What equation is solved by the graphed systems of equations?
-4-3-2
(-1,-4)
-4
4
Please help I am confused
Answers
The equations solved by the graph are y = x + 3 and y = x - 3
How to determine the equations solved by the graphFrom the question, we have the following parameters that can be used in our computation:
The graph of systems of equations
A linear equation is represented as
y = mx + c
Where
c = y when x = 0
Using the points on the graph, we have
Line 1: y = mx + 3
Line 2: y = mx - 3
Using the common points, we have
Line 1: -m + 3 = -4
Line 2: -m - 3 = -4
When evaluated, we have
Line 1: m = 1
Line 2: m = 1
So, we have
Line 1: y = x + 3
Line 2: y = x - 3
Hence, the equations solved by the graph are y = x + 3 and y = x - 3
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Related Questions
Find the savings plan balance after 12 months with an APR of 3% and monthly payments of $200
Answers
To find the savings plan balance after 12 months with an APR of 3% and monthly payments of $200, you can use the formula for the future value of an annuity. So the savings plan balance after 12 months with an APR of 3% and monthly payments of $200 is $2,492.80.
The formula for the future value of an annuity: FV = PMT * [(1 + r/n)^(n*t) - 1] / (r/n), where: FV is the future value of the annuity, PMT is the periodic payment, r is the annual interest rate (as a decimal), n is the number of times the interest is compounded per year, and t is the number of years.
Using this formula, we have r = 3% = 0.03n = 12 (monthly payments)t = 12/12 = 1 year PMT
= $200FV
= 200 * [(1 + 0.03/12)^(12*1) - 1] / (0.03/12)FV
= 200 * [(1.0025)^12 - 1] / (0.0025)FV
= 200 * 0.03115 / 0.0025FV = $2,492.80
Therefore, the savings plan balance after 12 months with an APR of 3% and monthly payments of $200 is $2,492.80.
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which of the following are solutions to the equation tan^2x-1=0?
Answers
The solutions to the equation tan²x - 1 = 0 are x = π/4 and x = 3π/4
How to determine the solutions to the equationFrom the question, we have the following parameters that can be used in our computation:
tan²x - 1 = 0
Add 1 to both sides of the equation
So, we have
tan²x = 1
Take the square root of both sides
tan(x) = ±1
Take the arc tan of both sidesof tan(x) = ±1
So, we have
x = π/4 and x = 3π/4
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Ciana earns an hourly wage of $30 at her job. In order to purchase her sneakers she will have to take time off work, so each hour away from her job
costs her $30 in lost Income. Assume that ciana travel time is the same each way (to and from the store) and that it will take her 30 minutes once
she reaches a store to complete her shopping. Assume throughout the question that ciara incurs no additional costs other than the sneakers, such as
gas.
complete the following table by computing the opportunity cost of ciana’s time and the total cost of shopping at each location
Answers
Ciana should purchase the skirt at the store across town because the total economic cost will be lowest.
How to determine the opportunity cost?Ciana makes $30 per hour at her work, and her purchase decision includes the opportunity cost of lost wages:
Total economic cost:
Local store = $114 + [1/4 hours x 2 (round trip) x $30] + (1/2 hours x $30 spent shopping) = $144
Across town = $86 + [1/2 hours x 2 (round trip) x $30] + (1/2 hours x $30 spent shopping) = $131
Neighboring city = $60 + [1 hour x 2 (round trip) x $30] + (1/2 hours x $30 spent shopping) = $135
Ciana should buy a skirt at the store across town. Because it has the lowest total economic cost ($131).
Opportunity cost is the lost benefit or additional cost of choosing one activity or investment over another. Economic costs include both accounting costs and opportunity costs.
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Andres Michael bought a new boat. He took out a loan for $24,000 at 2.75% interest for 4 years. He made a $4,210 partial payment at 4 months and another partial payment of $3,240 at 6 months. How much is due at maturity?
Answers
The total amount due at maturity is $23,495.73.Given:Principal amount (P) = $24,000 Rate of interest (R) = 2.75% Time (n) = 4 years First partial payment (A) = $4,210 Time after first partial payment (t) = 4/12 years = 1/3 year Second partial payment (B) = $3,240 Time after second partial payment (t) = 6/12 years = 1/2 year.
