find the value of m for which the simultaneous equations 3x my = 5 (m 2)x 5y = m a have infinitely many solutions b have no solution.
In the simultaneous equations the value of m is 3 and the equation have many solution.
The term called equation in math is known as a condition on a variable (or variables) such that two expressions in the variable have equal value.
Here we have know that the general solution is calculated as,
=> x = (m-5)/(m-3)
and
=> y = 2/(m-3)
When we equate the equations then we get,
=> (m - 5) / (m -3) = 2/(m - 3)
=> m - 5 = 2
=> m = 3
So this dependent system has infinitely many solutions.
When m=3 the equations become look like 3x+3y=5 and 5x+5y=3.
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the cubs are playing the red sox in the world series. to win the world series, a team must win 4 games before the other team does. if the cubs win each game with probability $\dfrac{3}{5}$ and there are no ties, what is the probability that the cubs will win the world series? express your answer as a percent rounded to the nearest whole percent.
The probability to win the world series by cubs as per given probability of each game with no ties is equal to 29% ( in percent nearest whole number).
Total number of games played by the teams 'n' = 7
Number of games required to win the world series 'x' = 4
Probability of winning a game by cubs 'p' ( success) = 0.60
1 - p = 1 - 0.60
= 0.40
Probability of winning the world series by Cubs
= ⁿCₓ pˣ ( 1 - p )ⁿ⁻ˣ
= ⁷C₄ ( 0.60 )⁴ ( 0.40 )⁷⁻⁴
= ( 7! / (4!) ( 7 -4 )! ) ( 0.60 )⁴ ( 0.40 )³
= ( 7 × 5 ) ( 0.0082944 )
= 0.29034
= 29% ( in percent )(nearest whole number)
Therefore, the probability of winning the world series by cubs is equal to 29%.
The above question is incomplete , the complete question is:
In the World Series, the team that wins 4 games out of seven is the champion. Suppose that the Cubs are playing the red Sox (and the world is about to end), and the Cubs have a probability of 3/5 of winning a game over the Sox and there is no ties. What is the probability that the Cubs will win the world series ? Express your answer as a percent rounded to the nearest whole percent.
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how to prove 431 divides 2^43 -1
By using Fermat's Little Theorem, we were able to prove through a series of mathematical steps that 431 divides 2⁴³⁻¹
In mathematics, proving that a number divides another number means finding out if it can evenly divide the other number with no remainder.
Here we need to prove that 431 divides 2⁴³⁻¹, we can use the mathematical property of modular arithmetic.
In modular arithmetic, we are interested in finding the remainder when a number is divided by another number.
In this case, we want to find the remainder when 2⁴³⁻¹ is divided by 431.
If the result is 0, then we can conclude that 431 divides 2^43 -1.
In order to do this, we can use Fermat's Little Theorem.
Since 431 is a prime number, we can use Fermat's Little Theorem to calculate
=> 2⁴³⁰ mod 431.
If the result is 1, then we know that
=> 2⁴³ mod 431
=> 2^(43 mod 430) * 2⁴³⁰ mod 431 = 2³ mod 431.
Finally, we can calculate 2³ mod 431 and if the result is 1, we can conclude that
=> 2⁴³⁻¹ mod 431 = 1.
This means that 2 is divisible by 431 with no remainder, so we can conclude that 431 divides 2⁴³⁻¹.
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Select the best response.
To display the distribution of grades (A, B, C, D, F) in a course, it would be correct to use A. a bar graph but not a pie chart. B. either a pie chart or a bar graph. C. a pie chart but not a bar graph. D. None of the above.
To display the distribution of grades (A, B, C, D, F) in a course, it would be correct to use histogram
a) We note that the given histogram contains a
very large number of bars and that most bars are quite low while also very narrow.
The display is then not appropriate, as the histogram should contain less bars to make the shape in the distribution more obvious.
(b) The distribution of grades is skewed to the left as the highest bars in the histogram are to the right in the graph.
The grades appear to be centered roughly at 170, while the graders vary from about 75 to 190.
Moreover, there appear to be a few outliers in the distribution, as there are gaps between the bars of the histogram.
since Histogram contains too many bars and Left-skewed histogram can be considered to display.
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Trey drove 315 miles in 5 hours.
At the same rate, how many miles would he drive in 11 hours?
Within 5 hours, Trey covered 315 miles. He would need to travel 504 miles in 11 hours at the same speed.
