All Answers 357330 Answers

1.  For the first....just  add  the 4  and 6  and append the radical...so we have...

10 ⁴√ [ 2x ]  ⇒  first answer

2.  A little trickier, NSS

Ratio  3 : 2     means that there are  3 + 2  = 5 equal parts......and the longer piece is  3 of these parts

So......the longer piece  =  

(3 /5) √250   =

(3/5) √ [ 125 * 2 ]    =

(3/5) √ ( 25 * 5 * 2 ]  =

(3/5) √ 25 * √5 * √2  =

(3/5) * 5  * √10  =

3 √ 10  units

Thank you!

Simple interest formula :

Interest earned  =  Principal * Interest Rate * Time (in years)

So......for the first one

Interest earned =  1450 * (.09)  * (5)  =  $652.50

The second one is a little trickier....we have...

10   =  250 *  .04  * Time             simplify

10  =  10  * Time                       divide both sides by 10

1  =  Time   

So  ...   1  year

Notice  that

AB  = AD

DC  = BC

and

AC  = AC

So.....by  SSS congruence,  triangle  ADC  is congruent to triangle ABC

So   angle  ADC  =  angle ABC   =   ( 180  - 32 - 41)°  =  107°    =  "B"

Thanks Cphill

Notice that a 5%  discount is the same thing as selling something for  (100 - 5)% = 95% of its original price

So....the first watch costs   50  x . 95  =  $47.50

And...a 20% discount means the same thing as selling something for (100 - 20) %  = 80% of its original price

So.....the second watch  costs    60 x .80   =  $48

So...the first watch is the better deal

Thanks for that good answer, hectictar.......here's another way to do this......

We can actually  use the same "formula" to figure both of these

% increase / decrease   =    (  l New Value - Old Value l  / Old Value  x 100 ) %

So...for the first one we have

(   l  180 - 120 l  /  120  x 100) %  =

(  l   60  l   /  120   x 100 ) %  =

(  60  / 120  x 100 ) %   =

( 1 / 2  x 100 ) %  =

50%  increase

For the second one, we have

(  l  8  - 10  l  /  10   x 100 ) %    =

(  l -2 l  / 10  x 100 ) %  =

(  2  / 10  x 100 ) %  =

( 1 / 5  x 100 ) %  =

20%  decrease

Thanks so much 

3)  We have a triangle at the top left with a base of 3 units  and a height of 4 units

So its area  is  (1/2) (3) (4)  =  6 inches^2

And we have a reatangle with dimensions 8 x 7   =  56 inches^2

So  the total area is   6 + 56   =    62 inches^2

Answered here :

https://web2.0calc.com/questions/in-the-figure-ad-cd-and-ab-cb

2.  Connect  ( 2, 1)  and (3, -3)....this creates a rectangle  with vertices (2,1) , (3, -3), (-5, -5) 

and (-6, -1)

The distance from  (2,1) to (3, -3)  forms one side of this rectangle  = √17 units

The distance  from (-6, -1) to (2,1)  forms the other side of this rectangle  = 2√17 units

So....the area of this rectangle is  √17 * 2√17  =  2 * 17  =  34 units^2   (1)

Connect (2, 1)  and (3, 1)......this forms a triangle at the bottom right with vertices  (2,1), (3,1) 

and (3, -3)

The base of this triangle  is  (2,1) to (3,1)  = 1 unit

And the height of this triangle is (3, 1)  to (3, -3)  =  4 units

So...the area of this triangle is  (1/2) (1) (4)  =  2 units^2   (2)

Connect (2,3)  and (3,3)....this creates a rectangle with vertices  (2,3) (3,3), (2,1) and (3,1)

The height of this rectangle is  (3,1) to (3, 3)  = 2 units

And the widthof this rectangle is from (2,1) to (3,1)  = 1 unit

So...the area of the rectangle is  2 * 1  = 2 units^2  (3)

Finally....we have a triangle with the vertices (2,3), (3,3)  and (2,7)

The base of this triangle is from (2,3) to (3,3)  = 1 unit

And the height of this triangle is from   (2,3) to (2,7) =  4 units

So....the area of this triangle  is  (1/2) (1) (4)  =  2 units^2   (4)

So  the total area =   (1)  + (2)  + (3) + (4)  =  [34 + 2 + 2 + 2]  =

40 units^2

1.     Draw VR....this creates triangle VMR  and rectangle VRDE

Let VR be the base of triangle VMR  = 6 units   and the height drawn from M  to the base  = 3 units

So...the area of this triangle is  (1/2)base* height  = (1/2) 6 *3  =  (1/2) 18  =

9 units^2   (1)

And rectangle VRDE has area VR * VE   =  6 * 5  =  30 units^2  (2)

So.....adding (1) and (2)  gives  the total area  of  39 units^2 

(x, y)  ⇒ reflected about  x axis  ⇒  ( x, -y)

⇒  rotated 90°  clockwise  about the origin  ⇒  ( -x , - y)

⇒  reflected in the y axis  ⇒  ( x, -y)

the first one is variable the second one is constant

Hope that helps

               thx,

               penquino21

i think its c-2 in algebra but yea thats right

 I think that there is something wrong with your question.

That division has a remainder .....

It would be very helpful if you gave the rest of the question too.

Hi Heureka and Chris,

You have each given a really good answer here.  

I would have done it similar to Chris but Heureka's answer is really incitful.

I am talking about the first half of Hereka's answer.  I never would have thought to do that and it is so simple!

