To integrate the function ∫(x² - 36) dx, we first need to factor out the expression inside the parentheses:
∫(x² - 36) dx = ∫(x - 6)(x + 6) dx
We can then use the power rule of integration to find the antiderivative:
∫(x - 6)(x + 6) dx = (1/3)x³ - 6x + C, where C is the constant of integration.
Since the original problem states X > 6, we can evaluate the definite integral using these limits:
∫(x² - 36) dx from 6 to X = [(1/3)X³ - 6X] - [(1/3)(6)³ - 6(6)]
= (1/3)X³ - 6X - 68
Therefore, the answer in exact form is (1/3)X³ - 6X - 68.
To integrate the given function, first note the correct notation for the function: ∫(x^2)/(x^2 - 36) dx for x > 6.
To solve this, we can use partial fraction decomposition. The given function can be rewritten as:
∫(A(x - 6) + B(x + 6))/(x^2 - 36) dx
Solving for A and B, we find that A = 1/12 and B = -1/12. Now we rewrite the integral as:
∫[(1/12)(x - 6) - (1/12)(x + 6)]/(x^2 - 36) dx
Next, separate the two terms and integrate them individually:
(1/12)∫[(x - 6)]/(x^2 - 36) dx - (1/12)∫[(x + 6)]/(x^2 - 36) dx
Now, notice that the integrals are of the form ∫u'/u dx. The integral of this form is ln|u|. So we have:
(1/12)[ln|(x - 6)| - ln|(x + 6)|] + C
Using the logarithm property, we can rewrite the answer as:
(1/12)ln|((x - 6)/(x + 6))| + C
That is the exact form of the antiderivative for the given function.
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A license plate is made of three letters and three numbers, how many different license plates are possible?
There are 17,576,000 different license plates possible, considering 26 letters (A-Z) and 10 numbers (0-9).
There are 26 options for each of the three letters and 10 options for each of the three numbers. Therefore, using the multiplication principle, the total number of possible license plates is 26 x 26 x 26 x 10 x 10 x 10 = 17,576,000.
Alternatively, we can use the permutation formula to calculate the number of arrangements: P(26,3) x P(10,3) = 15,600 x 720 = 11,251,200.
However, since order does not matter in a license plate, we need to divide by the number of permutations of three letters and three numbers, which is 3! x 3! = 36, resulting in 11,251,200 / 36 = 17,576,000 possible license plates.
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Molly has 250 trading cards. she gives n trading cards to her friend carol. marcus has 430 trading cards. he gives away three times as many cards as molly does. how many trading cards did molly give away if molly and marcus have the same number of trading cards left?
a.) 70 trading cards
b.) 80 trading cards
c.) 90 trading cards
d.) 180 trading cards
Molly gave away 90 trading cards, which is option (c).
Let's start by figuring out how many trading cards Marcus gave away. We know that Molly gave away n trading cards, so she has 250 - n cards left. Marcus gave away three times as many cards as Molly did, so he gave away 3n cards. That means he has 430 - 3n cards left.
We also know that Molly and Marcus have the same number of trading cards left, so:
250 - n = 430 - 3n
Simplifying and solving for n, we get:
2n = 180
n = 90
Therefore, Molly gave away 90 trading cards, which is option (c).
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Dean's family goes on a road trip every summer. This scatter plot shows the number of days
they traveled and how far they went during their last 7 road trips.
What was the most common distance?(miles)
The most common distance in miles would be = 1,200 miles.
How to determine the most common distance that was travelled?To determine the distance that is most travelled the following is considered;
The total number of road trips = 7
On day 3 the distance travelled = 600 and 1,200 miles
On day 4 the distance travelled = 1,000,1,100 and 1,200 miles
On day 5 the distance travelled = 800 miles.
On day 6 the distance travelled = 1,300 miles
Therefore the most travelled distance = 1,200 miles.
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express as a single simplified fraction. 3m^2-3n^2/m^2+mp divided by 6m-6n/p+m
The single simplified fraction is (m + n)(p + m) / 2m.
To simplify the expression
[tex](3m^2 - 3n^2) / (m^2 + mp)÷ (6m - 6n) / (p + m)[/tex]
we need to invert the second fraction and multiply by the first.
