Find the volume (don't forget your UNITS) *

At the same rate as his 100-meter freestyle swim in the 2012 Olympics, Nathan Adrian could have completed one loop of a 25-meter pool in 11.88 seconds.

The idea of proportionality can be applied. The time it would take Nathan Adrian to complete one lap in a 25-meter pool is known to be 47.52 seconds for the 100-meter freestyle. Since the tempo is constant while the distance varies, we can establish a ratio:

100 meters / 47.52 seconds = 25 meters / x seconds

where x is the unknown time for one lap in the 25-meter pool.

To solve for x, we can cross-multiply:

100 meters * x seconds = 47.52 seconds * 25 meters

100x = 1188

Dividing both sides by 100, we get:

x = 11.88 seconds

Therefore, at the same rate as his 100-meter freestyle swim in the 2012 Olympics, Nathan Adrian could have completed one loop of a 25-meter pool in 11.88 seconds.

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Integrating with shells is the easier method.

V = 2π ∫₁³ x (√x + 3x) dx

That is, at various values of x in the interval [1, 3], we take n shells of radius x, height y = √x + 3x, and thickness ∆x so that each shell contributes a volume of 2π x (√x + 3x) ∆x. We then let n → ∞ so that ∆x → dx and sum all of the volumes by integrating.

To compute the integral, just expand the integrand:

V = 2π ∫₁³ (x ³ʹ² + 3x ²) dx

V = 2π (2/5 x ⁵ʹ² + x ³) |₁³

V = 2π ((2/5 × 3⁵ʹ² + 3³) - (2/5 × 1⁵ʹ² + 1³))

V = 4π/5 (9√3 + 64)

keeping in mind that parallel lines have exactly the same slope, let's check for the slope of the equation above

\(x + y = 10\implies y = -x+10\implies y=\stackrel{\stackrel{m}{\downarrow }}{-1}x+10 \leftarrow \begin{array}{|c|ll} \cline{1-1} slope-intercept~form\\ \cline{1-1} \\ y=\underset{y-intercept}{\stackrel{slope\qquad }{\stackrel{\downarrow }{m}x+\underset{\uparrow }{b}}} \\\\ \cline{1-1} \end{array}\)

so, we're really looking for the equation of al ine whose slope is -1 and that passes through (-1 , 5)

\((\stackrel{x_1}{-1}~,~\stackrel{y_1}{5})\hspace{10em} \stackrel{slope}{m} ~=~ -1 \\\\\\ \begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{5}=\stackrel{m}{-1}( ~~ x-\stackrel{x_1}{(-1) ~~ }) \\\\\\ y-5 = -(x+1)\implies y-5=-x-1\implies y=-x+4\)

Answer:

the first one is not equivalent, the second one is equivalent.

Step-by-step explanation:

Answer:

x = 15

Step-by-step explanation:

1:3 is also known as 3.

Any scale factor that is greater than 1 will be an increased dilation of the shape.

Figure A to Figure B:

5 • 3 = x

x = 15

The coordinates of the point on the directed line segment from (-7, 9) to (3, -1) that partitions the segment into a ratio of 2 to 3 are (-3, 5).

To find the coordinates of the point that divides the directed line segment from (-7, 9) to (3, -1) into a ratio of 2 to 3, we can use the section formula.

Let's label the coordinates of the desired point as (x, y). According to the section formula, the x-coordinate of the point is given by:

x = (2 * 3 + 3 * (-7)) / (2 + 3) = (6 - 21) / 5 = -15 / 5 = -3

Similarly, the y-coordinate of the point is given by:

y = (2 * (-1) + 3 * 9) / (2 + 3) = (-2 + 27) / 5 = 25 / 5 = 5

Therefore, the coordinates of the point that divides the line segment in a ratio of 2 to 3 are (-3, 5).

To understand this conceptually, consider the line segment as a distance from the starting point (-7, 9) to the ending point (3, -1). The ratio of 2 to 3 means that the desired point is two-thirds of the way from the starting point and one-third of the way from the ending point. By calculating the x and y coordinates using the section formula, we find that the desired point is located at (-3, 5).

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The total surface area of the doghouse is 1452 ft²

The area occupied by a three-dimensional object by its outer surface is called the surface area.

The dog house has many surfaces including the roofs . The total surface area is the sum of all the area of the surfaces.

area of the roof part = 2( 13×11) + 2( 12× 10)×1/2

= 286 + 720

= 1006 ft²

surface area of the building

= 2( lb + lh + bh) -bh

= 2( 10× 11 + 10×8 + 11×8) - 10×11

= 2( 110+80+88)-110

= 2( 278) -110

= 556 -110

= 446ft²

The total surface area = 1006 + 446

= 1452 ft²

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Answer:

-8

Step-by-step explanation:

47 - 55

= 47 + (-55)

= -8

Hope it helps ⚜

Answer:

x = - 30

Step-by-step explanation:

x/3  + 8 = - 2     subtract 8 from both sides of the equation

x/3 = -2 - 8 = -10  now multiply both sides by 3

x = -30

Answer:

It is 1000

Step-by-step explanation:

To find the unit rate, we divide y/x. In this case our y is the height (ft) and the x is the time(min).

