answer the last one plss i give brain

Here are the general steps for using the u-substitution method to evaluate a definite integral:

1. Identify a part of the integrand that looks like the derivative of another function.

This is usually denoted as u, and u should be a function that is easy to integrate.

2. Replace the selected part of the integrand with u. In other words, substitute u for the expression inside the integral.

3. Take the derivative of u with respect to the variable of integration, and then multiply by dx to obtain du.

4. Replace dx with the expression in terms of du, using the relationship you found in step 3.

5. Rewrite the entire integral in terms of u and du, and simplify as much as possible.

6. Evaluate the integral using u and the limits of integration. Replace u with the original expression, and then evaluate the expression at the upper and lower limits of integration.

7. Substitute the values obtained in step 6 into the original equation to get the final answer.

The u-substitution method is a powerful technique for evaluating definite integrals.

The general idea behind the u-substitution method is to replace a complicated expression inside the integral with a simpler expression, which is equal to the original expression.

Here are the general steps for using the u-substitution method to evaluate a definite integral:

Let's take an example to illustrate these steps:

Evaluate the definite integral ∫[tex](2x + 1)^(3/2) dx from x = 1 to x = 2.[/tex]

Let u = 2x + 1.

Replace 2x + 1 with u in the integral to get ∫[tex]u^{3/2} dx.[/tex]

Take the derivative of u with respect to x to get du/dx = 2, and then multiply by dx to obtain du = 2dx.

Replace dx with 1/2 du.

Rewrite the entire integral in terms of u and du to get ∫[tex]u^{3/2} (1/2) du.[/tex]

Evaluate the integral using u and the limits of integration to get (2/5) [tex]u^{5/2}[/tex] from 3 to 5.

Substitute the values obtained in step 6 into the original equation to get the final answer, which is[tex](2/5) [(5^{5/2}- 3^{5/2}].[/tex]

Therefore, the value of the definite integral ∫[tex](2x + 1)^{3/2}[/tex]dx from x = 1 to x = 2 is approximately 6.243.

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Thus, the amount of the money in the bank account at the end of 2 years is found to be: $1337.5.

Simple interest denotes interest that is simply charged on the principal amount, which is the original amount borrowed or deposited. The interest charge is going to be applied once, regardless of how frequently it is applied. Many loans base their calculations on simple interest, but you should double-check before signing anything.

Given data:

Formula for the estimating simple interest:

SI = PRT/100

SI = 1250*3.5*2 / 100

SI = 87.5

Amount = principal + simple interest

A = 1250 + 87.5

A = 1337.5

Thus, the amount of the money in the bank account at the end of 2 years is found to be: $1337.5.

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Using simple interest his interest is $ 1165.50  

Simple interest is given by I = PRT/100 where

Given that Tucker deposits an amount of $7400 in an account that pays 3.15 % simple interest. Since he keep the amount in the account for 5 years without making any withdrawal or deposits, we need to find the amount of interest after 5 years.

Using the simple interest formula,

I = PRT/100

Given that

So, substituting the values of the variables into the equation, we have that

I = PRT/100

I = $ 7400 × 3.15 % × 5/100

= $ 74 × 315 × 5/100

= $ 116550/100

= $ 1165.50

So, his interest is $ 1165.50  

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The data set created from this survey is a categorical or nominal data set.

Categorical data is a type of data that can be divided into groups or categories based on specific characteristics or attributes. In this case, the groups are based on the mode of transportation used by the middle school students to get to school. The categories in this data set are "walk," "car," "bus," and "bike." Categorical data can be further classified as either nominal or ordinal.

Nominal data is a type of categorical data where the categories do not have any particular order or ranking, while ordinal data has a specific order or ranking. In this case, the categories "walk," "car," "bus," and "bike" do not have any specific order or ranking, so this data set is an example of nominal categorical data.

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The real values of "a" that make (x - a) a factor of the given polynomial are: a = 3, a = 2, and a = 3/14.

