CPhill

Last one before I head off to bed.....

(a)  They are similar because the corresponding sides are in the same ratio

36/30   = 48/40

6/5  = 6/5             

This is also the scale factor of the larger prism base to the smaller prism base  

(b)  If the prisms are the same height......the prism with base DEF will have a greater  volume

It will be (Scale Factor)^2   greater than the smaller prism  =  (6/5)^2  =  36/25  times as great 

1) 

a)   A =   f(n)  =  5 + .4 ( n )

b)   f^-1A   =  f^-1(n) 

Write  y   for  f(n)....and we have that

y  = 5 + .4(n)      subtract 5 from both sides

y - 5 = .400(n)        divide both sides by .4

[ y - 5] /. 4=  n     "swap"  y  and  n

[ n - 5] / .4= y       for y, write  f^-1 (n)

[ n - 5] / .4  = f-1(n)

c)  f(10)   is the amount of water present in the tank at 6PM  =

5 + .4(10)  =  5  + 4  =  9    ⇒  9000 gallons

d)  f^-1(10)   =   [ 10 - 5 ] / .4  =  12.5     this is the number of hours after 8AM  when the tank will contain 10,000 gallons of water   ⇒   8:30 PM

2.   

The perimeter  is 

2 ( 12 + 15)  =  2 (27)  =  54ft

The length of one diagonal  forms a hypotenuse of a right triangle  with legs of 12 and 15...its length is

sqrt  ( 12^2  + 15^2)  =   sqrt (144 + 225)  =  sqrt (289)  =  17 ft

And there are two diagonals so their combined length is  2 * 17  =  34 ft

So   .....  the total rolls she needs  is

[ 54 + 34 ]  / 20    =   88/ 20  = 4.4 rolls

But we can't purchase partial rolls...so.....she needs  5 rolls

In a  cyclic quadrilateral   [ a rectangle inscribed in a circle]  opposite angles sum to 180°

Therefore  ...this implies that

(x - 2)  + (x + 2)   =  180      simplify

2x  =  180      divide both sides by 2

x  =  90°

So   

Angle A  =  x + 2   =  90 + 2  =  92° 

Angle B  =  x - 2  = 90 - 2  =  88°

Angle D  =  x - 10  =  90 - 10  =  80° 

Note that  the sine of an angle  is just  opposite side / hypotenuse

Here.....the opposite side  = 11.8  and the hypotenuse  =12

So...we need to use the sine inverse  (arcsin) function to determine the angle this makes with the ground

arcsin ( 11.8 / 12)  =  angle measure  =  79.5°

So....the ladder is not safe with this angle

Here's a pic :

The angle across from the 5cm side  is 30°

In a 30-60-90  triangle like this, the side opposite the 60° will be √3  times as long as the side opposite the 30° angle

So  BC  =  height of the triangle  =  5√3

So....the area  =  (1/2) base * height  =  (1/2) (5) (5√3)  =

(25/2)* √3  cm^2   ⇒   21.7 cm^2

1)

a)  This is known as  the  Tangent-Secant Theorem....it says that :

"When a tangent and a secant are drawn from one single external point to a circle, the square of the length of tangent segment must be equal to the product of lengths of whole secant segment and the exterior portion of secant segment."

In plain English....it says that    TU^2  = TR* TS

b)  RT  =  9      ST  =  4

We can find  TU  thusly :

TU^2  = TR * TS

TU^2  = RT * ST

TU^2  =  (9) * (4)

TU^2   =  36       take the square root of both sides

TU  =   6

log_x a  -  log_1/x a^3

Using the change-of -base  rule, we have that

log a  / log x    -    log a  /  log (1/x)   =

log a / log x  -  log a^3  / log x^-1   =

log a / log x  -  3log a / [ - log x]   =

log a / log x   +  3log a   /log x  =

4 [log a  /  log x ]  =      [ reverse the change-of-base rule ]

4 log_x a

2)

a)  They are similar......each side of Rectangle 2 will be  k times as long as the corresponding side of Rectangle 1

b )   Let  the Perimeter of Rectangle 1 be    2 ( L + W) = (2L + 2W)......so  the  perimeter of Rectangle 2 will be   2( kL + kW )  =   2k (L + W)  =   k (2L + 2W).....note that the perimeter of  Rectangle 2 is k times that of Rectangle 1

c)  The area of Rectangle 1 is  ( L x W).....the area of Rectangle 2 is  kL x kW   since each side is k times that of Rectangle 1.....so   kL x kW  =  k * k * L * W  =  k^2 (L * W)....so Rectangle 2 has  an area k^2 times that of Rectangle 1

6)  In a triangle. a segment drawn parallel to a base always splits the sides in equal proportions

So we have that

48/ 40   = ? / 85   

6/ 5  =  ? / 85      cross-multiply

85 * 6   =  5 * ?

510  =  5 * ?      divide both sides by  5

102   =  ?

We  can check this

48 / 40  =  102 / 85

6 / 5  =   6/ 5