CPhill

These are a little tougher, RP...  

7) 42m^4  - 48 m^3  - 42m^2  + 48m

GCF 42m^4 and 48m^3 =  6m^3

GCF 42m^2 + 48  = 6m

So we have

6m^3 ( 7m - 8)  - 6m (7m - 8)

(7m - 8) ( 6m^3 - 6m)        ....we can factor 6m out of the second factor

(7m - 8 ) * 6m ( m^2 - 1)    .....m^2 - 1   factors as a difference of squares  = (m + 1) (m -1)

6m ( 7m - 8)(m + 1) (m - 1)

8) 56v^4  - 98v^3  - 64v^2 + 112v 

GCF   56v^4  and  98 v^3   =  14v^3

GCF  64 v^2 and 112v  = 16v

So we have

14v^3 ( 4v - 7)  - 16v ( 4v - 7)   

(4v - 7) ( 14v^3 - 16v)   ....we can factor  2v  out of the second factor

(4v - 7)  *2v *  (7v^2 - 8)

2v (4v - 7) (7v^2 - 8) 

No prob  !!!

% change  =   [   ( New Mass - Old Mass ) / (Old Mass ) * 100 ] %

So...for the first one we have

 [   ( 86.6 - 92.0 ) / 92.0   *   100  ] %  =   [ - 5.4 / 92  * 100 ] %  ≈ [ -5.87] %    (decrease )

Second one

[  (94.4 - 89.3 )  / 89.3   * 100 ] %  =  [ 5.1 / 89.3  * 100 ] %  ≈   [5.71 ]%    (increase)

No diagram is shown....

Excellent, RP  !!!

42) A trapezoidal prism of height 12 cm. The parallel sides of the base have lengths 12 cm and 6 cm. The other sides of the base are each 6 cm. The trapezoid's altitude measures 5.2 cm.

The key here is to  find the area of the trapezoidal base and then multiply this area by the height of the prism...note that info concerning "the other sides of the base" isn't really needed

Area of the trapezoid  =  [ base area ] =   (1/2) (altitude of the trapezoid) * (sum of the base lengths)  =

                                                                  (1/2) ( 5.2) (12 + 6) =

                                                                  (1/2) (5.2) (18) =

                                                                    9 * 5.2  =   

                                                                     46.8 cm

Now....the volume =   base are * height of the prism =   46.8 * 12   =  561.6 cm^3

45) A trapezoidal prism of height 6 yd. The parallel sides of the base have lengths 9 yd and 4 yd. The other sides of the base are each 5 yd. The trapezoid's altitude measures 4.3 yd

This one is similar

Area of the trapezoid =  (1/2) (4.3) (9 + 4)  = (1/2) (4.3) (13)   = 27.95 yd

Volume of the prism  =   6 (27.95)  = 167.7 yd^3

BTW...here's a website that might help : https://study.com/academy/answer/how-do-you-find-the-volume-of-trapezoidal-prism.html

5)  448n^4 - 56n^3 -56n^2 + 7n

Let's factor the last two terms, first  = -7n ( 8n - 1)

Now ,RP..let's guess that  (8n - 1)  might factor the first two terms in some manner

So  we have  (8n - 1)    and notice that if I multiply this  by 56n^3 ...   we have   56n^3 (8n -1)  = 448n^4 - 56n^3

So...the factorization becomes

56n^3 (8n - 1)  - 7n (8n - 1)  =

(8n- 1) (56n^3 - 7n)     and we can factor  a 7n  out of the second factor giving us

(8n - 1) * 7n * ( 8n^2 - 1)  =

7n (8n - 1) (8n^2 - 1)

6) 45a^5 + 54a^4 - 60a^3  - 72a^2

GCF  45a^5  and 54a^4  =  9a^4

GCF -60a^3 and - 72a^2  = -12a^2

So we have

9a^4 ( 5a + 6) - 12a^2 (5a + 6)

(5a + 6) (9a^4 - 12a^2)....we can factor  3a^2  out of the second factor

(5a + 6) * 3a^2 * ( 3a^2 - 4)  =

3a^2 (5a + 6) (3a^2  - 4)

First one :

We have a right triangle  as a base...the two legs are 6 and  8....the area of a right triangle  = (1/2) (product of the legs) =

(1/2) (6 * 8)  = (1/2) (48)  = 24 m^2  = base area

And the height of this triangular pyramid is 8m

So...the volume is  (1/3) (base area ) ( height) =  (1/3) (24) (8)  = (8) (8)  =  64 m^3

Second one :

If you look at the front face of this prism, it's just a trapezoid with a height of 4.3 cm and bases of 9cm  and 4 cm

The area of this trapezoid  is   (1/2) * height * (sum of the bases)  = (1/2)(4.3) ( 9 + 4)  =  (1/2) (4.3) (13)  = 27.95 cm^2

Now...if we consider all these similar trapezoids to  be  "stacked" to a depth of  6cm...the volume becomes

6 (27.95 )  =  167.7 cm^3

Ruan began painting a room at 9:00 AM. Leonardo, who can paint twice as fast as Ruan, started helping Ryan at 9:20 AM, and they worked together until the room was fully painted at 10:00am. What fraction of the room had been painted by 9:30 AM? Express your answer as a common fraction in lowest terms and clearly show/explain your reasoning.

Let the fraction of the room that Ruan can paint in one minute be  (1/x)

Let the faction of the room that Leonardo can paint in one minute  be 2 (1/x) =  (2/x)

So...fraction of the room painted in one minute * minutes on the job  =  part of the job done

Ruan paints for 60 minutes when the job is completed and  Leonardo paints for 40 minutes when the job is done....so we have this equation :

(1/x)(60) + (2/x)(40)  = 1

60/x  + 80/x  = 1

[60 + 80] / x   = 1

140 / x  = 1    ⇒   x  =  140

So....Ruan can paint (1/140) of the room in one minute 

So...in 30 minutes, Ruan paints  (1/140)(30)  =  30 / 140 of the room

And in  10 minutes, Leonardo paints  (2 / 140)(10) = 20 / 140 of the room

So...together....they paint   (30 + 20 ) / 140   =  50 / 140    = 5 / 14  of the room by 9:30 AM

Formula for sphere volume  = (4/3) pi *  (diameter / 2)^3...so we have....

Volume of outer sphere  =  (4/3) pi * ( 2.8 / 2)^3  

Volume of inner sphere  = (4/3) pi * ( 2.1 / 2)^3

So...area between the spheres is   (4/3) pi  [  (2.8 / 2)^3  -  (2.1 / 2)^3 ] ≈ 6.64 m^3