To find the values of "a" for which (x - a) is a factor of the given polynomial x⁴ + 3x³ - 6x² - 28x - 24, we can use the Remainder Theorem.

So, let's divide the given polynomial by (x - a) and set the remainder to zero to find the values of "a".

Using long division or synthetic division, we get:

       x³ + (a - 3)x² + (3a - 6)x + (6 - 28a)

   __________________________________________

x - a | x⁴ + 3x³ - 6x² - 28x - 24

Since the remainder is zero, we have;

x³ + (a - 3)x² + (3a - 6)x + (6 - 28a) = 0

Now, for (x - a) to be a factor of the given polynomial, the coefficients of x², x, and the constant term must be zero, because these are the coefficients of (a - 3)x², (3a - 6)x, and (6 - 28a), respectively.

So, we can set them to zero and solve for "a":

a - 3 = 0 (Coefficient of x²)

3a - 6 = 0 (Coefficient of x)

6 - 28a = 0 (Constant term)

Solving these equations, we get:

a - 3 = 0 => a = 3

3a - 6 = 0 => 3a = 6 => a = 2

6 - 28a = 0 => 28a = 6 => a = 6/28 => a = 3/14

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Yes, the project consists on single set of ordered pairs numbered 1 through 30 on coordinate planes for all class students is a suitable one.

Ordered pair is a composition of the x- coordinate (i.e., abscissa) and the y coordinate (i.e, ordinate), that's two values written in a fixed order within. In mathematics, an ordered pair ( pair means two objects) so, (a, b) is a pair of objects in particular order. The order in the pair is very significant, that is ordered pair (a, b) is different from the ordered pair (b, a). Now, Ms. todd provide a project to students which consists a single set of ordered pairs numbered 1 through 30, for all students in her class. We can easily graph a ordered pair on coordinate plane. Coordinate plane is a 2-D graph made using x-coordinate and y-coordinate. There is two axis x-axis and y-axis. So, first they collect all possible ordered pairs from 1 to 30 like ( 1,1),( 1,2),(1,3),..........(1,30), (30,1),.....,(30,30) ... etc. In this coordinate plane consists 30 numbers value on y-axis and 30 on x-axis. Then, draw on graph. Hence, it is suitable.

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The orthogonal projection of [tex]\vec{V}[/tex] =[−7−9] onto the line L through [26] and the origin is given by the vector [-69/10 -207/10].

What is algebra?

Algebra is a branch of mathematics that deals with mathematical operations and symbols used to represent numbers and quantities in equations and formulas.

To find the orthogonal projection of a vector  [tex]\vec{V}[/tex]  onto a line L, we need to follow these steps:

Find a vector [tex]\vec{n}[/tex]  that is orthogonal to the line L.

Find the projection of  [tex]\vec{v}[/tex] onto [tex]\vec{n}[/tex] . This projection will be a scalar.

The orthogonal projection of [tex]\vec{v}[/tex]  onto L is the vector that is obtained by multiplying the scalar projection of [tex]\vec{v}[/tex]  onto [tex]\vec{n}[/tex]  by the unit vector in the direction of L.

Let's apply these steps to the problem at hand:

Step 1: Find a vector [tex]\vec{n}[/tex] that is orthogonal to the line L.

The line L passes through [2 6] and the origin, so we can find the direction vector of the line by subtracting the two points:

[tex]\vec{d}[/tex] = [2 6]−[0 0]=[2 6].

To find a vector that is orthogonal to [tex]\vec{d}[/tex], we can take the cross product of  [tex]\vec{d}[/tex]with the vector [0 0 1]:

[tex]\vec{n}[/tex] = [tex]\vec{d}[/tex]  × [0 0 1] = [−6 2 0].

Step 2: Find the projection of  [tex]\vec{v}[/tex]onto [tex]\vec{n}[/tex]  .

The projection of  [tex]\vec{v}[/tex] onto  [tex]\vec{n}[/tex] is given by the formula:

proj n v = ( [tex]\vec{v}[/tex] ·  [tex]\vec{n}[/tex] ) / || [tex]\vec{n}[/tex] ||² *  [tex]\vec{n}[/tex]

where · denotes the dot product and ||  [tex]\vec{n}[/tex] || is the magnitude of  [tex]\vec{n}[/tex] .