[tex](3m^2 - 3n^2) / (m^2 + mp) \times (p + m) / (6m - 6n)[/tex]
We can then factor out a 3 from the numerator and the denominator, and cancel out the (m - n) terms. 3(m + n)(m - n) / 3m(m - n) x (p + m) / 6(m - n)
Simplifying further, we can cancel out the 3's and the (m - n) terms. (m + n) / m x (p + m) / 2
The simplified expression is (m + n)(p + m) / 2m.
To simplify the given expression, we invert the second fraction and multiply it by the first. Then we factor out common terms and cancel out like terms. We simplify the expression to obtain the single fraction (m + n)(p + m) / 2m.
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Austin spends a winter day recording the temperature once every three hours for science class. At 9 am, the temperature was -1.9°F. Between 9am and noon, the temperature rose 11.3°F. Between noon and 3pm, the temperature dropped 7.9°F. Between 3pm and 6pm, the temperature dropped 12.7°F. What was the temperature at 6pm?
To find the temperature at 6pm, we need to start with the temperature at 9am and then add or subtract the changes in temperature that occurred during the day.
We know that the temperature at 9am was -1.9°F. Between 9am and noon, the temperature rose 11.3°F, so at noon the temperature was:
-1.9 + 11.3 = 9.4°F
Between noon and 3pm, the temperature dropped 7.9°F, so at 3pm the temperature was:
9.4 - 7.9 = 1.5°F
Between 3pm and 6pm, the temperature dropped 12.7°F, so at 6pm the temperature was:
1.5 - 12.7 = -11.2°F
Therefore, the temperature at 6pm was -11.2°F.
Your brother traveled 115 miles in 2. 52 hours to come home for school break. What’s the average speed that he was traveling?
The average speed that your brother was traveling is approximately 45.63 miles per hour
The average speed is the total distance traveled divided by the total time taken. So we have:
Average speed = total distance ÷ total time
We are given the total distance as 115 miles and the total time as 2.52 hours. Therefore, the average speed is:
Average speed = 115 miles ÷ 2.52 hours
Average speed = 45.63 miles per hour
So, the average speed that your brother was traveling is approximately 45.63 miles per hour.
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What are all the possible rectangle with the perimeter 16cm,20cm,and 14cm, find the area of each rectangle
The area of a rectangle can be found by multiplying its length and width.
For perimeter 16cm: 15 cm² and 16 cm²For perimeter 20cm: 24 cm² and 25 cm²For perimeter 14cm: 12 cm²How to find area of rectangles?To find the area of rectangles with different perimeters, we need to use the formula:
Area = length x width
Perimeter = 16 cmPossible dimensions:
Length = 5 cm, Width = 3 cm
Length = 4 cm, Width = 4 cm
Area of the first rectangle = 5 cm x 3 cm = 15 cm²
Area of the second rectangle = 4 cm x 4 cm = 16 cm²
Perimeter = 20 cmPossible dimensions:
Length = 6 cm, Width = 4 cm
Length = 5 cm, Width = 5 cm
Area of the first rectangle = 6 cm x 4 cm = 24 cm²
Area of the second rectangle = 5 cm x 5 cm = 25 cm²
Perimeter = 14 cmPossible dimensions:
Length = 4 cm, Width = 3 cm
Area of the rectangle = 4 cm x 3 cm = 12 cm²
Therefore, the areas of the possible rectangles with perimeters 16cm, 20cm, and 14cm are:
For perimeter 16cm: 15 cm² and 16 cm²
For perimeter 20cm: 24 cm² and 25 cm²
For perimeter 14cm: 12 cm²
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Complete the sentences about the expressions 3x+4 –2x
, and 5x+2x+x
.
CLEAR CHECK
In the expression 3x+4 –2x
, you can combine
like terms, and the simplified expression is
.
In the expression 5x+2x+x
, you can combine
like terms, and the simplified expression is
For the expressions 3x+4 –2x, and 5x+2x+x the simplified expression after combining like terms is x+4 and 8x.
The given expressions are 3x+4 –2x, and 5x+2x+x
We have to simplify these expressions by combining the like terms
For the expression 3x+4 –2x
We have to combine like terms
x+4
Now for expression 5x+2x+x
Combine the like terms to get
8x
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For the class party, Josue and Pho each brought 1 3/5 liters of lemonade. How many liters of lemonade did they bring altogether?
