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how did you get x=68 and sqrt of 4624?

The volume of the apple juice in a glass whose shape is an inverted frustum of a cone is 1176π mm³. The depth of the juice is 18mm.

Find the radius of the lower base if the product of its radii is 60 mm². *

a.5 mm

b.6 mm

c.7 mm

d.8mm

Hello Guest!

\(V=\dfrac{1}{3}\pi h(R^2+Rr+r^2)\\ 1176mm^3=\dfrac{1}{3}\cdot 18mm\cdot (R^2+60mm^2+r^2)\\\)

\(Rr=60mm^2\\ R= \dfrac{60mm^2}{r}\)

\(1176mm^3=\dfrac{1}{3}\cdot 18mm\cdot ( (\frac{60mm^2}{r})^2+60mm^2+r^2)\)

\((\frac{60mm^2}{r})^2+60mm^2+r^2=196mm^2\\ 3600mm^4+60r^2mm^2+r^4=196r^2mm^2\\ r^2=x\)

\(3600mm^4+60x\ mm^2+x^2=196x\ mm^2\\ 3600mm^4-136x\ mm^2+x^2=0\)

\(x=68\pm \sqrt{4624-3600}\\ x=68\pm 32\\ r=\sqrt{x}\\ r=\sqrt{68-32}\)

\(r=6\ mm\)

The radius of the base of the frustum of a cone is 6mm.

b. it's right.

  !

Thank You

Here :   https://web2.0calc.com/questions/help_84203

Tony and his grandma went hiking together. Grandma was slow but steady and maintained the same speed throughout. Tony accompanied grandma at her pace forthe first 30 minutes. Then he walked ahead for the next 20 minutes at 4 miles an hour. And the next 40 minutes at 3 miles an hour. He then waited 15 minutes forgrandma, and once she caught up, they finished the hike (at her pace) in 20 minutes.What was the total distance of the hike in miles? Please give the answer rounded to the nearest tenth of a mile.

Mmmm......this one  looks  interesting...

30min = 1/2  hr

20 min = 1/3 hr

40 min = 2/3 hr

15 min = 1/4 hr

Call Gramdma's  pace =  P 

For the first  part of the  hike, the  distance  covered  =  (1/2)hr * P

For  the next part of the  hike  the distance that  grandma covers is grandma covers  =

(Tony's  time  hiked  when he walked alone + time he waited ) P  = 

( 1/3 hr + 2/3 hr + 1/4 hr) P

And this  equals the  distance that  Tony covers  =  ( 1/3)hr *4 + (2/3)hr *3  =  10/3 miles

So

10/3  = ( 1/3 + 2/3 + 1/4) P

10/3   =  ( 1 + 1/4) P

(10/3) = (5/4) P

P = (10/3) ( 4/5)  =  40/15 =  8/3  mi/hr

So   the total distance  hiked =   

(1/2)(8/3)  +  10/3  +  (1/3)(8/3)   = 

4/3  + 10/3  + 8/9  =

14/3 +  8/9  =

[42 + 8 ]  /  9  =

50/9  miles ≈  5.55 miles  =  5.6 miles 

We  just need to find  the  distance  between  the  lines   ....this will be the  side  of the  square

It's possible to derive this with a little work, but I prefer to  use a  formula to calculate the  distance between parallel lines

We  have  the forms

Ax + By  + C1  =  0

Ax  + By + C2   =  0

The  distance  is  given  by

l   C1  -C2  l                       l  7  - 11   l                        4

_______________     =   ______________  =       ________  

sqrt  ( A^2 + B^2)             sqrt ( 2^2 + 3^2)             sqrt (13)

Area of the  square =   side^2  =  ( 4/ sqrt (13) )^2  =    16/13    units^2

Call the altitude of triangle GCD , GH

And triangle  GHC    is similar to  triangle  FEC

Call  HC,  x

So  we have  that

FE/CE =  GH /  x

8/6  = GH /x

4/3 = GH /x

GH  = (4/3)x

And triangle  GHD  is similar to  triangle  ABD

So

AB/ BD  =  GH / HD

Let HD  = 3  - x

So

12/6 =  [ (4/3)x ]  /   ( 3 - x)