We need to find the amount due at maturity.Amount due at maturity can be calculated by the formula given below;A = P [ i ( 1 + i )n ] / [ ( 1 + i )n – 1 ] Where,A = amount due at maturity P = principal amount R = rate of interest in decimal n = time in yearsi = rate of interest per period For yearly interest, i = R / 100.For monthly interest, i = R / (12 × 100)For quarterly interest, i = R / (4 × 100)For half-yearly interest, i = R / (2 × 100)
Now, we will find the value of i first, which can be calculated by dividing the yearly interest rate by 100. Here, the interest is compounded half-yearly.i = R / (2 × 100)i = 2.75% / (2 × 100)i = 0.01375
Let's substitute the values of P, R, n, i, A in the formula and calculate the amount due at maturity.P = $24,000R = 2.75%n = 4 yearsi = 0.01375A = $4,210t = 1/3 yearsA = P [ i ( 1 + i )n ] / [ ( 1 + i )n – 1 ]4,210 = 24,000 [ 0.01375 ( 1 + 0.01375 )16 ] / [ ( 1 + 0.01375 )16 – 1 ]On solving the above equation, we get;Amount of loan due after 4 months is $23,188.45.
Now, we will calculate the second amount due at maturity after the second partial payment.B = $3,240t = 1/2 year.Let's find the balance due at the end of 6 months using the following formula;B = C ( 1 + i )n
Where,B = balance due at the end of 6 monthsC = amount due at the end of 4 monthsi = 0.01375n = (6 - 4) / 12 yearsB = C ( 1 + i )nB = $23,188.45 ( 1 + 0.01375 )1/2
On solving the above equation, we get;Amount due at maturity after the second partial payment is $22,201.98
Now, we will calculate the amount due at maturity.Amount due at maturity = balance due after second payment + interest for remaining periodAmount due at maturity = $22,201.98 + interest for 3.5 years.Let's calculate the interest for 3.5 years;Interest = P × i × nInterest = $24,000 × 0.01375 × 3.5 Interest = $1,293.75 Amount due at maturity = $22,201.98 + $1,293.75Amount due at maturity = $23,495.73.
Therefore, the total amount due at maturity is $23,495.73.
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. Which of the following is the maximum value of the equation y = −x2 − x + 6?
Answers
Answer:
x=6-x2-y
Step-by-step explanation:
Swap sides so that all variable terms are on the left hand side.
add x2 to both side
Subtract 6 from both sides.
Divide both sides by −1.
Dividing by −1 undoes the multiplication by −1.
Divide y+x2 −6 by −1.
is 7 1/5 x 49 7/5 equivalent to 49
Answers
7 1/5 becomes. 36/5
49 7/5 becomes. 252/5
Now multiply. 36/5. x. 252/5 = 9072/25
9072/25 becomes. 362.88 so
No it is not equivalent to 49
is this graph a minimum or a maximum pls help
Answers
The vertex of the graph is a maximum at (-2, 4)
Calculating if the vertex of the graph a maximum or a minimum?From the question, we have the following parameters that can be used in our computation:
The graph
The graph is a quadratic function
From the graph, we can see that the graph has a maximum value
This maximum value represents the vertex of the graph
And it is located at (-2, 4)
So, we have
Maximum = (-2, 4)
Hence, the maximum value of the function is (-2, 4)
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Add the following fractions and reduce them to the simplest form: 23/27 + 19/26
Answers
The sum of the fractions 23/27 and 19/26, in its simplest form, is 1111/702.
To add the fractions 23/27 and 19/26 and simplify the result, you need to find a common denominator for both fractions. The common denominator is the smallest number that both 27 and 26 can evenly divide into. In this case, the common denominator is 702 since both 27 and 26 can divide evenly into it.
Now, let's convert the fractions to have a denominator of 702:
23/27 = (23 * 26)/(27 * 26) = 598/702
19/26 = (19 * 27)/(26 * 27) = 513/702
Now, we can add the fractions:
598/702 + 513/702 = (598 + 513)/702 = 1111/702
To simplify this fraction, we can find the greatest common divisor (GCD) of the numerator and denominator, and then divide both by the GCD:
GCD(1111, 702) = 1
Dividing both the numerator and denominator by 1, we get:
1111/702 = 1111/702
Therefore, the sum of the fractions 23/27 and 19/26, in its simplest form, is 1111/702.
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Find the surface area
Answers
Answer: 120 yds
Step-by-step explanation:
48+30+24+18+120 yds
PLSSS HELP ME 14 POINTS
Answers
The transformation of f(x) to g(x) is :
Vertically compressed by a factor of 9Shifted down by 2 Describing the transformation of the functionsFrom the question, we have the following parameters that can be used in our computation:
The functions f(x) and g(x)
Where, we have
f(x) = x
g(x) = 1/9x - 2
The graph of the functions f(x) and g(x) are added as an attachment
And we have the transformations to be:
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Please help!!!! I will give points to correct answer !!!
