Find the volume of the solid. When appropriate, use 3.14 and round to the nearest hundredth.

Using Pythagorean Theorrem the length of AE = 46 units

What is a Right Angle Triangle?

A right triangle or right-angled triangle, or more technically an orthogonal triangle, formerly known as a rectangled triangle, is a triangle with one right angle, i.e. two perpendicular sides. Trigonometry is founded on the relationship between the sides and other angles of a right triangle.

Solution:

In Triangle ABC using Pythagorean Theorem

AC = √(21*21 + 20*20)

AC = √841

AC = 29

In Triangle CDE using Pythagorean Theorem

CE = √(8*8 + 15*15)

CE = √289

CE = 17

From the figure we can say that AE = AC + CE

AE = 29 + 17

AE = 46 units

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The point at which the given lines intersect is (6,0,2) and an equation of the plane that contains these lines is x+y=6.

(a) In the given question we have to find the point at which the given lines intersect.

r = [3,3,0] + t[3,-3,2]

r = [6,0,2] + s[-3,3,0]

The equations of the lines can be written as :

The points for first line is

L(1)⟹x=3+3t, y=3-3t, z=0+2t and

The points for second line is

L(2)⟹x=6-3s , y=0+3s , z=2+0s .

Set the z coordinates equal from the line one and two, and we get :

2t = 2

Divide by 2 we get

t=1 .

Substitute the value of t into the point of x of first line :

x=3+3(1)

x=3+3

x=6

Equating the value of x equal to the coordinate of x of line two

6=6-3s

Subtract 6 on both side, we get

-3s=0

s=0

Now substitute the value of t into the point of y of first line:

y=3-3(1)

y=3-3

y=0

Equating the value of y equal to the coordinate of y of line two

0=0+3s

s=0

Now substitute the value of t into the point of z of first line:

z=0+2(1)

z=2

Equating the value of z equal to the coordinate of z of line two

2=2+0(s=0)=2 .

We thus have the point of intersection is :

P=(x,y,z)

P=(6,0,2)

Hence, the point at which the given lines intersect is (6,0,2)

(b) The direction vector for the first line is

[tex]\vec{v_{1}}=3\hat{i}-3\hat{j}+2\hat{k}[/tex]

and the direction vector for the second line is

[tex]\vec{v_{2}}=-3\hat{i}+3\hat{j}+0\hat{k}[/tex]

We can find a normal vector to the line as :

[tex]\vec{N_{1}}=\vec{v_{1}}\times\vec{v_{2}}[/tex]

[tex]\vec{N_{1}}=\begin{vmatrix} \hat{i} &\hat{j} &\hat{k} \\-3 &3 &-2 \\ -3 &3 & 0 \end{vmatrix}[/tex]

[tex]\vec{N_{1}}=\hat{i}(0-(-6))-\hat{j}(0-6)+\hat{k}(-9-(-9))[/tex]

[tex]\vec{N_{1}}=6\hat{i}+6\hat{j}[/tex]

is a normal vector

But , we can also use

[tex]\vec{N}=\frac{\vec{N_{1}}}{6}[/tex]

[tex]\vec{N}=\hat i+\hat j[/tex]

as the normal vector to the plane .

We can thus use the point P=(6,0,2) and the normal vector [tex]\vec{N}=\hat i+\hat j[/tex] to write the equation of the plane :

1(x-6)+1(y-0)+0(z-2)=0

x-6+y+0=0

x+y=6

Hence, an equation of the plane that contains these lines is x+y=6.

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The right question is

(a) Find the point at which the given lines intersect.

r = [3,3,0] + t[3,-3,2] r = [6,0,2] + s[-3,3,0]

(x,y,z) = ( )

(b) Find an equation of the plane that contains these lines.

The value of (f  o g)(-6) will be;

⇒  (f  o g)(- 6) = - 157

What is an expression?

Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.