2 (3 - x)  = (4/3)x

6 - 2x  = (4/3)x

6   =  ( 4/3 + 2) x

6 =  ( 10/3) x

x = 18/10 =  9/5

So GH  =(4/3) (9/5)  =   36/15 =  12/5

So  the  area of   GCD   =  (1/2) CD * GH   =  (1/2) (3) ( 12/5)  =  18/5  units^2

Avg  rate  when the  distances are  equal  =   2 product  of the rates  / sum of the  rates

2 ( 40 * 24)  /  ( 40 + 24)   =

2 ( 960) / ( 64)   =

30 mph

Volume  _frustum  =     (1/3)  pi  * height  (  r^2  +Rr +  R^2)

height  = 18

Rr = 60

Note that  2Rr =   120  

We have  that

1176pi  =   (pi/3)   ( r^2  + Rr  +  R^2)

3528  =   r^2 + R^2   +  60

3468  =   r^2 + R^2

Note that

r^2 + 2Rr  + R^2 =  3468  + 2Rr             factor the  left  side

(r + R)^2   =   3468  + 120

(r + R)^2  =  3588       take  the positive  root

r + R   =   sqrt (3588)  

R  =  sqrt ( 3588)  -  r

The product of the  roots =  60

So

rR  = 60

r ( sqrt (3588)  - r)  = 60

-r^2  + sqrt (3588)r   - 60    =  0        mu;tply through  by  -1

r^2  - sqrt (3588)r + 60    =  0

A graph is the easiest way to solve this  : https://www.desmos.com/calculator/98tdpymtxf

Solving  this for r   gives that   r  ≈ 1.019   or  r ≈  58.881

Take the  smaller  value     r ≈ 1.019  mm

Call the intersection  of   AC  and  BD  =   M

Because   AC is  perpendicular to  BD,  then in triangle  BMC, angle  BMC is right

And   CM = BM   = 12/sqrt (2)=  legs  of  BMC

So the area of this triangle  is    product of the  legs / 2  =   (1/2)  (12/sqrt 2) ^2  =  144/ 4  =  36

But the area  of this triangle also  =  (1/2) 12  * altitude

36  = (1/2) 12  (altitude

36  = 6 * altitude

6  = altitude

And triangle  DMA  is similar to  to triangle BMC

So the height  of  triangle  DMA  =  4/12  *  6  =   2

So.....  the height of the trapezoid =    6 + 2   =   8

15/2 = 7.5

Adding multiples of 2 we can never get 1/2 unless we use an exponent of -1 at one point.

Having exponents smaller would result in longer decimals(.25,.125) So x has to be -1

Checking we get

2^-1+2^0+2^1+2^2

1/2+1+2+4

1/2+7

7.5

If EG is 5/3 the length of EF, then say EG is 5 centimeters and AF is 3 centimeters.

Then FG is 2 centimeters.

HF is 5 times that or 10 centimeters.

We know FG is 2 so HG is 8.

So EF/HG is 3/8

a + b  = -7   ⇒   b  = -7  - a    (1)

ab  = -7    (2)

Sub  (1)  into (2)

a ( -7 - a)   = -7  

-7a - a^2  =  -7       mutiply through  by  -1

a^2  + 7a   =   7        complete the square  on  a

a^2  + 7a  +  49/4   = 7 +  49/4          factor the left  side   and simplify the  right

(a + 7/2)^2   =  77/4          take the  positive  root

a + 7/2  =  sqrt (77) / 2

a =  sqrt (77)/2   -  7/2

b =  -7  - a  =   -7   - ( sqrt (77/2)  -  7/2)   =       -sqrt (77) / 2  - 7/2

Call the side of the  octagon, S

Area of  octagon  = 2(1 + sqrt (2) ) S^2

Blue  area  = 2 (1/2)  (S/sqrt2)^2    +  (S/sqrt (2))*S  = 

S^2 / 2 +  S^2 / sqrt (2)   =

( 1/2 + 1/sqrt (2)) S^2  =

(1/2)  ( 1 + sqrt (2) ) S^2

Blue Area  / Total Area =    (1/2)  / (2)   = 1/4

The constant of proportionality is 32/8^2 = 1/2.