Substituting the values we have:

proj n v = ([-7 -9] · [-6 2 0]) / ||[-6 2 0]||² * [-6 2 0]

= (-54 + (-18)) / (36 + 4) × [-6 2 0]

= -72/40 × [-6 2 0]

= [-27 9 0]/5

Step 3: The orthogonal projection [tex]\vec{v}[/tex] onto L is given by:

proj L v = (proj n v · [tex]\vec{d}[/tex]  / || [tex]\vec{d}[/tex]  ||²) × [tex]\vec{d}[/tex]  / || [tex]\vec{d}[/tex]  ||

where || [tex]\vec{d}[/tex] || is the magnitude of  [tex]\vec{d}[/tex]  .

Substituting the values we have:

proj L v = ([−27 9 0]/5 · [2 6]/(2² + 6²)) * [2 6]/(2² + 6²)

= -207/40 * [2 6]

= [-69/10 -207/10]

Therefore, the orthogonal projection of  [tex]\vec{v}[/tex] =[−7−9] onto the line L through [26] and the origin is given by the vector [-69/10 -207/10].

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The loan that will have a lower balance and the amount is D. The Stafford loan will have a lower balance by $274.54 at the time of repayment.

First, find the future balance of the PLUS Loan by the formula :

= Principal x ( 1 + monthly interest rate ) ^ number of months

= $ 11, 443.63 x ( 1 + 0. 005458 ) ⁶

= $ 11, 443.99

Then, find the future balance of the unsubsidized Stafford Loan with the same formula :

= 10, 720 x (1 + 0. 004958) ¹⁸

= $ 11, 169.46

This shows that the unsubsidized Stafford Loan would have a lower balance and the difference would be :

=  $11, 443.99 - $11, 169.46

=  $ 274.54

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Answer:

62.5%

Step-by-step explanation:

The number of states do not use nuclear: 20 + 12 = 32

The number of states that do not use nuclear and use coal: 20

so 20/32 = 5/8 = 62.5%

The result of rotating the point [-5, 2] by 90 degrees counterclockwise is the point [2, -5].

Rotation is a transformation in geometry that involves turning or spinning an object around a fixed point called the center of rotation. The center of rotation remains stationary while all the points of the object move along circular paths, at the same angle and distance from the center.

To rotate a point [x/y] counterclockwise by 90 degrees, we can use a 2x2 matrix of the form:

[ 0 -1 ]

[ 1 0 ]

To multiply this matrix by a point [x/y], we can use the following matrix multiplication:

[ 0 -1 ] [ x ] [ -y ]

[ 1 0 ] [ y ] = [ x ]

So, if we have a point [x/y] and we want to rotate it 90 degrees counterclockwise, we can multiply it by the matrix [ 0 -1 ; 1 0 ] to get the new point [-y/x].

To rotate a point 90 degrees counterclockwise using a matrix, we need to represent the point as a column matrix and then multiply it by the 2x2 rotation matrix.

Assuming that the point is [x, y] = [-5, 2], we can represent it as a column matrix:

| -5 |

| 2 |

To rotate the point counterclockwise by 90 degrees, we can use the following 2x2 rotation matrix:

| 0 -1 |

| 1 0 |

Multiplying the rotation matrix by the column matrix representing the point gives:

| 0 -1 | | -5 | | 2 |

| 1 0 | ×| 2 | = | -5 |

So the result of rotating the point [-5, 2] by 90 degrees counterclockwise is the point [2, -5].

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According to the banker's rule, Amber can afford to borrow a maximum of $96,900.

The banker's rule is a guideline used by some lenders to determine the maximum amount that an individual can borrow based on their annual income.

According to the banker's rule, the maximum amount that can be borrowed is typically a multiple of the borrower's annual income.

The exact multiple used in the banker's rule may vary depending on the lender and the borrower's financial situation, but a common multiple used is 3.