I will admit that I do not understand the second half of what you have done though Heureka. 

How did that combinatory maths sneak in there??

#1) If the price of an item is $130 and the discount is at 60%, this means the price of the sale price should be 60% of the original price, $130. 

What is 60% of 130? Well, the "of" in mathematics is an indicator of the multiplication operation, so the problem is effectively 60% * 130.

#2) A mark up means that the store is adding to the original price of the item. In this case, $45 with a 35% markup means that the sale price should increase by 35%. In other words, this means:

Sale price = $45 + (35% of 45)

Let's simplify this.

 If $f$ is a polynomial of degree $4$ such that $$f(0) = f(1) = f(2) = f(3) = 1$$ and $$f(4) = 0,$$ then determine $f(5)$.

$\small{ \begin{array}{lrrrrrrrrrrrrr} & f(0) && f(1) && f(2) && f(3) && f(4) && {\color{red}f(5)} && \cdots \\ & d_0=1 && 1 && 1 && 1 && 0 && {\color{red}-4} && \cdots \\ \text{1. Difference } && d_1=0 && 0 && 0 && -1 && {\color{red}-4} && \cdots \\ \text{2. Difference } &&& d_2=0 && 0 && -1 && {\color{red}-3} && \cdots \\ \text{3. Difference } &&&& d_3=0 && -1 && {\color{red}-2} && \cdots \\ \text{4. Difference } &&&&& {\color{red}d_4=-1 } && {\color{red}-1} && \cdots \\ \end{array} }$

$\text{ ${\color{red}d_4=-1}$ is constant, if a polynomial is of degree 4.}$

$f(5) = {\color{red}-1}+(-1)+(-1)+(-1)+0 = -4$

polynomial:

$\small{ \begin{array}{rcll} \hline a_n &=& \binom{n-1}{0}\cdot {\color{red}d_0 } + \binom{n-1}{1}\cdot {\color{red}d_1 } + \binom{n-1}{2}\cdot {\color{red}d_2 } + \binom{n-1}{3}\cdot {\color{red}d_3 } + \binom{n-1}{4}\cdot {\color{red}d_4 } \\\\ && \quad {\color{red}d_0 } = 1 \quad {\color{red}d_1}={\color{red}d_2}={\color{red}d_3} = 0 \quad {\color{red}d_4 } = -1 \\\\ a_n &=& \binom{n-1}{0}\cdot 1 + \binom{n-1}{1}\cdot 0 + \binom{n-1}{2}\cdot 0 + \binom{n-1}{3}\cdot 0 + \binom{n-1}{4}\cdot (-1) \\ a_n &=& \binom{n-1}{0}\cdot 1 - \binom{n-1}{4} \quad & | \quad \binom{n-1}{0} = 1 \\ a_n &=& 1 - \dbinom{n-1}{4} \quad & | \quad n = x+1 \\ \mathbf{f(x)} &\mathbf{=}& \mathbf{ 1 - \binom{x}{4} } \\\\ && \quad \dbinom{x}{4} = \left(\dfrac{x}{4}\right)\left(\dfrac{x-1}{3}\right)\left(\dfrac{x-2}{2}\right)\left(\dfrac{x-3}{1}\right) \\\\ \mathbf{f(x)} &\mathbf{=}& \mathbf{ 1 - \dfrac{x(x-1)(x-2)(x-3)}{24} } \\ \end{array} }$

1.

180  is bigger than  120  so this is an increase.

120  increased by what percent of  120  is  180   ?

Let  x  be the percent of increase.

120  +  120x   =   180

120x   =   60

x   =   60/120   =   0.5   =   50/100   =   50%

120  increased by  50%  of  120  is  180 .

2.

8  is smaller than  10  so this is a decrease.

10  decreased by what percent of  10  is  8    ?

10 - 10x  =  8

-10x  =  -2

x   =   -2 / -10   =   20 / 100   =   20%

10  decreased by  20%  of  10  is  8 .

O will be the center of the octagon

Draw OB, OC  and OD

Note that   area OAB  =  area  ODC =  area ODE

And area OBM  =  1/2 area OAB

And area OCM  = area OBM  =  1/2 area OAB

So

area  EDCMO  =   area ODE + area ODC  + area OCM

Substituting, we have that

area EDCMO  =  area OAB + area OAB + 1/2area OAB   =  2.5 areaOAB

And area ABMO  =  area OAB + area OBM

Substituting, we have that

area ABMO  =  area OAB + 1/2 area OAB   =   1.5 area OAB

So

[ ABMO] / [ EDCMO]  =

[ 1.5 area OAB ] / [ 2.5 area OAB ]  =

[1.5] / [ 2.5]  =

3 / 5 

Rewritten:

Let f(x) be a polynomial with integer coefficients.

Suppose there are four distinct integers p,q,r,s such that  f(p) = f(q) = f(r) = f(s) = 5.   (Can there be more??)

If t is an integer and f(t)>5,

what is the smallest possible value of f(t) ?

I'd like to see this one done too, it makes little sense to me .      

ANYWAYS... here's the answer if you don't want to do it

Simplify the original line: We get y=-(2/3)x+8

The line parallel to another line has the same slope.

Therefore, our line would be y=-(2/3)x+b, b being the y-intercept.

We also know that the line passes through point (12,-4). We can substitute the points into the line equation to solve b.

We get -4=-(2/3)*12+b => -4=-8+b => b=4

The line is y=-(2/3)x+4

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