 = 4). FP is looking basically the same size as HP, so FP = 4, not 2.
Step-by-step explanation:
These are all lies, your teacher is a liar and EF must be around 12 too.
Answer:
30
Step-by-step explanation:
Please help i need it quickly too please
If PQRS is a rectangle,
find the measure of ZQPT.
P
(16x - 17)
(10x + 1)°
T
R
S
Answer:
∠ QPT = 59°
Step-by-step explanation:
• the opposite sides are parallel
then
∠ SPT = ∠ QRT ( alternate angles ), that is
16x - 17 = 10x + 1 ( subtract 10x from both sides )
6x - 17 = 1 ( add 17 to both sides )
6x = 18 ( divide both sides by 6 )
x = 3
then
∠ SPT = 16x - 17 = 16(3) - 17 = 48 - 17 = 31°
• the vertices of a rectangle are right
∠ QPS = 90° , so
∠ QPT = 90° - ∠ SPT = 90° - 31° = 59°
3x + 2 > 2x - 3 > x - 12 solve the compound inequality
Answer:
We get that x > -5
Step-by-step explanation:
1. Split up the inequality into two parts
We think of it as a > b > c and know that it is the same as a > b and b > c.
Apply this to the compound inequality
3x + 2 > 2x-3 and 2x - 3 > x - 12
2. Start solving 3x + 2 > 2x-3
Subtract 2x from both sides
x + 2 >-3
Subtract 2 from both sides
x > -5
3. Start solving 2x - 3 > x - 12
Subtract x from both sides
x - 3 > -12
Add 3 to both sides
x > -9
4. Finish
Now we have to think how we started with 3x + 2 > 2x-3 and 2x - 3 > x - 12
Since it is an and we need to identify the interval in which it is true on both inequalities which is when the x overlap. In this case is from x > -5.
1. The normal temperature for the human body is 37 °C (degrees Celsius). A patient suffering from malaria has a temperature of 40,2 °C. How many °C must his temperature decrease to reach a normal temperature?
Answer:
3,2 °C
Step-by-step explanation:
40,2 °C - 37 °C = 3,2 °C
3 c 9 8. following is a statement of a theorem about certain cubic equations. for this theorem, b represents a real number. theorem a. if f is a cubic function of the form f .x/ d x 3 x c b and b > 1, then the function f has exactly one x-intercept. following is another theorem about x-intercepts of functions: theorem b. if f and g are functions with g.x/ d k f .x/, where k is a nonzero real number, then f and g have exactly the same x-intercepts. using only these two theorems and some simple algebraic manipulations, what can be concluded about the functions given by the following
Both f(x) and g(x) have one x-intercept at x = -1/3.
Theorem A states that for a cubic function of the form f(x) = x^3 + x + b, with b > 1, the function f has exactly one x-intercept. Using Theorem B, we can conclude that for any nonzero real number k, the function g(x) = kf(x) = kx^3 + kx + bk also has exactly one x-intercept.
For example, if f(x) = x^3 + x + 5 and k = 2, then g(x) = 2x^3 + 2x + 10. Both f(x) and g(x) have exactly one x-intercept. This can be shown by solving the quadratic equation formed by setting f(x) = 0 or g(x) = 0. Doing this yields x = -1/3. Therefore, both f(x) and g(x) have one x-intercept at x = -1/3.4
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HELP PLEASE someone ?????
Three friends each ordered a large cheese pizza. Shauntee ate $\frac{2}{3}$ of her pizza, Carlos ate $\frac{8}{9}$ of his pizza and Rocco ate $\frac{26}
{27}$ of his pizza. If the remaining portions from the three pizzas are put together, what fraction of a large pizza do they make? Express your answer as a common fraction.
The remaining portions from the three pizzas together make up 13/27 of a large pizza.
What is the fraction?A fraction is defined as a numerical representation of a part of a whole that represents a rational number.
Shauntee ate 2/3 of her pizza, so she has 1/3 left.
Carlos ate 8/9 of his pizza, so he has 1/9 left.
Rocco ate 26/27 of his pizza, so he has 1/27 left.
To determine the combined fraction of the remaining portions of the three pizzas, we add the fractions that are left:
1/3 + 1/9 + 1/27 = (1/3) + (1/9) + (1/27)
Simplifying this fraction by finding the least common denominator, which is 27:
(1/3) + (1/9) + (1/27) = (9/27) + (3/27) + (1/27) = 13/27
So they together make up 13/27 of a large pizza.