Therefore, to calculate the amount that Amber can afford to borrow using the banker's rule, we can multiply her annual income by 3.

Given that Amber makes $32,300 annually, we can calculate the amount that can be borrowed using the banker's rule as follows:

Maximum amount that can be borrowed = Annual income * Banker's rule multiple

Maximum amount that can be borrowed = $32,300 * 3

Maximum amount that can be borrowed = $96,900

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Answer: - 3x - 21

Step-by-step explanation:

All you have to do is distribute the -3

-3 multiplied by x is -3x

and -3 multiplied by +7 is -21

put it together

- 3x - 21

The probability of winning a giveaway with 73 entries and 2 winners is the fraction 2/73.

The probability of an event occurring is the fraction of the number of required outcome divided by the total number of possible outcomes.

The total possible outcome = 73

the event of winning = 2

probability of winning the giveaway = 2/73

This means that each winner has a 2/73 chance of winning the giveaway.

Therefore, the probability of winning a giveaway with 73 entries and 2 winners is the fraction 2/73.

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The approximate area of the sketch for the new lamp is : A = 70

We have,

A trapezium is one pair of parallel sides and one pair of non-parallel sides.

from the diagram we can extract the necessary information needed to find the area of the trapezium.

for the first trapezium (lamp shade)

a = 4 in

b = 6 in

h = 6 in

A1 = a+b/2 * h

   = 5 * 6 = 30

For the area of the second trapezium (lamp base)

a = 4 in

b = 6 in

h = 4 in

A2  =a+b/2  * h

      = 5 * 4 = 20

For the area of the third trapezium (lamp base)

a = 6 in

b = 4 in

h = 4 in

A3 = a+b/2  * h

     =  5 * 4 = 20

The total Area of the trapezium (lamp)

A =  A1 +  A2  + A3

A = 30 + 20 + 20

A = 70

In conclusion, The approximate area of the sketch for the new lamp is :

A = 70

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we can be sure that 6c + 3d will have a higher value than 4c + 2d if c and d are positive decimal numbers.

A variable in mathematics is a symbol or letter that is used to indicate a quantity in a mathematical statement or equation that can have many values. It is a symbol that may be used to represent an arbitrary or undefined integer in algebraic expressions and equations.

Both expressions have the same common factor of 2, so we can simplify them as follows:

4c + 2d = 2(2c + d)

6c + 3d = 3(2c + d)

Since c and d are positive decimal numbers, we know that both 2c + d and 3(2c + d) are positive. Therefore, the expression with the higher value will be the one with the larger coefficient, which is 3 in the second expression.

So, we can be sure that 6c + 3d will have a higher value than 4c + 2d if c and d are positive decimal numbers.

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Using circle theorems CD =  9.17

Circle theorems are theorems that govern circle properties.

Given that form the figure EF = 8, ON = 3 and OM = 2, we need to find CD.

From the figure using Pythagoras' theorem

OE² = ON² + EN²

= ON² + (EF/2)²

= 3² + (8/2)²

= 3² + 4²

= 9 + 16

= 25

OE = √25

= 5

Also, OD² = OM² + DM²

Making DM subject of the formula, we have that

DM² = OD² - OM²

DM = √(OD² - OM²)

Now OD = OE = 5

So, we have that

= √(OD² - OM²)

= √(OE² - OM²)

= √(5² - 2²)

= √(25 - 4)

= √(25 - 4)

= √21

= 4.58

Now CD = 2DM

= 2 × 4.58

= 9.17

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Using Ariana's method to  work out 135 ÷  12 1/2 without a calculator will gives us:  54/5 or 10.8.