Answers
The equation that shows the Pythagorean identity is true for θ = 270° and is in the form sin²θ + cos²θ = 1 is option B. 0² + (-1)² = 1
The Pythagorean identity is a fundamental trigonometric identity that relates the sine and cosine functions. It states that for any angle θ, the sum of the squares of the sine and cosine of that angle is equal to 1: sin²θ + cos²θ = 1.
We are given θ = 270° and we need to select the equation that satisfies the Pythagorean identity in the given form.
Let's evaluate each option:
A. 0² + 1² = 1
In this case, sin²θ = 0² = 0 and cos²θ = 1² = 1. Adding them together, we get 0 + 1 = 1, which satisfies the Pythagorean identity.
B. 0² + (−1)² = 1
Here, sin²θ = 0² = 0 and cos²θ = (−1)² = 1. Adding them, we have 0 + 1 = 1, which satisfies the Pythagorean identity.
C. (−1)² + 0² - 1
In this equation, sin²θ = (−1)² = 1 and co
s²θ = 0² = 0. However, the equation does not satisfy the Pythagorean identity because 1 + 0 - 1 ≠ 1.
D. 1² + 0² = 1
For this option, sin²θ = 1² = 1 and cos²θ = 0² = 0. Adding them together, we get 1 + 0 = 1, which satisfies the Pythagorean identity.
Based on our evaluation, options A and B both satisfy the Pythagorean identity for θ = 270°. Therefore, either A or B can be selected as the correct equation.The correct answer is b.
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The Question was Incomplete, Find the full content below :
Which equation shows that the Pythagorean identity is true for θ = 270°?
Select the equation that is in the form sin²θ+ cos²θ = 1.
A. 0² + 1² = 1
B. 0² + (−1)² = 1
C. (-1)² + 0² - 1
D. 1² + 0² = 1
There are 201 staplers in stock. Normally, 1624 are sold per year. How many days are on hand are there in inventory? Assume 350 days per year.
Answers
There are approximately 43.36 days on hand in inventory.
To calculate the number of days on hand in inventory, we need to divide the number of staplers in stock by the average number of staplers sold per day.
First, let's calculate the average number of staplers sold per day:
Average staplers sold per year = 1624
Number of days per year = 350
Average staplers sold per day = Average staplers sold per year / Number of days per year
= 1624 / 350
≈ 4.64 (rounded to two decimal places)
Now, we can determine the number of days on hand in inventory:
Number of staplers in stock = 201
Number of days on hand in inventory = Number of staplers in stock / Average staplers sold per day
= 201 / 4.64
≈ 43.36 (rounded to two decimal places)
Therefore, there are approximately 43.36 days on hand in inventory.
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Which of the lines or segments below is tangent to circle P?
A) <—>
ZM
B)——
EX
C)——
LE
D)<—>
TX
Answers
Answer:
A) Line ZM is tangent to circle P.
Roberto made a line plot to show the weight in pounds of the bags of granola in his store he concluded that the total weight of the granola was 2/1/2+2/3/4+3=8/1/4 pounds
Answers
The correct total weight of the bags of granola is 8 1/4 pounds.
One thing that can be done to improve Roberto's reasoning is to ensure the accuracy of the calculations.
In his conclusion, Roberto added the weights of the bags of granola (2 1/2, 2 3/4, and 3) and claimed that the total weight was 8 1/4 pounds. However, the sum of these weights does not equal 8 1/4 pounds.
To address this, Roberto should recheck his calculations. Adding mixed numbers involves adding the whole numbers separately and then adding the fractions separately. In this case, 2 1/2 + 2 3/4 + 3 can be calculated as follows:
2 + 2 + 3 = 7 (sum of whole numbers)
1/2 + 3/4 = 2/4 + 3/4 = 5/4 = 1 1/4 (sum of fractions)
Thus, the correct sum is 7 + 1 1/4 = 8 1/4 pounds.
By double-checking the calculations and providing the accurate sum, Roberto's reasoning would be more precise, reliable, and free from errors.
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The probable question may be:
Roberto made a line plot to show the weight in pounds of the bags of granola in his store he concluded that the total weight of the granola was 2 1/2+2 3/4+3=8 1/4 pounds.
what is one thing that you could do to Roberto's Reasoning
Pls help!