If we divide 1000/1, we get 1000

If we divide 2000/2, we get 1000

and so on...

Hope it helps :^)

Answer: 156.38 m

Step-by-step explanation:

Using the arc length formula, the answer is \(2\pi(8^2) \cdot \frac{140}{360} \approx 156.38\)

Answer:

  (c)  y = 3^x

Step-by-step explanation:

A function is described as exponential if the independent variable is found in the exponent.

  y = x^(1/2) . . . a square root function, not exponential

  y = 2x^3 . . . . a cubic (polynomial) function, not exponential

  y = 3^x . . . . . an exponential function

For which 161 square feet of carpet would be required.

We can use the algebraic formula for the area of a rectangle.

The formula to find the area (A) of a rectangle is:

A = length × width

We may enter the bedroom's dimensions—14 feet long by 11.5 feet wide—into the calculation to determine the room's area and, thus, the required amount of carpeting.

Let:

length = 14 feet

width = 11.5 feet

Then, the formula becomes:

A = 14 feet × 11.5 feet

Now, calculate the area:

A = 161 square feet

Therefore, for which 161 square feet of carpet would be required.

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Answer:

a represent 1

and number is 11448.

Step-by-step explanation:

We know that a number is divisible by 9 if the sum of its digit is divisible by 9.

Given number aa448

sum of 4+4+8 = 16

number divisible by 9, 18 , 27,36

as we see sum of 448 is 16 , hence sum of a+a+4+4+8 should be  number divisible by 9 and greater than 16 which is 18, 27,36

Let see of if sum of digits of aa448 is divisible by  18

a+a+4+4+8 = 18

2a + 16 = 18

2a = 18-16 = 2

a =2/2 = 1

Thus, number will be 11448

________________________________________

Let see of if sum of digits of aa448 is divisible by  27

a+a+4+4+8 = 27

2a  = 27 - 16

2a = 11 = 2

a =11/2 = 5.5

as digit cannot be fraction of decimal, it can take value from 1 to 9 hence

sum of digits of aa448 is not divisible by  27

______________________________________________

Let see of if sum of digits of aa448 is divisible by  36

a+a+4+4+8 = 36

2a + 16 = 36

2a = 36-16 = 20

a =20/2 = 10

as digit can take value from 1 to 9 hence sum of digits of aa448 cannot  be divisible by  36 or any higher number than 36 which is divisible by 9.

Thus , a represent 1

and number is 11448.

Answer:

In this context, unique means existence of only one value for that element and no more than one value

Step-by-step explanation:

Answer:

below

Step-by-step explanation:

Period = monthly  ( 12 periods per year)

Periodic interest in decimal form = i = .09/12

Periods ( months ) = n

FV = future value = 12000

PV = present Value = 7000

Formula

    FV = PV(1+i)^n =

12000=7000(1+.09/12)^n     solve for 'n' months

 n= 72.14 months = ~6 years

To find the mean, also known as the average, we need to add up all the sales and divide by the total number of sales:

Mean = (Total sales) / (Number of sales)

Total sales = $600 + $1,500 + $800 + $1,300 + $800 = $5,000

Number of sales = 5

Mean = $5,000 / 5 = $1,000

Therefore, the mean of the director's sales for the week is $1,000.

The property you are referring to is called the associative property of multiplication. According to this property, when multiplying three numbers, the grouping of the numbers does not affect the result. In other words, you can change the grouping of the factors without changing the product.

In the equation you provided: 3x(5x7) = (3x5)7

The associative property allows us to group the factors in different ways without changing the result. So, whether we multiply 5 and 7 first, or multiply 3 and 5 first, the final product will be the same.

The football is approximately 96.4 feet from the base of the goal post.

The tangent function in trigonometry is used to determine the proportion between the lengths of the adjacent and opposite sides in a right triangle. Where theta is the angle of interest, the tangent function is defined as:

tan(theta) = opposing / adjacent.

When the lengths of one side and one acute angle are known, the tangent function is used to solve for the unknown lengths or angles in right triangles. In order to utilise the tangent function, we must first determine the angle of interest, name the triangle's adjacent and opposite sides in relation to that angle, and then calculate the ratio of those sides using the tangent function.

Given, the angle of elevation is 16 degrees 42'.