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The quadratic functions f(x) and g(x) are described in the table.
x f(x) g(x)
−6 36 4
−5 25 1
−4 16 0
−3 9 1
−2 4 4
−1 1 9
0 0 16
1 1 25
2 4 36
In which direction and by how many units should f(x) be shifted to match g(x)?
Left by 4 units
Right by 4 units
Left by 8 units
Right by 8 units
In the horizontal direction and by 4 units to the left, f(x) should be shifted to match g(x).
The correct option is A.
What is transformation on the graphs?The function provided by the basic function f(x) and the constant
g(x) = f(x - k),
may be drawn by horizontally moving the f(x) k unit coordinates.
The shift's direction depends on the value of k. Specifically,
The base graph moves k units to the right if k > 0, and
The base graph moves k units to the left if k < 0.
Given:
The quadratic functions f(x) and g(x) are described in the table.
From the table, the functions,
f(x) = x².
And g(x) = (x + 4)².
In the attached image,
the blue curve is the function of f(x).
And red line is the function of g(x).
From the graph,
Left by 4 units, f(x) should be shifted to match g(x).
Therefore, left by 4 units, f(x) should be shifted to match g(x).
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Answer: A
Step-by-step explanation: took the test
Simplify.
Rewrite the expression in the form xn.
The answer would be x^-10, since the question asks you to rewrite it with a single variable.
I hope this helps! o(〃＾▽＾〃)o
Freshman and sophomores were surveyed during lunch about their favorite professional football team.
A bar graph with the first bar labeled Jaguars, divided into freshmen from 0 to 40 percent and sophomores from 40 to 100 percent. The second bar is labeled Dolphins, with freshman from 0 to 45 percent and sophomores from 45 to 100 percent, and the third bar is labeled Buccaneers, with freshmen from 0 to 45 percent and sophomores from 45 to 100 percent.
If 140 students selected Jaguars, how many of those students are freshman?
56 students
63 students
77 students
84 students
There are 56 students as freshmen. The correct answer would be an option (A).
What is the percentage?The percentage is defined as a ratio expressed as a fraction of 100.
A bar graph with the first bar labeled Jaguars, divided into freshmen from 0 to 40 percent and sophomores from 40 to 100 percent.
This can be determined by multiplying the percentage of freshmen who selected Jaguars (40%) by the total number of students surveyed (140).
⇒ 40% × 140 = 56
So there are 56 students as freshmen.
Hence, the correct answer would be option (A).
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A local art group wishes to raise $8,000. Currently, they have raised $4,200. what percent has the art group raised
Answer:
raised 53%
Step-by-step explanation:
goal = $8,000
raised = $4,200
$4,200/$8,000 = 0.525 = about 53%
need the answer please
Answer:
Step-by-step explanation:
Answer:
1.91 lb
Step-by-step explanation:
1 3/4 cups of pecans per pie = 1.75 cups.
For 5 pies, 5 × 1.75 cups = 8.75 cups are needed.
1 cup = 99 g
8.75 cup × (99 g) / (1 cup) = 866.25 g
Exact conversion: 453.59237 g = 1 lb
866.25 g × (1 lb)/(453.59237 g) = 1.91 lb
Answer: 1.91 lb
suppose phone calls arrive at an office according a poisson process at a mean rate of 5 calls per hour. the expected waiting time in hours until the first call arrived is
The expected waiting time in hours until the first call arrived is 0.00045.
The formula for poisson distribution is expressed as
P(x = r) = (e^- µ × µ^r)/r!
Where ,
µ= represents the mean of the theoretical distribution.
r = represents the number of successes of the event.
According to the question,
µ = 5
For the probability that there are one in one hour, it is expressed as
P(x = 1) ,
Therefore,
P(x = 1)
= (e^- 10 × 10^1)/1! = 0.00045
Therefore,
P(x = 1)
= 0.00045
Poisson distribution
A Poisson distribution is a discrete probability distribution. It gives the probability of an event happening a certain number of times (k) within a given interval of time or space.
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Simplify the equation 6i - (16 - 7i)
Angela plays soccer and golf for a
total of 125 minutes every day, She
plays soccer for 45 minutes longer
than she plays goif
Part A: Witte a pair of linear
equations to show the relationship
between the number of minutes
Angela plays soccer (x ) and the
number of minutes she plays golf (y)
every day.
Part B: How much does angela
spend playing golf everyday? Show
your work.