I’ll mark whoever gets this right in four hours brainiest!
The area below is a field (inside the track). What is the amount of money necessary to re-sod the field at $4.99/m (small two beside m)?
Answers
[tex]{\huge{\bold{\underline{\pink{\mathfrak{Answer}}}}}}[/tex]
______________________________________
→ To calculate the area of the field:
A = length x height
A = 100 m x 50 m
A = 5000 m²
______________________________________
→ To calculate the amount of money:
Cost = A x cost per square meter
Cost = 5000 m² x $4.99/m²
Cost = $24,950
______________________________________
→ Therefore, it would cost $24,950.
5.4.3
Question 8 of 10
Using the graphing function on your calculator, find the solution to the system
of equations shown below.
OA. x= -8, y = 2
OB. No solution
OC. More than 1 solution
OD. x= 12, y = 3
3y-12x=18
2y-8x=12
SUBMIT
Answers
Answer:
C) More than 1 solution
Step-by-step explanation:
[tex]3y-12x=18\\2y-8x=12\\\\3(y-4x)=18\\2(y-4x)=12\\\\y-4x=6\\y-4x=6\\\\6=6[/tex]
Therefore, since both sides will always be equal to each other, then there are infinitely many solutions (so C is the best choice)
Give the digits in the tens place and the hundredths place 98.56
Answers
Answer:
5 is the tens, 6 is the hundred
Answer:
Step-by-step explanation:
10s - 9
100ths - 6
What is the value of x?
Answers
Answer:
x = 34.5
Step-by-step explanation:
According to the Vertical Angles Theorem, when two straight lines intersect, the opposite vertical angles are congruent.
Therefore, the vertical angle opposite the angle labelled 40° is also 40°.
Assuming lines m and n are parallel, we can apply the Same-side Interior Angles Theorem to find the value of x.
According to the Same-side Interior Angles Theorem, when two parallel lines are intersected by a transversal, the angles that are interior to the parallel lines and on the same side of the transversal line sum to 180°.
Therefore, the angle labelled (4x + 2)° and the angle vertically opposite to the one labelled 40° sum to 180°:
[tex](4x + 2)^{\circ} + 40^{\circ} = 180^{\circ}[/tex]
Solve for x:
[tex]4x + 2 + 40 = 180[/tex]
[tex]4x + 42 = 180[/tex]
[tex]4x + 42-42 = 180-42[/tex]
[tex]4x =138[/tex]
[tex]\dfrac{4x}{4}=\dfrac{138}{4}[/tex]
[tex]x=34.5[/tex]
Therefore, the value of x is 34.5.
2+2=
a. 3
b. 5
c. 4
d. 7
Answers
Answer:
C
Step-by-step explanation:
2
+2
-----
4
A steep mountain is inclined 74 degree to the horizontal and rises to a height of 3400 ft above the surrounding plain. A cable car is to be installed running to the top of the mountain from a point 940 ft out in the plain from the base of the mountain. Find the shortest length of cable needed.
Answers
Answer:
3902.2 ft
Step-by-step explanation:
You want the length of cable required to span the distance from the top of a mountain 3400 ft above the plain from a location 940 ft from the base of the mountain, which rises at an angle of 74°.
Horizontal distanceThe edge of the base of the mountain will be at a horizontal distance from the point below the peak given by ...
Tan = Opposite/Adjacent
Adjacent = Opposite/Tan
width to center = (3400 ft)/tan(74°) ≈ 974.93 . . . . feet
Cable lengthThe cable is the hypotenuse of a right triangle with one leg 3400 ft and the other (974.93 +940) = 1941.93 ft. The length of that is ...
c = √(3400² +1914.93²) ≈ 3902.2 . . . . feet
The shortest length of cable needed is about 3902.2 feet.
<95141404393>
Can u answer this as it is very hard thank you very much
Answers
Answer:
180 houses
Step-by-step explanation:
For every 10 flats, there are 30 bungalows. So, if there are 30 bungalows then there will be 100 flats in the village.
For every 5 flats there are in the village, there are 9 houses. There are 100 flats in the village. So, we can divide 100 by 5.
[tex]\frac{100}{5}[/tex]=20
20(9)=180
Therefore, there are 180 houses in the village.
Good luck on your homework!
A popular restaurant has 48 tables. On each table are 3 different types of salsa. In one day, all of the tables are used for 9 different sets of customers. Which expression can be used to estimate how many containers of salsa are needed for all the tables in one day?