How to calculate speed?Speed exist calculated as distance times speed. You need to be aware of the units for distance and time in order to calculate the units for speed. The units in this example will be metres per second (m/s), since the distance is measured in metres (m) and the time is measured in seconds (s). Speed (or rate, r) is a scalar number that represents the amount of time (Δt) spent traveling over the distance (d), as shown by the equation r = d/Δt.In order to compute speed, the distance traveled must be divided by the amount of time spent traveling.You must ascertain the unit rate. Divide the miles by the hours to arrive at that.
The unit rate should be found to be 63 miles per hour.
You need to double that by 8 (because it takes 8 hours),
Which is 504 miles.
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Im Stuck on this question
The amount of the next 50 blogs that are expected to be of pop culture is given as follows:
20 blogs.
How to obtain the amount?The first step in obtaining the expected amount is obtaining the proportion out of the blogs that are of pop culture, dividing the number of blogs that are of pop culture by the total number of blogs.
Considering the data in the table, the proportion is obtained as follows:
p = 26/(20 + 26 + 7 + 12)
p = 0.4.
Then, considering the proportion of 0.4, out of the next 50 blogs, the expected amount of pop culture blogs is given as follows:
0.4 x 50 = 20 blogs.
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if a1 = 4 and an =2an-1-1 then find the value of a4
The value of [tex]a_{4} = 37[/tex]
We need to find value of [tex]a_{4}[/tex].
What is Sequence?A sequence is a collection of elements that are arranged according to a specific pattern.
Arithmetic sequence is a sequence where each consequtive term has a common constant difference.
Now,
Given that value of [tex]a_{1} = 3[/tex]
[tex]a_{n} = 4 a_{n-1} -1[/tex]
Then to find value of [tex]a_{4}[/tex], we can find [tex]a_{2} , a_{3}[/tex]as;
[tex]a_{2} = -2(a_{1} ) - 1\\a_{2} = -2(4) - 1\\a_{2} = -9[/tex]
[tex]a_{3} = -2(a_{2} ) - 1\\a_{3} = -2(9) - 1\\a_{3} = -19[/tex]
Hence, the value of [tex]a_{4}[/tex] as;
[tex]a_{4} = -2(a_{3} ) - 1\\a_{4} = -2(-19) - 1\\a_{4} = 37[/tex]
Therefore, The value of [tex]a_{4}[/tex]= 37.
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-10
#1.
A system of equations is graphed on the grid. Which
system of equations is best represented by the graph?
10
A.
y = x + 2
y = -x - 3
y = - 3x + 1
y = 2x + 2
B.
D.
y = -x + 2
y = 2x - 3
y =
y =
x + 2
-x + 4
The equations that correspond to this situation, that is, the equations that have the value of c as 2 and -3 are: y = - 4/3x + 2 y = 2x -3. Hence, option B is the correct answer.
What is y-intercept?The point where a graph intersects the y-axis is known as the y intercept. Any point on the y-axis has an x-coordinate of 0, is known as y intercept. Therefore, a x-coordinate of y intercept is 0.
The equation of the line is given by the following formula:
y = mx + c
where, c is the y-intercept of the line.
From the graph we can observe that the y-intercept of the red line is +2 and the y-intercept of the blue line is -3.
Hence, the equations that correspond to this situation, that is, the equations that have the value of c as 2 and -3 are:
y = - 4/3x + 2
y = 2x -3
Hence, option B is the correct answer.
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Less than the quotient of 10 and a number b
The quotient of 10 and a number b is the number you get when you divide 10 by b.
Less than the quotient of 10 and a number b?For example, if b is 2, then the quotient of 10 and a number b would be 5 (10÷2=5). If b is 5, then the quotient of 10 and a number b would be 2 (10÷5=2).The answer to the question "Less than the quotient of 10 and a number b?" is any number less than the quotient of 10 and b.For example, if b is 4, then the quotient of 10 and b is 2.5 (10÷4=2.5). Any number less than 2.5 would be a valid answer, such as 1.5, 0.5, 0.1, etc.It is important to note that the quotient of 10 and b will change as b changes.Therefore, the answer to the question of "Less than the quotient of 10 and a number b?" will also change.For example, if b is 8, then the quotient of 10 and b is 1.25 (10÷8=1.25). Any number less than 1.25 would be a valid answer, such as 0.5, 0.1, etc.In conclusion, the answer to the question of "Less than the quotient of 10 and a number b?" is any number that is less than the quotient of 10 and b.The quotient of 10 and b will change depending on the value of b, and therefore the answer to the question will also change.To learn more about The quotient refer to:
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The alternating current in an electric inductor is I = E/Z , where E is the voltage and Z = R+XLi is impendance. If E = 7(cos 50 degrees +i sin 50 degrees), R=6, and XL=4, find the current.
The value of the current 'I' for the given relation of current , impedance, and voltage I = E/ Z is equal to I = ( 4.5 + 5.4i ) / ( 6 + 4i ).
Here R=6, and XL=4
Impedance Z = R+ XL i
⇒ Z = 6 + 4i __(1)
Where
E = 7(cos 50 degrees +i sin 50 degrees)
Simplify the above expression to get the value of voltage E.
⇒ E = 7 ( 0.6428 + i 0.7660 )
⇒ E = 4.4996 + i( 5.362 )
⇒ E = 4. 5 + 5.4i ____(2)
Now alternating current ,
I = E / Z
Substitute the value of E and Z from (1) and (2) we get,
⇒ I = ( 4.5 + 5.4i ) / ( 6 + 4i )
Therefore, the value of the current I is equal to ( 4.5 + 5.4i ) / ( 6 + 4i ).
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You have one x2-block, twelve x-blocks, and as many 1-blocks as you wish. Create an algebra-block model that shows how many 1-blocks are required to form a square. What are the dimensions of the square?
The dimensions of the square are √2n by √2n
How to determine the dimensions of the squareLet's call the number of 1-blocks required to form a square "n".
The dimensions of the square can be represented as x^2 + x * (2 * n / x)
This is so because:
Each x^2 block represents a side length of x units. So, to form a square, x^2 blocks would represent x units on both sides, or x^2 units in total.Each x block represents a side length of 1 unit. To form the other two sides of the square, we need 2n/ x x blocks, each representing 1 unit, for a total of 2n units.So, we have a total of x^2 + x * (2n/x)Since the square is a perfect square, both sides of the equation must be equal:
x^2 = x * (2 * n / x)
Solving for x:
x^2 = 2n
Take the square root of both sides
x = √2n
So the dimensions of the square are √2n by √2n
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Please someone help!
The area of parallelogram
ABCD is 24 square units.
Find x.
The value of x of the parallelogram is 7 units
How to find the side x of the parallelogram?The product of the base and height gives the total area of the parallelogram.
The area of a parallelogram can be calculated using the formula:
Area = base * height
A = b * h
where the base is the length of one side of the parallelogram, and height is the perpendicular distance between the two parallel sides.
Given that A = 24 square units, b = 8
A = b * h
28 = 8 * h
28 = 8h
h = 28/8
h = 3.5 units
sin 30 °= h/x (opposite / hypotenuse)
0.5 = 3.5/x
0.5x = 3.5
x = 3.5/0.5
x = 7 units
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A tree that is 12 feet tall is growing at a rate of 2 1/2
feet each year. A tree that is 15 feet tall is growing at a rate of 1 foot each year.
Enter the number of years it will take the two trees to reach the same height.
HELP please
The number of years taken for the tree to reach the same height found using mathematical expression is 2 years.
What is an expression?
Mathematical statements are called expressions if they have at least two terms that are related by an operator and contain either numbers, variables, or both. Addition, subtraction, multiplication, and division are all possible mathematical operations.
Let x be the number of years taken to reach the same height.
Height of the first tree after x years,
12 + 2.5(x)
Height of the second tree after x years,
15 + 1(x)
Given that both heights are the same after x years.
So,
12 + 2.5x = 15 + x
1.5x = 3
x = 2
Therefore the number of years taken for the tree to reach the same height is 2 years.
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A number, x, rounded to 2 decimal places is 15.38 Write down the error interval for x.
The error interval needed is [15.375, 15.385)
Given that:
A number 'x' rounded to 2 decimal places = 15.38
According to the rules of the rounding of number, we surely know that
x must be in [ 15.375, 15.385) since if x is outside of this interval, then rounding to two places of decimal won't lead to 15.38.
Since x can't be 15.385 itself since that rounds up to 15.39
Or:
[tex]15.375 \leq x\leq 15.385[/tex]
Thus we have :
Error Interval = [15.375 , 15.385)
H(x)=(-4x-5)(-x+5)
Lesser x:
Greater x:
Answer:
The equation H(x) = (-4x-5)(-x+5) can be simplified to H(x) = 4x^2 + 25.
The lesser value for x is -5, and the greater value for x is 5.
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verify by substitution that the function y(x) = x cx−2is a solution of the differential equation xy′ 2y = 3x. in addition, find the value of c ∈rso that we have y(1) = 5.
c = log_10(25) is the value of c that makes y(1) = 5.
What is differential ?
A differential is a gear train with three drive shafts that has the property that the rotational speed of one shaft is the average of the speeds of the others, or a fixed multiple of that average.
y'(x) = c * x^(c-1) * x^(-2) + (-2) * x^c * x^(-2) = (c - 2) * x^(c-3)
Now, we can substitute y(x) and y'(x) into the differential equation:
xy' + 2y = 3x
x * (c - 2) * x^(c-3) + 2 * x^c * x^(-2) = 3x
(c - 2) * x^(c) + 2 * x^c * x^(-2) = 3x
(c - 2) * x^c + 2 * x^(-2) = 3x
This confirms that y(x) = x^c * x^(-2) is indeed a solution of the differential equation xy' + 2y = 3x.
To find the value of c so that y(1) = 5, we need to substitute x = 1 and y = 5 into the solution y(x) = x^c * x^(-2):
y(1) = 1^c * 1^(-2) = 5
1^c = 5 * 1^2 = 25
c = log_1(25)
c = log_10(25) / log_10(1) = log_10(25)
So, c = log_10(25) is the value of c that makes y(1) = 5.
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what is the area of the shaded portion in the figure? If, it is a square and the lines are parts of a circle and the side of the square is 10 unit. Give answers in terms of pi.
The area of the shaded part is 50(2-π)
What is area?Area is the measure of a region's size on a surface. The area of a plane region or plane area refers to the area of a shape or planar lamina, while surface area refers to the area of an open surface or the boundary of a three-dimensional object.
The area of the shaded part = area of a square - area of a semi circle
area of square = l× l = 10 × 10 = 10²
=100 squared units
the area of the semi circle = πr²/2
= π ×10²/2
= 50π
Therefore the area of the shaded part in term of π is 100-50π
= 50( 2-π) units
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A new gaming chair costs $239.87. You have already saved $106.37 and earn $44.50 each week babysitting. Write and solve an equation to determine how many weeks, w, you must babysit to earn enough money to buy the new gaming chair.
A. 44.5w + 106.37 = 239.87; w = 3
B. 44.5w − 106.37 = 239.87; w = 9
C. 44.5w − 239.87 = 106.37; w = 9
D. 44.5 + 106.37w = 239.87; w = 3
Answer:
A. 44.5w + 106.37 = 239.87; w = 3
Step-by-step explanation:
You want an equation and solution for finding the number of weeks (w) required to earn enough for a $239.87 gaming chair if you have saved $106.37 and earn $44.50 per week.
SavedThe total amount saved will be the sum of the amount already saved ($106.37) and the amount of earnings. We want that total to be $239.87.
Earnings are $44.50 per week. After w weeks, they total 44.50w. Then the amount saved will be ...
earnings + already saved = total savings
44.50w + 106.37 = 239.87
SolutionWe can find the value of w by subtracting 106.37 from this equation:
44.50w = 133.50
w = 3 . . . . . . . . . . . . . . . divide by 44.50
The equation and solution are ...
44.5w +106.37 = 239.87; w = 3 . . . . . . . matches choice A
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simplify. 5+√3 / √2-√3
please give step by step! ty!
The simplified expression is (- 5√2 - 5√3 - √6 - 3).
What are algebraic expressions?In mathematics, an expression or mathematical expression is a finite combination of symbols that is well-formed according to rules that depend on the context.Mathematical symbols can designate numbers (constants), variables, operations, functions, brackets, punctuation, and grouping to help determine order of operations and other aspects of logical syntax.Given is the expression as -
(5 + √3) / (√2 - √3)
The given expression is -
(5 + √3) / (√2 - √3)
We can write -
{(5 + √3) x (√2 + √3)} / {(√2 - √3) x (√2 + √3)}
{5(√2 + √3) + √3(√2 + √3)} / (√2² - √3²)
(5√2 + 5√3 + √6 + 3) / (- 1)
- (5√2 + 5√3 + √6 + 3)
- 5√2 - 5√3 - √6 - 3
Therefore, the simplified expression is (- 5√2 - 5√3 - √6 - 3).