Josue and Pho brought 3 1/5 liters of lemonade altogether
Josue and Pho brought 1 3/5 liters of lemonade each, so the total amount of lemonade they brought is:
1 3/5 + 1 3/5 = 3 1/5
To add the two mixed numbers, we first need to find a common denominator. In this case, the common denominator is 5. Then we convert both mixed numbers into fractions with a denominator of 5:
1 3/5 = (5 × 1 + 3) / 5 = 8/5
1 3/5 = (5 × 1 + 3) / 5 = 8/5
Now we can add the fractions:
8/5 + 8/5 = (8 + 8) / 5 = 16/5
Finally, we can convert the fraction back to a mixed number:
16/5 = 3 1/5
Therefore, Josue and Pho brought 3 1/5 liters of lemonade altogether.
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The sum of two numbers is 30. Determine the two numbers of their product is a maximum.
Answer:
Step-by-step explanation:
Let's call the two numbers x and y. We know that:
x + y = 30 (since the sum of the two numbers is 30)
We want to find the values of x and y that maximize their product, which is given by:
P = xy
To solve for x and y, we can use the fact that the sum of the two numbers is 30, so we can rewrite one of the numbers in terms of the other:
y = 30 - x
Substituting this into the equation for the product, we get:
P = x(30 - x)
Expanding this expression, we get:
P = 30x - x^2
To find the maximum value of P, we can take the derivative of this expression with respect to x and set it equal to zero:
dP/dx = 30 - 2x = 0
Solving for x, we get:
x = 15
So one of the numbers is x = 15, and the other is y = 30 - x = 15.
To confirm that this gives the maximum product, we can take the second derivative of P with respect to x:
d2P/dx2 = -2
Since the second derivative is negative, this means that the function P = 30x - x^2 has a maximum at x = 15.
Therefore, the two numbers are 15 and 15, and their product is maximized at P = 15 * 15 = 225.
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Mrs. galicia has a cupcake company. the amount of money earned is represented by ()=2√+4యwhere x is the number of years since 2015. (a) write the transformations that have occurred from the original parent function, ()=√య(b) mrs. galicia changes the purchase price and the new function, ℎ()=2√+2య+4. what transformations have occurred from the original cupcake company function, g(x)?
The transformations that have occurred from the original parent function ()=√య to the given function ()=2√x+4 are: vertical stretch by a factor of 2 and a vertical shift upward by 4 units.
(a) Transformations of original parent function?The transformations that have occurred from the original parent function ()=√x to the given function ()=2√x+4 are: vertical stretch by a factor of 2 and a vertical shift upward by 4 units. The square root function (√x) has been multiplied by 2, resulting in a steeper curve, and then shifted vertically upwards by 4 units.
(b) Transformations of new cupcake function?From the original cupcake company function g(x), the new function h(x)=2√x+2య+4 involves additional transformations. It starts with the transformations from part (a), which are a vertical stretch by a factor of 2 and a vertical shift upward by 4 units.
Annndditionally, the function is further transformed by a horizontal compression by a factor of 1/2, achieved by dividing the x-values by 2. Finally, a vertical shift upward by 2 units is applied. These transformations modify the shape, position, and scale of the original function to represent the changes in Mrs. Galicia's cupcake company's earnings.
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If KJ=10, find the length of arc HJ
Answer:
Step-by-step explanation:
The arc length formula is
[tex]L=\frac{\theta}{360}*2\pi r[/tex]
Theta is the angle that intersects arc HJ. That measure is 180-122 which is 58 degrees. We put that into the formula along with the radius measure of 10 to get:
[tex]L=\frac{58}{360}*2\pi (10)[/tex] which gives us, rounded to the nearest hundredth,
L = 10.12 units
Rote tells the little monsters to do an overhead press every $12$ seconds and a squat every $30$ seconds. (For example, they should do their first squat $30$ seconds into the drill.)
How many times during the $200$ second drill should the little monsters do an overhead press and a squat at the same instant?
The number of times that the little monsters will do an overhead press and a squat at the same instant during the drill is: 3 times
How to solve Prime Factorization Problems?We are told that, Rote tells the little monsters to carry out an overhead press every 12 seconds and then also a squat every 30 seconds.
Thus, prime factorization of 12: 2² x 3.
Thus, prime factorization of 30: 2 x 3 x 5.