A 50 × 9
B 16 × 3 × 9
C 50 × 3 × 10
D 40 × 5 × 5
Answers
The expression to estimate the number of containers of salsa needed is: 48 × 3 × 9. none of the option is correct.
To estimate how many containers of salsa are needed for all the tables in one day, we need to consider the total number of tables and the number of salsa containers required for each table.
Given that there are 48 tables and each table has 3 different types of salsa, we can estimate the total number of containers needed by multiplying the number of tables by the number of salsa types.
However, we also need to account for the fact that there are 9 different sets of customers throughout the day. Each set of customers will use all the tables, so we need to multiply the estimated number of containers by the number of sets of customers to get an accurate estimation for the day.
Let's analyze the options provided:
A) 50 × 9: This option assumes there are 50 tables, which is incorrect based on the given information.
B) 16 × 3 × 9: This option assumes there are 16 tables, which is incorrect based on the given information.
C) 50 × 3 × 10: This option assumes there are 50 tables and 10 different sets of customers. Although the number of tables is incorrect, this option accounts for the number of salsa types and the number of sets of customers. However, it does not accurately represent the given scenario.
D) 40 × 5 × 5: This option assumes there are 40 tables and 5 different sets of customers. It also considers the number of salsa types. However, it does not accurately represent the given scenario as the number of tables is incorrect.
None of the options provided accurately represent the given scenario. The correct expression to estimate the number of containers of salsa needed for all the tables in one day would be:48 × 3 × 9
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A group of friends orders a pizza. One person ate 1/4 of the pizza, one ate 1/8 of the pizza, and another ate 1/3 of the pizza. How much is left??
Answers
Answer:
Step-by-step explanation:
PLEASE HELP
A formula connecting speed (s), distance (d) and time (t) is s = d/t d = 160 to 2 significant figures t=7,2 to 2 significant figures Work out the upper and lower bounds for s. Give your answers to 3 significant figures.
Answers
Answer:
upper: 23.1lower: 21.4Step-by-step explanation:
You want the upper and lower bounds for speed, given that distance 160 was traveled in time 7.2 (both to 2 significant figures).
SpeedAs the problem statement tells you, speed is proportional to distance and inversely proportional to time.
BoundsThe upper bound for speed will be the upper bound for distance divided by the lower bound for time.
165/7.15 ≈ 23.1 . . . . . speed units (maximum)
The lower bound will be the lower bound for distance divided by the upper bound for time.
155/7.25 ≈ 21.4 . . . . . speed units (minimum)
__
Additional comment
Dividing the nominal values, the nominal speed is 22.22... speed units. Rounded to 2 sf, this would be 22 speed units. The implied bounds are 22±0.5. You can see that the upper and lower bounds computed here are actually about 22.25±0.85.
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solve the following simultaneous equation. 2x+2y+3z=210. 2x+3y+4z=270. 3x+4y+3z=300.
Answers
To solve the simultaneous equation: 2x+2y+3z=210, 2x+3y+4z=270, and 3x+4y+3z=300, we can use the elimination method. Here's how:
Step 1: Multiply the first equation by 2, and the second equation by -1 to eliminate x.4x + 4y + 6z = 420-2x - 3y - 4z = -270
Step 2: Add the two equations to eliminate x.2y + 2z = 150
Step 3: Multiply the first equation by -3, and the third equation by 2 to eliminate x.-6x - 6y - 9z = -6306x + 8y + 6z = 600
Step 4: Add the two equations to eliminate x.2y - 3z = -30
Step 5: Multiply the second equation by 2, and the fourth equation by 3 to eliminate y.4x + 6y + 8z = 540-6y + 9z = 90
Step 6: Add the two equations to eliminate y.4x + 17z = 630
Step 7: Substitute z = 2 into equation 2y + 2z = 150 to find y.2y + 4 = 150y = 73
Step 8: Substitute y = 73 and z = 2 into equation 4x + 17z = 630 to find x.4x + 34 = 630x = 149Therefore, the solution to the simultaneous equation 2x+2y+3z=210, 2x+3y+4z=270, and 3x+4y+3z=300 is x = 149, y = 73, and z = 2.
write an equation of the form y=mx for the line shown below (-1,4)
Answers
The equation of the Line of the form y = mx is y = -x + 3.
To write an equation of the form y = mx for the line shown below (-1,4), we need to determine the slope (m) of the line first.