Ariana's method for division involves breaking down the divisor into a more manageable fraction and adjusting the dividend accordingly. Here's how we can use this method to solve 135 ÷ 12 1/2 without a calculator:

Step 1: Rewrite the mixed number as an improper fraction:

12 1/2 = 25/2

Step 2: Determine the reciprocal of the fraction:

25/2 → 2/25 (reciprocal)

Step 3: Rewrite the division problem as multiplication by the reciprocal:

135 ÷ 12 1/2 = 135 x 2/25

Step 4: Simplify the expression:

135 x 2/25 = (27 x 5) x 2/25 = 54/5

Therefore, 135 ÷ 12 1/2 = 54/5 or 10.8 when rounded to one decimal place.

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the values of $a$ and $b$,is So, $b - a = 6$.using the smallest and largest integers within this interval. Since "integer" refers to whole numbers (both positive and negative), we can identify $a$ and $b$

let's first solve the given inequality $x^2 - 15 < 2x$. We can rewrite this inequality by moving all terms to one side:

$x^2 - 2x - 15 < 0$

Now, we want to factor the quadratic:

$(x - 5)(x + 3) < 0$

From this factored form, we can see that the quadratic changes sign at x = -3 and x = 5. This means the inequality holds between these values:

-3 < x < 5

Now, we want to find the smallest and largest integers within this interval. Since "integer" refers to whole numbers (both positive and negative), we can identify $a$ and $b$ as follows:

$a = -2$ (the smallest integer greater than -3)
$b = 4$ (the largest integer less than 5)

Finally, we need to find the difference $b - a$:

$b - a = 4 - (-2) = 4 + 2 = 6$

So, $b - a = 6$.

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The solutions to the equation are x = 7/2 or x = 2. This is because if either of the factors (2x - 7) or (x - 2) equals zero, then the whole expression will equal zero.

To solve the quadratic equation 2x²-11x=-14 by factorization, we need to rearrange the equation to bring all terms to one side so that it is in the form ax² + bx + c = 0.

First, we add 14 to both sides of the equation, which gives us:

2x² - 11x + 14 = 0

To factorize this quadratic equation, we need to find two numbers that multiply to give the constant term (14) and add to give the coefficient of the middle term (-11). In this case, the two numbers are -2 and -7, as (-2) x (-7) = 14 and (-2) + (-7) = -9.

Using these two numbers, we can rewrite the middle term as -2x - 7x, which gives us:

2x² - 2x - 7x + 14 = 0

We can then factorize the first two terms and the last two terms separately, which gives us:

2x(x - 1) - 7(x - 2) = 0

We can then use the distributive property to simplify the equation:

2x² - 2x - 7x + 14 = 0

2x(x - 1) - 7(x - 2) = 0

(2x - 7)(x - 2) = 0

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Answer:

Step-by-step explanation:

This graph represents the line of best fit for some dataset.

It is reduced to a linear function in this instance with a constant slope, where x represents the independent variable and y, the dependent variable. This shows us quiz score is supposedly directly proportional to the time spent on homework per week in hours.

To find the amount of time someone should expect to study to get a 96% on their quiz you need to resolve the linear function modeled by y=mx+b which is in slope-intercept form. where m is the slope and b is the y-intercept. the y-intercept can be found when x = 0  which is shown in the graph to be 63, to solve for slope you can use the equation [tex]\frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex] which is rise/ run. then plug in any 2 points I will use (1,69) and (2,75) [tex]\frac{75-69 }{2-1 }[/tex] or 6 which means m = 6, therefore, our equation is y = 6x + 63 and because we know that y must be 96 for this question, we can solve for x using inverse operations. 96 = 6x + 63 subtract 63 from both sides and 6x = 33 then divide by 6 and 33/6 = 5.5 so a person should expect to study 5.5 hours to get a quiz score of 96%

On solving the provided question we can say that As a result, the credit equation limit of a client who has paid on time every year for the past ten years is $7,000.

A mathematical equation is a formula that links two statements and uses the equals sign (=) to indicate equality. In algebra, an equation is a statement that demonstrates the equality of two mathematical expressions. The equal sign divides the variables 3x + 5 and 14 in the equation 3x + 5 = 14, for instance.

The relationship between the two sentences that are located on opposite sides of a letter is explained by a mathematical formula. Frequently, the symbol and the single variable are identical. like in 2x - 4 = 2, for example.