For us to get the least common multiple, we will have to find the highest power of each of the prime factors that show up in either factorization. Therefore, we can say that the smallest common multiple of 12 and 30 are: 2² x 3 x 5 = 60
This tells us that the little monsters will carry out an overhead press and then a squat at the same instant for every 60 seconds.
For us to find how many times this will occur during the 200-second drill, we will then divide 200 by 60:
200 ÷ 60 = 3 remainder 20
This tells us that there will exist 3 complete cycles of both of the exercises during the 200-second drill, with an additional 20 seconds left over.
Therefore, we can say that the little monsters will do an overhead press and a squat at the same instant 3 times during the drill.
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Write the algebraic expression that matches each graph.
Graph inserted below via image.
Answer: 7
Step-by-step explanation:
Answer:
y=|x-2|-2
Step-by-step explanation:
Go onto desmos and you can ask it to graph an equation to test your answers.
Patricia bought 4 apples and 9 bananas for $12. 70 Jose bought 8 apples and I bananas for $17. 70 at the same grocery store What is the cost of one apple?
The cost of one apple is $2.15.
To find the cost of one apple, we can set up a system of equations with the given information. Let's use A for the cost of one apple and B for the cost of one banana:
1) 4A + 9B = $12.70
2) 8A + B = $17.70
Now, we can solve this system of equations. We can multiply equation 1 by 2 to match the number of apples in equation 2:
1) 8A + 18B = $25.40
Now subtract equation 2 from the modified equation 1:
(8A + 18B) - (8A + B) = $25.40 - $17.70
17B = $7.70
Now, divide by 17 to find the cost of one banana:
B = $7.70 / 17 = $0.45
Now that we know the cost of one banana, we can substitute B in equation 2 to find the cost of one apple:
8A + ($0.45) = $17.70
8A = $17.25
A = $17.25 / 8 = $2.15
So, the cost of one apple is $2.15.
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a carpnter had a piece of wood that is 15 feet long. he cut the wood into pieces that are 1/4 of a foot long . how many pieces did he cut?
Therefore , the solution of the given problem of unitary method comes out to be the carpenter divided the wood into 60 pieces.
Definition of a unitary method.The well-known minimalist approach, current variables, and any crucial elements from the initial Diocesan tailored query can all be used to accomplish the work. In response, you can be granted another chance to utilise the item. If not, important impacts on our understanding of algorithms will vanish.
Here,
Divide the overall length of the wood by the length of each piece to get how many pieces the carpenter cut.
Each piece measures 0.25 feet, or 1/4 of a foot.
By dividing the overall length of the wood by the length of each component, we can determine the number of pieces:
=> Quantity = Length overall / Length of each element
=> Piece count is 15 feet divided by 0.25 feet. or 15/0.25
=> There are 60 parts total.
So, the carpenter divided the wood into 60 pieces.
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Ms. Frank is going to wallpaper a living room with dimensions 24 feet long, 18 feet wide, and 8 feet high. How much wallpaper will Ms. Frank need if she is only putting it on the four walls?
Answer:
Step-by-step explanation:
you take 24 multiplied by 18 then by 8 and that'll equal
24 x 18 x 8 = 3456
I NEED HELPPPPPPPPPPPP
Answer: V = 2527.2 in^3
Step-by-step explanation:
V = Bh
that is, Volume = base area x height
the base area is the hexagon, and the height is given as 12.
Think of dividing the hexagon into 6 equal triangles, with height 7.8
so the area of all 6 triangles, (effectively the area of the hexagon), will be:
6(0.5 x 9 x 7.8) = 210.6 in^2
multiply this by the height to get the volume:
210.6 x 12 = 2527.2 in^3
thats it!
V = 2527.2 in^3
You are building a fence for your pasture. the length is three more than four times the width
The perimeter of the fence for the pasture is 10W + 6.
To find the perimeter of the fence, we need to add up the lengths of all four sides. Let's start by using the given information to find the length and width of the pasture.
Let's say the width of the pasture is W. Then, according to the problem, the length of the pasture is 3 more than 4 times the width, which can be expressed as:
Length = 4W + 3
Now that we have the length and width, we can find the perimeter by adding up all four sides:
Perimeter = 2(Length + Width)
Perimeter = 2(4W + 3 + W)
Perimeter = 2(5W + 3)
Perimeter = 10W + 6
Therefore, the perimeter of the fence is 10W + 6.