Let (x₁, y₁) = (-1, 4) be a point on the line. Now let's find another point on the line. Let's say we have another point (x₂, y₂) = (1, 2).The slope (m) of the line can be calculated using the formula:m = (y₂ - y₁) / (x₂ - x₁)Substituting the values,
we get:m = (2 - 4) / (1 - (-1))= -2 / 2= -1
Now that we know the slope of the line, we can use the point-slope form of the equation of a line to write the equation of the line:y - y₁ = m(x - x₁)Substituting the values, we get:y - 4 = -1(x - (-1))y - 4 = -1(x + 1)y - 4 = -x - 1y = -x - 1 + 4y = -x + 3
Therefore, the equation of the line is y = -x + 3 in slope-intercept form. Since the question specifically asks for the equation of the form y = mx, we can rewrite the equation in this form by factoring out the slope:y = -x + 3y = (-1)x + 3
Thus, the equation of the line of the form y = mx is y = -x + 3.
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which of the following are solutions to the equation sin x cos x= -1/4
Answers
Answer:
x=-pie/12
sinxcosx=-1/4
multiply both sides by 2
sin2x = 2sinxcosx we know
now equation will be
sin2x = -1/2
2x=-pie/6
x=-pie/12
A = 11 + 2 (x-6) + 4 (-3-6)
Answers
Answer:
A = 2x - 37
Step-by-step explanation:
A
= 11 + 2(x - 6) + 4(-3 - 6)
= 11 + 2x - 12 - 12 - 24
= 2x - 37
For the given values of n and d, find integers q and r such that
n = dq + r
and
0 ≤ r < d.
n = 26, d = 50
q=
r=
Answers
Answer:
q = 0r = 26Step-by-step explanation:
You want integers q and r such that 26 = 50q +r, and 0 ≤ r < 50.
QSolving for q, we have ...
(26 -r)/50 = q
For r in the range 0–49, possible values of q are in the range ...
(26 -0)/50 ≥ q > (26 -49)/50
0.52 ≥ q > -0.46
The only integer in that range is ...
q = 0
RThen the value of r is ...
26 = 50·0 +r
r = 26
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Find the volume of the region bounded above by the paraboloid z=x^{2} +y^{2} and below by the triangle enclosed by the lines y=x, x=0 and x+y=6 in the xy-plane.\\
what is the volume under the paraboloid
Answers
The volume of the region bounded by the paraboloid and the triangle is 96 cubic units.
To find the volume of the region bounded by the paraboloid and the triangle, we can set up a double integral in the xy-plane. The paraboloid is represented by the equation z = x^2 + y^2, which forms a circular surface that extends infinitely in the positive z-direction. The triangle is defined by the lines y = x, x = 0, and x + y = 6 in the xy-plane.
To set up the integral, we need to determine the limits of integration for x and y. From the equations of the triangle, we can see that x ranges from 0 to 6, and y ranges from x to 6 - x. This means that for each value of x, y will vary within the corresponding range.
The volume can be calculated by integrating the function z = x^2 + y^2 over the given region. This gives us the double integral:
V = ∫∫[x^2 + y^2] dA,
where dA represents the differential area element in the xy-plane.
Integrating over the limits of integration for x and y, the volume can be expressed as:
V = ∫[0 to 6] ∫[x to 6 - x] (x^2 + y^2) dy dx.
To evaluate this double integral, we need to perform the integration step by step. First, we integrate with respect to y, treating x as a constant:
V = ∫[0 to 6] [xy + (y^3)/3] evaluated from y=x to y=6-x dx.
Simplifying the expression inside the square brackets, we have:
V = ∫[0 to 6] [x(6-x) + ((6-x)^3)/3 - x(x) - (x^3)/3] dx.
Combining like terms, we get:
V = ∫[0 to 6] [(6x - x^2) + (216 - 36x + 3x^2 - x^3)/3 - x^2 - (x^3)/3] dx.
Simplifying further, we have:
V = ∫[0 to 6] [(216 - 36x + 3x^2 - x^3)/3] dx.
Now, we integrate with respect to x:
V = [(72x - 18x^2 + x^3/3) / 3] evaluated from x=0 to x=6.
Substituting the limits of integration, we get:
V = [(72(6) - 18(6^2) + (6^3)/3) / 3] - [(72(0) - 18(0^2) + (0^3)/3) / 3].
Simplifying the expression, we find:
V = [(432 - 216 + 72) / 3] - [0 / 3].
V = 288 / 3.
V = 96.
Therefore, the volume of the region bounded by the paraboloid and the triangle is 96 cubic units.
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