PartC: Is it possible for Angela to
have spent 80 minutes playing
soccer if she plays for a total of
exactly 125 minutes and plays
soccer for 45 minutes longer than
she plays golf?
It is possible for Angela to have spent 80 minutes playing soccer if she plays for a total of exactly 125 minutes and plays soccer for 45 minutes longer than she plays golf because 80+45 adds up to 125.
What is a linear equation?A linear equation is an equation that has the variable of the highest power of 1. The standard form of a linear equation is of the form Ax + B = 0.
Angela plays soccer and golf for a total of 125 minutes every day. She plays soccer for 45 minutes longer than she plays golf.
Part A: A pair of linear equations to show the relationship between the number of minutes Angela plays soccer (x) and the number of minutes she plays golf (y) every day.
x = y + 45
Part B: A pair of linear equations to show how much Angela spends playing golf every day.
125 - 45 = 80,
80/2 = 40
Part C: Yes, it is possible for Angela to have spent 80 minutes playing soccer if she plays for a total of exactly 125 minutes and plays soccer for 45 minutes longer than she plays golf because 80+45 adds up to 125.
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What is the circumference of a circle with a radius of 21 feet? Use 22 over 7 for π.
132 feet
66 feet
5,544 feet
1,386 feet
The circumference of a circle whose radius is 21 feet, is 132 feet, thus the first option is the correct one.
How to get the circumference of the circle?We know that for a circle of radius R, the circumference is given by:
C = 2*π*R
Where π = 22/7
And here we know that the radius is 21 feet, then we can replace that in the formula above and we will get:
C = 2*(22/7)*21 feet = 132 feet
The circumference of that triangle is 132 feet, thus, the correct option is the first one.
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Farrah borrowed $155 from her brother. She has already paid back $15. She plans to pay back $35 each month until the debt is paid off. Which describes the number of months it will take to pay off the debt? Select three options. x + 15 + 35 = 155 35 x + 15 = 155 35 x = 155 minus 15 It will take 8 months to pay off the debt. It will take 4 months to pay off the debt.
It will take 4 months to pay off the debt for Farah.
What is an expression?Expression in maths is defined as the collection of numbers variables and functions by using signs like addition, subtraction, multiplication, and division.
Numbers (constants), variables, operations, functions, brackets, punctuation, and grouping can all be represented by mathematical symbols, which can also be used to indicate the logical syntax's order of operations and other features.
Given that Farrah borrowed $155 from her brother. She has already paid back $15. She plans to pay back $35 each month until the debt is paid off.
The expression can be written as:-
35x + 15 = 155
35x = 155 - 15
35x = 140
x = 140 / 35
x = 4 months
Hence, the number of months will be 4.
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Which statements regarding Triangle E F G are true? Select three options. EF + FG > EG EG + FG > EF EG - FG < EF EF- FG > EG EG + EF < FG
we can conclude that option (A) , (B) and (C) are true.
We know that in a triangle,
• Sum of any to sides is greater than the third side.
• Difference between any two sides is smaller than the third side.
So, in ∆EFG, we have,
• EF + FG > EG
• FG + EG> EF
• EF + EG > FG
and,
• EF - FG <EG or FG - EF <EG
• FG - EG < EF or EG - FG < EF
• EF - EG <FG or EG - EF < FG
From options now we get,
A) EF + FG > EG is true
B)EG + FG > EF is true
C) EG - FG < EF is true
D) EF - FG > EG is false
E) EG + EF < FG is false
Therefore, we can conclude that option () , () and () are true.
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We can conclude that option (A) , (B) and (C) are true.
We know that in a triangle,
• Sum of any to sides is greater than the third side.
• Difference between any two sides is smaller than the third side.
So, in ∆EFG, we have,
• EF + FG > EG
• FG + EG> EF
• EF + EG > FG
and,
• EF - FG <EG or FG - EF <EG
• FG - EG < EF or EG - FG < EF
• EF - EG <FG or EG - EF < FG
From options now we get,
A) EF + FG > EG is true
B)EG + FG > EF is true
C) EG - FG < EF is true
D) EF - FG > EG is false
E) EG + EF < FG is false
Therefore, we can conclude option (A) , (B) and (C) are true.
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Provide explanation for how to solve this problem.
The numeric value of the function is given as follows:
2x + h + 9.
How to find the numeric value of a function or of an expression?To find the numeric value of a function or of an expression, we replace each instance of the variable in the function or in the expression by the value at which we want to find the numeric value.