That is,

Angle of elevation = 16 degrees 42' = 16 + 42/60 = 16.7 degrees

Using tangent function we have:

tan(16.7) = 30/x

x = 30 / tan(16.7)

x = 96.4 feet

Hence, the football is approximately 96.4 feet from the base of the goal post.

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As a result, the extended form of (-8k+1)(-8k+1) is 64k²-16k+1, which is a polynomial in standard form.

A polynomial is an algebraic expression consisting of one or more terms, where each term contains a variable raised to a non-negative integer power, and may include coefficients, constants, and exponents. Polynomials can have one or more variables, and they can be of various degrees, depending on the highest exponent in the expression. Polynomials can be added, subtracted, multiplied, and divided, just like numbers. They are an important concept in algebra, and are used in many different areas of mathematics and science. Polynomials are often written in standard form, where the terms are arranged in descending order of degree, and the coefficients are written in front of the corresponding variable, separated by the multiplication symbol.

Here,

To expand the expression (-8k+1)(-8k+1), we can use the FOIL method, which stands for "First, Outer, Inner, Last".

First, we multiply the first terms of each binomial:

(-8k) x (-8k) = 64k²

Next, we multiply the outer terms of each binomial:

(-8k) x (1) = -8k

Then, we multiply the inner terms of each binomial:

(1) x (-8k) = -8k

Finally, we multiply the last terms of each binomial:

(1) x (1) = 1

Putting all these products together, we get:

64k² - 16k + 1

So, the expanded form of the expression (-8k+1)(-8k+1) is 64k² - 16k + 1, which is a polynomial in standard form.

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Problem

Solution

for this case we know that when x =2/3 , y= 6

Then we want to find the value of y= ? when x= 10 1/3 = 31/3 and we can do this:

6/(2/3) = y/(31/3)

And solving for y we got:

y = (3/2) * 6 * (31/3) = 31*3 = 93

C. y= 93

Let [r, s] denote the least common multiple of positive integers r and s. Find the number of ordered triples (a, b, c) of positive integers for which [a, b] = 1000, [b, c] = 2000, and [c, a] = 2000.

Solution 1

It's clear that we must have a = 2^j5^k, b = 2^m^6 and c = 2^p5^q for some nonnegative integers j, k, m, n, p, q. Dealing first with the powers of 2: from the given conditions, max(j, m)=3, max(m, p) = max(p, j) = 4. Thus we must have p = 4 and at least one of m, j equal to 3. This gives 7 possible triples (j,m,p): (0, 3, 4), (1, 3, 4), (2, 3, 4), (3, 3, 4), (3, 2, 4), (3, 1, 4) and (3, 0, 4).

Now, for the powers of 5: we have max (k, n) = max(n, q) = max(q, k) = 3. Thus, at least two of k, n, g must be equal to 3, and the other can take any value between 0 and 3. This gives us a total of 10 possible triples: (3, 3, 3) and three possibilities of each of the forms (3, 3, n), (3, n, 3) and (n, 3, 3).

Since the exponents of 2 and 5 must satisfy these conditions independently, we have a total of 7. 10= 70 possible valid triples.

Solution 2

1000=2³5³ and 2000= 2⁴5³. By looking at the prime factorization of 2000, c must have a factor of 2¹. If c has a factor of 53, then there are two cases: either (1) a or b = 5³2³, or (2) one of a and has a factor of 5³ and the other a factor of 2³. For case 1, the other number will be in the form of 2^x5^y, so there are 4 4 16 possible such numbers; since this can be either a orb there are a total of 2(16)-1=31 possibilities. For case 2, a and b are in the form of 2³5^x and 2^y5³, with a x<3 and y<3 (if they were equal to 3, it would overlap with case 1). Thus, there are 2(3-3)= 18 cases.

If c does not have a factor of 5³, then at least one of a and b must be 2³5³, and both must have a factor of 5³. Then, there are 4 solutions possible just considering a = 2353, and a total of 4.2-17 possibilities. Multiplying by three, as 0<=c<=2, there are 7.3 = 21.

Together, that makes 31 +18+21= 70 solutions for (a, b, c).

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Answer:

Step-by-step explanation:

area=length×width

=(2x+3)(3x-5)

Area=6x²-10x+9x-15

=6x²-x-15

The degree of the polynomial is 5 and the leading coefficient is -1.

The given polynomial is f(x)

f(x) = \(-x^{5} +2x^{4} +3x^{3} +2x^{2} -8x +9\)

The degree of the polynomial is the term with the highest power,

So, in the given polynomial, the term with the highest term is \(-x^{5}\)

The highest power is 5,

So, its degree is 5.

The leading coefficient is the coefficient of the term with the degree,

So, the leading coefficient is -1.

Hence, the degree of the polynomial is 5 and the leading coefficient is -1.

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