CPhill

1.  v^2 -8v  - 59  = -2    add 59 to both sides

v^2  - 8v     =  57       take 1/2 of 8  = 4...square it  = 16....add to both sides

v^2  - 8x + 16  = 57 + 16     factor the left side, simplify the right

(v - 4)^2  =  73      take both roots

x - 4 =  sqrt73          x  - 4  =  -sqrt73

add 4 to both sides

x = 4 + sqr73     or      x =  4 - sqrt73

Foe convenience sake, let the first number drawn  be the numerator of the fraction and the second the denominator.....

We have the following possibilities  (16)

1/1   2/2   3/3    4/4

1/2   1/3   1/4

2/3   2/4

3/4

2/1   3/1   4/1

3/2   4/2

4/3

The ones in red are whole numbers   (7)

So

7 / 16

Answered here :  https://web2.0calc.com/questions/similar-triangles_9

Question 2

I am NOT  sure about this one...but I'll venture an answer

I believe that we have the frusum of a triangular prism.....

Assuming that CP /CQ    = DN /MD   ...we have that

CP =  CQ * DN / MD  =  8  * 2/4 =  16/ 4  = 4

The area of MDN  =  A   =  (1/2) (DN)(MD)sqrt3 / 2  =  (8/4) * sqrt 3 = 2sqrt 3

The area  of QCP  = B  =   (1/2) (CP)(CQ)sqrt3 /2  = (32/4)sqrt3 = 8sqrt3

The fomula for the volume  a prismatic frustum is similar  [ I think ] to that of a conical frustum

The volume is given by :

(1/3) ( height)  [ A + B  +  sqrt (AB)  ] = 

(1/3) (12) ( 10sqrt 3  + sqrt [ 16 * 3] )  =

4 ( 10 sqrt 3  +  sqrt 48  )  =

4 [ 10srt 3 + 4sqr3  ]  =

4 [ 14 sqrt 3 ]  =

56sqrt 3    units^3 

Can  anyone double-check this  ???

Question 1: What is the total surface area, in square inches, of a cube with a volume of 1 cubic foot?

 L = W  = H...and all of these must  = 1   since  1 * 1 *  1  =  1 ft^3

So...each side is a 1 x 1 square

So...the total surface area  is

6 * [ 1 x 1  ]  =   6 ft^2

Question 3: A round pizza is 1/3 of an inch thick and has a diameter of 12 inches. It is cut into 12 congruent pieces. What is the number of cubic inches in the volume of one piece? Express your answer in terms of pi.

Volume  =  (1/3)  * pi * (diameter / 2)^2  =  (pi / 3 ) * [ 6 ] ^2 =  12 pi in^3

If it is cut into  12 congruent pieces...each piece must have a volume of pi in^3

10)At a party at the middle school over on third avenue next to the bank, 60% of the students are girls. After these students are joined by 20 more boy students, all of whom like to dance, the party is now 58% girls. How many students are now at the party?

Call the original  number of students, S...and the number of girls  = .60S

Total students  after 20 boys arrive = S + 20

And we have that

Number of  girls / total number of students  =  58%

.60S  / [ S + 20 ]  =   .58    multiply both sides by  S +  20

.60S  = .58 [ S + 20]

.60S  = .58S + 11.6   subtract  .58S from  both sides

.02S = 11.6   divide both sides  by .02

S  = 11/6 / .02 =    580

So...the total number of students = 580 + 20  = 600

Proof :

.60 * 580 = 348  girls originally

And  .40 * 580 =  232 boys originally

After  20  boys arrive their %  =  [ 232 + 20 ] / 600  = 42%

So...the girls % after this  = 1  - 42%  =  1 - .42   =  .58   = 58%

9)Winston grows green dragon apples. This year, he shipped 80%of these apples to sell at supermarkets and kept the rest to make cider. Of all the green dragon apples he sold to the supermarkets, only 60% of them were bought. If a total of 420 green dragon apples were bought, how many green dragon apples did Winston keep to make cider?

Call the number he shipped to the store, N

So

.60N  = 420   divide both sides by  ,60

420 / .60  = N  = 700 were shipped to the stores

But this number was  80%  of the total he grew = T

So

.80T  = 700   divide both sides by  .80

700/.80 = T  = 875

But..he kept 20% of these =   875 * .20  = 175

7)A city has an assessed valuation of $8,000,000. The rate for school taxes is 60 cents per $100 valuation. If all but 4% of the school taxes were collected, how many dollars were still owed?

The amount that should have been collected  is

[ 8000000 / 100] * .60    = $48000

If all but 4% have been collected, then   .04 * 48000  = $1920  is still owed

8)On Tuesday, a radio store reduces all of its prices by 20% from what they were on Monday.

On Wednesday, the store again reduces all of its prices by some percentage from what they were on Tuesday. If the resulting price of a radio is half of the original Monday price, then by what percentage did the store reduce its prices from Tuesday to Wednesday?

Call Monday's price, P

Tuesday's price must just be .80P   [ 20% reduction]

Call Wednesday's price reduction ( 1 - R)

So...we have that

Tuesday's price  *  Wednesday's reduction  = (1/2)P

(.80)P  *( 1 - R)  = (1/2)P    divide through by  P  and note that 1/2  = .50

.80 (1 - R)  = .50        divide both sides by .80

1 - R  =  .50/.80

1 - R  = 5/8

1 - 5/8 = R

3/8   = R = .375  = 37.5%  = the reduction from Tues. to Wed.

Proof of this...let Monday's price  = 100

Tuesdays price  = .80 * 100   = 80

Wed price  = 80 (1 - R)  = 80 (1 - .375)  = 50 =  half of  Mon. price

5)Three generations of the Wen family are going to the movies, two from each generation. The two members of the youngest generation receive a 50% discount as children. The two members of the oldest generation receive a 25% discount as senior citizens. The two members of the middle generation receive no discount. Grandfather Wen, whose senior ticket costs $6.00, is paying for everyone. How many dollars must he pay?

Call the price of the middle generation ticket, P

Call the price of the youngest generation ticket, .50P   [ 50% discount ]

Call the price of  the senior ticket, .75P  [ 25% discount ]

So.....we know that .75P  = 6    divide both sides by .75  

P  = 6/.75  = $8

So....the yoiungest generation pays  8 * .50   = $4 per ticket

And the middle generation pays  $8

So...the total amount paid   =  2 [ 4 + 8  + 6]  = 2 [ 18]   = $36

6)The city imposed a 0.08% tax on the value of dinosaur collections. Sheila had to pay $12 in dinosaur tax. How much is her dinosaur collection worth?

Call the value of the collection, V

And .08%  =  .0008  as a decimal

So  .08% of V  =  12

V * .0008  =  12   divide both sides  by .0008

V =  12  / .0008   = $15000

1) If 15% of x is 42, what is x?

.15x  =  42     divide both sides by .15

x  = 42 / .15  =  280

2)I need to make $500 per week after tax in order to pay all my bills. The income tax is 20%. What is the smallest pre-tax weekly salary I can earn and still be able to pay my bills after I pay my income tax? 

Call  A the amount you need to make

So......   A  -  .20A  = 500

.80A  = 500     divide both sides by .80

A = 500 / .80   = $625

3)After my salary was increased by 12%, it was 62,720. What was my salary before the increase?

Call S  the original salary...so..

1S + .12S  = 62720

1.12S = 62720     divide both sides by 1.12

S  = 62720 / 1.12  = $56000

4)After some bad news (due to a revolutionary but ultimately doomed process for making toast in a microwave oven), stock in Seraj Food Industries fell by 40%, to $324 a share. What was the original price of a share?

Call the original value of the stock, V

If the shares fell by 40%, they must be worth only 60%  of their original value

So

.60V  = 324    divide both sides by .60

324 / .60  = V  = $540