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Jenny drives from London to Swindon at an average speed of 54 miles per hour.
She drives for 1 1/2 hours
Work out the distance from London to Swindon.
The distance from London to Swindon is 81 miles.
What is Speed- distance Formula?Speed = Distance/TimeDistance = Speed X Time, and Time = Distance / Speed.We have,
Jenny drives from London to Swindon at an average speed of 54 miles per hour.
Time = 1 1/2 hour = 3/2 hour
Using Speed = distance / Time
Distance = speed x time
Distance = 54 x 3/2
Distance = 27 x 3
Distance = 81 miles
Thus, the distance is 81 miles.
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Caron want to put up a glow in the dark contellation diplay in hi bedroom he need a pack of two ided tape which cot $12 the tar ticker cot $14 per pack and he want to buy 3 pack what i a good etimation of how much change Caron will get back if he give $75?
If Caron gives $75, he will get $21 as a change, when he buys a constellation display.
To estimate the amount of change Caron will get back, we need to calculate the total cost of the two-sided tape and the star stickers. The two-sided tape costs $12 and Caron wants to buy 3 packs of star stickers which cost $14 each, so the total cost of the star stickers is 3 * $14 = $42. The total cost of both the two-sided tape and the star stickers is $12 + $42 = $54.
If Caron gives $75, he will get $75 - $54 = $21 as change. This is a rough estimate of the amount of change Caron will get back, and it's always a good idea to have a little extra money in case the actual cost is slightly higher.
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Please answer fully and answer
the inequality model given by eva is wrong, the correct answer is 5/3(y-1) = x.
What is inequality?
Inequalities are mathematical expressions where neither side is equal. In inequality, as opposed to equations, we compare two values. Less than (or less than or equal to), greater than (or greater than or equal to), or not equal to signs are used in place of the equal sign in between. Sometimes it can be about a "not equal to" relationship, where one thing is more than the other or less than. In mathematics, an inequality is a relationship that results in a non-equal comparison between two numbers or other mathematical expressions.
The given inequality is y =3/5x +1
Now subtract 1 from both sides y-1 = 3/5x
further taking 5/3 in both side we should have
5/3(y-1) = x
Hence, the inequality model given by eva is wrong, the correct answer is 5/3(y-1) = x.
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Special right triangle
Using properties of right angle, The value of a will be 32 units.
What is the special right triangle rule?Particular Right Triangle Formula
The right triangle formula, which states that the square of a right-angled triangle's right triangle is equal to the product of its other two squares, is the fundamental Pythagoras theorem formula.
How long are the other two legs if a hypotenuse of a triangle with a 45°, 45°, and 90° angle is 32 units.
We are aware that Leg: Ankle: Length x = x: x: x2 is the formula for a triangle with a 45°, 45°, and 90° angle.
We can state that x2 = 32 because it is a known that the hypotenuse equals 32. As a result, x = 3 is determined.
After replacing this number of x = 3, the lengths of the empty sides may now be determined.
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Complete question -
Find values of a and b -
a hypothesis is a a) prediction of results. b) tentative statement that something may be true. c) fact. d) all of these
A hypothesis is a) prediction of results and b) tentative statement that something maybe true.
A hypothesis is a statement or assumption made about a phenomenon or relationship, used as a starting point for further investigation. It is a tentative explanation for an observation, phenomenon or problem that can be tested through experimentation or additional data collection.
So a hypothesis can be true of false. A hypothesis is not certainly a fact too since it need to be tested.
If a hypothesis is proven to be true through experimentation and data analysis, it becomes a confirmed explanation or prediction. This means that the evidence supports the hypothesis and it can be used as a basis for further study or explanation. However, a true hypothesis is always subject to revision or rejection if new information or evidence becomes available.
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Start at 2⅓, add 2¼. I need the first 4 multiples
The multiple of 2⅓, add 2¼. is 2 7/12.
What are multiple?A multiple in mathematics is created by multiplying any number by an integer. In other words, if b = na for some integer n, known as the multiplier, it can be said that b is a multiple of a given two numbers, a and b. This is equivalent to stating that b/a is an integer if an is not zero.
In other words, the product of 4 and any natural integer is the multiple of 4. For instance, the result of multiplying 4 by 4 is 16, making 16 a multiple of 4. Examples of multiples of four are 4, 12, 20, 24, and so forth.
Start at 2⅓, add 2¼. I need the first 4 multiples?