Note: The question is incomplete. The complete question probably is: You are building a fence for your pasture. The length is three more than four times the width. What is the perimeter of the fence.
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Please help!!! 10 points
The solution is, Options 1, 3, and, 5 are true.
The statements given are,
1.) The product of reciprocals is 1.
Let the fraction be a/b , and reciprocal be b/a ,
The product of the two will be 1.
Hence, the statement is true.
2.) To divide fractions, multiply the divisor by the reciprocal of the dividend.
Let the fraction be a/b , now,
a/b*1/a = a^2/b,
Hence, the statement is false.
3.) The reciprocal of a whole number is 1 over the number.
Let the number be 3, now,
The reciprocal of number 3 is 1/3 .
Hence, the statement is true.
4.) Reciprocals are used to multiply fractions.
The statement is false, Reciprocals are not used to multiply fractions.
5.) To find the reciprocal of a fraction, switch the numerator and denominator.
Let the fraction be a/b , then the reciprocal will be b/a .
Hence, the statement is true.
Therefore, Options 1, 3, and, 5 are true.
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complete question:
Select all that apply. Determine which of the following statements are true of reciprocals. Select all that apply. The product of reciprocals is 1 To divide fractions, multiply the divisor by the reciprocal of the dividend The reciprocal of a whole number is 1 over the number Reciprocals are used to multiply fractions To find the reciprocal of a fraction, switch the numerator and denominator
Doug is filing singly. his net taxable income is $80,575. every week, $304 is withheld from his earnings for income tax. based on the table below, what can doug expect when his taxes are due? between 80,550 and 80,600 dollars, for filing single, the amount of taxes is 16,539 dollars. a. doug will receive a refund of $123. b. doug will receive a refund of $2,977. c. doug will owe an additional $1,125. d. doug will owe an additional $731.
Doug is filing singly, and his net taxable income is $80,575. The tax amount for this income range is $16,539. Every week, $304 is withheld from his earnings for income tax. Doug can expect he will owe an additional $731. So option d is the correct answer.
Calculate the total amount withheld for the year.Since the result is a positive number, Doug will owe an additional $731 when his taxes are due. Therefore, the correct answer is d. Doug will owe an additional $731.
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For a science experiment Corrine is adding hydrochloric acid to distilled
water. The relationship between the amount of hydrochloric acid, x, and the
amount of distilled water, y, is graphed below. Which inequality best
represents this graph?
The best inequality that represents the relationship between the amount of hydrochloric acid (x) and the amount of distilled water (y) in the given graph is 3y - 2x > 0, option D is correct.
The graph shows a straight line with a negative slope passing through the origin. As the amount of hydrochloric acid, x, increases, the amount of distilled water, y, decreases
To see why, let's use a point on the line, such as (2, 3), and plug it into the inequality. We get:
3(3) - 2(2) > 0
9 - 4 > 0
This is true, so the point (2, 3) is a solution to the inequality. Any point on the line will also satisfy this inequality since it represents all possible combinations of x and y that Corrine can use in her experiment.
Alternatively, we can rewrite the inequality in slope-intercept form:
y < (2/3)x
This means that the y-values on the line are less than the corresponding values of (2/3)x. So as x increases, y must decrease to stay below the line. This confirms that 3y - 2x > 0 is the correct inequality.
Hence, option D is correct.
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The correct question is:
For a science experiment, Corrine is adding hydrochloric acid to distilled water. The relationship between the amount of hydrochloric acid, x, and the amount of distilled water, y, is graphed below. Which inequality best represents this graph?
A. 2y - 3x < 0
B. 3y - 2x < 0
C. 2y - 3x > 0
D. 3y - 2x > 0
Town Hall is located 4. 3 miles directly east of the middle school. The fire station is located 1. 7 miles directly north of Town Hall.
Part A
What is the length of a straight line between the school and the fire station? Round to the nearest tenth. Enter your answer in the box.
Part B
The hospital is 3. 1 miles west of the fire station. What is the length of a straight line between the school and the hospital? Round to the nearest
tenth. Enter your answer in the box.
Using Pythagorean theorem the distance between the school and the fire station is approximately 4.7 miles, while the distance between the school and the hospital is approximately 7.8 miles.
To solve this problem, we can use the Pythagorean theorem to find the distances and then add them up to get the total distance between the school and the fire station, and then between the school and the hospital.
Part A:
Let's call the middle school point A, Town Hall point B, and the fire station point C. We can draw a right triangle with AB as the base, BC as the height, and AC as the hypotenuse. Using the Pythagorean theorem, we have:
AC^2 = AB^2 + BC^2
AC^2 = 4.3^2 + 1.7^2
AC^2 = 18.98 + 2.89
AC^2 = 21.87
AC ≈ 4.7 miles (rounded to the nearest tenth)
Therefore, the length of the straight line between the school and the fire station is approximately 4.7 miles.
Part B:
Let's call the hospital point D. We can draw another right triangle with CD as the base, BC as the height, and BD as the hypotenuse. Using the Pythagorean theorem, we have:
BD^2 = BC^2 + CD^2
BD^2 = 1.7^2 + 3.1^2
BD^2 = 2.89 + 9.61
BD^2 = 12.5
BD ≈ 3.5 miles (rounded to the nearest tenth)
Now, we can add the distance between A and B (4.3 miles) to the distance between B and D (3.5 miles) to get the total distance between A and D:
AD ≈ 7.8 miles (rounded to the nearest tenth)
Therefore, the length of the straight line between the school and the hospital is approximately 7.8 miles.
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How many solutions?
4x - y = 18 and -4x + y = -18
a. one
b. infinitely many
c. no solutions
The given equation 4x - y = 18 and -4x + y = -18 has b. infinitely many solutions.
To determine how many solutions there are for the system of equations 4x - y = 18 and -4x + y = -18, follow these steps:
Step 1: Notice that the second equation is just the negative of the first equation:
4x - y = 18
(-1)(4x - y) = (-1)(18)
-4x + y = -18
Step 2: Since the second equation is just the negative of the first, the two equations are dependent and represent the same line.
Step 3: When two equations represent the same line, there are infinitely many points where they intersect, as they overlap completely.
So, the answer is: b. infinitely many solutions.
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Mollie drew mol and ted drew ted. they measured a few parts of their triangles
and found that ml = td, ol = ed, and l = d. what postulate can mollie and ted
use to justify why their triangles must be congruenta
Mollie and Ted can use the Side-Side-Side (SSS) postulate to justify why their triangles must be congruent.
According to the given information, the two triangles share three corresponding sides of equal length: ML = TD, OL = ED, and L = D.
The SSS postulate states that if three corresponding sides of two triangles are congruent, then the triangles are congruent. Therefore, because Mollie's triangle and Ted's triangle share three corresponding sides of equal length, they are congruent by the SSS postulate.
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To the nearest whole cubic centimeter, what is the volume of the prism?
The volume of the prism in 120 cubic centimeter.
What is volume?Volume is a measure of the amount of space occupied by a three-dimensional object. It is a scalar quantity that characterizes the "size" or "capacity" of an object in three dimensions.
What is prism?A prism is a three-dimensional geometric shape that has two parallel and congruent bases connected by lateral faces that are typically rectangular or parallelogram-shaped. Prisms are classified based on the shape of their base, such as rectangular prisms, triangular prisms, pentagonal prisms, hexagonal prisms, and so on. The lateral faces of a prism are perpendicular to the bases, and the bases are usually parallel to each other.
According to the given information:
Given the measures of prism is length = 8cm, breadth = 6 cm, height = 5 cm
The formula for volume of prism = 1/2 l*b*h
= 1/2 8*6*5
= 120 cubic centimeter
Hence the volume of prism is 120 cubic centimeter.
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10 foot ladder is leaning against a vertical wall when Jack begins
pulling the foot of the ladder away from the wall at a rate of 0.5
fr/s. how fast is the top of the ladder sliding down the wall?
We can use the Pythagorean theorem to relate the distances between the ladder, wall, and ground. Let's call the distance from the foot of the ladder to the wall "x", and the distance from the top of the ladder to the ground "y". Then, we know that:
x^2 + y^2 = 10^2
We can differentiate this equation with respect to time to get:
2x(dx/dt) + 2y(dy/dt) = 0
We're interested in finding dy/dt, the rate at which the top of the ladder is sliding down the wall. We know that dx/dt = 0.5 ft/s, so we can plug in these values and solve for dy/dt:
2x(dx/dt) + 2y(dy/dt) = 0
2(8)(0.5) + 2y(dy/dt) = 0 (since x = 8 based on the Pythagorean theorem)
dy/dt = -4 ft/s
So the top of the ladder is sliding down the wall at a rate of 4 ft/s.