We take all the odd numbers, and take out all the multiples of odd squares.  This gives us an answer of 50 - 21 = 29.

Pls help I really need this.

Since f(x) is divisible by x^3, f(x) is of the form ax^5 + bx^4 + cx^3.

You then want ax^5 + bx^4 + cx^3 + 2 to be divisible by (x + 1)^3.  Using long division, you get the equations

-10a  + 6b - 3c = 0

4a - 3b + 2c = 0

-a + b - c + 2 = 0

==> a = 6, b = 16, c = 12

So f(x) = 6x^5 + 16x^4 + 12x^3.

First one 

BC   = HK

-3x + 11    =  87 + x

11 - 87 =   4x

-76   = 4x

x   = -19

So  GH  = -2x + 41   =   -2(-19)  + 41  =  79

Second  one

EF  =   XZ

x - 7 = 39  - x

2x  =  46

x  = 23

EG  = 5 + x   =   5 + 23  =  28

Last one

BD  = XZ

2n - 4   =  n + 4

n  =   8

ZW  =3n  -10   = 3(8)   - 10    =    14

4. Linda drove 50 miles from San Jose to San Francisco in 90 minutes. For the first 30 minutes, without much traffic, her average speed was 60 miles an hour. In the second 30 minutes, due to a traffic accident along US 101, her average speed dropped to 35 miles an hour. She was stuck in traffic for the last 30 minutes near San Francisco.What was her average speed, in miles per hour, during the last 30 minutes?

Trick question.....she didn't move  in the  last  30 min.....avg speed during this time  =   0  mph

3. On average customers order one pizza every 3 minutes and 20 seconds from Mathni-fient Pizza, and today (due to a mishap) they only have enough raw materials tomake 45 pizzas. They started selling pizza at 05:10pm. At what time does Math-nificent expect to sell out all pizzas?Please write the answer in the 12-hour hh:mm format. For example,9:30pm would be 09:30.

3 min +  20 sec =    3 + 1/3 min =     10/3  min    for  one pizza

So  45  pizzas take        45 * 10 /3 =  150 min  =    2 hrs 30   min

So

5:10  PM    +   2hrs 30  min  =       7: 40  PM   when they sell out

1. Sophia and Melody were biking together. along a straight bikeway from two positions.At one point Sophia stopped to rest, while Melody continued to bike. Sophia resumed biking again but at a faster pace of 22 miles per hour, while Melody continued at asteady pace of 15 miles per hour to allow Sophia to catch up. Sophia finally caughtup with Melody after 6 minutes. How far behind (in miles) was Sophia when she resumed biking? Please express the answer in decimal points

Notice  that the difference  in rates =  (22  - 15)   = 7 mph =  Sophia's  effective rate

6 min =     1/10 hr

So        rate  * time  = distance

7 (1/10)  =     7/10      =    .7   miles behind 

2. One mile is 5280 feet, and 3 feet is one yard. A professional soccer player runs upto 7 miles in a 90-minute match. It averages to how many yards every 9 seconds?Please round the answer to the nearest yard.

No. of yards run  =        5280 / 3  *  7   =  12320 yards

90 min  =1.5 hrs

No. of seconds  in 1.5  hrs  =   1.5  *  60  min  * 60sec  =  5400 sec

Yds run per second =  12320 /  5400  = 308/135 yds

Yds run in 9 sec =     (308/ 135) * 9    ≈  21 yds in 9 sec

s = kt    k = proportionality konstant

14 = k(18)

k = 14/18 = 7/9

sooo    s = 7/9 t

            s = 7/9 (7)

               = 49/9

Suppose s and t are directly proportional. If s=14 when t=18, what is s  when t=7?

14 / 18 = s / 7      s = (14 * 7) / 18

4x^2 + 15x + 2 =0 

4 (x^2 + 15/4 x) + 2

4 ( x + 15/8)²   - 225/16 + 2

4 (x + 15/8)² -193/16