The following equation determines the credit limit, Ln, of customers who have made on-time payments each year and are in the nth year of card ownership:

Ln = $2,000 + $500n

where Ln stands for the credit limit in the nth year and n is the number of years the cardholder has had it.

In the equation above, we substitute n=10 to get a customer's credit limit after ten years of card use:

L10 = $2,000 + $500(10)

L10 = $2,000 + $5,000

L10 = $7,000

As a result, a client with a ten-year history of on-time payments has a credit limit of $7,000.

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C21 represents 4 senior boys.

A matrix, which is used to represent a mathematical object or an attribute of one, is a rectangular array or table containing numbers, symbols, or expressions that are organized in rows and columns. is a matrix having two rows and three columns, for instance

Given:

[tex]\left[\begin{array}{ccc}5&6\\4&3\\\end{array}\right][/tex], [tex]\left[\begin{array}{ccc}c11&c12\\c21&c22\\\end{array}\right][/tex]

[tex]\left[\begin{array}{ccc}Junior boy&Junior girl\\Senior boy&Senior girl\\\end{array}\right][/tex]

4 Senior boys is c21 represented.

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If the base of an isosceles triangle is 50 cm and the length of one of its legs is 65 cm, the height of the isosceles triangle is 60 cm.

To find the height of the isosceles triangle, we need to use the Pythagorean theorem and the properties of isosceles triangles.

First, we draw a diagram of the triangle and label the known values. We know that the base is 50 cm and one of the legs is 65 cm. Since the triangle is isosceles, the other leg is also 65 cm.

Next, we can draw a perpendicular line from the top vertex to the base, which represents the height of the triangle. We can label this height as "h".

Now, we have a right triangle with a base of 50 cm, a hypotenuse of 65 cm, and a height of "h". We can use the Pythagorean theorem to find the value of "h":

h² + 25² = 65²

h² + 625 = 4225

h² = 3600

h = 60

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The estimated probability that the golfer will hit at least 4 times in his next 10 attempts is approximately 40%. So correct option is B.

Probability is a branch of mathematics that deals with the study of random events and their likelihood of occurrence. It involves quantifying the chance or likelihood of an event happening, and it is expressed as a number between 0 and 1, with 0 indicating that the event is impossible, and 1 indicating that the event is certain to happen.

Probability is often used in real-world situations to predict the likelihood of future events, make informed decisions, and understand patterns in data. It has many applications in various fields such as finance, physics, biology, and engineering.

To simulate the situation, we can use the table of random numbers to generate 10 digits, and count the number of times a digit is less than or equal to 3, which represents a hit for the golfer.

Using the table, we get the following sequence of digits:

26 53 48 69 55 23 44 72 89 55

Counting the number of hits, we get 4 hits out of 10 attempts. Therefore, the estimated probability that the golfer will hit at least 4 times in his next 10 attempts is approximately 40%.

So, the answer is B. Possible answer: 40%.

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The length of segment TU is given as follows:

TU = 24 units.

The length of segment TU is obtained applying the trapezoid midsegment theorem, which states that the length of the midsegment of the trapezoid is equals to the mean of the length of the bases of the trapezoid.

The parameters for this problem are given as follows:

Hence the length of TU is obtained as follows:

36 = (x + 48)/2

x + 48 = 72

x = 24.

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to convert 25min into hours we divide 25 by 60 so the answer would be 0.416 hours

the formula to find the distance is we multiply speed and time so

distance=4km/p X 0.416 hours =1.664

so now we have the first answer

now we do the same thing for the other part of the question

distance= timeXspeed

distance=0.416X3=1.248

then we add to find total distance

1.248+1.644=2.912

2.912 is the total distance

Answer:

x < 1.175

Step-by-step explanation:

-0.4x + 1.27 > 0.8

-0.4x > -0.47

x < 1.175

The simplification that accurately explains the statement 73 = 713A is A = 73/713

From the question, we have the following parameters that can be used in our computation:

7 3 = 7 1 3 A.