When the 10-foot ladder is leaning against a vertical wall, it forms a right-angled triangle with the wall and the ground. As Jack pulls the foot of the ladder away from the wall at a rate of 0.5 ft/s, the top of the ladder slides down the wall. To find the rate at which the top of the ladder slides down, we can use the Pythagorean theorem:
a^2 + b^2 = c^2
where a is the distance from the foot of the ladder to the wall, b is the height of the ladder's top from the ground, and c is the length of the ladder (10 feet).
Differentiating both sides with respect to time (t), we get:
2a(da/dt) + 2b(db/dt) = 0
We know that da/dt = 0.5 ft/s. We need to find db/dt, which is the rate at which the top of the ladder slides down the wall. To do this, we need to find the values of a and b at a given moment. Since the problem doesn't provide this information, it's not possible to determine the exact value of db/dt. However, if you have the values of a and b, you can plug them into the equation and solve for db/dt.
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Find an equation in slope-intercept form for the line passing through each pair of points: (-4, 4), (-5, -3)
The equation in slope-intercept form for the line passing through each pair of points, (-4, 4) and (-5, -3) is expressed as: y = 7x + 32
What is the Equation of a Line in Slope-Intercept Form?Given the points, (-4, 4) and (-5, -3), first find the slope of the line.
Slope (m) = change in y / change in x = -3 - 4 / -5 -(-4)
m = -7/-1
m = 7
Substitute m = 7, a = -4, and b = 4 into y - b = m(x - a):
y - 4 = 7(x + 4)
Rewrite im slope-intercept form:
y - 4 = 7x + 28
y - 4 + 4 = 7x + 28 + 4
y = 7x + 32
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With all the responses in, Jayda found the Mean Absolute Deviation (MAD), rounded to the nearest tenth. Select the correct Mean Absolute Deviation and what it tells you about the data set.
The numbers 0, 1, 3, 10, 12, 12, 15, 17, 18, 22, 66
From the given data set, the mean absolute deviation is 10.72
What is the mean absolute deviationTo determine the mean absolute deviation of the data set, we need to find the mean first.
mean = (0 + 1 + 3 + 10 + 12 + 12 + 15 + 17 + 18 + 22 + 66) / 11
mean = 16
Now, let's calculate the mean absolute deviation
|0 - 16| = 16
|1 - 16| = 15
|3 - 16| = 13
|10 - 16| = 6
|12 - 16| = 4
|12 - 16| = 4
|15 - 16| = 1
|17 - 16| = 1
|18 - 16| = 2
|22 - 16| = 6
|66 - 16| = 50
MAD = (16 + 15 + 13 + 6 + 4 + 4 + 1 + 1 + 2 + 6 + 50) / 11
MAD = 10.72
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The Worthington Family is applying for a loan to purchase a new home. In order to qualify they
must have a net worth greater than $100,000.
• Mr. Worthington is a dentist, so he has $85,400 in student loans to pay off.
•Mrs. Worthington earns $42,500 per year at her job.
•The Worthington family has a savings account with a balance of $18,800.
•They own two vehicles that are worth a combined total of $64,600.
• Mrs. Worthington owns an antique necklace valued at $6,200.
"Run.
• The Worthingtons have been saving for their children's college fund.
currently has a balance of $24,700
The net worth of the Worthington family is $28,900, which is greater than the requirement of $100,000 for the loan. Therefore, they meet the net worth requirement for the loan.
To calculate the net worth of the Worthington family, we need to add up all their assets and subtract their liabilities (debts). Let's start by listing them:
Assets:
- Savings account: $18,800
- Vehicles: $64,600
- Antique necklace: $6,200
- College fund: $24,700
Total assets: $114,300
Liabilities:
- Mr. Worthington's student loans: $85,400
Total liabilities: $85,400
To calculate the net worth, we subtract the liabilities from the assets:
Net worth = Total assets - Total liabilities
Net worth = $114,300 - $85,400
Net worth = $28,900
The net worth of the Worthington family is $28,900, which is greater than the requirement of $100,000 for the loan. Therefore, they meet the net worth requirement for the loan.
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