The function in this problem is given as follows:
f(x) = x² + 9x.
At x = x + h, the function is defined as follows:
f(x + h) = (x + h)² + 9(x + h)
f(x + h) = x² + 2xh + h² + 9x + 9h
f(x + h) = x² + 2xh + h² + 9x + 9h.
Subtracting by f(x), we have that:
f(x + h) - f(x) = h² + 2xh + 9h.
f(x + h) - f(x) = h(h + 2x + 9).
Dividing by h, the numeric value is given as follows:
2x + h + 9.
Meaning that the first option is the correct option.
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Determine the value of x.
john maintains two separate lines of lobster traps in penobscot bay. the east bay trap produces a mean of 12 pounds of lobsters per day with a standard deviation of 7 pounds. the west bay trap produces a mean of 10 pounds of lobsters a day with a standard deviation of 4.5 pounds. the weight of lobsters produced by each trap is independent of the other. which of the following are the mean and approximate standard deviation of the total weight of lobsters john traps in a day? (a) Mean 11; Standard deviation-2.50 (b) Mean 2; Standard deviation-2.50 (c) Mean-2; Standard deviation- 5.36 (d) Mean- 2; Standard deviation -8.32 (e) Mean 2; Standard deviation 11.50
The mean and approximate standard deviation of the difference in the total weight of lobster john traps in a day are 2 pounds and 8.32 pounds respectively. Hence, option d is the right answer.
Since the weight of lobsters produced by each trap is independent of the other, we can find the mean and standard deviation of the difference in the total weight of lobsters trapped in a day by subtracting the mean and standard deviation of the west bay trap from the mean and standard deviation of the east bay trap.
Mean = 12 pounds (east bay) - 10 pounds (west bay) = 2 pounds
Standard deviation = √(7^2 + 4.5^2) = √69.25 ≈ 8.32 pounds
So, the mean and approximate standard deviation of the difference in the total weight of lobsters trapped in a day is -
Mean = 2 pounds
Standard deviation = 8.32 pounds
Therefore, the answer is (d) Mean=2; Standard deviation= 8.32.
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The complete question is -
John maintains two separate lines of lobster traps in Penobscot bay. the east bay trap produces a mean of 12 pounds of lobsters per day with a standard deviation of 7 pounds. the west bay trap produces a mean of 10 pounds of lobsters a day with a standard deviation of 4.5 pounds. the weight of lobsters produced by each trap is independent of the other. which of the following are the mean and approximate standard deviation of the difference in the total weight of lobsters john traps in a day?
(a) Mean=11; Standard deviation=2.50
(b) Mean=2; Standard deviation=2.50
(c) Mean=2; Standard deviation= 5.36
(d) Mean=2; Standard deviation =8.32
(e) Mean=2; Standard deviation=11.50
8.498 ? 8.478 use >,<,=
The inequality equation is 8.498 > 8.478
What is an Inequality Equation?Inequalities are the mathematical expressions in which both sides are not equal. In inequality, unlike in equations, we compare two values. The equal sign in between is replaced by less than (or less than or equal to), greater than (or greater than or equal to), or not equal to sign.
In an inequality, the two expressions are not necessarily equal which is indicated by the symbols: >, <, ≤ or ≥.
Given data ,
Let the inequality equation be represented as A
Now , the value of A is
Let the first number be represented as = p
Let the second number be represented as = q
Now , the value of p = 8.498
The value of q = 8.478
And , the hundredths value in the number 8.498 is greater than the hundredths value in the number 8.478
So , the value of p is greater than q
Therefore , 8.498 > 8.478
Hence , the inequality is 8.498 > 8.478
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Parallelogram EFGH is shown below. Give the coordinates of G.
←*
F (0,s)
E (-1,0)
G (?, ?)
H (u,0)
Answer:
G(u+t, s)
Step-by-step explanation:
Given coordinates E(-1, 0), F(0, s), and H(u, 0) of parallelogram EFGH, you want the coordinates of point G.
Segment lengthSegment FG is a horizontal line. Its length will be the same length as horizontal line EH. Since the lines are horizontal, the length is the difference of x-coordinates.
Point H is (u -(-t)) = u+t units to the right of point E. The x-coordinate of point G will be the same number of units to the right of point F.
The x-coordinate of G is 0 +(u+t) = u+t, the length of EH added to the x-coordinate of F.
Horizontal lineThe y-coordinates of any point on a horizontal line are the same. The y-coordinate of point F is 's'. Point G lies on that same horizontal line, so has the same y-coordinate.
The coordinates of point G are (u+t, s).
__
Additional comment
The midpoints of the diagonals are the same. This means ...
(E+G)/2 = (F+H)/2
E+G = F+H . . . . . . multiply by 2
G = F +H -E . . . . . subtract E
Using the given coordinates:
G = (0, s) +(u, 0) -(-t, 0) = (0+u+t, s+0+0) = (u+t, s) . . . . as above
All of your "parallelogram" problems are worked the same way. The fourth point has coordinates that are the sum of the coordinates of the end points of the known diagonal, less the known point on the unknown diagonal.
If you work the problem any of several other ways, you get the same answer.
Point E is (u -(-t)) = (u +(-t)) units away from point H. Point G will be located the same amount of units to the right of point F in terms of its x-coordinate. G(u+t, s)
What are the possible coordinates of Parallelogram?You need to know the coordinates of point G in the parallelogram EFGH given the coordinates E(-1, 0), F(0, s), and H(u, 0).
FG is a horizontal segment. It will have a length equal to that of a horizontal line, EH. The length is determined by the difference in x-coordinates because the lines are horizontal.
The length of EH multiplied by the x-coordinate of F yields the x-coordinate of G, which is 0 +(u+t) = u+t.
Horizontal line
In a horizontal line, all points have the same y-coordinates. Point F has the y-coordinate of's'. Point G's y-coordinate is the same because it is located on the same horizontal line.
The coordinates of point G are (u+t, s).
__
Additional comment
The midpoints of the diagonals are the same. This means ...
(E+G)/2 = (F+H)/2
E+G = F+H . . . . . . Multiply by 2
G = F +H -E. . . . . Subtract E
Using the given coordinates:
G = (0, s) +(u, 0) (u, 0) (-t, 0) = (u+t, s) + (u+t, s) + (u+t, s) . . . . As mentioned earlier
Your “parallelogram” issues are all solved in the same manner. The fourth point's coordinates are equal to the sum of the coordinates of the known diagonal's end points, minus the known diagonal's known point.
The solution is the same regardless of how you approach the issue.
Vertical line
Any point on a horizontal line has the same y-coordinates. Point F's y-coordinate is's'. Point G has the same y-coordinate since it is located on the same horizontal line.
Point G's coordinates are (u+t, s).
Further commentary
The diagonals' midpoints are identical. This implies...
(E+G)/2 = (F+H)/2
E+G = F+H, multiply by 2, and so on.
G = F + H -E.... Remove E.
Using the coordinates provided:
G = (0, s) +(u, 0) (u, 0) (-t, 0) = (u+t, s) + (u+t, s) + (u+t, s) . . . . as mentioned earlier
Therefore, Your “parallelogram” issues are all solved in the same manner. The fourth point's coordinates are equal to the sum of the coordinates of the known diagonal's end points, minus the known diagonal's known point.
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B. In square EFGH, find the length of EG.
Answer:
6√2
Step-by-step explanation:
You want one side of a square whose half-diagonal has length 6.
Right triangleIf X is the center point of the square where the diagonals cross at right angles, triangle EXG is an isosceles right triangle with leg length 6. The hypotenuse of an isosceles right triangle is √2 times the leg length, so ...
EG = 6√2
__
Additional comment
If you like, you can use the Pythagorean theorem to prove that.
EG² + EX² +GX² = 6² +6² = 72
EG = √72 = √(36·2) = 6√2
A square is a rhombus, so the diagonals cross at right angles. A square is a rectangle, so the diagonals are the same length. A square is a parallelogram, so the diagonals bisect each other.
36
Solve for x.
48
24
will give brainliest!!
which of the following graphs best shows a strong negative association between dd and tt ?question 5 of 38reportchoose 1 answer:a scatterplot is graphed on the dt-plane. the data points form a loose diagonal cluster that trends downward.a scatterplot is graphed on the dt-plane. the data points form a tight diagonal cluster that trends shallowly upward.a scatterplot is graphed on the dt-plane. the data points form a tight diagonal cluster that trends upward.a scatterplot is graphed on the dt-plane. the data points form a tight diagonal cluster that trends downward.
The formula for determining the strength of a negative association between two variables is Pearson's Correlation coefficient (r).
This coefficient measures the degree of linear relationship between two variables and ranges from -1 to +1. If two variables have a negative association, the correlation coefficient will be negative.