→2 1/3+2 1/4
⇒2 3+4/12
⇒2 7/12
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please help in due in like 30 mins and I dont know how to work it out
The fraction by which the income decreased in August is given as follows:
1/5.
How to obtain the fraction?The total income is given by a proportional relationship of the cost per item and the number of items sold, hence:
y = kx.
In which:
y is the total income.k is the price per item.x is the number of items sold.For August, the parameters are given as follows:
k: 4k/3, as 1/3 + 1 = 4/3.x: 3x/5, as 1 - 2/5 = 3/5.Hence the August expression is given as follows:
y = 4k/3 x 3x/5
y = 12kx/15
y = 4kx/5.
Hence the decrease is given as follows:
1 - 4/5 = 5/5 - 4/5 = 1/5.
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A new refrigerator comes in a box shaped like a rectangular prism. The base of the box measures 4 feet by 5 feet. The total surface area of the box is 148 square feet.
What is the height of the box in feet?
The height of the box in feet is 6 feet.
What is surface area of rectangular prism?The surface area of a rectangular prism is the overall region or area that is covered by all of its faces. A three-dimensional shape is a rectangular prism. It has six faces, each of which is rectangular in shape. Consequently, a rectangular prism's two bases must likewise be rectangles.
The surface area of the rectangular prism is given by the formula:
A = 2(wl + hl + hw)
Substitute the value of A = 148, l = 4 and w = 5:
148 = 2((5)(4) + (4)h + (5)h)
148 = 40 + 8h + 10h
148 = 40 + 18h
108 = 18h
h = 6
Hence, the height of the box in feet is 6 feet.
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The rectangle shown has a perimeter of 52 cm and the given area. Its length is 6 more than four times its width. Write and solve a system of equations to find the dimensions of the rectangle. The length of the rectangle is cm and the width of the rectangle is cm. (...) W L 2 A 88 cm
a fully amortizing mortgage is made for $136,000 at 6.5 percent interest. required: if the monthly payments are $1,180 per month, when will the loan be repaid
The loan will be (ln(Time) - ln(136,000) + ln(1,180)) / ln(1 + 6.5 / 12) repaid.
What is interest ?
Interest is payment from a borrower or deposit-taking financial institution to a lender or depositor of an amount above repayment of the principal sum, at a particular rate.
Time = Loan Amount * (1 + Interest Rate / 12)^(12 * Number of Payments) / Monthly Payment
Given the loan amount ($136,000), interest rate (6.5%), and monthly payment ($1,180), we can plug these values into the formula:
Time = 136,000 * (1 + 6.5 / 12)^(12 * Number of Payments) / 1,180
To find the number of payments, we need to take the natural logarithm of both sides:
ln(Time) = ln(136,000 * (1 + 6.5 / 12)^(12 * Number of Payments) / 1,180)
ln(Time) = ln(136,000) + ln((1 + 6.5 / 12)^(12 * Number ofPayments)) - ln(1,180)
ln(Time) = ln(136,000) + 12 * Number of Payments * ln(1 + 6.5 / 12) - ln(1,180)
Next, we can isolate the number of payments on one side:
12 * Number ofPayments * ln(1 + 6.5 / 12) = ln(Time) - ln(136,000) + ln(1,180)
The loan will be (ln(Time) - ln(136,000) + ln(1,180)) / ln(1 + 6.5 / 12) repaid.
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Which substitution should best be used in solving the recurrence T(n) = 5T(4n/3) + 2?a) n = (3/4)^kb) n = (4/3)^kc) n = (15/4)^kd ) n = (20/3)^ke) n = (3/20)^kf ) n = (3/15)^kg ) n = 2^kh ) n = 3^ki ) n = 4^kj ) n = 5^k
The substitution that should best be used in solving the recurrence T(n) = 5T(4n/3) + 2 is "n = (4/3)^k".
The idea behind choosing a substitution for a recurrence relation is to simplify the relation and make it easier to solve. In this case, we want to simplify T(4n/3) to something involving T(n). By setting n = (4/3)^k, we can rewrite T(4n/3) as T((4/3)^(k+1)).
This substitution allows us to express T(4n/3) in terms of T(n) and simplify the relation.
T(n) = 5T(4n/3) + 2
becomes
T(n) = 5T((4/3)^(k+1)) + 2
We can then use this substitution to solve the relation using, for example, the Master Theorem.
In conclusion, the substitution "n = (4/3)^k" is the best option because it simplifies the relation and makes it easier to solve.
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