Given that;

The values are,

⇒ f (x) = - 4x - 1

And,  g (x) = x² + 3

Now,

Since, The values are,

⇒ f (x) = - 4x - 1

And,  g (x) = x² + 3

Hence, The value of function  (f  o g)(x) is,

⇒  (f  o g)(x) = f (g (x) )

                  = f (x² + 3)

                  = - 4 (x² + 3) - 1

                  = - 4x² - 12 - 1

                  = - 4x² - 13

Thus, (f  o g)(- 6) = - 4 (-6)² - 13

                         = - 4 × 36 - 13

                         = - 144 - 13

                         = - 157

So, The value of (f  o g)(-6) will be;

⇒  (f  o g)(- 6) = - 157

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The area under the graph using first two and then four rectangles f(x) = x³ between x = 2 and x = 4 is 0.2421875.

Here a=0, b=1 and n=2( two rectangles), so

Δx = (1 - 0)/2

Δx = 1/2

Δx = 0.5

M2 = (0.5) * ( f((0+0.5)/2)) + f((0.5+1)/2)) )

= (0.5) * ( f(0.25) + f(0.75) )

Using  f(x) = x^3

=(0.5) * ( 0.015625 + 0.421875 )

=0.21875

Here a=0, b=1 and n=4( four rectangles), so

Δx = (1 - 0)/4

Δx = 1/4

Δx = 0.25

M4 = (0.25) * ( f((0+0.25)/2)) + f((0.25+0.5)/2)) + f((0.5+0.75)/2)) + f((0.75+1)/2)) )

=(0.25) * ( f(0.125) + f(0.375) + f(0.625) + f(0.875) )

Using  f(x) = x^3

= (0.25) * ( 0.001953125 + 0.052734375 + 0.244140625 + 0.669921875 )

= 0.2421875

Hence,  the area under the graph using first two and then four rectangles f(x) = x³ between x = 2 and x = 4 is 0.2421875.

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For the given data

(a) sample mean is 17 and sample standard deviation = 8.91 .

(b) the 95% confidence interval is (9.551 , 24.449)  .

(c) the 90% confidence interval is (11.032 , 22.968) .  .

In the question ,

the data is given as 25, 11, 12, 27,26, 22, 8,5.

So , the number of terms (n) = 8

Part(a)

the sample mean(μ)  is = (25+11+12+27+26+22+8+5)/8 = 17

the standard deviation is calculated as :

= √[(25 - 17)² + (11 - 17)² + (12 - 17)² + (27 - 17)² + (26  - 17)² + (22 - 17)² + (8 - 17)² + (5 - 17)²)]/7

Simplifying the above data further ,

we get

= 8.91

Part(b)

we know that at 95% confidence interval , the critical value is t₀.₀₂₅,₇ = 2.3646

So , the 95% confidence interval is :

= μ ± t₀.₀₂₅,₇*s/√n

= 17 ± 2.3646*8.91/√8

= 17 ± 7.449

So , the interval is ⇒ (9.551 , 24.449) .

Part(c)

we know that at 90% confidence interval , the critical value is t₀.₀₅,₇=1.8946

So , the 90% confidence interval is :

= μ ± t₀.₂₅,₇*s/√n

= 17 ± 1.8946*8.91/√8

= 17 ± 5.968

So , the interval is ⇒ (11.032 , 22.968) .

Therefore ,(a)  the sample mean and standard deviation are 17 and 8.91

(b) the 95% interval is (9.551 , 24.449) .

(c) the 90% interval is (11.032 , 22.968) .

The given question is incomplete , the complete question is

Let the following sample of 8 observations be drawn from a normal population with unknown mean and standard deviation: 25, 11, 12, 27,26, 22, 8,5.

(a) Calculate the sample mean and the sample standard deviation.

(b) Construct the 95% confidence interval for the population mean.

(c) Construct the 90% confidence interval for the population mean.

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The total cost of installing a fence around all sides of the rectangular yard would be $38250.

What is the area of the rectangle?

For the dimensions of the rectangle of length 'l' and breadth 'b', the area can be computed by using the formula,

           Area = Length ×  Breadth

Given data:

The length of the yard is 90 feet. and the breadth is 50 feet.

Area of the yard = length * breadth

                           = 90 * 50

                           = 4500 square foot

Now per square foot, the cost is $8.50 so the total cost of installing a fence around all sides of the yard would be = 4500 * 8.5

                                                                          = $38250.

Hence, the total cost of installing a fence around all sides of the rectangular yard would be $38250.

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Answer:

70

Step-by-step explanation:

Substitute the value -3 into the expression and simplify:

10(4 - -3)

10(4 + 3)

10(7)

70

Write each power as repeated multiplication. Then evaluate and compare their values is the correct option to compare the value of each expression.

What is the relation between power and base?

Compare  7² and 2⁷.

7 ≠ 2

So, write the power as repeated multiplication of the given expression:

7²  = 7 * 7

2⁷ = 2*2*2*2*2*2*2

Evaluate:

7² = 49

2⁷ = 128

Compare the values:

128 > 49

2⁷ > 7²

Hence, Write each power as repeated multiplication. Then evaluate and compare their values is the correct option to compare the value of each expression.

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The angles of the cyclic quadrilateral and the arc angle is as follows:

A cyclic quadrilateral is a quadrilateral drawn inside a circle. In other words, a cyclic quadrilateral is a quadrilateral which has all its four vertices lying on a circle.

A cyclic Quadrilateral's opposite angles add to 180°.

Therefore,

∠PSR = 180 - 93 = 87 degrees.

An inscribed angle is one-half of the measure of the arc that it intercepts.

Therefore,

87 = 1 / 2 arc PQR

cross multiply

arc PQR = 87 × 2

arc PQR  = 174 degrees

Hence,

arc QR = 174 - 126

arc QR = 48 degrees

arc RS = 360 - 90 - 174

arc RS = 96 degrees.

∠QPS = 1 / 2 (96 + 48)

∠QPS = 1 / 2 (144)

∠QPS = 144 / 2

∠QPS = 72 degrees

∠SRQ = 180 - 72

∠SRQ = 108 degrees

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Answer:    

   48°

   96°

    72°

    87°

   108°

Step-by-step explanation:

   arc QR = 48°

   arc RS = 96°

   ∠QPS = 72°

   ∠PSR = 87°

   ∠SRQ = 108°

The equation of the line that passes through the point (3,4) and has an undefined slope is y = 3

An undefined slope is an infinite slope. This is the slope of a vertical line. The x values never changes no matter the changes in the y values.

The equation of a line in slope intercept form can be represented as follows:

y = mx + c

where

m = slope

c = y-intercept

Since the slope, m is undefined, the expression that has an undefined slope is a vertical line

Therefore,

y = 3

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The next time they will meet again is 08:44 (next day)

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

EZPlanes Airways depart = every 264 minutes

Bryanair Airlines depart = every 198 minutes

Both airlines depart = at 06:20

Using the above as a guide, we have the following departure times

EZPlanes Airways depart = 264, 528, 792, 1056, 1320, 1584, 1848, 2112, 2376 and 2640 minutes,

Bryanair Airlines depart = 198, 396, 594, 792, 990, 1188, 1386, 1584, 1782 and 1980 minutes

The common departure time in the above is

Time = 1584 minute

So, we have

Common departure time = 06:20 + 1584 minutes

Evaluate the sum

Common departure time = 06:20 + 26 hours + 24 minutes

Evaluate the sum

Common departure time = 08:44 (next day)

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Answer

hello I have provided a picture of the answer

Step-by-step explanation:

ur welcome and have a nice day

The time taken for the population to reach 26.6 million is 2 years while the population in year 2000 is 18.5 million.

We can be able to  define population growth as the process by which the population of an area is increasing. We know that the increase or the decrease of population would have to be an exponential function.

Given that we have the function A =18.5e^0.1708t where A is the population after a given time from year 2000 and t is the number of years after year 2000. It then follows that 18.5 million refers to the population of the united states as at year 2000.

We now have;

26.6 =18.5e^0.1708t

26.6/18.5 = e^0.1708t

1.44 = e^0.1708t

ln(1.44) = 0.1708t

t = ln(1.44)/ 0.1708

t = 2 years

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The required solution (1, 4) of a linear system of the equation is determined by the elimination method.

Given the linear system of equations:

8x - 2y = 0 or 4x - y = 0 …(i)

-6x + 14y = 50 or -3x + 7y = 25 …(ii)

Equations (i) and (ii) constitute a system of two first-degree equations in the two variables x and y.

So, we have to find out the value of 'x’ and 'y’.

By multiplying 7 by equation (i) and adding both equations

28x - 7y -3x + 7y = 25

25x = 25

x = 25/25

x = 1

Substitute the value of x = 1 in the equation (i), and solve for y

4(1) - y = 0

y = 4

Hence, the required solutions are x = 1 and y = 4.

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Answer:

Step-by-step explanation:

i believe the total would be 4

Answer: The set of integers greater than or equal to 2 is {2, 3, 4, 5, ...}. This set can be represented using set notation as:

{y: y is an integer and y ≥ 2}

This set includes all the integers that are greater than or equal to 2, including 2 itself. It does not include negative integers or fractions.

For example, the following are all elements of this set:

2, 3, 4, 5, 6, 7, 8, ...

The following are not elements of this set:

-1, 0, 1, 1.5, -3.14, ...

This set is infinite, because there are an infinite number of integers greater than or equal to 2.

The equivalent expressions are 16a + b, 17b + 11a, and 17b.

A mathematical expression made up of variables, coefficients, constants, and operations like addition, subtraction, multiplication, and division is called an algebraic expression.

In general, if anything is considered equal then the two of them are said to be equivalent. Similarly to this, if any two or more algebraic equations have identical solutions or roots then the equations are Equivalent expressions even they look different.        

Here we have

Expressions

3a +b +13a

20b +11a -3b

20b -5b +2b  

To simplify given expressions add or subtract like terms as given below

3a +b +13a => (3a+13a) +b = 16a + b

Therefore,

The equivalent expression of 3a +b +13a is 16a + b        

20b +11a -3b => (20b-3b) + 11a = 17b + 11a

Therefore

The equivalent expression of 20b +11a -3b is 17b + 11a  

20b-5b+2b => 20b+2b - 5b = 22b -5b = 17b

Therefore,

The equivalent expression of 20b-5b+2b is 17b

Therefore,

The equivalent expressions are 16a + b, 17b + 11a, and 17b.

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Answer:

5, -11. -3, 8. 10, 17. -15, 13.

Step-by-step explanation:

All you have to do is switch x and y

The number of One-person tubes and two-person tubes is 3 and 12 respectively.

It is the basic arithmetic operation, in which you are separating the number into some parts.

Given:

A group spends $277.50 to rent 15 tubes for a float down the Lchetucknee River. One-person tubes cost $12.50 and two-person tubes cost $20.

The equation can be written as,

12.5x + 20y = 277.5

Here, x is the number of One-person tubes and y is two-person tubes.

Also, x + y = 15

Solve the above equation by elimination method,

x = 3 and y = 15 - 3 = 12

Therefore, the number of One-person tubes and two-person tubes is 3 and 12 respectively.

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Answer:

Step-by-step explanation:

If a number is squared and then divided by 15, the resulting expression can be written as (x^2)/15, where x is the number. For example, if the number is 5, then the expression can be written as (5^2)/15 = 25/15. In general, the value of the expression (x^2)/15 will be the square of the number x divided by 15.

Answer:

The possible values of k are -0.56, -3.56, -2 and -1.

Step-by-step explanation:

We have the equation of the parabola, .

Substituting x=0, we get that  i.e. y = -1.5k

So, the y-intercept is (0,-1.5k).

Thus, the height of the triangle becomes |-1.5k| = 1.5k

Again, substituting y=0,  i.e.  i.e. x=3 and x=-k

So, the x-intercepts are (3,0) and (-k,0).

Thus, the base of the triangle is 3+k if k>-3 or -3-k if k<-3.

We see that 'k' cannot be 0 or -3 as if k=0, then height = 0, which is not possible. If k= -3, then the base = 0, which is also not possible.

As, area of a triangle = .

Substituting the values, we get,

1.5=

i.e.

i.e. k = -0.56 and k = -3.56

or

1.5=

i.e.

i.e. k = -2 and k = -1.

Thus, the possible values of k are -0.56, -3.56, -2 and -1.

Answer:

1: 50

2: 1

3: 2

4: 36

5: 0

6: -12

7: -2

8: 3

Step-by-step explanation: Use PEMDAS

P: Parentheses

E: Exponents

M: Multiplication

D: Division

A: Addition

S: Subtraction

Answer: 4.5 years or 54 months

Step-by-step explanation: Divide $4,000 by $75 a month to get 53.33 then round to 54. 54 months = 4.5 years.

The equation of the tangent to the given curve x = t⁴ + 3 , y = t³ + t at the corresponding point is y = x - 2   .

In the question ,

it is given that ,

the parametric equation is x = t⁴ + 3 , y = t³ + t ,

differentiating with respect to t ,

we get ,

dx/dt = 4t³  and   dy/dt = 3t² + 1

On dividing dx/dt by dy/dt to get the slope ,

we get

dy/dx = (3t² + 1)/ (4t³)

Substituting in t=1

dy/dx = (3 + 1)/4 = 1

Slope (m) = 1 ;

Substituting t = 1 , in the parametric equation ,

we have , x = 4 and y = 2 .

So , (y - 2) = 1(x - 4)

y - 2 = x - 4

y = x - 2

Therefore , the equation of the tangent is  y = x - 2  .

The given question is incomplete , the complete question is

Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter. x = t⁴ + 3 , y = t³ + 3 ; t = 4 ?

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Evan's error was that he subtracted  -2 1/2 from 4, and thus he got  6 1/2.  On adding he would be getting  1 1/2

A straight line of numbers is shown visually as a number line. This line is used to compare integers on an endless line that extends on both sides, either horizontally or vertically, at equal intervals. On a horizontal number line, the numbers grow as we walk to the right and decrease as we move to the left.

It is given that Evan added -2 1/2 and 4 but got 6 1/2

The mistake that Evan did was instead of adding, he substracted -2 1/2 from 4, and thus he got 6 1/2. This is because

4 -  (-2 1/2)

= 4 + 2 1/2

=  6 1/2

This is wrong as he was asked to add -2 1/2 and 4 not to subtract.

Therefore, on adding -2 1/2 and 4, we get,

= 4 - 2 1/2

= 1 1/2

Evan's error was that he subtracted  -2 1/2 from 4, and thus he got 6 1/2         .  On Evan's error was that he subtracted  from 4, and thus he got          .  On adding he would be getting  1 1/2

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In 11.61 years the given principal amount of $ 900 will be double.

Compound interest is interest charged on a loan and is determined by both the principal amount and the rate of interest compounding per year.

The formula for the Amount when compound interest is calculated is given by

P = Principal amount

r = Rate of interest

n = Number of compounds

t = Time period in years

Here from the given data,

Principal amount = $ 900

Rate of interest = 6% = 6/100 = 0.06

Number of compounds = 12    [ ∵ compounded monthly]

Here we need to find the time period

Assume that after 't' years the amount will be doubled

From the given data and formula,

Amount, A = 900(1+0.06/12)^12t        

= 900(1+0.06/12)^12t

= 900(1 + 0.005)^12t

= 900 (1.005)^12t  

As we assumed after 't' years the amount is double

=> 900 (1.005)^12t  = 2 × 900

=> (1.005)^12t  = 2    

Applying log on both sides

=> log (1.005)^12t  = log 2

=> 12t log(1.005) = log 2

=> 12t = log 2/log(1.005)

[ we can use a calculator for log (1.005) and log 2]

=> 12t = 0.30108/ 0.00216  

=> 12t = 139.888

=>  t =  11.61

Therefore,

In 11.61 years the given principal amount of $ 900 will be double.

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The required distance between the points  (−1,6) and (−1,7). is 1 unit.

Given that,
using the distance formula to evaluate the distance between the points  (−1,6) and (−1,7).

Distance is defined as the object traveling at a particular speed in time from one point to another.

Here,
The distance formula is given as,
D = √[[x₂ - x₁]² + [y₂+ - y₁]²]
Substitute the values in the above equation,
D = √[[-1 + 1]² + [7 - 6]²]
D = √[0 + 1]
D = 1

Thus, the required distance between the points  (−1,6) and (−1,7). is 1 unit.

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Answer:

64/100

Step-by-step explanation:

Answer: 16/25or64/100

Step-by-step explanation: 64/100 and divide it by 4 . To change a percent to a fraction, put the percentage over 100 (after removing the % sign) and simplify if necessary