Express properly

So, we have

73 = 713A

Divide both sides of the equation by 713

So, we have

A = 73/713

Hence, the simplification that accurately explains the statement is A = 73/713

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Answer: 48.76 square inches.

Step-by-step explanation: A regular pentagonal prism has 7 faces: 2 pentagonal bases and 5 rectangular faces.

To find the surface area, we need to find the area of each face and add them up.

The area of one pentagonal base can be found using the formula:

A = (5/4) * (edge length)^2 * cot(π/5)

Substituting the given values, we get:

A = (5/4) * (2)^2 * cot(π/5)

≈ 6.8819 square inches (rounded to the nearest hundredth)

Since there are two pentagonal bases, their total area is:

2A ≈ 13.7638 square inches

The area of one rectangular face can be found using the formula:

A = (edge length) * (height)

Substituting the given values, we get:

A = 2 * 3.5

= 7 square inches

Since there are five rectangular faces, their total area is:

5A = 5 * 7 = 35 square inches

Therefore, the total surface area of the regular pentagonal prism is approximately:

13.7638 + 35 = 48.7638 square inches

Rounding to the nearest hundredth, we get:

Surface area ≈ 48.76 square inches.

The surface area of a regular pentagonal prism is 193.82 square inches.

A prism is a three-dimensional object.

There are triangular prism and rectangular prism.

We have,

The formula for the surface area of a regular pentagonal prism is:

SA = 5 × base area + 5 × lateral face area

where the base area is the area of one of the pentagonal bases, and the lateral face area is the area of one of the rectangular faces.

Since the base is a regular pentagon, we can use the formula for the area of a regular pentagon:

Area of pentagon = (1/4) × n × s² × tan(180°/n)

where n is the number of sides of the pentagon (which is 5 since it's a regular pentagon), and s is the length of one of its sides.

In this case,

s = 2 inches

Area of pentagon = (1/4) × 5 × 2² × tan(180°/5)

Area of the pentagon = 6.8819 square inches

This is the area of one of the pentagonal bases.

Since there are two bases,

The total base area is:

Base area = 2 × 6.8819

                 = 13.7638 square inches

Now,

Each lateral face is a rectangle with a width equal to the base edge length (2 inches) and a height equal to the height of the prism (3.5 inches).

Lateral face area.

= base edge length × height

= 2 inches × 3.5 inches

= 7 square inches

Since there are five lateral faces, the total lateral face area is:

Lateral face area

= 5 × 7

= 35 square inches

Now we can add up the base area and lateral face area to get the total surface area:

SA = 5 × base area + 5 × lateral face area

     = 5(13.7638) + 5(35)

     = 193.819 square inches

Rounding this to the nearest hundredth.

SA = 193.82 square inches

Thus,

The surface area of a regular pentagonal prism is 193.82 square inches.

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Answer: Let the side length of the square rug be x, and the dimensions of the rectangular rug be l and w.

In the first arrangement, the three rugs are arranged side by side, with the rectangular rugs aligned vertically and the square rug aligned horizontally. The length of this arrangement is l + 2x, and the width is w. We know that the length is 3 times the width, so we can write:

l + 2x = 3w

In the second arrangement, the two rectangular rugs are arranged side by side, with the square rug placed on top. The length of this arrangement is l, and the width is w + x. We know that the length is twice the width, so we can write:

l = 2(w + x)

We now have two equations with two unknowns, l and w:

l + 2x = 3w

l = 2(w + x)

Substituting the second equation into the first, we get:

2(w + x) + 2x = 3w

4x = w

Substituting this value of w into the second equation, we get:

l = 2(4x) = 8x

So the dimensions of the rectangular rug are 4x by x, and the dimensions of the square rug are x by x.

To find the value of x, we use the measurements given in the diagram:

l + 2x = 60

l = 36

Substituting into the equation l = 2(w + x), we get:

36 = 2(4x + x)

36 = 10x

x = 3.6

So the side length of the square rug is 3.6 feet.

Step-by-step explanation: