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Posted in Mathematics

Calculate the moments Mx, My, and the center of mass (x bar, y bar) of a lamina with the given density p=5 and the shape:


M x = 1/2 p ∫ ( f ( x )² - g ( x )² ) d x
f ( x ) = √( 1 - x²),  g ( x ) = - 2
M x = 1/2 * 5  \int\limits^1_{-1} (- x^{2} -3)\, dx= \\ =-5/2 * [ x^{3}/3 + 3 x]^1 _{-1} = \\ =-5/2 * (1/3+3+1/3+3)=
= - 50/3
My = p ∫ x * ( f ( x ) ) dx
My = p ∫ x ( √(1+x²)) dx
Substitution: 1 - x² = u,  x dx = - du/2
M y = 5 *  \int\limits^0_0 { \sqrt{u} } \, du = 0
M x = - 50/3, M y = 0
M = 5 *  \int\limits^1_{-1}  { (\sqrt{1+ x^{2} }+2)} \, dx  = \\ 
=5* [1/2  \sqrt{1+ x^{2} }  *x + 2 x + 1/2 *sinh ^{-1} x]^1_{-1}
M ≈ 5 * 6.3 ≈ 31.2  
x = M y / M = 0 / 31.5 = 0
y = M x / M =  -50/3 : 31.5 ≈ - 0.529
The center of mass is ( 0, -0.529 )



Related Questions

Posted in Mathematics

The binomials shown below will be multiplied to produce a quadratic trinomial. (x + p)(x + q) Which part of the trinomial will equal the sum of p and q? A. The constant term B. The coefficient of the x-term C. The degree D. The coefficient of the x2-term

I believe the correct answer from the choices listed above is option B. The part of the trinomial will equal the sum of p and q would be the coefficient term. We do it as follows:

(x + p)(x + q) 
x^2 + px + 1x +pq
x^2 + (p+q)x + pq

Hope this answers the question. Have a nice day.

Posted in Mathematics

If a polynomial is divided by (x-a) and the remainder equals zero then (x -a) is a factor of the polynomial. true or false

The Factor theorem states that  for any polynomial f(x) if f(c)=0 then x-c is a factor of the polynomial f(x).

If any polynomial f(x) is divided by x-a and remainder is 0 that means f(a)= 0 .In other words we can say x-a is a factor of the polynomial f(x).So the statement :If a polynomial is divided by (x-a) and the remainder equals zero then (x -a) is a factor of the polynomial is True.

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Posted in Mathematics

For f(x)=2x+1 and g(x)=x^2-7, find (f-g)(x)

I hope this helps you



(f-g)(x)=2x+1-(x^2-7)



(f-g)(x)=2x+1-x^2+7


(f-g)(x)= -x^2+2x+8

Posted in Mathematics

1. Which set of ordered pairs in the form of (x,y) does not represent a function of x? (1 Point) A. (1, 1.5), (2, -1.5), (3, 1.5), (4, 1.5) B. (0, 1.5), (3, 2.5), (1, 3.3), (1, 4.5) C. (1, 1.5), (-1, 1.5), (2, 2.5), (-2, 2.5) D. (1, 1.5), (-1, -1.5), (2, 2.5), (-2, 2.5)

Based on the given set of ordered pairs above, the set of ordered pairs in the form of (x,y) that does not represent a function of x would be option B. Why is this so? if you notice, the number 1 in the x is being repeated so this means this is not a function. Hope this answer helps.

Posted in Mathematics

What is f(x) = 2x2 + 28x – 5 written in vertex form?

To convert a quadratic from y = ax2 + bx + c form to vertex form, y = a(x - h)2+ k, you use the process of completing the square. We do as follows:

f(x) = 2x^2 + 28x – 5
f(x) = 2(x^2 + 14x) – 5
f(x) = 2(x^2 + 14x + 49) – 5 - 98
f(x) = 2(x + 7)^2 – 103

Hope this answers the question. Have a nice day.

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Posted in Mathematics

Given f(x)=17-x^2, what is the average rate of change in f(x) over the interval [1, 5]?

For the answer to the question above, I'll show my to solution to the answer below.
average rate of change is 
(f(5) - f(1))/(5 - 1) = 

(17 - 5^2 - (17 - 1^2))/4 = 

(17 - 25 - 17 + 1)/4 = 
So the answer for this problem is
-6
I hope my answer helped you. Feel free to ask for more questions.

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Posted in Mathematics

A certain forest covers an area of 2,000 square kilometers. Suppose that each year this area decreases by 6%. What is the function that best represents the area of the forest each year and how much area remains after 12 years? Round your answer to the nearest square kilometer. Hint: Use the formula, f(x) = P(1 + r)x

For the answer to the question above, I think the formula you is wrong. The x must be in exponential form. It must look like this

f(x) = P(1 + r)^x
Now we can solve.

Given:
P = 2000
r = -0.06

f(x)  =  2000 (1 - 0.06)^x
       =  2000 (0.94)^x


         
2000 (0.94)¹² = 

2000 (0.476)  = 
The answer is 
951.8 square kilometers.
                   

Posted in Mathematics

What is the range of the function f(x) = 3x2 + 6x – 9

I have a solution here however with a slight change in the equation:

 f(x)=3x^2-6x+1

My solution is:

The domain is all real numbers--there are no restrictions like a square root or variable in the denominator 

this is a U shape parabola and you want to find the bottom point, it is at the vertex 

x=-b/2a =- (-6) /2 (3) =1 
f(1) =3(1)^2 -6(1) +1 =3-6+1 =-2 
the minimum is (1, -2) 
so the range is all real numbers >= -2

By examining my solution, you could just answer the problem on your own! I hope it helps!

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Posted in Mathematics

Solve for x. round to nearest tenth if needed. 1. 7r-7=2r+18 a.r=-5 b.r=5 c.r=2.2 d.r=1.2 2. 2x+12=18-x a.x=3 b.x=10 c.x=6 d.x=2 3. 8x-3=15x+18 a.x=-3 b.x=3 c.x=2.1 d.x=0.9 4. 6y-6=4y+16 a.y=2.2 b.y=-2.2 c.y=11 d.y=5 5. 3(x-4)=5(x+2) a.x=11 b.x=-11 c.x=1 d.x=-1

1)  The answer is: [B]: r = 5 .
__________________________
Explanation:
__________________________ 
Given: 7r − 7 = 2r + 18 ; Round your answer to the nearest tenth, if necessary.
____________________________
Since "r" is the only variable given, let us assume we want to solve for "r" (instead of "x").
___________________________
→ Subtract "2r" from EACH SIDE of the equation; and & add "7" to EACH SIDE of the equation:
_____________
→ 7r − 7 − 2r + 7 = 2r + 18 − 2r + 7 ;  to get: → 5r = 25 ;
_____________
→ Now, divide EACH SIDE of the equation by "5"; to isolate "r" on one side of the equation; and to solve for "r" :
______________
→ 5r / 5 = 25 / 5 → r = 5 → which is: "Answer choice: [B]".
_________________
Let us check our answer, by plugging in "5" for "r" in the original equation:
_________________
 → 7r − 7 = 2r + 18 ;  →  7(5) − 7 =? 2(5) + 18? ;
______________________
→ 35 − 7 =? 10 + 18 ?;     → 28 =? 28? Yes!
______________________
 2) The answer is: [D]: x = 2 .
_____________
Explanation: 
_____________
Given: 2x + 12 = 18 − x ; Solve for "x" (round to nearest tenth, if necessary).
_______________
→ Add "x" to EACH SIDE of the equation, & subtract "12" from EACH SIDE of the equation:  → 2x + 12 + x − 12 = 18 − x + x − 12 ; 
______________
→ To get: 3x = 6 ;  → Divide EACH SIDE of the equation by "3";
to isolate "x" on one side of the equation; and to solve for "x":
_____________
→ 3x / 3 = 6 / 3 ; → x = 2 ; which is: "Answer choice: [D]".
______________
Let us check our answer, by plugging in "2" for "x" in the original equation:
________________
→ 2x + 12 = 18 − x ; → 2(2) + 12 =? 18 − 2 ?
________________
→ 4 + 12 =? 18 − 2 ? ;   → 16 =? 16?  Yes!
________________________________
3)  The answer is: [A]: x = -3 . 
_____________
Explanation:
________________
Given:  8x − 3 = 15x + 18 ; Solve for "x". Round your answer to the nearest tenth, if necessary.
_________________
→ Subtract "8x" from EACH SIDE of the equation, & add "3" to EACH SIDE of the equation:
_______________
→ 8x − 3 − 8x + 3 = 15x + 18 − 8x + 3 ; to get:
_______________
→ 0 = 7x + 21 ; → Subtract "21" from EACH SIDE of the equation;
_______________
→ 0 − 21 = 7x + 21 − 21 ; to get:
_______________
→  -21 = 7x ; Now divide EACH SIDE of the equation by "7";
    to isolate "x" on one side of the equation; & to solve for "x":
_______________
→ = -21 / 7 = 7x / 7 ; →  -3 = x ; which is "Answer choice: [A]."
_________________
Let us check our answer, by plugging in "-3" for "x" in the original equation:
________________
→  8x − 3 = 15x + 18 ;  → 8(-3) − 3 =?  15(-3) + 18 ?;
________________________
→ -24 − 3 =?  -45 + 18 ? ;   →  -27 =? -27?  Yes!
___________________________
4)  The answer is: [C]: y = 11 .
_____________
Explanation:
____________
Given: 6y − 6 = 4y + 16 ; Solve for "y"; Round to the nearest tenth, if necessary.
____________
(Note: Since "y" is the only variable given; assume we are to solve for "y" instead of "x").
____________
→ Subtract "4y" from EACH SIDE of the equation, & add "6" to EACH SIDE of the equation; → 6y − 6 − 4y + 6 = 4y + 16 − 4y + 6 ; to get:
_______________
→ 2y  = 22 ; Now, divide EACH SIDE of the equation by "2"; to isolate "y" one side of the equation; and to solve for "y" ;
_______________
→ 2y / 2 = 22 / 2 ; →  y = 11 → which is "Answer choice: [C]".
_______________________________
Let us check our answer, by plugging in "11" for "y" in the original equation:
___________________
→  6y − 6 = 4y + 16 ; → 6(11) − 6 =? 4(11) + 16 ?
_______________________
→ 66 − 6  =? 44 + 16 ?  → 60 =? 60 ?  Yes!
__________________
5)  The answer is: [B]: x = -11 .
_____________________
Explanation:
_________________
Given: 3(x − 4) = 5(x + 2) ; Solve for "x". Round to the nearest tenth, if necessary.
___________
→Note the "distributive property of multiplication": 
_____________
a*(b + c) = ab + ac ;  and: a*(b − c) = ab − ac ;
_______________
→ Let us expand EACH SIDE of our given equation.
→Start with the "left-hand side":
____________
3(x − 4) = (3*x)  − (3*4) = 3x − 12;
______________________________
→Now let us expand the "right-hand side" of the given equation:
____________
→  5(x + 2) = (5*x) + (5*2) = 5x + 10 ;
______________
→Now, we can rewrite the original equation:
_______________
→ 3(x − 4) = 5(x + 2) ; by substituting the expanded values for EACH SIDE of the question:   →  3x − 12 = 5x + 10 ;
__________________
→ Subtract "3x" from EACH SIDE of the equation; and add "12" to EACH SIDE of the equation: →  3x − 12 − 3x + 12 = 5x + 10 − 3x + 12 ; to get:
________________
→  0 = 2x + 22;  → Now subtract "22" from EACH SIDE of the equation:
______________
→  0 − 22 = 2x + 22 − 22 ; to get:  →  -22 = 2x ;
__________
→ Divide EACH SIDE of the equation by "2"; to isolate "x" on one side of the equation; & to solve for "x" ;
_____________
→  -22 / 2 = 2x /2 ;  →  -11 = x ; which is "Answer choice: [B]".
______________
Let us check our answer, by plugging in "-11" for "x" in the original equation:
___________
→ 3(x − 4) = 5(x + 2) ; →  3(-11 − 4) =? 5(-11 + 2) ? ; 
_______________________
→3(-15) =? 5(-9) ? ; → -45 =? -45 ?  Yes!
_____________________________________________
Hope these answers and explanations are helpful. Best of luck!

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Posted in Mathematics

Which rule represents the translation of hexagon DEFGHI to hexagon D'E'F'G'H'I'? (x, y)→(x - 8, y - 7) (x, y)→(x - 7, x - 8) (x, y)→(x - 4, x - 5) (x, y)→(x - 5, y - 4)

Correct answer is A.

Each vertex of the hexagon is translated 8 units left and 7 units down. So, the x-coordinate is 8 units smaller (x - 8), and the y-coordinate is 7 units smaller (y - 7).

Therefore (x, y) → (x - 8, y -7)

Posted in Mathematics

Calculate the length of the circumference of a circle with a diameter of 4cm

Length of circumference is 2πr
2 π r and r is equal to 2 cm
I.e 12.5 cm

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Posted in Mathematics

What is the radius of a circle whose equation is (x – 7)2 + (y – 10)2 = 4? 2 units 4 units 8 units 16 units

Answer:

Option (a) is correct.

The radius of circle with equation (x-7)^2+(y-10)^2=4 is 2 units.          

Step-by-step explanation:

 Given : The equation of circle as (x-7)^2+(y-10)^2=4

We haave to determine the radiusof th circle whose equation is given as (x-7)^2+(y-10)^2=4.

Consider the gicen equation (x-7)^2+(y-10)^2=4

The standard equation of circle with centre (h, k) and radius r is given by (x-h)^2+(y-k)^2=r^2

Thus, Comparing the gicen equation h = 7 , k = 10 and r = 2

Thus, The radius of circle with equation (x-7)^2+(y-10)^2=4 is 2 units.

Posted in Mathematics

What is (x-9)(x-8) simplified?

Dear HopellennibH, you need to FOIL. It is an algorithm to use when multiplying two binomials. Here are the steps. The "F" means to multiply the first two terms which gives you x squared. The "O" means to multiply the first term by the outer term, which gives you  -8x. The "I" means to multiply the inner terms, which  -9 times x which gives you -9x. The "L" means to multiply the last terms, which are -9 times -8, which gives you 72. Write the terms in standard form and you have x squared  -17x +72.

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Posted in Mathematics

Which function is the inverse of f(x) = 2x + 3? f-1(x) = –2x + 3 f-1(x) = 2x + 3

Answer with Step-by-step explanation:

Let f(x)=y

⇒ x=f^{-1}(y)

f(x)=2x+3=y

⇒ 2x+3=y

⇒ 2x=y-3

⇒ x=(y-3)/2

f^{-1}(y)=\dfrac{y-3}{2}

Replacing y by x

f^{-1}(x)=\dfrac{x-3}{2}

Hence, inverse of the function f(x)=2x+3 is:

f^{-1}(x)=\dfrac{x-3}{2}

Posted in Mathematics

Figure EFGHK is to be transformed to Figure E'F'G'H'K' using the rule (x, y)→(x + 8, y + 5): Which coordinates will best represent point K'? (3, 5) (2, 6) (6, 2) (5, 3)

The rule that has been given means that x coordinate of some point we increase by 8 and y coordinate of some point we increase by 5.

Coordinates of K point on original Figure are:
(-2,-3)

once we implement rule on this we get K':

K' ( -2+8,-3+5) 
or 
K' ( 6,2)

Answer is third option.

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Posted in Mathematics

Which of the graphs below represent the function f(x) = x3 - 5x2 + 2x + 8? You may sketch the graph to compare.

Answer:

The answer in the attached figure

Step-by-step explanation:

we have

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

we know that

The y-intercept is the value of the function when the value of x is equal to zero

The x-intercept is the value of x when the value of the function is equal to zero

Using a graphing tool-------> Find the  intercepts of the function

see the attached figure

The y-intercept is 8

The x-intercepts are -1,2,4

therefore

the answer in the attached figure


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