For example, let's say we are looking at the relationship between daily consumption of donuts (dd) and daily temperature (tt). To calculate the Pearson's Correlation coefficient, we would first calculate the covariance of the two variables using the following formula:
Cov(dd,tt) = (∑ (dd - dd_mean) * (tt - tt_mean)) / n
Next, we would calculate the standard deviation of both variables using the following formula:
Stdev_dd = √(∑ (dd - dd_mean)^2/n)
Stdev_tt = √(∑ (tt - tt_mean)^2/n)
Finally, we would calculate the Pearson's Correlation coefficient using the following formula:
r = Cov(dd,tt) / (Stdev_dd * Stdev_tt)
If r is close to -1, then this indicates a strong negative association between dd and tt.
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Write an algebraic expression that Marshall can use to determine the total cost of buying a watermelon that weigh w pounds and some tomatoes that weigh t pounds. How much will it cost to buy a watermelon that weighs 18.5 pounds and 5 pounds of tomatoes? Tomatoes are $3.25 / lb and Watermelon is $0.68 / lb
The algebraic expression which represents the total cost of buying watermelons and tomatoes as described is; C = 0.68w + 3.25t.
The cost of buying a watermelon that weighs 18.5 pounds and 5 pounds of tomatoes is; $28.83.
What is the cost of buying watermelons and tomatoes?It follows from the task content that the cost of buying Tomatoes are $3.25 / lb and Watermelon is $0.68 / lb.
Hence, the algebraic expression which represents the cost of buying a watermelon that weighs, w pounds and tomatoes that weigh t pounds is;
C = 0.68w + 3.25t.
Hence, if the watermelon bought weighs 18.5 pounds and tomatoes weigh 5 pounds,
C = 0.68 (18.5) + 3.25 (5)
C = 12.58 + 16.25
C = $28.83.
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let $s$ be a subset of $\mathbb{r}$ that is closed under multiplication (that is, if $a$ and $b$ are in $s$, then so is $ab$). let $t$ and $u$ be disjoint subsets of $s$ whose union is $s$. given that the product of any three (not necessarily distinct) elements of $t$ is in $t$ and that the product of any three elements of $u$ is in $u$, show that at least one of the two subsets $t$, $u$ is closed under multiplication.
As per the given subset, the set T and U are not closed under multiplication.
In math the term subset is defined through the example set A is a subset of another set B if all elements of the set A are elements of the set B.
Here let us consider that S be a subset of R that is closed under multiplication
Here we have also consider that T and U be disjoint subsets of S whose union is S.
Here we have given that the product of any three elements of T is in T and that the product of any three elements of U is in U, now we have to show that at least one of the two subsets T, U is closed under multiplication.
Here we have to start out with the assumption that neither T nor U is closed under multiplication and use proof by contradiction to prove that at least one of T or U must be closed under multiplication.
Here let us write this as,
=> t1,t2,t3∈T1,2,3 and
=> u1,u2,u3∈U1,2,3∈,
While it is given by the problem that, then T and U are disjoint subsets of S and therefore,
=> T∩U = ∅.
At this time we have take that
=> t3 = u1⋅u2
=> 3 = 1⋅2 and
=> u3=t1⋅t2
=> 3=1⋅2.
While we looking these statements are valid because T and U are not closed under multiplication, where as the product of u1⋅u21⋅2 must not be in the set U and the product of t1⋅t21⋅2 must not be in set T.
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Mason is working two summer jobs, making $7 per hour babysitting and making $10 per hour clearing tables. In a given week, he can work at most 12 total hours and must earn no less than $90. If xx represents the number of hours babysitting and yy represents the number of hours clearing tables, write and solve a system of inequalities graphically and determine one possible solution.
Answer: We know that:
x + y = 12 (because he can work at most 12 hours in total)
7x + 10y ≥ 90 (because he must earn no less than $90)
We can graph these inequalities on the coordinate plane.
The equation x + y = 12 represents a line with a slope of -1 and y-intercept of 12. This is the line that separates the region where he works more hours babysitting and the region where he works more hours clearing tables.
The inequality 7x + 10y ≥ 90 represents the region where he earns more than $90.
You can find the solution by finding the intersection of the line and the shaded region. The solution will be a point that lies on both the line and the shaded region.
One possible solution would be for Mason to work 8 hours babysitting and 4 hours clearing tables, earning him $56 from babysitting and $40 from clearing tables, for a total of $96.
Step-by-step explanation: